doubleSYcomputational(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

💖 doubleSYcomputational(3) — LAPACK 💖

Okay bestie, welcome to the manual page for doubleSYcomputational(3), part 1 of 21. We're diving into the SY matrix computational functions, and it's giving iconic LAPACK energy. No cap. Let's get into it. 💅

✨ NAME ✨

doubleSYcomputational — double

🌈 SYNOPSIS 🌈

🛠️ Functions 🛠️

  • subroutine dla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) — DLA_SYAMV does a matrix-vector product with a symmetric indefinite matrix, and it's specifically for calculating error bounds. 💅
  • double precision function dla_syrcond (UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK) — DLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix. no cap.
  • subroutine dla_syrfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO) — DLA_SYRFSX_EXTENDED improves the computed solution to a linear system for symmetric indefinite matrices by doing extra-precise iterative refinement, and it gives you error bounds and backward error estimates for the solution. slay.
  • double precision function dla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK) — DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. iconic.
  • subroutine dlasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) — DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method. periodt.
  • subroutine dlasyf_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK) — DLASYF_AA — that's the whole description, bestie.
  • subroutine dlasyf_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO) — DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method. 💖
  • subroutine dlasyf_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) — DLASYF_ROOK *> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. (Yes, the *> is literally in the original, so we're keeping it.)
  • subroutine dsycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) — DSYCON — just the name, and that's the tea.
  • subroutine dsycon_3 (UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, IWORK, INFO) — DSYCON_3 — name only, bestie.
  • subroutine dsycon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) — DSYCON_ROOK — name only.
  • subroutine dsyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO) — DSYCONV — name only.
  • subroutine dsyconvf (UPLO, WAY, N, A, LDA, E, IPIV, INFO) — DSYCONVF — name only.
  • subroutine dsyconvf_rook (UPLO, WAY, N, A, LDA, E, IPIV, INFO) — DSYCONVF_ROOK — name only.
  • subroutine dsyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO) — DSYEQUB — name only.
  • subroutine dsygs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) — DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
  • subroutine dsygst (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) — DSYGST — name only.
  • subroutine dsyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) — DSYRFS — name only.
  • subroutine dsyrfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) — DSYRFSX — name only.
  • subroutine dsytd2 (UPLO, N, A, LDA, D, E, TAU, INFO) — DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
  • subroutine dsytf2 (UPLO, N, A, LDA, IPIV, INFO) — DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
  • subroutine dsytf2_rk (UPLO, N, A, LDA, E, IPIV, INFO) — DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
  • subroutine dsytf2_rook (UPLO, N, A, LDA, IPIV, INFO) — DSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
  • subroutine dsytrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO) — DSYTRD — name only.
  • subroutine dsytrd_2stage (VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO) — DSYTRD_2STAGE — name only.
  • subroutine dsytrd_sy2sb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO) — DSYTRD_SY2SB — name only.
  • subroutine dsytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — DSYTRF — name only.
  • subroutine dsytrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — DSYTRF_AA — name only.
  • subroutine dsytrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO) — DSYTRF_AA_2STAGE — name only.
  • subroutine dsytrf_rk (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO) — DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
  • subroutine dsytrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — DSYTRF_ROOK — name only.
  • subroutine dsytri (UPLO, N, A, LDA, IPIV, WORK, INFO) — DSYTRI — name only.
  • subroutine dsytri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — DSYTRI2 — name only.
  • subroutine dsytri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO) — DSYTRI2X — name only.
  • subroutine dsytri_3 (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO) — DSYTRI_3 — name only.
  • subroutine dsytri_3x (UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO) — DSYTRI_3X — name only.
  • subroutine dsytri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO) — DSYTRI_ROOK — name only.
  • subroutine dsytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) — DSYTRS — name only.
  • subroutine dsytrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO) — DSYTRS2 — name only.
  • subroutine dsytrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO) — DSYTRS_3 — name only.
  • subroutine dsytrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO) — DSYTRS_AA — name only.
  • subroutine dsytrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO) — DSYTRS_AA_2STAGE — name only.
  • subroutine dsytrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) — DSYTRS_ROOK — name only.
  • subroutine dtgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) — DTGSYL — name only.
  • subroutine dtrsyl (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO) — DTRSYL — name only.

🌟 Detailed Description 🌟

This is the whole squad of double computational functions for SY matrices. iykyk.

👑 Function Documentation 👑

💅 subroutine dla_syamv (integer UPLO, integer N, double precision ALPHA, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) X, integer INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY) 💅

DLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.

🎯 Purpose 🎯

DLA_SYAMV performs the matrix-vector operation

y := alpha*abs(A)*abs(x) + beta*abs(y),

where alpha and beta are scalars, x and y are vectors and A is an n by n symmetric matrix.

This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, 'symbolically' zero components are not perturbed. A zero entry is considered 'symbolic' if all multiplications involved in computing that entry have at least one zero multiplicand.

🔧 Parameters 🔧

  • UPLO — UPLO is INTEGER. On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows:
    • UPLO = BLAS_UPPER — Only the upper triangular part of A is to be referenced.
    • UPLO = BLAS_LOWER — Only the lower triangular part of A is to be referenced.

Unchanged on exit.

  • N — N is INTEGER. On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.
  • ALPHA — ALPHA is DOUBLE PRECISION . On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
  • A — A is DOUBLE PRECISION array, dimension ( LDA, n ). Before entry, the leading m by n part of the array A must contain the matrix of coefficients. Unchanged on exit.
  • LDA — LDA is INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). Unchanged on exit.
  • X — X is DOUBLE PRECISION array, dimension ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the vector x. Unchanged on exit.
  • INCX — INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.
  • BETA — BETA is DOUBLE PRECISION . On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit.
  • Y — Y is DOUBLE PRECISION array, dimension ( 1 + ( n - 1 )*abs( INCY ) ). Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y.
  • INCY — INCY is INTEGER. On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.

✍️ Author ✍️

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

📚 Further Details: 📚

Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. -- Modified for the absolute-value product, April 2006 Jason Riedy, UC Berkeley

💖 dla_syrcond

double precision function dla_syrcond (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, integer CMODE, double precision, dimension( * ) C, integer INFO, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK)

okay so bestie, meet DLA_SYRCOND — she's a double precision function and her whole thing is estimating the Skeel condition number for a symmetric indefinite matrix. Not just any condition number. Skeel. She's built different, fr fr. 💅

Purpose:

So here's the vibe. DLA_SYRCOND estimates the Skeel condition number of op(A) * op2(C), where op2 gets decided by CMODE like this:

CMODE =  1    op2(C) = C
CMODE =  0    op2(C) = I
CMODE = -1    op2(C) = inv(C)

The Skeel condition number cond(A) = norminf( |inv(A)||A| ) is computed by finding scaling factors R such that diag(R)*A*op2(C) is row equilibrated, and then computing the standard infinity-norm condition number. It's giving ✨*methodical*✨ and honestly? obsessed. 😭

Parameters:

  • UPLO — CHARACTER*1. Tells her which triangle of A we're living in:
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. (Zero is allowed, don't panic 🫶)
  • A — DOUBLE PRECISION array, dimension (LDA,N). On entry, the N-by-N matrix A.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — DOUBLE PRECISION array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.
  • LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF.
  • CMODE — INTEGER. Determines op2(C) in the formula op(A) * op2(C) as follows:
    • CMODE = 1 → op2(C) = C
    • CMODE = 0 → op2(C) = I
    • CMODE = -1 → op2(C) = inv(C)
  • C — DOUBLE PRECISION array, dimension (N). The vector C in the formula op(A) * op2(C).
  • INFO — INTEGER. The receipts:
    • = 0: Successful exit. 🎉
    • i > 0: The ith argument is invalid. 🚩
  • WORK — DOUBLE PRECISION array, dimension (3*N). Workspace.
  • IWORK — INTEGER array, dimension (N). Workspace.

Author:

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🦋 dla_syrfsx_extended

subroutine dla_syrfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldy, * ) Y, integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, double precision, dimension( * ) RES, double precision, dimension( * ) AYB, double precision, dimension( * ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)

DLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by doing extra-precise iterative refinement, and she also hands you error bounds and backward error estimates for the solution. She's the glow-up arc. 🌸

Purpose:

DLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. This subroutine is called by DSYRFSX to perform iterative refinement.

And ngl, in addition to the normwise error bound, the code provides the maximum componentwise error bound if possible. See the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on those error bounds. Note that this subroutine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. That's it. That's her job. Don't ask her for more. 💀

Parameters:

  • PREC_TYPE — INTEGER. Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and
    • = 'S': Single
    • = 'D': Double
    • = 'I': Indigenous
    • = 'X' or 'E': Extra
  • UPLO — CHARACTER*1.
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B.
  • A — DOUBLE PRECISION array, dimension (LDA,N). On entry, the N-by-N matrix A.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — DOUBLE PRECISION array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.
  • LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF.
  • COLEQU — LOGICAL. If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly. So like, if you skipped it, don't lie about it, bestie. 🙈
  • C — DOUBLE PRECISION array, dimension (N). The column scale factors for A. If COLEQU = .FALSE., C is not accessed. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates.
Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. ⚠️ So yeah, do it right or the whole vibe check fails.
  • B — DOUBLE PRECISION array, dimension (LDB,NRHS). The right-hand-side matrix B.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • Y — DOUBLE PRECISION array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by DSYTRS. On exit, the improved solution matrix Y. She literally levels you up. 👑
  • LDY — INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
  • BERR_OUT — DOUBLE PRECISION array, dimension (NRHS). On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by DLA_LIN_BERR.

  • N_NORMS — INTEGER. Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP).
    • If N_NORMS >= 1 return normwise error bounds.
    • If N_NORMS >= 2 return componentwise error bounds.
  • ERR_BNDS_NORM — DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

  • err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field above.

See Lapack Working Note 165 for further details and extra cautions. 📓

  • ERR_BNDS_COMP — DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

  • err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. ✨

RES

RES is DOUBLE PRECISION array, dimension (N)

Workspace to hold the intermediate residual. 🫶

AYB

AYB is DOUBLE PRECISION array, dimension (N)

Workspace. This can be the same workspace passed for Y_TAIL. reuse era 💅

DY

DY is DOUBLE PRECISION array, dimension (N)

Workspace to hold the intermediate solution.

Y_TAIL

Y_TAIL is DOUBLE PRECISION array, dimension (N)

Workspace to hold the trailing bits of the intermediate solution. the little leftover crumbs 🍓

RCOND

RCOND is DOUBLE PRECISION

Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. 😭

ITHRESH

ITHRESH is INTEGER

The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. 👀

RTHRESH

RTHRESH is DOUBLE PRECISION

Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.

DZ_UB

DZ_UB is DOUBLE PRECISION

Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we define as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details. 🌸

IGNORE_CWISE

IGNORE_CWISE is LOGICAL

If .TRUE. then ignore componentwise convergence. Default value is .FALSE..

INFO

INFO is INTEGER

 = 0:  Successful exit.
 < 0:  if INFO = -i, the ith argument to DLA_SYRFSX_EXTENDED had an illegal
       value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

double precision function dla_syrpvgrw

double precision function dla_syrpvgrw (character*1 UPLO, integer N, integer
    INFO, double precision, dimension( lda, * ) A, integer LDA, double
    precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( *
    ) IPIV, double precision, dimension( * ) WORK)

DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. 💜

Purpose:

DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. no cap, this is the vibe check for whether you can trust anything downstream 😭

Parameters

UPLO

UPLO is CHARACTER*1

 = 'U':  Upper triangle of A is stored;
 = 'L':  Lower triangle of A is stored.

N

N is INTEGER

The number of linear equations, i.e., the order of the matrix A. N >= 0.

INFO

INFO is INTEGER

The value of INFO returned from DSYTRF, .i.e., the pivot in column INFO is exactly 0.

A

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the N-by-N matrix A.

LDA

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

AF

AF is DOUBLE PRECISION array, dimension (LDAF,N)

The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.

LDAF

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D as determined by DSYTRF.

WORK

WORK is DOUBLE PRECISION array, dimension (2*N)

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dlasyf

subroutine dlasyf (character UPLO, integer N, integer NB, integer KB, double
    precision, dimension( lda, * ) A, integer LDA, integer, dimension( * )
    IPIV, double precision, dimension( ldw, * ) W, integer LDW, integer
    INFO)

DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method. 🎀

Purpose:

DLASYF computes a partial factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). ✨

Parameters

UPLO

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

 = 'U':  Upper triangular
 = 'L':  Lower triangular

N

N is INTEGER

The order of the matrix A. N >= 0.

NB

NB is INTEGER

The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks.

KB

KB is INTEGER

The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

A

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, A contains details of the partial factorization.

LDA

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If UPLO = 'U':

  • Only the last KB elements of IPIV are set.
  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

  • Only the first KB elements of IPIV are set.
  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

W

  • W — a DOUBLE PRECISION array, dimension (LDW,NB) ✨
  • LDW — LDW is INTEGER, and it's the leading dimension of the array W. So like, LDW >= max(1,N), no cap 💅
  • INFO — INFO is INTEGER, and here's the tea:
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero.  The factorization
     has been completed, but the block diagonal matrix D is
     exactly singular.

so yeah, INFO > 0 means the whole thing finished but D is singular at that spot — iconic but also a little tragic 🫶

🎀 Author 🎀

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🦋 Contributors: 🦋

  • November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley

💖 subroutine dlasyf_aa (character UPLO, integer J1, integer M, integer NB, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WORK) 💖

DLASYF_AA

Purpose:

okay so DLATRF_AA factorizes a panel of a real symmetric matrix A using the Aasen's algorithm 💫 The panel is a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L.

To actually factorize the panel, Aasen's algorithm needs the last row (or column) of the previous panel. And the very first row (or column) of A gets set to be the first row (or column) of an identity matrix, which is what's used to factorize that first panel. Cute, right?

The resulting J-th row of U (or J-th column of L) gets stored in the (J-1)-th row (or column) of A — and no, without the unit diagonals — while the diagonal and subdiagonal of A get overwritten by those of T. slay 💅

Parameters:

  • UPLO — UPLO is CHARACTER*1:
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.
  • J1 — J1 is INTEGER. It's where the first row (or column) of the panel lives inside the submatrix of A that got passed to this routine. Like, when DSYTRF_AA calls it: for the first panel, J1 is 1, and for all the remaining panels, J1 is 2. iykyk 🌸
  • M — M is INTEGER. The dimension of the submatrix. M >= 0.
  • NB — NB is INTEGER. The dimension of the panel to be facotorized. (yes, that's the original spelling bestie, we respect the typo 😭)
  • A — A is a DOUBLE PRECISION array, dimension (LDA,M) for the first panel, and dimension (LDA,M+1) for the remaining panels.

On entry, A holds the last row (or column) of the previous panel, plus the trailing submatrix of A to be factorized — except for the first panel, where only the panel is passed. 🙈

On exit, the leading panel is factorized. periodt.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • IPIV — IPIV is an INTEGER array, dimension (M). It's all the details of the row and column interchanges — the row and column k were interchanged with the row and column IPIV(k).
  • H — H is DOUBLE PRECISION workspace, dimension (LDH,NB).
  • LDH — LDH is INTEGER. The leading dimension of the workspace H. LDH >= max(1,M).
  • WORK — WORK is DOUBLE PRECISION workspace, dimension (M).

🌟 Author 🌟

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

👑 subroutine dlasyf_rk (character UPLO, integer N, integer NB, integer KB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( ldw, * ) W, integer LDW, integer INFO) 👑

DLASYF_RK computes a partial factorization of a real symmetric
indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting
method.

Purpose:

DLASYF_RK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method 🦋 The partial factorization has this form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB 💖 The actual order comes back in the argument KB, and it's either NB or NB-1, or N if N <= NB.

DLASYF_RK is an auxiliary routine called by DSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). no cap 🔥

Parameters:

  • UPLO — UPLO is CHARACTER*1. Tells you whether the upper or lower triangular part of the symmetric matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NB — NB is INTEGER. The max number of columns of A that should be factored. And bestie, NB should be at least 2 to allow for 2-by-2 pivot blocks. fr fr.
  • KB — KB is INTEGER. The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.
  • A — A is a DOUBLE PRECISION array, dimension (LDA,N).

On entry, it's the symmetric matrix A:

  • If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. 💀
  • If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, it contains:

  1. ONLY the diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (the superdiagonal, or subdiagonal, elements of D are stored on exit in array E), and
  2. If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • E — E is a DOUBLE PRECISION array, dimension (N).

On exit, it contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:

If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases. 🌸
  • IPIV — IPIV is an INTEGER array, dimension (N). IPIV describes the permutation matrix P in the factorization of matrix A like this: the absolute value of IPIV(k) is the index of the row and column that got interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. And the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, which correspond to 1 or 2 interchanges at each factorization step. 💫

If UPLO = 'U' (in factorization order, k decreases from N to 1):

  1. A single positive entry IPIV(k) > 0 means:
    • D(k,k) is a 1-by-1 diagonal block.
    • If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,N-KB+1:N);
    • If IPIV(k) = k, no interchange occurred.
  2. A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means:
    • D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    • (NOTE: negative entries in IPIV appear ONLY in pairs). 🫶
      1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N). If -IPIV(k) = k, no interchange occurred.
      2. If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N). If -IPIV(k-1) = k-1, no interchange occurred.
  3. In both cases a) and b) it's always ABS( IPIV(k) ) <= k.
  4. NOTE: Any entry IPIV(k) is always NONZERO on output. 💅

If UPLO = 'L' (in factorization order, k increases from 1 to N):

  1. A single positive entry IPIV(k) > 0 means:
    • D(k,k) is a 1-by-1 diagonal block.
    • If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,1:KB).
    • If IPIV(k) = k, no interchange occurred.
  2. A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means:
    • D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    • (NOTE: negative entries in IPIV appear ONLY in pairs). 🫶
      1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k) = k, no interchange occurred.
      2. If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k+1) = k+1, no interchange occurred.
  3. In both cases a) and b) it's always ABS( IPIV(k) ) >= k.
  4. NOTE: Any entry IPIV(k) is always NONZERO on output. 💅
  • W — W is a DOUBLE PRECISION array, dimension (LDW,NB).
  • LDW — LDW is INTEGER. The leading dimension of the array W. LDW >= max(1,N).
  • INFO
  • = 0 — successful exit, we love a clean win ✨
  • < 0 — if INFO = -k, bestie, the k-th argument had an illegal value 🙈
  • > 0 — if INFO = k, the matrix A is singular, and the reason is giving very specific:
    • if UPLO = 'U': column k in the upper triangular part of A contains all zeros 💀
    • if UPLO = 'L': column k in the lower triangular part of A contains all zeros 💀

So D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. That's not a vibe, that's a crash. 🚨

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes. So INFO is not the gossip log, it's just the first text in the group chat 💅

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Contributors:

  • December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine dlasyf_rook (character UPLO, integer N, integer NB, integer KB, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldw, * ) W, integer LDW, integer INFO)

DLASYF_ROOK *> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. Okay so that's a lot of words but the vibe is: it does the math, and it does it the rook way ♜✨

Purpose:

DLASYF_ROOK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. It's giving "you get what you get and it's documented, so it's fine." 🦋

DLASYF_ROOK is an auxiliary routine called by DSYTRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). Level 3 BLAS?? she's fast, she's blocked, she ate. 🔥

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NB — NB is INTEGER. The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks. Two. At least two. Don't be out here passing 1 and expecting 2-by-2 blocks, that's not how any of this works. 😭
  • KB — KB is INTEGER. The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, A contains details of the partial factorization. She gets a whole glow-up in place, we stan an in-place queen. 💖
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D.

If UPLO = 'U':

  • Only the last KB elements of IPIV are set.
  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. Two whole swaps, a matched set, iconic. 💅

If UPLO = 'L':

  • Only the first KB elements of IPIV are set.
  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • W — W is DOUBLE PRECISION array, dimension (LDW,NB).
  • LDW — LDW is INTEGER. The leading dimension of the array W. LDW >= max(1,N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit ✅
    • > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular. Singular means division by zero incoming, do NOT solve with it. 🚩

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Contributors:

  • November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine dsycon (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYCON

Purpose:

DSYCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). So basically she's asking "how sketchy is this matrix, on a scale from 'fine' to 'do not touch'?" and reporting back a number. 🌸

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF.
  • ANORM — ANORM is DOUBLE PRECISION. The 1-norm of the original matrix A.
  • RCOND — RCOND is DOUBLE PRECISION. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.
  • WORK — WORK is DOUBLE PRECISION array, dimension (2*N).
  • IWORK — IWORK is INTEGER array, dimension (N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 🙈

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

subroutine dsycon_3 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYCON_3

Purpose:

DSYCON_3 estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. Permutation matrix, transposes, blocks on blocks — she's doing the most and doing it correctly. 👑

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). This routine uses BLAS3 solver DSYTRS_3. BLAS3 solver, so she's fast, fr fr. 🔥

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix:
    • = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
    • = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). Diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK:
    1. ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and
    2. If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
  • LDA
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). Bestie, this is not a suggestion, it's the law. 💅
  • E — E is DOUBLE PRECISION array, dimension (N). On entry, it contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:
    • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
    • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases. Periodt. 💖
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK. Slay. ✨
  • ANORM — ANORM is DOUBLE PRECISION. The 1-norm of the original matrix A. Iconic. 👑
  • RCOND — RCOND is DOUBLE PRECISION. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine. Math queen behavior. 🧮
  • WORK — WORK is DOUBLE PRECISION array, dimension (2*N). Yes, 2*N. No cap. 💅
  • IWORK — IWORK is INTEGER array, dimension (N). 💖
  • INFO — INFO is INTEGER.
    • = 0: successful exit. We love a win. ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value. Not the vibe. 😭

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Contributors:

  • June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

✨ subroutine dsycon_rook ✨

   subroutine dsycon_rook (character UPLO, integer N, double precision,
       dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double
       precision ANORM, double precision RCOND, double precision, dimension( *
       ) WORK, integer, dimension( * ) IWORK, integer INFO)
        DSYCON_ROOK

Purpose

DSYCON_ROOK estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF_ROOK.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). It's giving "I know how ill-conditioned you are". 💅

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0. Simple. ✨
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF_ROOK. Main character energy. 👑
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). No cap. 💅
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF_ROOK. 🦋
  • ANORM — ANORM is DOUBLE PRECISION. The 1-norm of the original matrix A. 🌈
  • RCOND — RCOND is DOUBLE PRECISION. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine. 🧮
  • WORK — WORK is DOUBLE PRECISION array, dimension (2*N). 💖
  • IWORK — IWORK is INTEGER array, dimension (N). ✨
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Contributors:

  • April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

✨ subroutine dsyconv ✨

   subroutine dsyconv (character UPLO, character WAY, integer N, double
       precision, dimension( lda, * ) A, integer LDA, integer, dimension( * )
       IPIV, double precision, dimension( * ) E, integer INFO)
       DSYCONV

Purpose

DSYCONV convert A given by TRF into L and D and vice-versa. Get Non-diag elements of D (returned in workspace) and apply or reverse permutation done in TRF. It's giving conversion therapy but for matrices. 💅

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • WAY — WAY is CHARACTER*1
    • = 'C': Convert
    • = 'R': Revert
  • N — N is INTEGER. The order of the matrix A. N >= 0. ✨
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF. 👑
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💅
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF. 🦋
  • E — E is DOUBLE PRECISION array, dimension (N). E stores the supdiagonal/subdiagonal of the symmetric 1-by-1 or 2-by-2 block diagonal matrix D in LDLT. (Yes, "supdiagonal" is exactly what it says — don't come for me, I'm just the messenger. 💀)
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

✨ subroutine dsyconvf ✨

   subroutine dsyconvf (character UPLO, character WAY, integer N, double
       precision, dimension( lda, * ) A, integer LDA, double precision,
       dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)
       DSYCONVF

Purpose

If parameter WAY = 'C': DSYCONVF converts the factorization output format used in DSYTRF provided on entry in parameter A into the factorization output format used in DSYTRF_RK (or DSYTRF_BK) that is stored on exit in parameters A and E. It also converts in place details of the intechanges stored in IPIV from the format used in DSYTRF into the format used in DSYTRF_RK (or DSYTRF_BK).

If parameter WAY = 'R': DSYCONVF performs the conversion in reverse direction, i.e. converts the factorization output format used in DSYTRF_RK (or DSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in DSYTRF that is stored on exit in parameter A. It also converts in place details of the intechanges stored in IPIV from the format used in DSYTRF_RK (or DSYTRF_BK) into the format used in DSYTRF.

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix A.
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • WAY — WAY is CHARACTER*1
    • = 'C': Convert
    • = 'R': Revert
  • N — N is INTEGER. The order of the matrix A. N >= 0. ✨
  • A — A is DOUBLE PRECISION array, dimension (LDA,N).
    1. If WAY ='C':

On entry, contains factorization details in format used in DSYTRF: a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.

On exit, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

  1. If WAY = 'R':

On entry, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

On exit, contains factorization details in format used in DSYTRF: a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💅
  • E — E is DOUBLE PRECISION array, dimension (N).
    1. If WAY ='C':
  1. If WAY = 'R':

On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where

  • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
  • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.

On exit, is not changed

  • IPIV — IPIV is INTEGER array, dimension (N)
    1. If WAY ='C': On entry, details of the interchanges and the block structure of D in the format used in DSYTRF. On exit, details of the interchanges and the block structure of D in the format used in DSYTRF_RK ( or DSYTRF_BK).
    2. If WAY ='R': On entry, details of the interchanges and the block structure of D in the format used in DSYTRF_RK ( or DSYTRF_BK). On exit, details of the interchanges and the block structure of D in the format used in DSYTRF.

bestie the IPIV array is literally out here doing a whole costume change mid-call and honestly? iconic. same array, different format, no notes. 💅

  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

✨ Author ✨

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 Contributors: 💖

November 2017,  Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

🦋 subroutine dsyconvf_rook 🦋

subroutine dsyconvf_rook (character UPLO, character WAY, integer N, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)

DSYCONVF_ROOK

Purpose:

If parameter WAY = 'C':

DSYCONVF_ROOK converts the factorization output format used in DSYTRF_ROOK provided on entry in parameter A into the factorization output format used in DSYTRF_RK (or DSYTRF_BK) that is stored on exit in parameters A and E. IPIV format for DSYTRF_ROOK and DSYTRF_RK (or DSYTRF_BK) is the same and is not converted.

If parameter WAY = 'R':

DSYCONVF_ROOK performs the conversion in reverse direction, i.e. converts the factorization output format used in DSYTRF_RK (or DSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in DSYTRF_ROOK that is stored on exit in parameter A. IPIV format for DSYTRF_ROOK and DSYTRF_RK (or DSYTRF_BK) is the same and is not converted.

Parameters

  • UPLO — UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix A.
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • WAY — WAY is CHARACTER*1
    • = 'C': Convert
    • = 'R': Revert
  • N — N is INTEGER The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N)
    1. If WAY ='C':

On entry, contains factorization details in format used in DSYTRF_ROOK:

  • a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and
  • b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.

On exit, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK:

  • a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
  • b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
  1. If WAY = 'R':

On entry, contains factorization details in format used in DSYTRF_RK or DSYTRF_BK:

  • a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
  • b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

On exit, contains factorization details in format used in DSYTRF_ROOK:

  • a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and
  • b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.

okay the A array is switching outfits depending on which way WAY is pointing and it's giving ✨dual citizenship✨ — read the direction carefully bestie, it matters. 🫶

  • LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • E — E is DOUBLE PRECISION array, dimension (N)
    1. If WAY ='C':

On entry, just a workspace.

On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

  1. If WAY = 'R':

On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

On exit, is not changed

  • IPIV — IPIV is INTEGER array, dimension (N) On entry, details of the interchanges and the block structure of D as determined:
    1. by DSYTRF_ROOK, if WAY ='C';
    2. by DSYTRF_RK (or DSYTRF_BK), if WAY ='R'. The IPIV format is the same for all these routines.

On exit, is not changed.

  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

✨ Author ✨

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 Contributors: 💖

November 2017,  Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

🌸 subroutine dsyequb 🌸

subroutine dsyequb (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, double precision, dimension( * ) S, double
    precision SCOND, double precision AMAX, double precision, dimension( * )
    WORK, integer INFO)

DSYEQUB

Purpose:

DSYEQUB computes row and column scalings intended to equilibrate a symmetric matrix A (with respect to the Euclidean norm) and reduce its condition number. The scale factors S are computed by the BIN algorithm (see references) so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of the smallest possible condition number over all possible diagonal scalings.

it's giving glow-up for matrices — DSYEQUB is out here picking the perfect scale factors so your condition number stops being a drama queen. 💅✨

Parameters

  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N) The N-by-N symmetric matrix whose scaling factors are to be computed.
  • LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • S — S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A.
  • SCOND — SCOND is DOUBLE PRECISION If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.

ngl this is the "don't bother bestie" check — if SCOND >= 0.1 and AMAX is chill, scaling by S is literally not worth it. 💀

  • AMAX — AMAX is DOUBLE PRECISION Largest absolute value of any matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
🚨 if AMAX is flirting with overflow or underflow, you should scale. that's not a vibe, that's a warning.
  • WORK — WORK is DOUBLE PRECISION array, dimension (2*N)
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the i-th diagonal element is nonpositive.

✨ Author ✨

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

📚 References: 📚

Livne, O.E. and Golub, G.H., 'Scaling by Binormalization',
 Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
 DOI 10.1023/B:NUMA.0000016606.32820.69
 Tech report version:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

🎀 subroutine dsygs2 🎀

subroutine dsygs2 (integer ITYPE, character UPLO, integer N, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldb, * ) B, integer LDB, integer INFO)

DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).

Purpose:

DSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form.

okay so ITYPE is lowkey running the whole vibe check here, and it goes like this:

  • if ITYPE = 1, the problem is A*x = lambda*B*x, and A gets overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
  • if ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A gets overwritten by U*A*U**T or L**T *A*L.

And B?? bestie it must have been previously factorized as U**T *U or L*L**T by DPOTRF — that's a prerequisite, not a suggestion, fr fr 💀

Parameters 📋

  • ITYPE — ITYPE is INTEGER. = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T); = 2 or 3: compute U*A*U**T or L**T *A*L.
  • UPLO — UPLO is CHARACTER*1. Says whether the upper or lower triangular part of the symmetric matrix A is stored, and how B has been factorized. = 'U': Upper triangular. = 'L': Lower triangular.
  • N — N is INTEGER. The order of the matrices A and B. N >= 0. (no negative orders, we don't do that here 🫶)
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, it's the symmetric matrix A. If UPLO = 'U', the leading n by n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, if INFO = 0, you get the transformed matrix, stored in the same format as A. iconic ✨

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is DOUBLE PRECISION array, dimension (LDB,N). The triangular factor from the Cholesky factorization of B, exactly as returned by DPOTRF.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🦋 subroutine dsygst 🦋

subroutine dsygst (integer ITYPE, character UPLO, integer N, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldb, * ) B, integer LDB, integer INFO)

DSYGST

Purpose: 🌸

DSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. the glow-up pipeline

If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T).

If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.

B must have been previously factorized as U**T*U or L*L**T by DPOTRF. no shortcuts, bestie 🎀

Parameters 📋

  • ITYPE — ITYPE is INTEGER. = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T); = 2 or 3: compute U*A*U**T or L**T*A*L.
  • UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored and B is factored as U**T*U. = 'L': Lower triangle of A is stored and B is factored as L*L**T.
  • N — N is INTEGER. The order of the matrices A and B. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, if INFO = 0, the transformed matrix, stored in the same format as A. 💅

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is DOUBLE PRECISION array, dimension (LDB,N). The triangular factor from the Cholesky factorization of B, as returned by DPOTRF.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🎀 subroutine dsyrfs 🎀

subroutine dsyrfs (character UPLO, integer N, integer NRHS, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,
    double precision, dimension( ldb, * ) B, integer LDB, double precision,
    dimension( ldx, * ) X, integer LDX, double precision, dimension( * )
    FERR, double precision, dimension( * ) BERR, double precision,
    dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYRFS

Purpose: 🌟

DSYRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and hands you error bounds plus backward error estimates for the solution. like, it's not just gonna hand you an answer and ghost — it tells you how much to trust it 🫶

Parameters 📋

  • UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored. = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, aka the number of columns of the matrices B and X. NRHS >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — AF is DOUBLE PRECISION array, dimension (LDAF,N). The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
  • LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). The details of the interchanges and the block structure of D, as determined by DSYTRF.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). The right hand side matrix B.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — X is DOUBLE PRECISION array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by DSYTRS. On exit, the improved solution matrix X. glow up, bestie ✨
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • FERR — FERR is DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. ngl it's a little dramatic, it overestimates 😭
  • BERR — BERR is DOUBLE PRECISION array, dimension (NRHS). The componentwise relative backward error of each solution vector X(j) — that's the smallest relative change in any element of A or B that makes X(j) an exact solution.
  • WORK — WORK is DOUBLE PRECISION array, dimension (3*N)
  • IWORK — IWORK is INTEGER array, dimension (N)
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Internal Parameters: 🔍

ITMAX is the maximum number of steps of iterative refinement.

Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

💜 subroutine dsyrfsx 💜

subroutine dsyrfsx (character UPLO, character EQUED, integer N, integer
    NRHS, double precision, dimension( lda, * ) A, integer LDA, double
    precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( *
    ) IPIV, double precision, dimension( * ) S, double precision, dimension(
    ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X,
    integer LDX, double precision RCOND, double precision, dimension( * )
    BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * )
    ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP,
    integer NPARAMS, double precision, dimension( * ) PARAMS, double
    precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer
    INFO)

DSYRFSX

Purpose: 🌈

DSYRFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and provides error bounds and backward error estimates for the solution. and — on top of the normwise error bound — the code also provides the maximum componentwise error bound if possible. see the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on those error bounds 👀

The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. In that case, the solution and error bounds returned are for the original unequilibrated system. it's giving ✨transformation✨

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but ngl, the defaults are usually giving exactly what you need and you can just vibe. If the defaults are acceptable, users can pass NPARAMS = 0, which prevents the source code from even touching the PARAMS argument. Zero touching, zero drama. 💅

✨ Parameters ✨

  • UPLO — which triangle are we even talking about bestie
UPLO is CHARACTER*1
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
  • EQUED — the receipts on what equilibration happened
EQUED is CHARACTER*1

Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly. Like, you can't do the math right if you don't know what already got scaled, iykyk.

  • = 'N': No equilibration
  • = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). The right hand side B has been changed accordingly.
  • N — the order of the matrix
N is INTEGER

The order of the matrix A. N >= 0.

  • NRHS — how many right hand sides we're serving
NRHS is INTEGER

The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

  • A — the actual symmetric matrix, main character
A is DOUBLE PRECISION array, dimension (LDA,N)

The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced (she's off duty). If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

  • LDA — leading dimension of A
LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

  • AF — the factored form, the glow-up version of A
AF is DOUBLE PRECISION array, dimension (LDAF,N)

The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.

  • LDAF — leading dimension of AF
LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

  • IPIV — the swap lore
IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D as determined by DSYTRF.

  • S — the scale factors, she's got range
S is DOUBLE PRECISION array, dimension (N)

The scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S). S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. If S is output, each element of S is a power of the radix. If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. So like, keep it to powers of the radix, bestie. Trust the process. 🦋

  • B — right hand side matrix, our beloved
B is DOUBLE PRECISION array, dimension (LDB,NRHS)

The right hand side matrix B.

  • LDB — leading dimension of B
LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

  • X — solution matrix, entering the chat
X is DOUBLE PRECISION array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by DGETRS. On exit, the improved solution matrix X. She got a glow-up mid-call, iconic behavior. ✨

  • LDX — leading dimension of X
LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

  • RCOND — reciprocal scaled condition number
RCOND is DOUBLE PRECISION

Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. Yeah she's giving singular energy but the error can still be cute, don't panic.

  • BERR — componentwise relative backward error
BERR is DOUBLE PRECISION array, dimension (NRHS)

Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).

  • N_ERR_BNDS — how many error bounds per right hand side
N_ERR_BNDS is INTEGER

Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below. 👇

  • ERR_BNDS_NORM — the normwise error tea
ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

  • err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon').
  • err = 2 — 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. Only trust her if the boolean said so, fr fr.
  • err = 3 — Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 📓

  • ERR_BNDS_COMP — the componentwise error tea, the sequel
ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned. Not requested? Then she simply doesn't get accessed. Boundaries, periodt.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

  • err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon').
  • err = 2 — 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 📓

  • NPARAMS — how many params you actually set
NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Set it to zero and she literally will not look at PARAMS. Ghosted. 💀

  • PARAMS — the algorithm parameter control panel
PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)

Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

  • PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not.
    • Default: 1.0D+0
      • = 0.0: No refinement is performed, and no error bounds are computed.
      • = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
      • (other values are reserved for future use)
  • PARAMS(LA_LINRX_ITHRESH_I = 2) — the maximum number of residual computations allowed for refinement. bestie this is your budget, spend it wisely 💸
    • Default: 10
    • Aggressive: set it to 100 to permit convergence using approximate factorizations or factorizations other than LU. okay but real talk — if the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. 😬 proceed accordingly.
  • PARAMS(LA_LINRX_CWISE_I = 3) — flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. it's giving yes/no energy 🫶
    • Default: 1.0 (attempt componentwise convergence)

WORK

WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

INFO is INTEGER

  • = 0: Successful exit. The solution to every right-hand side is guaranteed. slay ✨ every single RHS came through
  • < 0: If INFO = -i, the i-th argument had an illegal value. not the vibe 💀
  • > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
  • = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP. 👑

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine dsytd2 🦋

subroutine dsytd2 (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, double precision, dimension( * ) D, double
    precision, dimension( * ) E, double precision, dimension( * ) TAU,
    integer INFO)

DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). unblocked = it's doing the whole thing itself, no helpers, main character energy 💅

🎯 Purpose

DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.

🎀 Parameters

  • UPLO
    • UPLO is CHARACTER*1
    • Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
      • = 'U': Upper triangular
      • = 'L': Lower triangular
  • N
    • N is INTEGER
    • The order of the matrix A. N >= 0.
  • A
    • A is DOUBLE PRECISION array, dimension (LDA,N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
    • On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
  • LDA
    • LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • D
    • D is DOUBLE PRECISION array, dimension (N)
    • The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
  • E
    • E is DOUBLE PRECISION array, dimension (N-1)
    • The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
  • TAU
    • TAU is DOUBLE PRECISION array, dimension (N-1)
    • The scalar factors of the elementary reflectors (see Further Details).
  • INFO
    • INFO is INTEGER
      • = 0: successful exit
      • < 0: if INFO = -i, the i-th argument had an illegal value.

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔍 Further Details

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  d   e   v2  v3  v4 )              (  d                  )
  (      d   e   v3  v4 )              (  e   d              )
  (          d   e   v4 )              (  v1  e   d          )
  (              d   e  )              (  v1  v2  e   d      )
  (                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). 🌈

🌸 subroutine dsytf2 🌸

subroutine dsytf2 (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). pivoting but make it unblocked ✨

🎯 Purpose

DSYTF2 computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method:

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**T is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS. Level 2 BLAS bestie, she's not doing the fancy block stuff, she's just built different 🫶

🎀 Parameters

  • UPLO
    • UPLO is CHARACTER*1
    • Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
      • = 'U': Upper triangular
      • = 'L': Lower triangular
  • N
    • N is INTEGER
    • The order of the matrix A. N >= 0.
  • A
    • A is DOUBLE PRECISION array, dimension (LDA,N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
    • On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
  • LDA
    • LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • IPIV
    • IPIV is INTEGER array, dimension (N)
    • Details of the interchanges and the block structure of D.
    • If UPLO = 'U':
      • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
      • If IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    • If UPLO = 'L':
      • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
      • If IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • INFO
    • INFO is INTEGER
      • = 0: successful exit
      • < 0: if INFO = -k, the k-th argument had an illegal value
      • > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. 😭 do not do that

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔍 Further Details

If UPLO = 'U', then A = U*D*U**T, where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    v    0   )   k-s
   U(k) =  (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    0     0   )  k-1
   L(k) =  (   0    I     0   )  s
           (   0    v     I   )  n-k-s+1
              k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors 🫶✨

09-29-06 — patch from Bobby Cheng, MathWorks

Replace l.204 and l.372

        IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN

by

        IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN

01-01-96 — Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

1-96 — Based on modifications by J. Lewis, Boeing Computer Services Company

subroutine dsytf2_rk 💖

subroutine dsytf2_rk (character UPLO, integer N, double precision,
    dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E,
    integer, dimension( * ) IPIV, integer INFO)

DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). bestie, that's the whole vibe right there 💅

Purpose ✨

DSYTF2_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS. For more info, go peek at the Further Details section 👀

Parameters 🌸

UPLO

UPLO is CHARACTER*1

Tells you whether the upper or lower triangular part of the symmetric matrix A is stored:

  • = 'U': Upper triangular
  • = 'L': Lower triangular

N

N is INTEGER

The order of the matrix A. N >= 0.

A

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the symmetric matrix A.

  • If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
  • If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, contains:

  1. ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
  2. If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

LDA

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

E

E is DOUBLE PRECISION array, dimension (N)

On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where

  • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
  • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases. 💖

IPIV

IPIV is INTEGER array, dimension (N)

IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see the Further Details section 👑

If UPLO = 'U', ( in factorization order, k decreases from N to 1 ):

  • a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
  • b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
    1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
    2. If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
  • c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
  • d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L', ( in factorization order, k increases from 1 to N ):

  • a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
  • b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
    1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
    2. If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
  • c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
  • d) NOTE: Any entry IPIV(k) is always NONZERO on output.

INFO

INFO is INTEGER

  • = 0: successful exit 🎉
  • < 0: If INFO = -k, the k-th argument had an illegal value 😭
  • > 0: If INFO = k, the matrix A is singular, because:
    • If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
    • If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. 🚨

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.

Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Further Details 🔍

TODO: put further details

Contributors 🫶

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

01-01-96 — Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

subroutine dsytf2_rook 💅

subroutine dsytf2_rook (character UPLO, integer N, double precision,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
    integer INFO)

DSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). 🦋

Purpose ✨

okay so DSYTF2_ROOK is out here computing the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting method, and bestie, it's giving:

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**T is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. 💅

This is the unblocked version of the algorithm, calling Level 2 BLAS. lowkey humble of it.

🌸 Parameters 🌸

  • UPLO — UPLO is CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. 💭

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details, bestie, we get there).

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D. 🎀

If UPLO = 'U':

  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • INFO — INFO is INTEGER.
    • = 0: successful exit ✨
    • < 0: if INFO = -k, the k-th argument had an illegal value
    • > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. not the vibe, be careful bestie 😭

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 Further Details:

If UPLO = 'U', then A = U*D*U**T, where

   U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    v    0   )   k-s
   U(k) =  (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

   L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    0     0   )  k-1
   L(k) =  (   0    I     0   )  s
           (   0    v     I   )  n-k-s+1
              k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

💖 Contributors:

November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

🧚 subroutine dsytrd 🧚

subroutine dsytrd (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, double precision, dimension( * ) D, double
    precision, dimension( * ) E, double precision, dimension( * ) TAU,
    double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRD

Purpose:

DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation:

Q**T * A * Q = T.

periodt. ✨

🌈 Parameters 🌈

  • UPLO — UPLO is CHARACTER*1.
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. 👀

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • D — D is DOUBLE PRECISION array, dimension (N). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
  • E — E is DOUBLE PRECISION array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
  • TAU — TAU is DOUBLE PRECISION array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. 🔥

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

  • INFO — INFO is INTEGER.
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🍓 Further Details:

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors

   Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

   H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors

   Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

   H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U': if UPLO = 'L':

  (  d   e   v2  v3  v4 )              (  d                  )
  (      d   e   v3  v4 )              (  e   d              )
  (          d   e   v4 )              (  v1  e   d          )
  (              d   e  )              (  v1  v2  e   d      )
  (                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). 💜

dsytrd_2stage ✨

subroutine dsytrd_2stage (character VECT, character UPLO, integer N, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( * ) D, double precision, dimension( * ) E, double precision,
    dimension( * ) TAU, double precision, dimension( * ) HOUS2, integer
    LHOUS2, double precision, dimension( * ) WORK, integer LWORK, integer
    INFO)

DSYTRD_2STAGE 💖

Purpose 🎀

okay so bestie, what does she actually do. DSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric tridiagonal form T by a orthogonal similarity transformation:

Q1**T Q2**T* A * Q2 * Q1 = T.

that's the whole vibe. two-stage, two Q's, one tridiagonal queen 👑

Parameters 🌸

  • VECT — CHARACTER*1
    • = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
    • = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).
  • UPLO — CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER
    • The order of the matrix A. N >= 0. (bestie if it's negative we're not doing this 💀)
  • A — DOUBLE PRECISION array, dimension (LDA,N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
    • On exit, if UPLO = 'U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are over- written by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors. See Further Details.
  • LDA — INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • D — DOUBLE PRECISION array, dimension (N)
    • The diagonal elements of the tridiagonal matrix T. the main character of the whole operation fr 🫶
  • E — DOUBLE PRECISION array, dimension (N-1)
    • The off-diagonal elements of the tridiagonal matrix T.
  • TAU — DOUBLE PRECISION array, dimension (N-KD)
    • The scalar factors of the elementary reflectors of the first stage (see Further Details).
  • HOUS2 — DOUBLE PRECISION array, dimension (LHOUS2)
    • Stores the Householder representation of the stage2 band to tridiagonal.
  • LHOUS2 — INTEGER
    • The dimension of the array HOUS2.
    • If LWORK = -1, or LHOUS2 = -1, then a query is assumed; the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA.
    • If VECT='N', LHOUS2 = max(1, 4*n); if VECT='V', option not yet available.
  • WORK — DOUBLE PRECISION array, dimension (LWORK)
  • LWORK — INTEGER
    • The dimension of the array WORK. LWORK = MAX(1, dimension)
    • If LWORK = -1, or LHOUS2=-1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
    • LWORK = MAX(1, dimension) where
dimension   = max(stage1,stage2) + (KD+1)*N
            = N*KD + N*max(KD+1,FACTOPTNB)
              + max(2*KD*KD, KD*NTHREADS)
              + (KD+1)*N
  • where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice, NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.
  • INFO — INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

Author 🎀

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details: 💅

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

dsytrd_sy2sb 🦋

subroutine dsytrd_sy2sb (character UPLO, integer N, integer KD, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * )
    TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRD_SY2SB ✨

Purpose 🌈

DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric band-diagonal form AB by a orthogonal similarity transformation:

Q**T * A * Q = AB.

she's taking you from full symmetric to band-diagonal, no cap 💅

Parameters 💖

  • UPLO — CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER
    • The order of the matrix A. N >= 0.
  • KD — INTEGER
    • The number of superdiagonals of the reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
    • The reduced matrix is stored in the array AB.
  • A — DOUBLE PRECISION array, dimension (LDA,N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
    • On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
  • LDA — INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • AB — DOUBLE PRECISION array, dimension (LDAB,N)
    • On exit, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
  • LDAB — INTEGER
    • The leading dimension of the array AB. LDAB >= KD+1.
  • TAU — DOUBLE PRECISION array, dimension (N-KD)
    • The scalar factors of the elementary reflectors (see Further Details).
  • WORK — DOUBLE PRECISION array, dimension (LWORK)
    • On exit, if INFO = 0, or if LWORK=-1, WORK(1) returns the size of LWORK.
  • LWORK — INTEGER
    • The dimension of the array WORK which should be calculated by a workspace query. LWORK = MAX(1, LWORK_QUERY)
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
    • LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
    • where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting LWORK=-1 will provide the size of WORK.
  • INFO
and that's literally where this part cuts off bestie 😭 INFO's whole description is waiting for you in the next bit 🫶
  • INFO is INTEGER — the little status report at the end 💌
    • = 0: successful exit, we ate and left no crumbs ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭 (rude of it, honestly)

Author 🫶

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Further Details: 👑

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

okay bestie, now the shape of Q 💅 — if UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
  (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
  (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
  (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
  (                            ab    )              (  v1     v2     v3     ab/v4 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i)..fi

🌸 subroutine dsytrf 🌸

subroutine dsytrf (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision,
    dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF — say hi to the icon 👑

Purpose: ✨

DSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is

A = U**T*D*U  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS. (iykyk, that's the fast one 🔥)

Parameters 🎀

  • UPLO UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N N is INTEGER The order of the matrix A. N >= 0.
  • A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).

  • LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • IPIV IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • WORK WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK LWORK is INTEGER The length of WORK. LWORK >= 1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

  • INFO INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
bestie not the singular matrix 💀 if you see INFO > 0 do NOT go solving systems with that thing, division by zero is not the vibe.

Author 🫶

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Further Details: 👑

If UPLO = 'U', then A = U**T*D*U, where

   U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    v    0   )   k-s
   U(k) =  (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

   L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    0     0   )  k-1
   L(k) =  (   0    I     0   )  s
           (   0    v     I   )  n-k-s+1
              k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

🦋 subroutine dsytrf_aa 🦋

subroutine dsytrf_aa (character UPLO, integer N, double precision,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double
    precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF_AA — same energy, different algorithm, we love a sister routine 💖

Purpose: ✨

DSYTRF_AA computes the factorization of a real symmetric matrix A using the Aasen's algorithm. The form of the factorization is

A = U**T*T*U  or  A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a symmetric tridiagonal matrix.

This is the blocked version of the algorithm, calling Level 3 BLAS. (again with the Level 3 BLAS, iconic 🔥)

Parameters 🎀

  • UPLO UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N N is INTEGER The order of the matrix A. N >= 0.
  • A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the tridiagonal matrix is stored in the diagonals and the subdiagonals of A just below (or above) the diagonals, and L is stored below (or above) the subdiaonals, when UPLO is 'L' (or 'U').

  • LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • IPIV IPIV is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k).
  • WORK WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK LWORK is INTEGER The length of WORK. LWORK >= MAX(1,2*N). For optimum performance LWORK >= N*(1+NB), where NB is the optimal blocksize.
  • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value.

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine dsytrf_aa_2stage 🦋

subroutine dsytrf_aa_2stage (character UPLO, integer N, double precision,
    dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TB,
    integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
    IPIV2, double precision, dimension( * ) WORK, integer LWORK, integer
    INFO)

DSYTRF_AA_2STAGE

✨ Purpose: ✨

DSYTRF_AA_2STAGE computes the factorization of a real symmetric matrix A using the Aasen's algorithm. The form of the factorization is

A = U**T*T*U  or  A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a symmetric band matrix with the bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is LU factorized with partial pivoting).

This is the blocked version of the algorithm, calling Level 3 BLAS. periodt 💅

🌸 Parameters 🌸

  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER
    • The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
    • On exit, L is stored below (or above) the subdiagonal blocks, when UPLO is 'L' (or 'U').
  • LDA — LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • TB — TB is DOUBLE PRECISION array, dimension (LTB)
    • On exit, details of the LU factorization of the band matrix.
  • LTB — LTB is INTEGER
    • The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N.
    • If LTB = -1, then a workspace query is assumed; the routine only calculates the optimal size of LTB, returns this value as the first entry of TB, and no error message related to LTB is issued by XERBLA.
  • IPIV — IPIV is INTEGER array, dimension (N)
    • On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k).
  • IPIV2 — IPIV2 is INTEGER array, dimension (N)
    • On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV2(k).
  • WORK — WORK is DOUBLE PRECISION workspace of size LWORK
  • LWORK — LWORK is INTEGER
    • The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: if INFO = i, band LU factorization failed on i-th column

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎀 subroutine dsytrf_rk 🎀

subroutine dsytrf_rk (character UPLO, integer N, double precision,
    dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E,
    integer, dimension( * ) IPIV, double precision, dimension( * ) WORK,
    integer LWORK, integer INFO)

DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). bestie this one is A LOT, buckle up 🫶

✨ Purpose: ✨

DSYTRF_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section.

🌸 Parameters 🌸

  • UPLO — UPLO is CHARACTER*1
    • Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • N — N is INTEGER
    • The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N)
    • On entry, the symmetric matrix A.
      • If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
      • If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
    • On exit, contains:
      1. ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
      2. If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
  • LDA — LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • E — E is DOUBLE PRECISION array, dimension (N)
    • On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where
    • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
    • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
    • NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.
  • IPIV — IPIV is INTEGER array, dimension (N)
    • IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section.
    • If UPLO = 'U', ( in factorization order, k decreases from N to 1 ):
      1. A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
      2. A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
        • If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
        • If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
      3. In both cases a) and b), always ABS( IPIV(k) ) <= k.
      4. NOTE: Any entry IPIV(k) is always NONZERO on output.
    • If UPLO = 'L', ( in factorization order, k increases from 1 to N ):
      1. A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
      2. A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
        • If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
        • If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
      3. In both cases a) and b), always ABS( IPIV(k) ) >= k.
  • d) NOTE: any entry IPIV(k) is always NONZERO on output. like, always. bestie you can trust that one. 💅
  • WORK — WORK is a DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ). On exit, if INFO = 0, WORK(1) hands you back the optimal LWORK. ✨
  • LWORK — LWORK is INTEGER, and it's the length of WORK. Rule one: LWORK >=1. But if you want it to actually perform, LWORK >= N*NB, where NB is the block size returned by ILAENV. obsessed with that.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. it's giving "just window shopping, don't yell at me" 🛍️

  • INFO — INFO is INTEGER
    • = 0: successful exit 🎉
    • < 0: If INFO = -k, the k-th argument had an illegal value 😭
    • > 0: If INFO = k, the matrix A is singular, because:
      • If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
      • If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. division by zero. do not do that. that is not the vibe. 💀

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes. so it's like, it tells you about the first red flag and then just keeps going. 🚩

Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details: 🦋

 TODO: put correct description

not the TODO in the manpage 😭 they really just left it there fr fr. iconic honestly.

Contributors: 👑

  • December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine dsytrf_rook (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF_ROOK

Purpose: 🎀

DSYTRF_ROOK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. The form of the factorization is

   A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS. blocked = fast = we love her. 🔥

Parameters 🌈

  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N).

On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). she comes back different. glow up. ✨

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D.

If UPLO = 'U':

  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 💖
  • LWORK — LWORK is INTEGER. The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

  • INFO — INFO is INTEGER
    • = 0: successful exit 🫶
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. again: do NOT divide by that zero. she's singular and she means it.

Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details: 🧚

If UPLO = 'U', then A = U*D*U**T, where

   U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    v    0   )   k-s
   U(k) =  (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

   L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    0     0   )  k-1
   L(k) =  (   0    I     0   )  s
           (   0    v     I   )  n-k-s+1
              k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors: 👑

  • April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

subroutine dsytri (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer INFO)

DSYTRI

Purpose: 🦋

DSYTRI computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. she's the follow-up act. the encore. the "we already did the hard part, now we flip it." 🔥

Parameters 💜

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N).

On entry, the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.

On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.

bestie, she does not read the half of the matrix she doesn't need. she's efficient like that. 💅
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF.
  • WORK — WORK is DOUBLE PRECISION array, dimension (N).
  • INFO — INFO is INTEGER
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. no inverse for you. she said no. respectfully. 😭

Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌸 dsytri2 — the little setup bestie that hands off the real work

subroutine dsytri2 (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision,
    dimension( * ) WORK, integer LWORK, integer INFO)

Purpose 💖

DSYTRI2 computes the inverse of a DOUBLE PRECISION symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. And then — plot twist — DSYTRI2 sets the LEADING DIMENSION of the workspace before calling DSYTRI2X, because DSYTRI2X is the one that actually computes the inverse. So like, dsytri2 is the hype girl setting up the stage, and dsytri2x walks out and eats. 💅

Parameters 🎀

  • UPLO — it's a CHARACTER*1, bestie. It says whether the factorization details are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0. (Zero is allowed, no gatekeeping here. 👑)
  • A — A is a DOUBLE PRECISION array, dimension (LDA,N).
    • On entry: the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF. So it walks in carrying the receipts from DSYTRF. 🧾
    • On exit, if INFO = 0: the (symmetric) inverse of the original matrix. And here's the vibe check — if UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L', the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced. So like, half of A is literally just vibing and nobody's looking at it. 🙈
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is an INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF.
  • WORK — WORK is a DOUBLE PRECISION array, dimension (N+NB+1)*(NB+3). That's the exact shape, don't come at me with anything else. ✨
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. WORK is size >= (N+NB+1)*(NB+3).
    • If LWORK = -1, then a workspace query is assumed; the routine calculates:
      • the optimal size of the WORK array, returns this value as the first entry of the WORK array,
      • and no error message related to LWORK is issued by XERBLA.
    • Like, it's the "asking for a friend" mode. -1 and she just tells you the size, no drama, no XERBLA screaming. 🫶
  • INFO — INFO is INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 💀
    • > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. Yeah so that's the "it's giving singular, no inverse for you" situation. 😭

Author ✨

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 dsytri2x — the one who actually does the inverse

subroutine dsytri2x (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision,
    dimension( n+nb+1,* ) WORK, integer NB, integer INFO)

Purpose 💜

DSYTRI2X computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. No hand-off, no middleman, she just does it. Main character energy fr fr. 👑

Parameters 🌈

  • UPLO — CHARACTER*1, same deal as always. It specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • A — DOUBLE PRECISION array, dimension (LDA,N).
    • On entry: the NNB diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.
    • On exit, if INFO = 0: the (symmetric) inverse of the original matrix. And again — if UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L', the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced. Half of it is just there for the aesthetic, we don't read it. 🙈
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — INTEGER array, dimension (N). Details of the interchanges and the NNB structure of D as determined by DSYTRF.
  • WORK — DOUBLE PRECISION array, dimension (N+NB+1,NB+3).
  • NB — INTEGER. Block size. That's it, that's the whole description. Short queen. 🎀
  • INFO — INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 💀
    • > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. 😭

Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌟 dsytri_3 — the blocked version, Level 3 BLAS girlie

subroutine dsytri_3 (character UPLO, integer N, double precision, dimension(
    lda, * ) A, integer LDA, double precision, dimension( * ) E, integer,
    dimension( * ) IPIV, double precision, dimension( * ) WORK, integer
    LWORK, integer INFO)

Purpose ⭐

DSYTRI_3 computes the inverse of a real symmetric indefinite matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

And okay so — DSYTRI_3 sets the leading dimension of the workspace before calling DSYTRI_3X that actually computes the inverse. This is the blocked version of the algorithm, calling Level 3 BLAS. She's optimized, she's fast, she's THAT girl. 🔥

Parameters 🧚

  • UPLO — CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • A — DOUBLE PRECISION array, dimension (LDA,N).
    • On entry: the diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK:
      • a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and
      • b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
    • On exit, if INFO = 0: the symmetric inverse of the original matrix.
      • If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced;
      • If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • E — DOUBLE PRECISION array, dimension (N).
    • On entry: contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where
      • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
      • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
    • NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases. She's just sitting there unread, minding her business. 💀
  • IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK.
  • WORK — DOUBLE PRECISION array, dimension (N+NB+1)*(NB+3). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — INTEGER. The length of WORK. LWORK >= (N+NB+1)*(NB+3).
    • If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. Yep, it says "the optimal size of the optimal size" — we don't fix it, we report it, that's the whole brand. 🫶
  • INFO — INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 💀
    • > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. 😭

Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors: 🎀

November 2017,  Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

💜 dsytri_3x — blocked algorithm, part two, the sequel nobody asked for but everybody needed

subroutine dsytri_3x (character UPLO, integer N, double precision,
    dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E,
    integer, dimension( * ) IPIV, double precision, dimension( n+nb+1, * )
    WORK, integer NB, integer INFO)

Purpose ✨

DSYTRI_3X computes the inverse of a real symmetric indefinite matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS. Same energy as her sister, just doing the actual computing. 💅

Parameters 🌈

  • UPLO — CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • A — bestie hold that thought, this one keeps going in the next part 💅

On entry, the diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK:

  • a) ONLY the diagonal elements of the symmetric block diagonal matrix D sit on the diagonal of A, i.e. D(k,k) = A(k,k); (and the superdiagonal — or subdiagonal — elements of D should be handed to you on entry in the array E), and
  • b) if UPLO = 'U': factor U lives in the superdiagonal part of A. If UPLO = 'L': factor L lives in the subdiagonal part of A.

On exit, if INFO = 0, you get the symmetric inverse of the original matrix. 💅

  • If UPLO = 'U': the upper triangular part of the inverse is formed, and the part of A below the diagonal is not referenced;
  • If UPLO = 'L': the lower triangular part of the inverse is formed, and the part of A above the diagonal is not referenced.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • E — E is DOUBLE PRECISION array, dimension (N). On entry, it contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:
    • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
    • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
NOTE: for a 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both the UPLO = 'U' and UPLO = 'L' cases. like, neither. ok? 🙈
  • IPIV — IPIV is INTEGER array, dimension (N). All the tea on the interchanges and the block structure of D, as determined by DSYTRF_RK or DSYTRF_BK.
  • WORK — WORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3).
  • NB — NB is INTEGER. Block size. ✨
  • INFO — INFO is INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. she's singular bestie, no inverse for you 😭

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors: 👑

June 2017,  Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

subroutine dsytri_rook 🦋

subroutine dsytri_rook (character UPLO, integer N, double precision,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double
    precision, dimension( * ) WORK, integer INFO)

DSYTRI_ROOK 💜

Purpose:

DSYTRI_ROOK computes the inverse of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF_ROOK. iconic behavior honestly

Parameters 🎀

  • UPLO — UPLO is CHARACTER*1. Tells you whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF_ROOK.

On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced. ✨

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF_ROOK.
  • WORK — WORK is DOUBLE PRECISION array, dimension (N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. rip 😭

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors: 👑

 April 2012, Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                School of Mathematics,
                University of Manchester

subroutine dsytrs 🌸

subroutine dsytrs (character UPLO, integer N, integer NRHS, double
    precision, dimension( lda, * ) A, integer LDA, integer, dimension( * )
    IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer
    INFO)

DSYTRS 🧚

Purpose:

DSYTRS solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. solving systems, we love to see it

Parameters 🎀

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. the glow-up arc of arrays fr fr
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrs2 🦋

subroutine dsytrs2 (character UPLO, integer N, integer NRHS, double
    precision, dimension( lda, * ) A, integer LDA, integer, dimension( * )
    IPIV, double precision, dimension( ldb, * ) B, integer LDB, double
    precision, dimension( * ) WORK, integer INFO)

DSYTRS2 💅

Purpose:

DSYTRS2 solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF and converted by DSYCONV. note the plot twist, bestie

Parameters 🎀

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF.
Note that A is input / output. This might be counter-intuitive, and one may think that A is input only. A is input / output. This is because, at the start of the subroutine, we permute A in a 'better' form and then we permute A back to its original form at the end. she leaves, she comes back, she's the same girl 😌
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrs_3 🌈

subroutine dsytrs_3 (character UPLO, integer N, integer NRHS, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( * ) E, integer, dimension( * ) IPIV, double precision,
    dimension( ldb, * ) B, integer LDB, integer INFO)

DSYTRS_3 👑

Purpose:

DSYTRS_3 solves a system of linear equations A * X = B with a real symmetric matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK:

   A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is a unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This algorithm is using Level 3 BLAS. fast girl summer 🔥

Parameters 🎀

  • UPLO —
  • UPLO — it's CHARACTER*1, bestie. It specifies whether the details of the factorization are stored as an upper or lower triangular matrix:
    • = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
    • = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). Diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK:
    1. ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and
    2. If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • E — E is DOUBLE PRECISION array, dimension (N). On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where
    • If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
    • If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Contributors:

  • June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

DSYTRS_AA ✨

subroutine dsytrs_aa (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

Purpose:

DSYTRS_AA solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U**T*T*U or A = L*T*L**T computed by DSYTRF_AA.

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U**T*T*U;
    • = 'L': Lower triangular, form is A = L*T*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). Details of factors computed by DSYTRF_AA.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges as computed by DSYTRF_AA.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,3*N-2).
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

DSYTRS_AA_2STAGE 🎀

subroutine dsytrs_aa_2stage (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

Purpose:

DSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U**T*T*U or A = L*T*L**T computed by DSYTRF_AA_2STAGE.

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U**T*T*U;
    • = 'L': Lower triangular, form is A = L*T*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). Details of factors computed by DSYTRF_AA_2STAGE.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • TB — TB is DOUBLE PRECISION array, dimension (LTB). Details of factors computed by DSYTRF_AA_2STAGE.
  • LTB — LTB is INTEGER. The size of the array TB. LTB >= 4*N.
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges as computed by DSYTRF_AA_2STAGE.
  • IPIV2 — IPIV2 is INTEGER array, dimension (N). Details of the interchanges as computed by DSYTRF_AA_2STAGE.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

DSYTRS_ROOK 🦋

subroutine dsytrs_rook (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

Purpose:

DSYTRS_ROOK solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF_ROOK.

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
    • = 'U': Upper triangular, form is A = U*D*U**T;
    • = 'L': Lower triangular, form is A = L*D*L**T.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF_ROOK.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by DSYTRF_ROOK.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Contributors:

  • April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

DTGSYL 🌸

subroutine dtgsyl (character TRANS, integer IJOB, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( ldd, * ) D, integer LDD, double precision, dimension( lde, * ) E, integer LDE, double precision, dimension( ldf, * ) F, integer LDF, double precision SCALE, double precision DIF, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

Purpose:

bestie, the details for DTGSYL are literally about to start in the next part — stay ready 💅

so like, DTGSYL solves the generalized Sylvester equation:

            A * R - L * B = scale * C                 (1)
            D * R - L * E = scale * F

where R and L are the unknown m-by-n matrices, and (A, D), (B, E) and (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n, respectively, with real entries. bestie, listen: (A, D) and (B, E) must be in generalized (real) Schur canonical form, i.e. A, B are upper quasi triangular and D, E are upper triangular. no exceptions, that's the vibe check 🫶

The solution (R, L) overwrites (C, F). And 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow. protecting your numbers, as it should 💅

In matrix notation (1) is equivalent to solve Zx = scale b, where Z is defined as

                       Z = [ kron(In, A)  -kron(B**T, Im) ]         (2)
                           [ kron(In, D)  -kron(E**T, Im) ].

Here Ik is the identity matrix of size k and X**T is the transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. iconic math behavior, honestly ✨

If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b, which is equivalent to solve for R and L in

                        A**T * R + D**T * L = scale * C           (3)
                        R * B**T + L * E**T = scale * -F

This case (TRANS = 'T') is used to compute an one-norm-based estimate of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) and (B,E), using DLACON. iykyk 🌸

If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the reciprocal of the smallest singular value of Z. See [1-2] for more information. 📚

This is a level 3 BLAS algorithm. level 3 BLAS?! she ate. 🔥

🫶 Parameters 🫶

  • TRANS — TRANS is CHARACTER*1
    • = 'N': solve the generalized Sylvester equation (1).
    • = 'T': solve the 'transposed' system (3).
  • IJOB — IJOB is INTEGER. Specifies what kind of functionality to be performed.
    • = 0: solve (1) only.
    • = 1: The functionality of 0 and 3.
    • = 2: The functionality of 0 and 4.
    • = 3: Only an estimate of Dif[(A,D), (B,E)] is computed. (look ahead strategy IJOB = 1 is used).
    • = 4: Only an estimate of Dif[(A,D), (B,E)] is computed. ( DGECON on sub-systems is used ).
    • Not referenced if TRANS = 'T'.
  • M — M is INTEGER. The order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
  • N — N is INTEGER. The order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
  • A — A is DOUBLE PRECISION array, dimension (LDA, M). The upper quasi triangular matrix A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1, M).
  • B — B is DOUBLE PRECISION array, dimension (LDB, N). The upper quasi triangular matrix B.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1, N).
  • C — C is DOUBLE PRECISION array, dimension (LDC, N).
    • On entry, C contains the right-hand-side of the first matrix equation in (1) or (3).
    • On exit, if IJOB = 0, 1 or 2, C has been overwritten by the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R, the solution achieved during the computation of the Dif-estimate. she contains multitudes tbh
  • LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1, M).
  • D — D is DOUBLE PRECISION array, dimension (LDD, M). The upper triangular matrix D.
  • LDD — LDD is INTEGER. The leading dimension of the array D. LDD >= max(1, M).
  • E — E is DOUBLE PRECISION array, dimension (LDE, N). The upper triangular matrix E.
  • LDE — LDE is INTEGER. The leading dimension of the array E. LDE >= max(1, N).
  • F — F is DOUBLE PRECISION array, dimension (LDF, N).
    • On entry, F contains the right-hand-side of the second matrix equation in (1) or (3).
    • On exit, if IJOB = 0, 1 or 2, F has been overwritten by the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L, the solution achieved during the computation of the Dif-estimate.
  • LDF — LDF is INTEGER. The leading dimension of the array F. LDF >= max(1, M).
  • DIF — DIF is DOUBLE PRECISION. On exit DIF is the reciprocal of a lower bound of the reciprocal of the Dif-function, i.e. DIF is an upper bound of Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2). IF IJOB = 0 or TRANS = 'T', DIF is not touched.
  • SCALE — SCALE is DOUBLE PRECISION. On exit SCALE is the scaling factor in (1) or (3). If 0 < SCALE < 1, C and F hold the solutions R and L, resp., to a slightly perturbed system but the input matrices A, B, D and E have not been changed. If SCALE = 0, C and F hold the solutions R and L, respectively, to the homogeneous system with C = F = 0. Normally, SCALE = 1.
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 👑
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK > = 1.
    • If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. workspace query my beloved 🎀
  • IWORK — IWORK is INTEGER array, dimension (M+N+6).
  • INFO — INFO is INTEGER
    • =0: successful exit
    • <0: If INFO = -i, the i-th argument had an illegal value.
    • >0: (A, D) and (B, E) have common or close eigenvalues. not the slay we wanted, bestie — them eigenvalues are getting too close 😭

💖 Author 💖

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🌈 Contributors: 🌈

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. the OGs fr fr 🫶

📚 References: 📚

 [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
     for Solving the Generalized Sylvester Equation and Estimating the
     Separation between Regular Matrix Pairs, Report UMINF - 93.23,
     Department of Computing Science, Umea University, S-901 87 Umea,
     Sweden, December 1993, Revised April 1994, Also as LAPACK Working
     Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
     No 1, 1996.

 [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
     Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
     Appl., 15(4):1045-1060, 1994

 [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
     Condition Estimators for Solving the Generalized Sylvester
     Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
     July 1989, pp 745-751.

🔥 subroutine dtrsyl 🔥

subroutine dtrsyl (character TRANA, character TRANB, integer ISGN, integer
    M, integer N, double precision, dimension( lda, * ) A, integer LDA,
    double precision, dimension( ldb, * ) B, integer LDB, double precision,
    dimension( ldc, * ) C, integer LDC, double precision SCALE, integer
    INFO)

DTRSYL 💅

✨ Purpose: ✨

DTRSYL solves the real Sylvester matrix equation:

   op(A)*X + X*op(B) = scale*C or
   op(A)*X - X*op(B) = scale*C,

where op(A) = A or A**T, and A and B are both upper quasi-triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X. safety first, always 🦋

A and B must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. that's the law, bestie, no cap

🎀 Parameters 🎀

  • TRANA — TRANA is CHARACTER*1. Specifies the option op(A):
    • = 'N': op(A) = A (No transpose)
    • = 'T': op(A) = A**T (Transpose)
    • = 'C': op(A) = A**H (Conjugate transpose = Transpose)
  • TRANB — TRANB is CHARACTER*1. Specifies the option op(B):
    • = 'N': op(B) = B (No transpose)
    • = 'T': op(B) = B**T (Transpose)
    • = 'C': op(B) = B**H (Conjugate transpose = Transpose)
  • ISGN — ISGN is INTEGER. Specifies the sign in the equation:
    • = +1: solve op(A)*X + X*op(B) = scale*C
    • = -1: solve op(A)*X - X*op(B) = scale*C
  • M — M is INTEGER. The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.
  • N — N is INTEGER. The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,M). The upper quasi-triangular matrix A, in Schur canonical form.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B — B is DOUBLE PRECISION array, dimension (LDB,N). The upper quasi-triangular matrix B, in Schur canonical form.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • C — C is DOUBLE PRECISION array, dimension (LDC,N).
    • On entry, the M-by-N right hand side matrix C.
    • On exit, C is overwritten by the solution matrix X. glow up: right-hand side to solution ✨
  • LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M)
  • SCALE — SCALE is DOUBLE PRECISION. The scale factor, scale, set <= 1 to avoid overflow in X.
  • INFO — INFO is INTEGER
  • INFO is INTEGER — okay bestie, this is the lil' status report the routine hands back, and here's the whole vibe check 💅
    • = 0: successful exit ✨ everything slayed, no notes.
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭 like, argument number i was simply not it.
    • = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged).

👑 Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

✨ Author ✨

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