realSYcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

okay bestie, sit down, hydrate, because we're doing realSYcomputational(3) from the LAPACK fam and it's a LOT 💖 this is part 1 of 15 so buckle up, we're not even close to done. no cap.

✨ NAME ✨

realSYcomputational — real

🌈 SYNOPSIS 🌈

Functions

subroutine sla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) — SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
real function sla_syrcond (UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK) — SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.
subroutine sla_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) — SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
real function sla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK) — SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
subroutine slasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) — SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine slasyf_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK) — SLASYF_AA
subroutine slasyf_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) — SLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine ssycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) — SSYCON
subroutine ssycon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) — SSYCON_ROOK
subroutine ssyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO) — SSYCONV
subroutine ssyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO) — SSYEQUB
subroutine ssygs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) — SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
subroutine ssygst (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) — SSYGST
subroutine ssyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) — SSYRFS
subroutine ssyrfsx (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) — SSYRFSX
subroutine ssytd2 (UPLO, N, A, LDA, D, E, TAU, INFO) — SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
subroutine ssytf2 (UPLO, N, A, LDA, IPIV, INFO) — SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine ssytf2_rook (UPLO, N, A, LDA, IPIV, INFO) — SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
subroutine ssytrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO) — SSYTRD
subroutine ssytrd_2stage (VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO) — SSYTRD_2STAGE
subroutine ssytrd_sy2sb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO) — SSYTRD_SY2SB
subroutine ssytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — SSYTRF
subroutine ssytrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — SSYTRF_AA
subroutine ssytrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO) — SSYTRF_AA_2STAGE
subroutine ssytrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — SSYTRF_ROOK
subroutine ssytri (UPLO, N, A, LDA, IPIV, WORK, INFO) — SSYTRI
subroutine ssytri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) — SSYTRI2
subroutine ssytri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO) — SSYTRI2X
subroutine ssytri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO) — SSYTRI_ROOK
subroutine ssytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) — SSYTRS
subroutine ssytrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO) — SSYTRS2
subroutine ssytrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO) — SSYTRS_AA
subroutine ssytrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO) — SSYTRS_AA_2STAGE
subroutine ssytrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) — SSYTRS_ROOK
subroutine stgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) — STGSYL
subroutine strsyl (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO) — STRSYL

💅 Detailed Description 💅

This is the group of real computational functions for SY matrices. that's the whole vibe. that's the assignment. 🫶

🔥 Function Documentation 🔥

💖 subroutine sla_syamv 💖

subroutine sla_syamv (integer UPLO, integer N, real ALPHA, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y, integer INCY)

SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. error bounds girlie 💅

Purpose:

SLA_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.

okay so this one is primarily used in calculating error bounds. and here's the tea — to protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. but 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. iykyk 🧚

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 REAL. On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
A — A is REAL 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 REAL 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 REAL. 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 REAL 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

💜 real function sla_syrcond 💜

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

SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix. she's fancy, she's measuring things, we love her 🎀

Purpose:

        SLA_SYRCOND estimates the Skeel condition number of  op(A) * op2(C)
        where op2 is determined by CMODE as follows
        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 computing scaling factors R such that
        diag(R)*A*op2(C) is row equilibrated and computing the standard
        infinity-norm condition number.

so basically: she's out here computing scaling factors R so that diag(R)*A*op2(C) is row equilibrated, then she computes the standard infinity-norm condition number. 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 number of linear equations, i.e., the order of the matrix A. N >= 0.
A — A is REAL 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 REAL array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF.
LDAF
LDAF — INTEGER</p>, bestie, and it's the leading dimension of the array AF. Rules are rules: LDAF >= max(1,N). 💅
IPIV — INTEGER array, dimension (N). It's holding all the deets about the interchanges and the block structure of D, exactly as SSYTRF figured them out. ✨
CMODE — INTEGER. This one decides what op2(C) is in the formula op(A) * op2(C), and the vibes are:

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

C — REAL array, dimension (N). It's literally the vector C in the formula op(A) * op2(C). 🎀
INFO — INTEGER, the little status report we all wait for:

  = 0:  Successful exit.
i > 0:  The ith argument is invalid.

WORK — REAL array, dimension (3*N). Workspace. It's giving "background character but essential". 🫶
IWORK — INTEGER array, dimension (N). Also workspace. Respect the grind. ✨

👑 Author

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

🦋 subroutine sla_syrfsx_extended

Okay bestie, deep breath, because this signature is a lot. Here it is, character for character, no notes:

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

SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. 😤 Iconic behaviour, honestly.

✨ Purpose ✨

SLA_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 SSYRFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. Note that this subroutine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. That's the whole assignment, bestie, don't do extra. 🎯

🌸 Parameters 🌸

PREC_TYPE — INTEGER. Says which intermediate precision we're using in refinement. The value comes from ILAPREC(P) where P is a CHARACTER, and:

P = 'S':  Single
    = 'D':  Double
    = 'I':  Indigenous
    = 'X' or 'E':  Extra

UPLO — CHARACTER*1, and it's a pick-your-triangle moment:

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

N — INTEGER. The number of linear equations, aka the order of the matrix A. And N >= 0, obviously. 📏
NRHS — INTEGER. The number of right-hand-sides, i.e. the number of columns of the matrix B. So many sides, so little time. 😭
A — REAL array, dimension (LDA,N). On entry, the N-by-N matrix A. It's the main character. 👑
LDA — INTEGER. Leading dimension of the array A. LDA >= max(1,N). 💅
AF — REAL array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF. It did its lil factorization arc and now it's here. ✨
LDAF — INTEGER. Leading dimension of the array AF. LDAF >= max(1,N). Periodt.
IPIV — INTEGER array, dimension (N). All the deets on the interchanges and the block structure of D, as determined by SSYTRF. 🫶
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 don't be sloppy with it. 🙈
C — REAL 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. Okay so that's a whole warning, noted and respected. ⚠️
B — REAL array, dimension (LDB,NRHS). The right-hand-side matrix B. Slay. 🎀
LDB — INTEGER. Leading dimension of the array B. LDB >= max(1,N). ✨
Y — REAL array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by SSYTRS. On exit, the improved solution matrix Y. Glow-up arc, literally. 🦋
LDY — INTEGER. Leading dimension of the array Y. LDY >= max(1,N). 💅
BERR_OUT — REAL 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 SLA_LIN_BERR. Give credit where it's due. 💜

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. More norms, more knowledge. 🌈
ERR_BNDS_NORM — REAL 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 — REAL 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.

okay so this next bit is the tail end of the SLA_SYRFSX_EXTENDED parameter squad — keep scrolling bestie, we're almost at the next function 💅

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

RES — REAL array, dimension (N). It's the workspace that holds the intermediate residual. 🫧
AYB — REAL array, dimension (N). Workspace! And here's the tea: this can literally be the same workspace you passed for Y_TAIL. Reuse queen ♻️
DY — REAL array, dimension (N). Workspace to hold the intermediate solution.
Y_TAIL — REAL array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. It's giving leftovers but make it fashion 🎀
RCOND — REAL. Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that even happened). If this is less than the machine precision — and yes, especially if it's straight-up zero — the matrix is singular to working precision. BUT note this: the error may still be small even if this number is very small and the matrix appears ill-conditioned. Plot twist 😭
ITHRESH — INTEGER. The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set it to 100 to permit convergence using approximate factorizations or factorizations other than LU. Heads up though: 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. no cap ⚠️
RTHRESH — REAL. 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. And RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set it to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details 📚
DZ_UB — REAL. Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, and "stable" here means the relative change in each component is less than DZ_UB. The default value is 0.25, which requires the first bit to be stable. See LAWN 165 for more details 🦋
IGNORE_CWISE — LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. She's a simple girl, she doesn't want the extra math 💅
INFO — INTEGER.
- = 0: Successful exit. 🎉
- < 0: if INFO = -i, the ith argument to SLA_SYRFSX_EXTENDED had an illegal value 💀

Author

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

🌟 `sla_syrpvgrw` 🌟

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

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

Purpose:

SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is the one being used here. And if this comes out much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. Which also means the solution X, the estimated condition numbers, and the error bounds could all be unreliable. Red flag, bestie 😬

Parameters

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.
INFO — INTEGER. The value of INFO returned from SSYTRF, .i.e., the pivot in column INFO is exactly 0. (Not me leaving that punctuation exactly how I found it ✋)
A — REAL 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 — REAL array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF.
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 SSYTRF.
WORK — REAL array, dimension (2*N).

Author

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

🧚 `slasyf` 🧚

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

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

Purpose:

SLASYF 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.

SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). Blocked code is the moment, honestly 🔥

Parameters

UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — INTEGER. The order of the matrix A. N >= 0.
NB — 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 — INTEGER. The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.
A — REAL 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — 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 — REAL array, dimension (LDW,NB).
LDW — INTEGER. The leading dimension of the array W. LDW >= max(1,N).
INFO — 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 💀

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 🫶

🌈 `slasyf_aa` 🌈

   subroutine slasyf_aa (character UPLO, integer J1, integer M, integer NB,
       real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
       real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WORK)

SLASYF_AA

Purpose:

DLATRF_AA factorizes a panel of a real symmetric matrix A using the Aasen's algorithm. The panel consists of a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L. (yes, it says DLATRF_AA in there — that's what the page says, and I am not touching it 💅)

In order to factorize the panel, the Aasen's algorithm requires the last row, or column, of the previous panel. The first row, or column, of A is set to be the first row, or column, of an identity matrix, which is used to factorize the first panel. Girl is borrowing from the panel before her, it's giving hand-me-downs but make it linear algebra 🎀

The resulting J-th row of U, or J-th column of L, is stored in the (J-1)-th row, or column, of A (without the unit diagonals), while the diagonal and subdiagonal of A are overwritten by those of T. Overwrite core ✨

Parameters

UPLO — CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
J1
J1 is INTEGER — bestie, this is the location of the first row (or column) of the panel inside the submatrix of A that gets passed to this routine. Like, when SSYTRF_AA calls it, for the first panel J1 is 1, while for the remaining panels J1 is 2. iykyk 💅
M is INTEGER — the dimension of the submatrix. M >= 0. periodt.
NB is INTEGER — the dimension of the panel to be facotorized. (their spelling, not mine 🫶)
A is REAL array, dimension (LDA,M) for the first panel, while dimension (LDA,M+1) for the remaining panels.

On entry, A contains the last row, or column, of the previous panel, and the trailing submatrix of A to be factorized, except for the first panel, only the panel is passed.

On exit, the leading panel is factorized.

LDA is INTEGER — the leading dimension of the array A. LDA >= max(1,M).
IPIV is INTEGER array, dimension (M) — details of the row and column interchanges, the row and column k were interchanged with the row and column IPIV(k).
H is REAL workspace, dimension (LDH,NB).
LDH is INTEGER — the leading dimension of the workspace H. LDH >= max(1,M).
WORK is REAL workspace, dimension (M).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`subroutine slasyf_rook (character UPLO, integer N, integer NB, integer KB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldw, * ) W, integer LDW, integer INFO)`

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

Purpose:

SLASYF_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.

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

Parameters:

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 is INTEGER — The order of the matrix A. N >= 0.
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 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 is REAL 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 is INTEGER — The leading dimension of the array A. LDA >= max(1,N).
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.

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 is REAL array, dimension (LDW,NB).
LDW is INTEGER — The leading dimension of the array W. LDW >= max(1,N).
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.

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 ssycon (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)`

SSYCON

Purpose:

SSYCON 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 SSYTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters:

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 is INTEGER — The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N) — The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF.
LDA is INTEGER — The leading dimension of the array A. LDA >= max(1,N).
IPIV is INTEGER array, dimension (N) — Details of the interchanges and the block structure of D as determined by SSYTRF.
ANORM is REAL — The 1-norm of the original matrix A.
RCOND is REAL — 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 is REAL array, dimension (2*N).
IWORK is INTEGER array, dimension (N).
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 ssycon_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)`

SSYCON_ROOK

Purpose:

SSYCON_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 SSYTRF_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))).

Parameters:

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 is INTEGER — The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N) — The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF_ROOK.
LDA is INTEGER — The leading dimension of the array A. LDA >= max(1,N).
IPIV is INTEGER array, dimension (N) — Details of the interchanges and the block structure of D as determined by SSYTRF_ROOK.
ANORM is REAL — The 1-norm of the original matrix A.
RCOND is REAL — 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 is REAL array, dimension (2*N).
IWORK is INTEGER array, dimension (N).
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:

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 `ssyconv` 🫶

subroutine ssyconv (character UPLO, character WAY, integer N, real,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real,
    dimension( * ) E, integer INFO)

SSYCONV

Purpose:

SSYCONV 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.

So basically: you already ran TRF and got a factorization, and now you need it to hop between the L-and-D form and back, bestie. It grabs the off-diagonal bits of D (they get handed back to you in the workspace), and it either applies or undoes the permutation that TRF did. ✨

Parameters

UPLO — 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.
WAY — CHARACTER*1. = 'C': Convert; = 'R': Revert. That's it, that's the whole direction situation. 💅
N — INTEGER. The order of the matrix A. N >= 0.
A — REAL array, dimension (LDA,N). It's the block diagonal matrix D and the multipliers used to obtain the factor U or L, exactly as computed by SSYTRF. Iconic little package deal.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by SSYTRF.
E — REAL array, dimension (N). E stores the supdiagonal/subdiagonal of the symmetric 1-by-1 or 2-by-2 block diagonal matrix D in LDLT.
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.

🦋 subroutine `ssyequb` 🦋

subroutine ssyequb (character UPLO, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) S, real SCOND, real AMAX, real,
    dimension( * ) WORK, integer INFO)

SSYEQUB

Purpose:

SSYEQUB 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.

tl;dr it's the glow-up routine for your matrix 🔥 — it finds row/column scalings that balance a symmetric matrix out (Euclidean-norm style) and calm its condition number down. The scale factors S come from the BIN algorithm (peep the references below), and the scaled matrix B lands with elements B(i,j) = S(i)*A(i,j)*S(j), whose condition number is within a factor N of the smallest possible one across all diagonal scalings. Fr fr, that's the guarantee.

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.
A — REAL array, dimension (LDA,N). The N-by-N symmetric matrix whose scaling factors are to be computed.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
S — REAL array, dimension (N). If INFO = 0, S contains the scale factors for A.
SCOND — REAL. 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.
AMAX — REAL. Largest absolute value of any matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
WORK — REAL array, dimension (2*N).
INFO — 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 `ssygs2` 🌸

subroutine ssygs2 (integer ITYPE, character UPLO, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer
    LDB, integer INFO)

SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).

Unblocked algorithm — meaning no blocking, no fancy tricks, just the straightforward path. Respect it. 🎀

Purpose:

SSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form.

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 SPOTRF.

Parameters

ITYPE — 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 — CHARACTER*1. Specifies 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 — INTEGER. The order of the matrices A and B. N >= 0.
A — REAL 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — REAL array, dimension (LDB,N). The triangular factor from the Cholesky factorization of B, as returned by SPOTRF.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
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.

🧚 subroutine `ssygst` 🧚

subroutine ssygst (integer ITYPE, character UPLO, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer
    LDB, integer INFO)

SSYGST

Purpose:

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

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 SPOTRF.

Sibling energy with SSYGS2, except this one is the blocked version. Same goal, more efficient execution. 👑

Parameters

ITYPE — 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 — 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 — INTEGER. The order of the matrices A and B. N >= 0.
A — REAL 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — REAL array, dimension (LDB,N). The triangular factor from the Cholesky factorization of B, as returned by SPOTRF.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
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.

💜 subroutine `ssyrfs` 💜

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

SSYRFS

Purpose:

SSYRFS 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.

Bestie this is the vibe check routine 😭 — you already solved the system, and now SSYRFS comes in to polish that solution up when the coefficient matrix is symmetric indefinite. Plus it hands you error bounds and backward error estimates, so you actually know how much to trust the answer. Periodt.

Parameters

UPLO
UPLO — bestie, UPLO is CHARACTER*1 and it tells us which half of A we're even looking at:

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

N — N is INTEGER, and it's the order of the matrix A. So like, how big is she. N >= 0. No negative sizes in this house 💅
NRHS — NRHS is INTEGER, the number of right hand sides, i.e. the number of columns of the matrices B and X. NRHS >= 0.
A — A is a REAL array, dimension (LDA,N), and it's the symmetric matrix A itself 🫶 If UPLO = 'U', then 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 (we literally don't look at it, she's not invited). 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. Pick a side, commit to the bit.
LDA — LDA is INTEGER, the leading dimension of the array A. LDA >= max(1,N). It's giving "please allocate enough rows" energy.
AF — AF is a REAL 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 SSYTRF. So like, SSYTRF already did the hard part and left AF here for us 💖
LDAF — LDAF is INTEGER, the leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is an INTEGER array, dimension (N). It's got the details of the interchanges and the block structure of D as determined by SSYTRF. Receipts, basically 🧾
B — B is a REAL array, dimension (LDB,NRHS). It's the right hand side matrix B. The other side of the equation, the bestie to A's main character.
LDB — LDB is INTEGER, the leading dimension of the array B. LDB >= max(1,N).
X — X is a REAL array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by SSYTRS. On exit, the improved solution matrix X. Glow-up arc ✨
LDX — LDX is INTEGER, the leading dimension of the array X. LDX >= max(1,N).
FERR — FERR is a REAL array, dimension (NRHS), the estimated forward error bound for each solution vector X(j) (that's the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), then 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). And ngl, the estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. She's dramatic like that.
BERR — BERR is a REAL array, dimension (NRHS), 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. Periodt.
WORK — WORK is a REAL array, dimension (3*N). Scratch space, do not perceive her.
IWORK — IWORK is an INTEGER array, dimension (N). More scratch. Also not for perceiving.
INFO — INFO is INTEGER, and she's the vibe check at the end:

= 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. That's it, that's the whole tea 🍵

👑 Author 👑

Absolute legends, all of them:

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

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

🌈 SSYRFSX 🌈

💖 Purpose 💖

Okay so SSYRFSX 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. 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 details of the error bounds — they spill the whole story there.

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. Important distinction, bestie 🚨

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. Lowkey the move when you just want the default vibe 💅

✨ Parameters ✨

UPLO — UPLO is CHARACTER*1:

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

EQUED — 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 — you can't skip this step, she needs to know what happened before she got here:

= '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 — 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 matrices B and X. NRHS >= 0.
A — A is a REAL 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. Same deal as before, no notes 🌸
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is a REAL 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 SSYTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is an INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by SSYTRF.
S — S is a REAL 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.

Okay this next bit matters, so listen up 🫶 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.

B — B is a REAL 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 a REAL array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix X. Another glow-up ✨
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is REAL. 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. So don't panic-spiral just because RCOND is tiny, bestie 😭

BERR — BERR is a REAL 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 — 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. Yes, both, she does both 💅
ERR_BNDS_NORM — ERR_BNDS_NORM is a REAL 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.

See Lapack Working Note 165 for further details and extra cautions. 📚 (yes bestie, there's homework, and it's actually worth reading)

ERR_BNDS_COMP 💅

ERR_BNDS_COMP is REAL 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 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.

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

NPARAMS

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Iconic "no thoughts, just defaults" behaviour, honestly. ✨

PARAMS 🎀

PARAMS is REAL 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: 1.0
- = 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) : Maximum number of residual computations allowed for refinement.
- Default: 10
- 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.

⚠️ Bestie, that "may no longer be trustworthy" is not a drill. Going aggressive here means the guarantees stop guaranteeing. 🫠

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.
- Default: 1.0 (attempt componentwise convergence)

WORK

WORK is REAL array, dimension (4*N) — it's giving "bring your own scratch space" 💀

IWORK

IWORK is INTEGER array, dimension (N)

INFO 🔥

INFO is INTEGER

= 0: Successful exit. The solution to every right-hand side is guaranteed. 👑
< 0: If INFO = -i, the i-th argument had an illegal value.
> 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 ssytd2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, integer INFO)

subroutine ssytd2 (character UPLO, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) D, real, dimension( * ) E, real,
    dimension( * ) TAU, integer INFO)

SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). Unblocked, so it's the slow-and-steady version — but she gets the job DONE. 💅

Purpose: ✨

SSYTD2 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 REAL 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 REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
E — E is REAL 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 REAL 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 stashed on exit in A(1:i-1,i+1), and tau lives in TAU(i). bestie, that's just where the pieces go, no cap.

If UPLO = 'L', then the matrix Q is a product of elementary reflectors, like:

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). iykyk 💅

The contents of A on exit are illustrated by these 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). literally the receipts of your factorization, right there in the array 💖

✨ subroutine ssytf2 ✨

subroutine ssytf2 (character UPLO, integer N, real, dimension( lda, * ) A,
    integer LDA, integer, dimension( * ) IPIV, integer INFO)

SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). iconic behavior honestly 👑

Purpose:

SSYTF2 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. so like, the low-key version, still ate tho 🔥

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 REAL 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. the tea on who swapped with who 🫖

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. 💀 like bestie, do not pass go, do not solve that system.

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. SISNAN(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

the lore ✨ the history ✨ the receipts 💜

✨ subroutine ssytf2_rook ✨

subroutine ssytf2_rook (character UPLO, integer N, real, dimension( lda, * )
    A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). same energy, different pivot strategy 👑

Purpose:

SSYTF2_ROOK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting method:

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

This is the unblocked version of the algorithm, calling Level 2 BLAS. 🦋

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 REAL 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) < 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 interchanged, 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 interchanged, 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. 💀

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    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 ssytrd` ✨

subroutine ssytrd (character UPLO, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) D, real, dimension( * ) E, real,
    dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer
    INFO)

SSYTRD

💅 Purpose 💅

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

🌸 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 REAL 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 REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
E — E is REAL 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 REAL array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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).

✨ `subroutine ssytrd_2stage` ✨

subroutine ssytrd_2stage (character VECT, character UPLO, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real,
    dimension( * ) E, real, dimension( * ) TAU, real, dimension( * ) HOUS2,
    integer LHOUS2, real, dimension( * ) WORK, integer LWORK, integer INFO)

SSYTRD_2STAGE

💅 Purpose 💅

SSYTRD_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.

🌸 Parameters 🌸

VECT — VECT is 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 — 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 REAL 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
D — D is REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T.
E — E is REAL array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix T.
TAU — TAU is REAL array, dimension (N-KD). The scalar factors of the elementary reflectors of the first stage (see Further Details).
HOUS2 — HOUS2 is REAL array, dimension (LHOUS2). Stores the Householder representation of the stage2 band to tridiagonal.
LHOUS2 — LHOUS2 is 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 — WORK is REAL array, dimension (LWORK)
LWORK
LWORK — INTEGER, the dimension of the array WORK. LWORK = MAX(1, dimension)

And listen — if LWORK = -1, or LHOUS2=-1, bestie, then we're in workspace query mode 🕵️‍♀️. The routine only calculates the optimal size of the WORK array, hands that value back as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. Iconic behaviour, honestly.

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 💅 — and NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

INFO — INTEGER, the receipts 📋

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

Implemented by Azzam Haidar. ✨

All details are available on technical report, SC11, SC13 papers. Like, go read them, they ate:

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

🦋 subroutine ssytrd_sy2sb

Ok new function bestie, get comfy 🛋️

subroutine ssytrd_sy2sb (character UPLO, integer N, integer KD, real,
    dimension( lda, * ) A, integer LDA, real, dimension( ldab, * ) AB,
    integer LDAB, real, dimension( * ) TAU, real, dimension( * ) WORK,
    integer LWORK, integer INFO)

SSYTRD_SY2SB

✨ Purpose

SSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric band-diagonal form AB by a orthogonal similarity transformation:

Q**T * A * Q = AB.

the way that it just... transforms ur whole matrix. obsessed 💖

🎀 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.
KD — KD is 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 — A is REAL 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).
AB — AB is REAL 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).

no cap, this indexing is a whole vibe check 🫠

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD+1.
TAU — TAU is REAL array, dimension (N-KD). The scalar factors of the elementary reflectors (see Further Details). 💅
WORK — WORK is REAL array, dimension (LWORK). On exit, if INFO = 0, or if LWORK=-1, WORK(1) returns the size of LWORK. 🧚
LWORK — LWORK is 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 — 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

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

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 ssytrf

ok ANOTHER one, we're on a roll bestie 🎢

subroutine ssytrf (character UPLO, integer N, real, dimension( lda, * ) A,
    integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK,
    integer LWORK, integer INFO)

SSYTRF

✨ Purpose

SSYTRF 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. she's fast, she's blocked, she's serving 🔥

🎀 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 a REAL 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). ✨

LDA — is INTEGER. It's the leading dimension of the array A. So like, LDA >= max(1,N), no cap. 💅
IPIV — is an INTEGER array, dimension (N). It's got the details of the interchanges and the block structure of D. So 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 — is a REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, then WORK(1) returns the optimal LWORK. 🛠️
LWORK — is INTEGER. It's the length of WORK. So 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 — is INTEGER. It's giving:
- = 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. 💀

Author 👑

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

Further Details 🔎

Okay so here's the tea on the factorization:

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 ssytrf_aa` (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)

SSYTRF_AA

Purpose 🎯

SSYTRF_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.

Parameters 📋

UPLO — is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 💖
N — is INTEGER. The order of the matrix A. N >= 0.
A — is a REAL 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 — is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
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 — is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal 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 — 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 ssytrf_aa_2stage` (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, real, dimension( * ) WORK, integer LWORK, integer INFO)

SSYTRF_AA_2STAGE

Purpose 🎯

SSYTRF_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.

Parameters 📋

UPLO — is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 💖
N — is INTEGER. The order of the matrix A. N >= 0.
A — is a REAL 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 subdiaonal blocks, when UPLO is 'L' (or 'U'). ✨
LDA — is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
TB — is REAL array, dimension (LTB). On exit, details of the LU factorization of the band matrix. 📦
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 — 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 — 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 IPIV(k). 🔄
WORK — is REAL workspace of size 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 `ssytrf_rook` 🦋

subroutine ssytrf_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)

SSYTRF_ROOK

Purpose 💖

SSYTRF_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. slay ✨

Parameters 🎀

UPLO — is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — is INTEGER
- The order of the matrix A. N >= 0.
A — is REAL 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 — is INTEGER
- The leading dimension of the array A. LDA >= max(1,N).
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 interchanged, 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 interchanged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

WORK — is REAL array, dimension (MAX(1,LWORK)).
- On exit, if INFO = 0, WORK(1) returns the optimal 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 — 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 listen — that INFO > 0 case is not a cute lil warning, it's a real one. The factorization did finish, but D is singular, and the moment you try to solve a system with it, you're dividing by zero. 🫠

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 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

Contributors 🌸

June 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 `ssytri` 🦋

subroutine ssytri (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer INFO)

SSYTRI

Purpose 💖

SSYTRI 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 SSYTRF.

Parameters 🎀

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 — is INTEGER
- The order of the matrix A. N >= 0.
A — is REAL 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 SSYTRF.
- 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 — is INTEGER
- The leading dimension of the array A. LDA >= max(1,N).
IPIV — is INTEGER array, dimension (N)
- Details of the interchanges and the block structure of D as determined by SSYTRF.
WORK — is REAL array, dimension (N)
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.

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine `ssytri2` 🦋

subroutine ssytri2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)

SSYTRI2

Purpose 💖

SSYTRI2 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 SSYTRF. SSYTRI2 sets the LEADING DIMENSION of the workspace before calling SSYTRI2X that actually computes the inverse.

Parameters 🎀

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 — is INTEGER
- The order of the matrix A. N >= 0.
A — is REAL 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 SSYTRF.
- 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 — is INTEGER
- The leading dimension of the array A. LDA >= max(1,N).
IPIV — is INTEGER array, dimension (N)
- Details of the interchanges and the block structure of D as determined by SSYTRF.
WORK — is REAL array, dimension (N+NB+1)*(NB+3)
LWORK
LWORK — INTEGER, and it's literally just the dimension of the array WORK. Size check, bestie: WORK is size >= (N+NB+1)*(NB+3). And if LWORK = -1, we're in a workspace query situation — the routine then calculates:
- the optimal size of the WORK array, and returns this value as the first entry of the WORK array,
- and no error message related to LWORK is issued by XERBLA. 🫶
INFO — INTEGER, the receipts:
- = 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.

✨ subroutine `ssytri2x` ✨

subroutine ssytri2x (character UPLO, integer N, real, dimension( lda, * ) A,
    integer LDA, integer, dimension( * ) IPIV, real, dimension( n+nb+1,* )
    WORK, integer NB, integer INFO)

SSYTRI2X — yes that's the whole name, no notes.

🎯 Purpose

SSYTRI2X 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 SSYTRF. It's giving "we already did the hard part, now we flip it." 💅

🎀 Parameters

UPLO — 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 — INTEGER. The order of the matrix A. N >= 0.
A — REAL 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 SSYTRF. 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. (Half of this array is basically just vibing in the background, unbothered. 🙈)
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 SSYTRF.
WORK — REAL array, dimension (N+NB+1,NB+3).
NB — INTEGER. Block size.
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.

🌸 subroutine `ssytri_rook` 🌸

subroutine ssytri_rook (character UPLO, integer N, real, dimension( lda, * )
    A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK,
    integer INFO)

SSYTRI_ROOK — iconic name, no cap.

🎯 Purpose

SSYTRI_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 SSYTRF_ROOK. 🦋

🎀 Parameters

UPLO — 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 — INTEGER. The order of the matrix A. N >= 0.
A — REAL 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 SSYTRF_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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by SSYTRF_ROOK.
WORK — REAL array, dimension (N).
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

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 `ssytrs` 💜

subroutine ssytrs (character UPLO, integer N, integer NRHS, real, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension(
    ldb, * ) B, integer LDB, integer INFO)

SSYTRS — the solver era begins. ✨

🎯 Purpose

SSYTRS 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 SSYTRF. Found the X, periodt. 🔥

🎀 Parameters

UPLO — 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 — INTEGER. 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. NRHS >= 0.
A — REAL array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by SSYTRF.
B — REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. (Glow up. 💅)
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
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.

🎀 subroutine `ssytrs2` 🎀

subroutine ssytrs2 (character UPLO, integer N, integer NRHS, real,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real,
    dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, integer
    INFO)

SSYTRS2 — the sequel, and honestly it ate. 🌈

🎯 Purpose

SSYTRS2 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 SSYTRF and converted by SSYCONV.

🎀 Parameters

UPLO — 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 — INTEGER. 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. NRHS >= 0.
A — REAL array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF.

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. (It's giving "she left, she came back, she's still her." 🦋)

LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by SSYTRF.
B — REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
WORK — REAL array, dimension (N).
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.

⭐ subroutine `ssytrs_aa` ⭐

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

SSYTRS_AA — the _AA girlies always have the most ✨interesting✨ factorizations.

🎯 Purpose

SSYTRS_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 SSYTRF_AA. 🌸

🎀 Parameters

UPLO —
UPLO — CHARACTER*1, bestie. Tells us whether the factorization details are chilling as an upper or a lower triangular matrix.

= 'U':  Upper triangular, form is A = U**T*T*U;
= 'L':  Lower triangular, form is A = L*T*L**T.

N — INTEGER. The order of the matrix A. Rules are N >= 0, no exceptions, periodt.
NRHS — INTEGER. How many right hand sides we're serving, aka the number of columns of the matrix B. NRHS >= 0.
A — REAL array, dimension (LDA,N). It's holding the details of the factors that SSYTRF_AA computed. iconic behavior tbh.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N), don't play with it.
IPIV — INTEGER array, dimension (N). All the deets of the interchanges, exactly as SSYTRF_AA computed them.
B — REAL array, dimension (LDB,NRHS). On entry, it's the right hand side matrix B. On exit? It's the solution matrix X. glow-up arc ✨
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
WORK — REAL array, dimension (MAX(1,LWORK)). It's the workspace, it does the vibes behind the scenes 🫶
LWORK — INTEGER. The dimension of the array WORK. And the rule is LWORK >= max(1,3*N-2). write that down bestie.
INFO — INTEGER. The receipts:

= 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 `ssytrs_aa_2stage`

subroutine ssytrs_aa_2stage (character UPLO, integer N, integer NRHS, real,
    dimension( lda, * ) A, integer LDA, real, dimension( * ) TB, integer
    LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, real,
    dimension( ldb, * ) B, integer LDB, integer INFO)

SSYTRS_AA_2STAGE — the two-stage girlie 💅

✨ Purpose

SSYTRS_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 SSYTRF_AA_2STAGE.

🌸 Parameters

UPLO — 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 — INTEGER. 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. NRHS >= 0.
A — REAL array, dimension (LDA,N). Details of factors computed by SSYTRF_AA_2STAGE.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
TB — REAL array, dimension (LTB). Details of factors computed by SSYTRF_AA_2STAGE.
LTB — INTEGER. The size of the array TB. LTB >= 4*N.
IPIV — INTEGER array, dimension (N). Details of the interchanges as computed by SSYTRF_AA_2STAGE.
IPIV2 — INTEGER array, dimension (N). Details of the interchanges as computed by SSYTRF_AA_2STAGE. yes there are two of them, she contains multitudes 🫶
B — REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
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.

🌈 subroutine `ssytrs_rook`

subroutine ssytrs_rook (character UPLO, integer N, integer NRHS, real,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real,
    dimension( ldb, * ) B, integer LDB, integer INFO)

SSYTRS_ROOK — she's serving rook pivoting energy 👑

✨ Purpose

SSYTRS_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 SSYTRF_ROOK.

🌸 Parameters

UPLO — 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 — INTEGER. 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. NRHS >= 0.
A — REAL array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF_ROOK.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by SSYTRF_ROOK.
B — REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
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.

💖 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 `stgsyl`

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

STGSYL — generalized Sylvester equation hours, and honestly? she ate 🔥

✨ Purpose

STGSYL solves the generalized Sylvester equation:

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

where R and L are unknown m-by-n matrices, (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. (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.

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow.

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.

If TRANS = 'T', STGSYL 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 SLACON.

If IJOB >= 1, STGSYL 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. flex but make it linear algebra 💅

🌸 Parameters

TRANS — CHARACTER*1.

= 'N': solve the generalized Sylvester equation (1).
= 'T': solve the 'transposed' system (3).

IJOB — 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.
     ( SGECON on sub-systems is used ).

Not referenced if TRANS = 'T'. so don't even worry about it in that case bestie

M — INTEGER. The order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
N — INTEGER. The order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
A — REAL array, dimension (LDA, M). The upper quasi triangular matrix A.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1, M).
B — REAL array, dimension (LDB, N). The upper quasi triangular matrix B.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1, N).
C — REAL 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's doing double duty and she's not even complaining 🫶
LDC — INTEGER. The leading dimension of the array C. LDC >= max(1, M).
D —
D — REAL array, dimension (LDD, M). The upper triangular matrix D. 💅
LDD — INTEGER. The leading dimension of the array D. LDD >= max(1, M).
E — REAL array, dimension (LDE, N). The upper triangular matrix E. ✨
LDE — INTEGER. The leading dimension of the array E. LDE >= max(1, N).
F — REAL 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 — INTEGER. The leading dimension of the array F. LDF >= max(1, M).
DIF — REAL. 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 — REAL. 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 — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — 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.
IWORK — INTEGER array, dimension (M+N+6).
INFO — 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.

✨ 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.

📚 References: 📚

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.
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
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 strsyl 🦋

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

STRSYL

✨ Purpose: ✨

STRSYL 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.

A and B must be in Schur canonical form (as returned by SHSEQR), 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.

🎀 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 REAL 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 REAL 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 REAL 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.
LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M).
SCALE — SCALE is REAL. The scale factor, scale, set <= 1 to avoid overflow in X.
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 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 🌈

Generated automatically by Doxygen for LAPACK from the source code.

That's the whole vibe, bestie. This was LAPACK Version 3.11.0, dated Sun Nov 27 2022, from realSYcomputational(3). Slay. 💅✨

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

realSYcomputational(3)

✨ NAME ✨

🌈 SYNOPSIS 🌈

Functions

💅 Detailed Description 💅

🔥 Function Documentation 🔥

💖 subroutine sla_syamv 💖

Purpose:

Parameters

Author

Further Details:

💜 real function sla_syrcond 💜

Purpose:

Parameters

👑 Author

🦋 subroutine sla_syrfsx_extended

✨ Purpose ✨

🌸 Parameters 🌸

🌟 sla_syrpvgrw 🌟

🧚 slasyf 🧚

🌈 slasyf_aa 🌈

Author

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

Author

Contributors:

subroutine ssycon (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

Author

subroutine ssycon_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

👑 Author 👑

🫶 subroutine ssyconv 🫶

🦋 subroutine ssyequb 🦋

🌸 subroutine ssygs2 🌸

🧚 subroutine ssygst 🧚

💜 subroutine ssyrfs 💜

✨ Internal Parameters ✨

👑 Author 👑

🌈 SSYRFSX 🌈

💖 Purpose 💖

✨ Parameters ✨

ERR_BNDS_COMP 💅

NPARAMS

PARAMS 🎀

WORK

IWORK

INFO 🔥

Author 🫶

subroutine ssytd2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, integer INFO)

Purpose: ✨

Parameters

Author 🫶

Further Details: 🧚

✨ subroutine ssytf2 ✨

Purpose:

Parameters

Author

Further Details:

Contributors:

✨ subroutine ssytf2_rook ✨

Purpose:

Parameters

Author

Further Details:

🫶 Contributors 🫶

✨ subroutine ssytrd ✨

💅 Purpose 💅

🌸 Parameters 🌸

👑 Author 👑

🦋 Further Details 🦋

✨ subroutine ssytrd_2stage ✨

💅 Purpose 💅

🌸 Parameters 🌸

🫶 Author

🔥 Further Details

🦋 subroutine ssytrd_sy2sb

✨ Purpose

🎀 Parameters

🫶 Author

🔥 Further Details

🌈 subroutine ssytrf

✨ Purpose

🌟 `sla_syrpvgrw` 🌟

🧚 `slasyf` 🧚

🌈 `slasyf_aa` 🌈

`subroutine slasyf_rook (character UPLO, integer N, integer NB, integer KB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldw, * ) W, integer LDW, integer INFO)`

`subroutine ssycon (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)`

`subroutine ssycon_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)`

🫶 subroutine `ssyconv` 🫶

🦋 subroutine `ssyequb` 🦋

🌸 subroutine `ssygs2` 🌸

🧚 subroutine `ssygst` 🧚

💜 subroutine `ssyrfs` 💜

✨ `subroutine ssytrd` ✨

✨ `subroutine ssytrd_2stage` ✨

`subroutine ssytrf_aa` (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)

`subroutine ssytrf_aa_2stage` (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, real, dimension( * ) WORK, integer LWORK, integer INFO)

🦋 subroutine `ssytrf_rook` 🦋

🦋 subroutine `ssytri` 🦋

🦋 subroutine `ssytri2` 🦋

✨ subroutine `ssytri2x` ✨

🌸 subroutine `ssytri_rook` 🌸

💜 subroutine `ssytrs` 💜

🎀 subroutine `ssytrs2` 🎀

⭐ subroutine `ssytrs_aa` ⭐

🦋 subroutine `ssytrs_aa_2stage`

🌈 subroutine `ssytrs_rook`

⭐ subroutine `stgsyl`