realPOcomputational(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

💖 realPOcomputational(3) 💖

okay so this is realPOcomputational from the LAPACK manual, section 3 — the library-calls girlies. bestie, it's giving real. that's the whole NAME, no cap. 💅

✨ SYNOPSIS ✨

🌈 Functions 🌈

  • real function sla_porcond (UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK, IWORK) — SLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix. 🎀
  • subroutine sla_porfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. literally the glow-up routine ✨
  • real function sla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF, WORK) — SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
  • subroutine spocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO) — SPOCON
  • subroutine spoequ (N, A, LDA, S, SCOND, AMAX, INFO) — SPOEQU
  • subroutine spoequb (N, A, LDA, S, SCOND, AMAX, INFO) — SPOEQUB
  • subroutine sporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) — SPORFS
  • subroutine sporfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) — SPORFSX
  • subroutine spotf2 (UPLO, N, A, LDA, INFO) — SPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
  • subroutine spotrf (UPLO, N, A, LDA, INFO) — SPOTRF
  • recursive subroutine spotrf2 (UPLO, N, A, LDA, INFO) — SPOTRF2
  • subroutine spotri (UPLO, N, A, LDA, INFO) — SPOTRI
  • subroutine spotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO) — SPOTRS

🌸 Detailed Description 🌸

This is the group of real computational functions for PO matrices. that's it, that's the tea ☕ — PO matrices, real computational functions, all together in one iconic little squad 🫶

👑 Function Documentation 👑

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

SLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix. 💖

🎯 Purpose

SLA_PORCOND 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 it's out here computing scaling factors R so that diag(R)*A*op2(C) is row equilibrated, then hitting it with the standard infinity-norm condition number. main character math behavior 💅

🔍 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF.
  • LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • CMODE — CMODE is INTEGER. Determines op2(C) in the formula op(A) * op2(C) as follows:
    • CMODE = 1 → op2(C) = C
    • CMODE = 0 → op2(C) = I
    • CMODE = -1 → op2(C) = inv(C)
  • C — C is REAL array, dimension (N). The vector C in the formula op(A) * op2(C).
  • INFO — INFO is INTEGER
    • = 0: Successful exit.
    • i > 0: The ith argument is invalid.
  • WORK — WORK is REAL array, dimension (3*N). Workspace.
  • IWORK — IWORK is INTEGER array, dimension (N). Workspace.

✍️ Author

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

🧚 subroutine sla_porfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, 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_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. 🦋

🎯 Purpose

SLA_PORFSX_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 SPORFSX 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.

okay so real talk: this one is called by SPORFSX to do the iterative refinement. it hands you a normwise error bound and, when it can, a maximum componentwise error bound too. and heads up — it's only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP, no more, no less. 🙈

🔍 Parameters

  • PREC_TYPE — PREC_TYPE is INTEGER. Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P
    • = 'S': Single
    • = 'D': Double
    • = 'I': Indigenous
    • = 'X' or 'E': Extra
  • 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.
  • NRHS — NRHS is INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B.
  • 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF.
  • LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • COLEQU — COLEQU is 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.
  • C — C is 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.
bestie listen — that last bit is a real warning, not a vibe. scale by anything that isn't a power of the radix and your error estimates may not be reliable. periodt. 💀
  • B — B is 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).
  • Y — Y is REAL array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by SPOTRS. On exit, the improved solution matrix Y.
  • LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
  • BERR_OUT — BERR_OUT is 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.

  • N_NORMS — N_NORMS is INTEGER. Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP).
    • If N_NORMS >= 1 return normwise error bounds.
    • If N_NORMS >= 2 return componentwise error bounds.
  • ERR_BNDS_NORM — ERR_BNDS_NORM 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 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 — the 'Guaranteed' error bound, bestie. It's the estimated forward error, and it's almost certainly within a factor of 10 of the true error as long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). Only trust this error bound if the previous boolean is true. periodt 💅
  • err = 3 — the reciprocal condition number: the estimated normwise reciprocal condition number. It gets compared with the threshold sqrt(n) * slamch('Epsilon') to decide 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. it's giving math girlie ✨

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. 📝 no cap.

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

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

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

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

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

  • RES — RES is REAL array, dimension (N). Workspace to hold the intermediate residual. 🛠️
  • AYB — AYB is REAL array, dimension (N). Workspace. This can be the same workspace passed for Y_TAIL. 🔁
  • DY — DY is REAL array, dimension (N). Workspace to hold the intermediate solution. 💡
  • Y_TAIL — Y_TAIL is REAL array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. ✨
  • 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. 😭 lowkey tragic.
  • ITHRESH — ITHRESH is INTEGER. The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. 💅
  • RTHRESH — RTHRESH is 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. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details. 📚
  • DZ_UB — DZ_UB is REAL. Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we define as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details. 🧐
  • IGNORE_CWISE — IGNORE_CWISE is LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. 🙈
  • INFO — INFO is INTEGER. = 0: Successful exit. < 0: if INFO = -i, the ith argument to SPOTRS had an illegal value. 💀

Author

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

real function sla_porpvgrw

   real function sla_porpvgrw (character*1 UPLO, integer NCOLS, real,
       dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF,
       integer LDAF, real, dimension( * ) WORK)

SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix. 🦋

Purpose:

SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. 😬

Parameters

  • UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 🔼🔽
  • NCOLS — NCOLS is INTEGER. The number of columns of the matrix A. NCOLS >= 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 🧮
  • LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N). 📏
  • WORK — WORK is REAL array, dimension (2*N). 🛠️

Author

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

subroutine spocon

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

SPOCON

Purpose:

SPOCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF.

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 — 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). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • ANORM — ANORM is REAL. The 1-norm (or infinity-norm) of the symmetric matrix A.
  • RCOND — 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 — WORK is REAL array, dimension (3*N).
  • IWORK — IWORK is INTEGER array, dimension (N).
  • INFO — INFO is INTEGER. = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

subroutine spoequ

   subroutine spoequ (integer N, real, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) S, real SCOND, real AMAX, integer INFO)

SPOEQU

Purpose:

Okay bestie, SPOEQU is the moment you need to compute row and column scalings to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm). 💅

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. ✨

Parameters 🧮

  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is REAL array, dimension (LDA,N). The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • S — S is REAL array, dimension (N). If INFO = 0, S contains the scale factors for A.
  • SCOND — SCOND is 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 — AMAX is REAL. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the i-th diagonal element is nonpositive.

Author 👑

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

subroutine spoequb — SPOEQUB 🦋

   subroutine spoequb (integer N, real, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) S, real SCOND, real AMAX, integer INFO)
       SPOEQUB

Purpose 💖

SPOEQUB computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

This routine differs from SPOEQU by restricting the scaling factors to a power of the radix. Barring over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix). 😭

Parameters 🧮

  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • A — A is REAL array, dimension (LDA,N). The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • S — S is REAL array, dimension (N). If INFO = 0, S contains the scale factors for A.
  • SCOND — SCOND is 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 — AMAX is REAL. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the i-th diagonal element is nonpositive.

Author 👑

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

subroutine sporfs — SPORFS 🔥

   subroutine sporfs (character UPLO, integer N, integer NRHS, real, dimension(
       lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF,
       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)
       SPORFS

Purpose 💖

SPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and provides error bounds and backward error estimates for the solution. No cap, it's giving accuracy. ✨

Parameters 🧮

  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • A — A is 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.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — AF is REAL array, dimension (LDAF,N). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF.
  • LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • B — B is 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 REAL array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by SPOTRS. On exit, the improved solution matrix X.
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • FERR — FERR is REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
  • BERR — BERR is 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).
  • WORK — WORK is REAL array, dimension (3*N).
  • IWORK — IWORK is INTEGER array, dimension (N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

Internal Parameters 🫶

ITMAX is the maximum number of steps of iterative refinement.

Author 👑

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

subroutine sporfsx — SPORFSX 🌈

   subroutine sporfsx (character UPLO, character EQUED, integer N, integer
       NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, *
       ) AF, integer LDAF, 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)
       SPORFSX

Purpose 💖

SPORFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and provides error bounds and backward error estimates for the solution. 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. 👑

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

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. ✨

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.
    • = '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

NRHS is INTEGER — the number of right hand sides, i.e., the number of columns of the matrices B and X. And it's gotta be NRHS >= 0, no negative nonsense allowed bestie 💅

A

A is REAL array, dimension (LDA,N) — this is your 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. So like, it's literally just sitting there unbothered, untouched, no thoughts head empty 🌸
  • 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.

So pick a side, babe. Half of it gets ignored either way 🙈

LDA

LDA is INTEGER — the leading dimension of the array A. Rule is LDA >= max(1,N). That's the vibe check, don't fail it 💀

AF

AF is REAL array, dimension (LDAF,N) — the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. She's already been through the factorization pipeline, she's ready 👑

LDAF

LDAF is INTEGER — the leading dimension of the array AF. And yes, LDAF >= max(1,N). Same energy, different array 🎀

S

S is REAL array, dimension (N) — the scale factors for A. Okay so this one's got layers, listen up:

  • If EQUED = 'Y', A is multiplied on the left and right by diag(S).
  • S is an input argument if FACT = 'F'; otherwise, S is an output argument.
  • If FACT = 'F' and EQUED = 'Y', each element of S must be positive.
  • If S is output, each element of S is a power of the radix.
  • If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows.
⚠️ Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. So like… it'll still run, it just might lie to you. And nobody wants a lying scale factor, fr fr 😭

B

B is REAL array, dimension (LDB,NRHS) — the right hand side matrix B. That's her. That's the one you're solving against 🦋

LDB

LDB is INTEGER — the leading dimension of the array B. LDB >= max(1,N), always, periodt 💅

X

X is REAL array, dimension (LDX,NRHS)

  • On entry: the solution matrix X, as computed by SGETRS.
  • On exit: the improved solution matrix X. ✨ glow up ✨

LDX

LDX is INTEGER — the leading dimension of the array X. LDX >= max(1,N). You know the drill by now 🎀

RCOND

RCOND is REAL — reciprocal scaled condition number. Basically it's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that was done).

If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision 💀

But bestie, hear me out — the error may still be small even if this number is very small and the matrix appears ill-conditioned. So don't panic-spiral immediately. It's giving "it's complicated" 😭

BERR

BERR is REAL array, dimension (NRHS) — componentwise relative backward error. Per solution vector X(j), this is the componentwise relative backward error (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). So like, how much would we have to fudge things for this to be perfect. Very real of it 🫶

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, they're coming, be patient 💖

ERR_BNDS_NORM

ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)

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

Normwise relative error in the *i*th 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 *i*th 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 📓

ERR_BNDS_COMP

ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)

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

Componentwise relative error in the *i*th 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. And if N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned. So like, sometimes she just doesn't show up to the party 🙈

The first index in ERR_BNDS_COMP(i,:) corresponds to the *i*th 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 📓 (yes, again — it's that important, go read it iykyk)

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. She said "I'll pick for myself, thanks" 💅

PARAMS

PARAMS is REAL array, dimension NPARAMS — specifies algorithm parameters.

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

  • 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. So like… turn it up to 100 if you're feeling brave, but the receipts get sketchy 😭
  • 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) — her little scratch pad, don't touch, she's working 🎀

IWORK

IWORK is INTEGER array, dimension (N) — same deal, but integer flavored ✨

INFO

okay so INFO is the very next thing and bestie… she has a lot to say. stay with me, it's coming 💖

  • INFO is INTEGER — bestie, this is the tea on what went down:
    • = 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 spotf2 ✨

   subroutine spotf2 (character UPLO, integer N, real, dimension( lda, * ) A,
       integer LDA, integer INFO)

SPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm). 💅

🎯 Purpose

SPOTF2 computes the Cholesky factorization of a real symmetric positive definite matrix A.

The factorization has the form

   A = U**T * U ,  if UPLO = 'U', or
   A = L  * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular.

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, if INFO = 0, the factor U or L from the Cholesky factorization A = U**T *U or A = L*L**T.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -k, the k-th argument had an illegal value
    • > 0: if INFO = k, the leading minor of order k is not positive definite, and the factorization could not be completed.

👑 Author

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

✨ subroutine spotrf ✨

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

SPOTRF SPOTRF VARIANT: top-looking block version of the algorithm, calling Level 3 BLAS.

🎯 Purpose

SPOTRF computes the Cholesky factorization of a real symmetric positive definite matrix A.

The factorization has the form

   A = U**T * U,  if UPLO = 'U', or
   A = L  * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS. 🔥

🔧 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 INFO = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.

👑 Author

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

🎯 Purpose (yes, again, bestie 💖)

SPOTRF computes the Cholesky factorization of a real symmetric positive definite matrix A.

The factorization has the form

   A = U**T * U,  if UPLO = 'U', or
   A = L  * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular.

This is the top-looking block version of the algorithm, calling Level 3 BLAS. 👑

🔧 Parameters (same energy, different day)

  • 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 INFO = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.

👑 Author (again)

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

📅 Date

December 2016

✨ recursive subroutine spotrf2 ✨

   recursive subroutine spotrf2 (character UPLO, integer N, real, dimension(
       lda, * ) A, integer LDA, integer INFO)

SPOTRF2

🎯 Purpose

SPOTRF2 computes the Cholesky factorization of a real symmetric positive definite matrix A using the recursive algorithm.

The factorization has the form

   A = U**T * U,  if UPLO = 'U', or
   A = L  * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular.

This is the recursive version of the algorithm. It divides the matrix into four submatrices:

                   [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
               A = [ -----|----- ]  with n1 = n/2
                   [  A21 | A22  ]       n2 = n-n1

The subroutine calls itself to factor A11. Update and scale A21 or A12, update A22 then call itself to factor A22. 🌀

🔧 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 INFO = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.

👑 Author

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

✨ subroutine spotri ✨

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

SPOTRI

🎯 Purpose

SPOTRI computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF. 🔄

🔧 Parameters

  • UPLO — UPLO is CHARACTER*1.
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N —
  • N — N is INTEGER. The order of the matrix A. N >= 0, bestie, always, no exceptions.
  • A — A is REAL array, dimension (LDA,N). On entry, it's the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, exactly as SPOTRF handed it over. On exit? The upper or lower triangle of the (symmetric) inverse of A just moves in and overwrites your input factor U or L. She said out with the old, in with the inverse. 💅
  • LDA — LDA is INTEGER. The leading dimension of the array A. Gotta be LDA >= max(1,N) or the whole thing falls apart.
  • INFO — INFO is INTEGER. The vibe report:
    • = 0: successful exit, we ate and left no crumbs ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value. Rude of you, honestly.
    • > 0: if INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. She's singular, she's not coming back, the math said no. 😭

🌸 Author 🌸

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

🦋 subroutine spotrs 🦋

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

SPOTRS

💖 Purpose 💖

SPOTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A, using the Cholesky factorization A = U**T*U or A = L*L**T that SPOTRF already computed for you. She's not doing the hard work again — she's just cashing in the factorization you already paid for. Periodt. 👑

✨ Parameters ✨

  • UPLO — UPLO is CHARACTER*1. Pick your lane:
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is REAL array, dimension (LDA,N). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. She's read-only energy, don't touch her.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. Glow up of the century, lowkey iconic. 🌈
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER:
    • = 0: successful exit, slay 🎀
    • < 0: if INFO = -i, the i-th argument had an illegal value. We've all been there.

🫶 Author 🫶

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

👑 Author 👑

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