complex16POcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

, in debian bookworm

💖 complex16POcomputational(3) — LAPACK 💖

🌸 NAME 🌸

complex16POcomputational — complex16

✨ SYNOPSIS ✨

Functions

Okay bestie, here's the whole squad of PO matrix functions. No cap, they're all iconic. 💅

zla_porcond_c — double precision function zla_porcond_c (UPLO, N, A, LDA, AF, LDAF, C, CAPPLY, INFO, WORK, RWORK) — ZLA_PORCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian positive-definite matrices.
zla_porcond_x — double precision function zla_porcond_x (UPLO, N, A, LDA, AF, LDAF, X, INFO, WORK, RWORK) — ZLA_PORCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian positive-definite matrices.
zla_porfsx_extended — subroutine zla_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) — ZLA_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.
zla_porpvgrw — double precision function zla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF, WORK) — ZLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
zpocon — subroutine zpocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO) — ZPOCON
zpoequ — subroutine zpoequ (N, A, LDA, S, SCOND, AMAX, INFO) — ZPOEQU
zpoequb — subroutine zpoequb (N, A, LDA, S, SCOND, AMAX, INFO) — ZPOEQUB
zporfs — subroutine zporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) — ZPORFS
zporfsx — subroutine zporfsx (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, RWORK, INFO) — ZPORFSX
zpotf2 — subroutine zpotf2 (UPLO, N, A, LDA, INFO) — ZPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
zpotrf — subroutine zpotrf (UPLO, N, A, LDA, INFO) — ZPOTRF
zpotrf2 — recursive subroutine zpotrf2 (UPLO, N, A, LDA, INFO) — ZPOTRF2
zpotri — subroutine zpotri (UPLO, N, A, LDA, INFO) — ZPOTRI
zpotrs — subroutine zpotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO) — ZPOTRS

🦋 Detailed Description 🦋

This is the group of complex16 computational functions for PO matrices. Periodt. 💖

👑 Function Documentation 👑

💅 `zla_porcond_c` 💅

double precision function zla_porcond_c (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( * ) C, logical CAPPLY, integer INFO, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK)

OMG bestie, ZLA_PORCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian positive-definite matrices. It's giving math girlie energy. 🧮✨

Purpose:

ZLA_PORCOND_C computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector. No cap. 💯

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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDAF,N) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by ZPOTRF.
LDAF — LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).
C — C is DOUBLE PRECISION array, dimension (N) The vector C in the formula op(A) * inv(diag(C)).
CAPPLY — CAPPLY is LOGICAL If .TRUE. then access the vector C in the formula above.
INFO — INFO is INTEGER
- = 0: Successful exit.
- i > 0: The ith argument is invalid.
WORK — WORK is COMPLEX*16 array, dimension (2*N). Workspace.
RWORK — RWORK is DOUBLE PRECISION array, dimension (N). Workspace.

Author

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

💖 `zla_porcond_x` 💖

double precision function zla_porcond_x (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, complex*16, dimension( * ) X, integer INFO, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK)

ZLA_PORCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian positive-definite matrices. Another baddie in the lineup. 💅

Purpose:

ZLA_PORCOND_X computes the infinity norm condition number of op(A) * diag(X) where X is a COMPLEX*16 vector.

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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDAF,N) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by ZPOTRF.
LDAF — LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).
X — X is COMPLEX*16 array, dimension (N) The vector X in the formula op(A) * diag(X).
INFO — INFO is INTEGER
- = 0: Successful exit.
- i > 0: The ith argument is invalid.
WORK — WORK is COMPLEX*16 array, dimension (2*N). Workspace.
RWORK — RWORK is DOUBLE PRECISION array, dimension (N). Workspace.

Author

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

🦋 `zla_porfsx_extended` 🦋

subroutine zla_porfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, logical COLEQU, double precision, dimension( * ) C, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldy, * ) Y, integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, complex*16, dimension( * ) RES, double precision, dimension( * ) AYB, complex*16, dimension( * ) DY, complex*16, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)

ZLA_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. It's giving extra in the best way. ✨

Purpose:

ZLA_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 ZPORFSX 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.

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 COMPLEX*16 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 COMPLEX*16 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 ZPOTRF.
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 DOUBLE PRECISION array, dimension (N) The column scale factors for A. If COLEQU = .FALSE., C is not accessed. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.
B — B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by ZPOTRS. On exit, the improved solution matrix Y.
LDY — LDY is INTEGER
LDY — INTEGER 💫 the leading dimension of the array Y. Rules are rules: LDY >= max(1,N).
BERR_OUT — DOUBLE PRECISION array, dimension (NRHS). On exit, BERR_OUT(j) holds the componentwise relative backward error for right-hand-side j, straight from this formula:

max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) means the componentwise absolute value of the matrix or vector Z. This lil number is computed by ZLA_LIN_BERR, no cap 🫶

N_NORMS — INTEGER. Tells you which error bounds come back (peep ERR_BNDS_NORM and ERR_BNDS_COMP for the receipts).
- If N_NORMS >= 1 you get normwise error bounds.
- If N_NORMS >= 2 you get componentwise error bounds.
ERR_BNDS_NORM — DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For every right-hand side, this array is stuffed with info about various error bounds and condition numbers for the normwise relative error, which is defined like this:

        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, described below. Currently there are up to three pieces of info 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 these 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.

Heads up bestie: this subroutine is only responsible for setting that second field above. Nothing else. 💅

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

ERR_BNDS_COMP — DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array holds info about various error bounds and condition numbers for the componentwise relative error, defined as:

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

The array is indexed by the right-hand side i (the componentwise relative error depends on it), and by the type of error information described below. There are currently up to three pieces of info returned for each right-hand side. If componentwise accuracy isn't requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed — like, don't even look at it. 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 these 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.

Again, and I cannot stress this enough: this subroutine is only responsible for setting that second field above. ✨

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

RES — COMPLEX*16 array, dimension (N). Workspace that holds the intermediate residual. 🎀
AYB — DOUBLE PRECISION array, dimension (N). Workspace. Just vibing, holding space.
DY — COMPLEX*16 PRECISION array, dimension (N). Workspace to hold the intermediate solution.
Y_TAIL — COMPLEX*16 array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. the little crumbs fr fr 🙈
RCOND — DOUBLE PRECISION. Reciprocal scaled condition number. It's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if equilibration happened). If this is less than the machine precision — and especially if it's zero — then the matrix is singular to working precision. 💀 And note this: the error may still be small even if this number is very small and the matrix looks ill-conditioned. Not me warning you twice.
ITHRESH — INTEGER. The maximum number of residual computations allowed for refinement. Default is 10. For 'aggressive' set it to 100 so convergence can happen using approximate factorizations or factorizations that aren't LU. If the factorization uses a technique other than Gaussian elimination, then the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. 😭
RTHRESH — DOUBLE PRECISION. Decides when to stop refinement if the error estimate stops decreasing. Refinement stops 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. 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 — DOUBLE PRECISION. Decides when to start considering componentwise convergence. Componentwise convergence only gets 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. 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..
INFO — INTEGER. The status check bestie:
- = 0: Successful exit. 🎉
- < 0: if INFO = -i, the ith argument to ZPOTRS had an illegal value.

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 double precision function zla_porpvgrw

double precision function zla_porpvgrw (character*1 UPLO, integer NCOLS,
    complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(
    ldaf, * ) AF, integer LDAF, double precision, dimension( * ) WORK)

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

Purpose:

ZLA_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. That also means the solution X, the estimated condition numbers, and the error bounds could be unreliable. 🫠

Parameters

UPLO — CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
NCOLS — INTEGER. The number of columns of the matrix A. NCOLS >= 0.
A — COMPLEX*16 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 — COMPLEX*16 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 ZPOTRF.
LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
WORK — DOUBLE PRECISION array, dimension (2*N).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine zpocon

subroutine zpocon (character UPLO, integer N, complex*16, dimension( lda, *
    ) A, integer LDA, double precision ANORM, double precision RCOND,
    complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK,
    integer INFO)

ZPOCON ✨

Purpose:

ZPOCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPOTRF.

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

Parameters ✨

UPLO — bestie, this is a CHARACTER*1. If it's 'U', the upper triangle of A is stored; if it's 'L', the lower triangle of A is stored. 👑
N — N is an INTEGER. It's the order of the matrix A. N >= 0. ✨
A — A is a COMPLEX*16 array, dimension (LDA,N). It's the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by ZPOTRF. 💅
LDA — LDA is an INTEGER. It's the leading dimension of the array A. LDA >= max(1,N). 🦋
ANORM — ANORM is DOUBLE PRECISION. It's the 1-norm (or infinity-norm) of the Hermitian matrix A. 🌸
RCOND — RCOND is DOUBLE PRECISION. It's 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 a COMPLEX*16 array, dimension (2*N). 🎀
RWORK — RWORK is a DOUBLE PRECISION array, dimension (N). 🧚
INFO — INFO is an 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 zpoequ` ✨

   subroutine zpoequ (integer N, complex*16, dimension( lda, * ) A, integer
       LDA, double precision, dimension( * ) S, double precision SCOND, double
       precision AMAX, integer INFO)

ZPOEQU

Purpose:

ZPOEQU computes row and column scalings intended to equilibrate a Hermitian 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 an INTEGER. The order of the matrix A. N >= 0. ✨
A — A is a COMPLEX*16 array, dimension (LDA,N). The N-by-N Hermitian positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced. 👑
LDA — LDA is an INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🦋
S — S is a DOUBLE PRECISION array, dimension (N). If INFO = 0, S contains the scale factors for A. 🌸
SCOND — SCOND is DOUBLE PRECISION. If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S. 💖
AMAX — AMAX is DOUBLE PRECISION. 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 an 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 zpoequb` 💅

   subroutine zpoequb (integer N, complex*16, dimension( lda, * ) A, integer
       LDA, double precision, dimension( * ) S, double precision SCOND, double
       precision AMAX, integer INFO)

ZPOEQUB

Purpose:

ZPOEQUB computes row and column scalings intended to equilibrate a Hermitian 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 ZPOEQU 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 an INTEGER. The order of the matrix A. N >= 0. ✨
A — A is a COMPLEX*16 array, dimension (LDA,N). The N-by-N Hermitian positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced. 👑
LDA — LDA is an INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🦋
S — S is a DOUBLE PRECISION array, dimension (N). If INFO = 0, S contains the scale factors for A. 🌸
SCOND — SCOND is DOUBLE PRECISION. If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S. 💖
AMAX — AMAX is DOUBLE PRECISION. 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 an 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 zporfs` 🔥

   subroutine zporfs (character UPLO, integer N, integer NRHS, complex*16,
       dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF,
       integer LDAF, complex*16, dimension( ldb, * ) B, integer LDB,
       complex*16, dimension( ldx, * ) X, integer LDX, double precision,
       dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16,
       dimension( * ) WORK, double precision, dimension( * ) RWORK, integer
       INFO)

ZPORFS

Purpose:

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

Parameters:

UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 👑
N — N is an INTEGER. The order of the matrix A. N >= 0. ✨
NRHS — NRHS is an 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 COMPLEX*16 array, dimension (LDA,N). The Hermitian 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 an INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💖
AF — AF is a COMPLEX*16 array, dimension (LDAF,N). The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by ZPOTRF. 🎀
LDAF — LDAF is an INTEGER. The leading dimension of the array AF. LDAF >= max(1,N). 🧚
B — B is a COMPLEX*16 array, dimension (LDB,NRHS). The right hand side matrix B. 💜
LDB — LDB is an INTEGER. The leading dimension of the array B. LDB >= max(1,N). 🌈
X — X is a COMPLEX*16 array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by ZPOTRS. On exit, the improved solution matrix X. ⭐
LDX — LDX is an INTEGER. The leading dimension of the array X. LDX >= max(1,N). 💫
FERR — FERR is a DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. 💖
BERR — BERR is a DOUBLE PRECISION 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 a COMPLEX*16 array, dimension (2*N). 🦋
RWORK — RWORK is a DOUBLE PRECISION array, dimension (N). 🌸
INFO — INFO is an 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.

`zporfsx` 🫶

   subroutine zporfsx (character UPLO, character EQUED, integer N, integer
       NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16,
       dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( * )
       S, complex*16, dimension( ldb, * ) B, integer LDB, complex*16,
       dimension( ldx, * ) X, integer LDX, double precision RCOND, double
       precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision,
       dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, *
       ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension(*) PARAMS,
       complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK,
       integer INFO)

ZPORFSX ✨

Purpose 💡

Bestie, listen. ZPORFSX takes your already-computed solution to a linear system and makes it better — it improves that computed solution when the coefficient matrix is Hermitian positive definite, and it also hands you error bounds and backward error estimates for the solution. Like, it's not just fixing your answer, it's also telling you how much you should trust it. iconic behavior, honestly 🌈

So besides the normwise error bound, this routine also gives you the maximum componentwise error bound when it can. If you wanna know the deets on those error bounds, go read the comments for ERR_BNDS_NORM and ERR_BNDS_COMP — that's where the tea is. 🫖

Also! Your original system of linear equations might have been equilibrated before you called this routine — that's exactly what the EQUED and S arguments are describing below. And in that case, the solution and error bounds you get back are for the original unequilibrated system. So no confusion there, bestie. 💅

And then there's the whole optional-parameters situation: some optional parameters are bundled up in the PARAMS array. These settings determine how the refinement is performed — but ngl, the defaults are usually perfectly fine. If the defaults are acceptable to you, you can pass NPARAMS = 0, which stops the source code from ever touching the PARAMS argument at all. Efficient. We love. 🎀

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. You need this so the solution and error bounds get computed correctly — it's not optional vibes, it's load-bearing.

= '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 COMPLEX*16 array, dimension (LDA,N)

The Hermitian 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. (So like… only give it the half it asked for, bestie. The other half is giving nothing and it will not look at it.) 💀

LDA — LDA is INTEGER

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

AF — AF is COMPLEX*16 array, dimension (LDAF,N)

The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by ZPOTRF. 🧚

LDAF — LDAF is INTEGER

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

S — S is DOUBLE PRECISION array, dimension (N)

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

So yeah — if you're supplying S yourself, keep it powers-of-the-radix. That's the whole trick. Don't freestyle it. 🙈

B — B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by ZGETRS. On exit, the improved solution matrix X. Glow-up arc complete. 🦋

LDX — LDX is INTEGER

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

RCOND — RCOND is DOUBLE PRECISION

Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. But heads up — the error may still be small even if this number is very small and the matrix appears ill-conditioned. So don't panic prematurely, ok? 😭

BERR — BERR is DOUBLE PRECISION array, dimension (NRHS)

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

N_ERR_BNDS — N_ERR_BNDS is INTEGER

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

ERR_BNDS_NORM — ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

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

Normwise relative error in the ith solution vector:

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

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

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

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

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

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

ERR_BNDS_COMP — ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

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

Componentwise relative error in the ith solution vector:

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

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

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

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

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

See Lapack Working Note 165 for further details and extra cautions. 📖 (yes, again — same note, it's got range)

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. Set it and forget it, bestie. 🎀

PARAMS —

*PARAMS</b> is a DOUBLE PRECISION array, dimension NPARAMS. It's giving *algorithm parameters, bestie. Here's the deal: if any entry is < 0.0, that entry gets filled in with the default value for that parameter. Only positions up to NPARAMS are ever accessed, and anything higher-numbered just uses defaults. 💅

PARAMS(LA_LINRX_ITREF_I = 1) — are we doing iterative refinement or not?
- Default: 1.0D+0
- = 0.0: no refinement happens, and no error bounds get computed. Zero. Nada. 🫥
- = 1.0: use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment doesn't support DOUBLE PRECISION. (other values are reserved for future use 👀)
PARAMS(LA_LINRX_ITHRESH_I = 2) — the max number of residual computations allowed for refinement.
- Default: 10
- 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. Like, at all. 😬
PARAMS(LA_LINRX_CWISE_I = 3) — flag deciding whether 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 COMPLEX*16 array, dimension (2*N)

🦋 RWORK 🦋

RWORK is DOUBLE PRECISION array, dimension (2*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. Rude.
> 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 `zpotf2` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 🎀

ZPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm). It's the lowkey version — no blocking, just vibes.

✨ Purpose ✨

ZPOTF2 computes the Cholesky factorization of a complex Hermitian positive definite matrix A.

The factorization has the form

   A = U**H * U ,  if UPLO = 'U', or
   A = L  * L**H,  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 Hermitian 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 COMPLEX*16 array, dimension (LDA,N).
- On entry, the Hermitian 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**H *U or A = L*L**H.
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 `zpotrf` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 🦋

ZPOTRF — the moment has arrived.

✨ Purpose ✨

ZPOTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A.

The factorization has the form

   A = U**H * U,  if UPLO = 'U', or
   A = L  * L**H,  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. She's fast. She ate. 🔥

🌈 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 COMPLEX*16 array, dimension (LDA,N).
- On entry, the Hermitian 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**H *U or A = L*L**H.
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.

💜 recursive subroutine `zpotrf2` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 💜

ZPOTRF2 — she's recursive, she contains multitudes.

✨ Purpose ✨

ZPOTRF2 computes the Cholesky factorization of a Hermitian positive definite matrix A using the recursive algorithm.

The factorization has the form

   A = U**H * U,  if UPLO = 'U', or
   A = L  * L**H,  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. It's giving main character energy, honestly — she does the whole thing by calling herself. 🙈

🌈 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 COMPLEX*16 array, dimension (LDA,N).
- On entry, the Hermitian 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**H*U or A = L*L**H.
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 `zpotri` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 🍓

ZPOTRI — the glow-up from factorized to inverted. ✨

✨ Purpose ✨

ZPOTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPOTRF.

🌈 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 — okay bestie, this one is a COMPLEX*16 array, dimension (LDA,N). On entry, it's the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by ZPOTRF. On exit, the upper or lower triangle of the (Hermitian) inverse of A is literally overwriting the input factor U or L. 💅
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). no cap.
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,i) element of the factor U or L is zero, and the inverse could not be computed. 💀

✨ Author ✨

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

✨ `subroutine zpotrs` ✨

   subroutine zpotrs (character UPLO, integer N, integer NRHS, complex*16,
       dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B,
       integer LDB, integer INFO)
       ZPOTRS

Purpose 🎯

ZPOTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H * U or A = L * L**H computed by ZPOTRF. it's giving linear algebra realness.

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 matrix B. NRHS >= 0.
A — A is COMPLEX*16 array, dimension (LDA,N). The triangular factor U or L from the Cholesky factorization A = U**H * U or A = L * L**H, as computed by ZPOTRF.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
INFO — INFO is INTEGER.
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value 😭

✨ Author ✨

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

✨ Author ✨ (Doxygen's version)

Generated automatically by Doxygen for LAPACK from the source code.

This page is complex16POcomputational(3), version 3.11.0, dated Sun Nov 27 2022. periodt. 💅✨🫶

📦 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

complex16POcomputational(3)

💖 complex16POcomputational(3) — LAPACK 💖

🌸 NAME 🌸

✨ SYNOPSIS ✨

Functions

🦋 Detailed Description 🦋

👑 Function Documentation 👑

💅 zla_porcond_c 💅

💖 zla_porcond_x 💖

🦋 zla_porfsx_extended 🦋

🌸 Author 🌸

💖 double precision function zla_porpvgrw

Purpose:

Parameters

Author

🦋 subroutine zpocon

Purpose:

Parameters ✨

Author 👑

subroutine zpoequ ✨

Purpose:

Parameters:

Author 👑

subroutine zpoequb 💅

Purpose:

Parameters:

Author 👑

subroutine zporfs 🔥

Purpose:

Parameters:

Internal Parameters:

Author 👑

zporfsx 🫶

ZPORFSX ✨

Purpose 💡

Parameters 📋

💖 WORK 💖

🦋 RWORK 🦋

🌸 INFO 🌸

👑 Author 👑

🎀 subroutine zpotf2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO) 🎀

✨ Purpose ✨

🌈 Parameters 🌈

👑 Author 👑

🦋 subroutine zpotrf (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO) 🦋

✨ Purpose ✨

🌈 Parameters 🌈

👑 Author 👑

💜 recursive subroutine zpotrf2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO) 💜

✨ Purpose ✨

🌈 Parameters 🌈

👑 Author 👑

🍓 subroutine zpotri (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO) 🍓

✨ Purpose ✨

🌈 Parameters 🌈

✨ Author ✨

✨ subroutine zpotrs ✨

Purpose 🎯

Parameters 🧚

✨ Author ✨

✨ Author ✨ (Doxygen's version)

💅 `zla_porcond_c` 💅

💖 `zla_porcond_x` 💖

🦋 `zla_porfsx_extended` 🦋

`subroutine zpoequ` ✨

`subroutine zpoequb` 💅

`subroutine zporfs` 🔥

`zporfsx` 🫶

🎀 subroutine `zpotf2` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 🎀

🦋 subroutine `zpotrf` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 🦋

💜 recursive subroutine `zpotrf2` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 💜

🍓 subroutine `zpotri` (character `UPLO`, integer `N`, complex16, dimension( `lda`, ) `A`, integer `LDA`, integer `INFO`) 🍓

✨ `subroutine zpotrs` ✨