complexPOcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

🎀 complexPOcomputational(3) 🎀

okay so bestie, welcome to the complex computational glow-up zone for PO matrices ✨ this is a LAPACK page, it's long, and we're doing it in 6 parts — this is part 1, so buckle up 💅

✨ NAME ✨

complexPOcomputational — complex

like. that's it. that's the name. she's mysterious, she's complex, she contains multitudes 🦋

🌸 SYNOPSIS 🌸

💜 Functions

real function cla_porcond_c (UPLO, N, A, LDA, AF, LDAF, C, CAPPLY, INFO, WORK, RWORK) — CLA_PORCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian positive-definite matrices.
real function cla_porcond_x (UPLO, N, A, LDA, AF, LDAF, X, INFO, WORK, RWORK) — CLA_PORCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian positive-definite matrices.
subroutine cla_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) — CLA_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.
real function cla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF, WORK) — CLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
subroutine cpocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO) — CPOCON
subroutine cpoequ (N, A, LDA, S, SCOND, AMAX, INFO) — CPOEQU
subroutine cpoequb (N, A, LDA, S, SCOND, AMAX, INFO) — CPOEQUB
subroutine cporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) — CPORFS
subroutine cporfsx (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) — CPORFSX
subroutine cpotf2 (UPLO, N, A, LDA, INFO) — CPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
subroutine cpotrf (UPLO, N, A, LDA, INFO) — CPOTRF
recursive subroutine cpotrf2 (UPLO, N, A, LDA, INFO) — CPOTRF2
subroutine cpotri (UPLO, N, A, LDA, INFO) — CPOTRI
subroutine cpotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO) — CPOTRS

🌈 Detailed Description 🌈

This is the group of complex computational functions for PO matrices. that's the whole squad, right there, all of them, iconic behavior 💖

👑 Function Documentation 👑

🦋 real function cla_porcond_c

real function cla_porcond_c (character UPLO, integer N, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,
    real, dimension( * ) C, logical CAPPLY, integer INFO, complex,
    dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_PORCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian positive-definite matrices. she's doing math and she's doing it flawlessly ✨

🍓 Purpose:

CLA_PORCOND_C Computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL 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 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 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 CPOTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
C — C is REAL 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 array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

🌸 Author

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

🧚 real function cla_porcond_x

real function cla_porcond_x (character UPLO, integer N, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,
    complex, dimension( * ) X, integer INFO, complex, dimension( * ) WORK,
    real, dimension( * ) RWORK)

CLA_PORCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian positive-definite matrices. her sister but make it diag 💅

🍓 Purpose:

CLA_PORCOND_X Computes the infinity norm condition number of op(A) * diag(X) where X is a COMPLEX 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 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 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 CPOTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
X — X is COMPLEX 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 array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

🌸 Author

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

🔥 subroutine cla_porfsx_extended

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

CLA_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. extra-precise?? she said "let me do it AGAIN but better" and honestly? ate 💅

🍓 Purpose:

CLA_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 CPORFSX 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 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 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 CPOTRF.
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.

⚠️ okay so this one is a whole warning paragraph, and it's real: if you scale by stuff that isn't a power of the radix, you get rounding errors, and then you're refining against a matrix that isn't the one you started with, and your error estimates might just be lying to you. bestie. respect the radix. 🫶

B — B is COMPLEX 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 array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by CPOTRS. 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 a REAL array, dimension (NRHS). On exit, BERR_OUT(j) holds 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 CLA_LIN_BERR.

N_NORMS is INTEGER. It decides 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 is a REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

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

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

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

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

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

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

ERR_BNDS_COMP is a REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the 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 is a COMPLEX array, dimension (N). Workspace to hold the intermediate residual.

AYB is a REAL array, dimension (N). Workspace.

DY is a COMPLEX array, dimension (N). Workspace to hold the intermediate solution.

Y_TAIL is a COMPLEX array, dimension (N). Workspace to hold the trailing bits of the intermediate solution.

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.

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

INFO is INTEGER

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

👑 Author 👑

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

🦋 real function `cla_porpvgrw` 🦋

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

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

🎯 Purpose 🎯

CLA_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 COMPLEX 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 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 CPOTRF.
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 `cpocon` 🌸

subroutine cpocon (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real ANORM, real RCOND, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CPOCON

🎯 Purpose 🎯

CPOCON 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 CPOTRF.

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 — N is INTEGER. it's the order of the matrix A. and like, N >= 0, bestie, no negative orders in this house 💅
A — A is COMPLEX array, dimension (LDA,N). it's holding the triangular factor U or L, the one you got from the Cholesky factorization A = U**H*U or A = L*L**H, exactly as computed by CPOTRF. that's the lore 🫶
LDA — LDA is INTEGER. the leading dimension of the array A. so LDA >= max(1,N), okay? that's the rule, don't fight it ✨
ANORM — ANORM is REAL. it's the 1-norm (or the infinity-norm, your pick) of the Hermitian matrix A 🌸
RCOND — RCOND is REAL. the reciprocal of the condition number of matrix A, and it's computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed right here in this routine. periodt 🎀
WORK — WORK is COMPLEX array, dimension (2*N) 🦋
RWORK — RWORK is REAL array, dimension (N) 💜
INFO — INFO is INTEGER:
- = 0: successful exit, we ate 💅
- < 0: if INFO = -i, the i-th argument had an illegal value, not iconic of it honestly

👑 Author

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

`subroutine cpoequ (integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)` 💖

CPOEQU

🎯 Purpose

CPOEQU 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. like, that's the whole glow-up plan ✨

Parameters

N — N is INTEGER. the order of the matrix A. N >= 0 🦋
A — A is COMPLEX array, dimension (LDA,N). the N-by-N Hermitian positive definite matrix whose scaling factors are getting computed. heads up bestie: 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). and if SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S. so like, don't do extra work for no reason, that's the vibe check 🌸
AMAX — AMAX is REAL. the absolute value of the largest matrix element. if AMAX is very close to overflow or very close to underflow, the matrix should be scaled. lowkey urgent 🚨
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 cpoequb (integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)` 🌈

CPOEQUB

🎯 Purpose

CPOEQUB 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.

and here's where it gets different, bestie — this routine differs from CPOEQU 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). it's giving "same goal, different font" 💅

Parameters

N — N is INTEGER. the order of the matrix A. N >= 0 🦋
A — A is COMPLEX array, dimension (LDA,N). the N-by-N Hermitian positive definite matrix whose scaling factors are to be computed. and again: 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 cporfs (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)` 💖

CPORFS

🎯 Purpose

CPORFS 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. it's literally the "okay but make it more accurate" routine 🫶

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 COMPLEX 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 half of it matters, the other half is just vibes 🌸
LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N) ✨
AF — AF is COMPLEX 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 CPOTRF 🎀
LDAF — LDAF is INTEGER. the leading dimension of the array AF. LDAF >= max(1,N) 💜
B — B is COMPLEX 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 array, dimension (LDX,NRHS). on entry, the solution matrix X, as computed by CPOTRS. on exit, the improved solution matrix X. glow-up complete ✨
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. so it's a lil dramatic but in a safe direction 🫶
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 COMPLEX array, dimension (2*N) 🎀
RWORK — RWORK is REAL 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. that's the grind limit, bestie 💅

👑 Author

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

subroutine cporfsx (character UPLO, character EQUED, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, real, dimension( * ) S, complex, dimension( ldb, * ) B, integer LDB, complex, 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, complex, dimension( ) WORK, real, dimension( * ) RWORK, integer INFO) 🌈

CPORFSX

🎯 Purpose

CPORFSX 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. 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. it's giving overachiever, and we love that for it 💅

Quick heads up before we get into the args, bestie: 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.

Also — 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. like, it's not optional info, it literally changes the math, okurrr

= '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 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.

(tl;dr only one half of her is showing and the other half is simply not invited 💅)

🫶 `LDA`

LDA is INTEGER

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

🌸 `AF`

AF is COMPLEX 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 CPOTRF.

💖 `LDAF`

LDAF is INTEGER

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

⭐ `S`

S is REAL array, dimension (N)

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

okay bestie this one is a whole situation 😭 power-of-the-radix scaling is the safe lane. If you roll in with a non-power-of-the-radix S as input, the scaling itself can introduce rounding errors — and then you're refining against a matrix that isn't equivalent to your input matrix, and the error estimates you get back may not be reliable. Not a vibe.

🧚 `B`

B is COMPLEX 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 array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix X.

🌟 `LDX`

LDX is INTEGER

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

🔥 `RCOND`

RCOND is REAL

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

so like... tiny RCOND = she's in trouble, but not necessarily doomed. read the note. 🙈

💀 `BERR`

BERR is REAL array, dimension (NRHS)

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

🎀 `N_ERR_BNDS`

N_ERR_BNDS is INTEGER

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

🌈 `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 — '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 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.

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

💖 `NPARAMS`

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.

🦋 `PARAMS`

PARAMS is REAL array, dimension NPARAMS

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

PARAMS(LA_LINRX_ITREF_I = 1) — Whether to perform iterative refinement or not.
- Default: 1.0
- = 0.0: No refinement is performed, and no error bounds are computed.
- = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
- (other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) — max number of residual computations you're allowed to do for refinement, bestie.
- Default: 10
- Aggressive: set it to 100 so it can converge using approximate factorizations or factorizations that aren't LU. > ⚠️ Heads up tho: if the factorization uses something 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 that decides whether the code will try to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
- Default: 1.0 (yes bestie, attempt componentwise convergence 💅)
WORK — WORK is COMPLEX array, dimension (2*N)
RWORK — RWORK is REAL 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
- > 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 cpotf2

subroutine cpotf2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).

✨ Purpose:

CPOTF2 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. no cap, it's the little one. 🐣

🎀 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 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 cpotrf

subroutine cpotrf (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

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

✨ Purpose:

CPOTRF 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. glow-up era fr fr 🔥

🎀 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 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.

✨ Purpose:

CPOTRF computes the Cholesky factorization of a real symmetric 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 top-looking 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 COMPLEX 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**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.

📅 Date

December 2016

🧚 recursive subroutine cpotrf2

recursive subroutine cpotrf2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CPOTRF2

✨ Purpose:

CPOTRF2 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 calls itself to factor A22. it's giving recursion, it's giving main character energy 👑

🎀 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 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 — LDA is INTEGER. It's the leading dimension of the array A, bestie. The rule is LDA >= max(1,N). No exceptions, no vibes, the math said what it said. 💅
INFO — INFO is INTEGER, and it's the little status report at the end, okay?
- = 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 cpotri (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CPOTRI

Purpose:

CPOTRI 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 CPOTRF. It's giving "undo the factorization, serve the inverse." 🎀

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. And N >= 0, periodt.
A — A is COMPLEX array, dimension (LDA,N).
- On entry, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by CPOTRF.
- On exit, the upper or lower triangle of the (Hermitian) inverse of A, overwriting the input factor U or L. It literally replaces what you gave it, no cap. 🦋
LDA — LDA is INTEGER. Leading dimension of the array A. LDA >= max(1,N), always. 🌸
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. So like, that inverse? Not happening. 💀

Author

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

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

CPOTRS

Purpose:

CPOTRS 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 CPOTRF. Solve the system, slay, go home. 🔥

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. And NRHS >= 0.
A — A is COMPLEX 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 CPOTRF.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💅
B — B is COMPLEX array, dimension (LDB,NRHS)
- On entry, the right hand side matrix B.
- On exit, the solution matrix X. Glow up. ✨
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 💖

Generated automatically by Doxygen for LAPACK from the source code. Yes bestie, the docs wrote themselves. Iconic behaviour honestly.

And that's the whole page, babe! 🌈 This was Version 3.11.0, dated Sun Nov 27 2022, and it lives at complexPOcomputational(3) — so if you ever need to find it again, that's your girl. mwah 💋

📦 where this page comes from

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complexPOcomputational(3)

🎀 complexPOcomputational(3) 🎀

✨ NAME ✨

🌸 SYNOPSIS 🌸

💜 Functions

🌈 Detailed Description 🌈

👑 Function Documentation 👑

🦋 real function cla_porcond_c

🍓 Purpose:

💅 Parameters

🌸 Author

🧚 real function cla_porcond_x

🍓 Purpose:

💜 Parameters

🌸 Author

🔥 subroutine cla_porfsx_extended

🍓 Purpose:

💜 Parameters

👑 Author 👑

🦋 real function cla_porpvgrw 🦋

🎯 Purpose 🎯

✨ Parameters ✨

👑 Author 👑

🌸 subroutine cpocon 🌸

🎯 Purpose 🎯

✨ Parameters ✨

👑 Author

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

🎯 Purpose

Parameters

👑 Author

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

🎯 Purpose

Parameters

👑 Author

🎯 Purpose

Parameters

Internal Parameters

👑 Author

🎯 Purpose

🌸 Parameters 🌸

💜 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

🫶 Author

🦋 subroutine cpotf2

✨ Purpose:

🎀 Parameters

🫶 Author

🌸 subroutine cpotrf

✨ Purpose:

🎀 Parameters

🫶 Author

✨ Purpose:

🎀 Parameters

🫶 Author

📅 Date

🧚 recursive subroutine cpotrf2

✨ Purpose:

🎀 Parameters

Author

subroutine cpotri (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CPOTRI

🦋 real function `cla_porpvgrw` 🦋

🌸 subroutine `cpocon` 🌸

`subroutine cpoequ (integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)` 💖

`subroutine cpoequb (integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)` 🌈

💜 `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`