SRC/zcposv.f(3)

okay bestie, welcome to SRC/zcposv.f(3) from the Library Functions Manual — this is the ZCPOSV page and it's literally about solving A * X = B for PO matrices. Buckle up, it's giving linear algebra but make it cute. 💅

✨ NAME ✨

SRC/zcposv.f — yes bestie, the page is named after the source file itself. 🫶

💅 SYNOPSIS 💅

Functions/Subroutines

subroutine zcposv (uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork,
           rwork, iter, info)
            ZCPOSV computes the solution to system of linear equations A * X =
           B for PO matrices

🦋 Function/Subroutine Documentation 🦋

subroutine zcposv (character uplo, integer n, integer nrhs, complex*16,
    dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,
    integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, complex*16,
    dimension( n, * ) work, complex, dimension( * ) swork, double precision,
    dimension( * ) rwork, integer iter, integer info)

ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices — iconic behavior, honestly.

Purpose

okay so here's the tea: ZCPOSV computes the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. ✨

First it attempts to factorize the matrix in COMPLEX and use this factorization within an iterative refinement procedure to produce a solution with COMPLEX*16 normwise backward error quality (see below). If the approach fails, the method switches to a COMPLEX*16 factorization and solve. No cap. 🫶

The iterative refinement is not going to be a winning strategy if the ratio COMPLEX performance over COMPLEX*16 performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement. 💖

The iterative refinement process is stopped if

ITER > ITERMAX

or for all the RHS we have:

RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX

where

• ITER is the number of the current iteration in the iterative refinement process
• RNRM is the infinity-norm of the residual
• XNRM is the infinity-norm of the solution
• ANRM is the infinity-operator-norm of the matrix A
• EPS is the machine epsilon returned by DLAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively. 💅

Parameters

• UPLO — is CHARACTER*1.
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
• NRHS — is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• 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. Note that the imaginary parts of the diagonal elements need not be set and are assumed to be zero. On exit, if iterative refinement has been successfully used (INFO = 0 and ITER >= 0, see description below), then A is unchanged; if double precision factorization has been used (INFO = 0 and ITER < 0, see description below), then the array A contains the factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H.
• LDA — is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• B — is COMPLEX*16 array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
• LDB — is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• X — is COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0, the N-by-NRHS solution matrix X.
• LDX — is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
• WORK — is COMPLEX*16 array, dimension (N,NRHS). This array is used to hold the residual vectors.
• SWORK — is COMPLEX array, dimension (N*(N+NRHS)). This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.
• RWORK — is DOUBLE PRECISION array, dimension (N).
• ITER — is INTEGER.
  • < 0: iterative refinement has failed, COMPLEX*16 factorization has been performed
    • -1: the routine fell back to full precision for implementation- or machine-specific reasons
    • -2: narrowing the precision induced an overflow, the routine fell back to full precision
    • -3: failure of CPOTRF
    • -31: stop the iterative refinement after the 30th iterations
  • > 0: iterative refinement has been successfully used. Returns the number of iterations.
• INFO — is INTEGER.
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, the leading principal minor of order i of (COMPLEX*16) A is not positive, so the factorization could not be completed, and the solution has not been computed.

👑 Author

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

📍 Definition at line 207 of file zcposv.f.

🌸 Author 🌸

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole page, bestie! Shoutout to LAPACK Version 3.12.0 and SRC/zcposv.f(3) — the footer doesn't show a date, so we're not inventing one. Periodt. 💖✨