SRC/dsposv.f(3)

💖 SRC/dsposv.f(3) — Library Functions Manual 💖

okay so bestie, welcome to SRC/dsposv.f(3) from the Library Functions Manual — this is the whole group chat for SRC/dsposv.f, and we are keeping every single detail. no cap. 💅

🌸 NAME 🌸

SRC/dsposv.f

✨ SYNOPSIS ✨

🧩 Functions/Subroutines 🧩

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

👑 Function/Subroutine Documentation 👑

💫 subroutine dsposv signature 💫

subroutine dsposv (character uplo, integer n, integer nrhs, double
    precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, *
    ) x, integer ldx, double precision, dimension( n, * ) work, real,
    dimension( * ) swork, integer iter, integer info)

DSPOSV computes the solution to system of linear equations A * X = B for PO matrices — yep, that is the whole one-liner, bestie. 💖

🦋 Purpose 🦋

okay so here is the entire point: DSPOSV computes the solution to a real system of linear equations

A * X = B,

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

DSPOSV first attempts to factorize the matrix in SINGLE PRECISION and use this factorization within an iterative refinement procedure to produce a solution with DOUBLE PRECISION normwise backward error quality (see below). If the approach fails the method switches to a DOUBLE PRECISION factorization and solve. not me being flexible or whatever 💅

The iterative refinement is not going to be a winning strategy if the ratio SINGLE PRECISION performance over DOUBLE PRECISION 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. periodt.

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 — 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. NRHS >= 0.
• A — A is DOUBLE PRECISION 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 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**T*U or A = L*L**T. 🦋
• LDA — LDA is INTEGER; the leading dimension of the array A. LDA >= max(1,N).
• B — B is DOUBLE PRECISION array, dimension (LDB,NRHS); the N-by-NRHS right hand side matrix B.
• LDB — LDB is INTEGER; the leading dimension of the array B. LDB >= max(1,N).
• X — X is DOUBLE PRECISION array, dimension (LDX,NRHS); if INFO = 0, the N-by-NRHS solution matrix X.
• LDX — LDX is INTEGER; the leading dimension of the array X. LDX >= max(1,N).
• WORK — WORK is DOUBLE PRECISION array, dimension (N,NRHS); this array is used to hold the residual vectors.
• SWORK — SWORK is REAL 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.
• ITER — ITER is INTEGER
  • < 0: iterative refinement has failed, double precision 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 SPOTRF
    • -31: stop the iterative refinement after the 30th iterations
  • > 0: iterative refinement has been successfully used. Returns the number of iterations
• 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 principal minor of order i of (DOUBLE PRECISION) 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 197 of file dsposv.f.

🌀 Author 🌀

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and that is the entire page, bestie 🫶 — LAPACK Version 3.12.0, SRC/dsposv.f(3). mwah 💋