SRC/dposvx.f(3)

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version 3.12.1 · 7 versions
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From the Library Functions Manual, this is the SRC/dposvx.f(3) page, bestie.

💅 NAME 💅

SRC/dposvx.f — that's literally the whole name entry, no cap. This is the page for that file, and we're about to get into it.

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

okay so here's the function roster: it's one subroutine, and she's the main character:

subroutine dposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b,
    ldb, x, ldx, rcond, ferr, berr, work, iwork, info)

DPOSVX computes the solution to system of linear equations A * X = B for PO matrices 💖

💻 Function/Subroutine Documentation

Okay bestie, here’s the actual signature, copied exactly because we do not play with argument order:

subroutine dposvx (character fact, character uplo, integer n, integer nrhs,
       double precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( ldaf, * ) af, integer ldaf, character equed, double
       precision, dimension( * ) s, double precision, dimension( ldb, * ) b,
       integer ldb, double precision, dimension( ldx, * ) x, integer ldx,
       double precision rcond, double precision, dimension( * ) ferr, double
       precision, dimension( * ) berr, double precision, dimension( * ) work,
       integer, dimension( * ) iwork, integer info)

And the one-liner is literally: DPOSVX computes the solution to system of linear equations A * X = B for PO matrices ✨

💖 Purpose

DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute 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.

Error bounds on the solution and a condition estimate are also provided. So yes, it’s not just “solve and bye” — it’s giving thorough 💅

🦋 Description

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  1. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',

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

  1. If the leading principal minor of order i is not positive, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
  2. The system of equations is solved for X using the factored form of A.
  3. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
  4. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

🎀 Parameters

  • FACT — CHARACTER*1. Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.
    • 'F': On entry, AF contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified.
    • 'N': The matrix A will be copied to AF and factored.
    • 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.
  • UPLO — CHARACTER*1.
    • 'U': Upper triangle of A is stored;
    • 'L': Lower triangle of A is stored.
  • N — INTEGER. The number of linear equations, i.e., the order of the matrix A. Constraint: N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. Constraint: NRHS >= 0.
  • A — DOUBLE PRECISION array, dimension (LDA,N). On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).
  • LDA — INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,N).
  • AF — DOUBLE PRECISION array, dimension (LDAF,N). If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).
  • LDAF — INTEGER. The leading dimension of the array AF. Constraint: LDAF >= max(1,N).
  • EQUED — CHARACTER*1. Specifies the form of equilibration that was done.
    • 'N': No equilibration (always true if FACT = 'N').
    • 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
  • S — DOUBLE PRECISION array, dimension (N). The scale factors for A; not accessed if EQUED = 'N'. 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.
  • B — DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B.
  • LDB — INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,N).
  • X — DOUBLE PRECISION array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
  • LDX — INTEGER. The leading dimension of the array X. Constraint: LDX >= max(1,N).
  • RCOND — DOUBLE PRECISION. The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.
  • FERR — 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 — 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 — DOUBLE PRECISION array, dimension (3*N).
  • IWORK — INTEGER array, dimension (N).
  • INFO — INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, and i is
      • <= N: the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned.
      • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

👑 Author

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

Definition at line 304 of file dposvx.f.

👑 AUTHOR

So like… nobody actually wrote this one by hand, bestie. It was generated automatically by Doxygen for LAPACK from the source code — the docs equivalent of a bot doing its lil thing and honestly? It ate. 💅✨

Which means no human author to credit here, just the machinery doing its job and the LAPACK source code being the real main character. 🫶

And that's the whole page, periodt! 🎀 This was LAPACK Version 3.12.0, living its best life as SRC/dposvx.f(3), auto-generated by Doxygen from the source code. Go forth and dposvx responsibly, icon. 🌈💖🔥

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