SRC/zposvx.f(3)

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version 3.12.1 · 7 versions
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🦋 SRC/zposvx.f 🦋

ok so bestie, welcome to the zposvx.f page from the Library Functions Manual — yes that's the vibe, we're in the library, we're being ✨productive✨, and honestly this one is a whole moment.

🌸 NAME 🌸

SRC/zposvx.f — literally the file, that's the name, no notes 💅

✨ SYNOPSIS ✨

🎀 Functions/Subroutines

ok so under here we've got one (1) icon doing all the work, and she is:

  • subroutine zposvx — described as ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices 🔥

and here's her full signature, copied exactly, do not touch, she is perfect as she is:

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

like… look at all those arguments. lda, ldaf, ldb, ldx — the lds really said "we're a family and we're showing up together" 🫶 and then rcond, ferr, berr, work, rwork, and info rounding out the group chat at the end. no cap, she's booked and busy 💀

and what does she do?? she computes the solution to system of linear equations A * X = B for PO matrices — that's the whole flex, that's the entire personality, and honestly? iconic 👑

💫 Function/Subroutine Documentation 💫

   subroutine zposvx (character fact, character uplo, integer n, integer nrhs,
       complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
       ldaf, * ) af, integer ldaf, character equed, double precision,
       dimension( * ) s, complex*16, dimension( ldb, * ) b, integer ldb,
       complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond,
       double precision, dimension( * ) ferr, double precision, dimension( * )
       berr, complex*16, dimension( * ) work, double precision, dimension( * )
       rwork, integer info)

Bestie, ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices. 💅

🌸 Purpose 🌸

Okay so ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute 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. ✨

Error bounds on the solution and a condition estimate are also provided. No cap. 🫶

🦋 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**H* U,  if UPLO = 'U', or
   A = L * L**H,  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 — FACT is 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 — 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 matrices B and X. NRHS >= 0. 🫶
  • A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the Hermitian 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💖
  • AF — AF is COMPLEX*16 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**H *U or A = L*L**H, 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**H *U or A = L*L**H 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**H *U or A = L*L**H of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). 🦋
  • LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N). ⭐
  • EQUED — EQUED is 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 — S is 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 — B is COMPLEX*16 array, dimension (LDB,NRHS). On entry, the N-by-NRHS righthand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * 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). 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 — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N). 💫
  • RCOND — RCOND is 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 — FERR is 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 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 COMPLEX*16 array, dimension (2*N). 🧚
  • RWORK — RWORK is DOUBLE PRECISION array, dimension (N). 🌸
  • INFO — INFO is 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 303 of file zposvx.f. 🌟

🌸 AUTHOR 🌸

Okay bestie, real talk — nobody hand-wrote this one. It's generated automatically by Doxygen for LAPACK, straight from the source code, zero human fingerprints. So like… the "author" is a build tool. And honestly? She ate. 💅

No cap, that means every word on this page got pulled out of the code itself, so if something looks crunchy and mechanical, that's why — it's giving ✨automation✨, not a person lovingly typing at 3 a.m. 🫶

And that's the whole page, bestie — you made it to the end! 🎀 The footer is literally just the glow-up credits: this is SRC/zposvx.f(3) from LAPACK, Version 3.12.0, with the generated-by-Doxygen timestamp doing its little cameo down there. 👑

Now go forth and zposvx responsibly — you're the main character of this linear algebra journey fr fr. 💖✨🦋

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