ZPPSVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

🎀 ZPPSVX(3) — Library Functions Manual

okay bestie, welcome to ZPPSVX(3) from the Library Functions Manual. this routine is lowkey the friend who solves your complex Hermitian positive-definite packed-matrix equations and also hands you condition estimates and error bounds. it’s giving linear algebra main character energy, ngl. 💅

📛 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zppsvx.f

✨ SYNOPSIS

🌸 Functions/Subroutines

   subroutine ZPPSVX (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x,
       ldx, rcond, ferr, berr, work, rwork, info)

ZPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices.

🧚 Function/Subroutine Documentation

   subroutine ZPPSVX (character fact, character uplo, integer n, integer nrhs,
       complex*16, dimension( * ) ap, complex*16, dimension( * ) afp, 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)

ZPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices.

🎯 Purpose

okay so ZPPSVX 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 stored in packed format and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. periodt. 💖

🪄 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, L is a lower triangular matrix, and **H indicates conjugate transpose.

  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, AFP contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified.
  • = 'N': The matrix A will be copied to AFP and factored.
  • = 'E': The matrix A will be equilibrated if necessary, then copied to AFP 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.

✨ AP

AP is COMPLEX*16 array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). The j-th column of A is stored in the array AP as follows:

      if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
      if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

See below for further details. 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).

🎀 AFP

AFP is COMPLEX*16 array, dimension (N*(N+1)/2).

If FACT = 'F', then AFP 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 AFP is the factored form of the equilibrated matrix A.

If FACT = 'N', then AFP 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 AFP 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 AP for the form of the equilibrated matrix).

💜 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 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

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.

Further Details 🌸

Okay so this is the visual aid for the packed storage scheme, bestie — it’s giving “trust the layout” energy. The source says it’s illustrated by this example when N = 4, UPLO = 'U':

!>
!>  The packed storage scheme is illustrated by the following example
!>  when N = 4, UPLO = 'U':
!>
!>  Two-dimensional storage of the Hermitian matrix A:
!>
!>     a11 a12 a13 a14
!>         a22 a23 a24
!>             a33 a34     (aij = conjg(aji))
!>                 a44
!>
!>  Packed storage of the upper triangle of A:
!>
!>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
!>

No cap, that’s the whole flex: the upper triangle gets squished into that one AP list. Then the page drops this exact location note: Definition at line 309 of file zppsvx.f. 🎀

Author 🫶

Generated automatically by Doxygen for LAPACK.

That’s literally it — Doxygen did the typing, LAPACK brought the math. ✨

Catch you later, bestie 💅 — this page is from LAPACK Version 3.12.0, source /home/abui...RC/zppsvx.f(3). 🦋

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