SRC/cspsvx.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

💖 SRC/cspsvx.f(3) — LAPACK manual page, bestie edition 💖

This is the LAPACK Library Functions Manual page for SRC/cspsvx.f, and honestly? It's giving “complex symmetric solver with receipts.” Let's get into it ✨

🌸 NAME 🌸

SRC/cspsvx.f

✨ SYNOPSIS ✨

Functions/Subroutines

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

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

🎀 Function/Subroutine Documentation 🎀

subroutine cspsvx (character fact, character uplo, integer n, integer nrhs,
     complex, dimension( * ) ap, complex, dimension( * ) afp, integer,
     dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,
     complex, dimension( ldx, * ) x, integer ldx, real rcond, real,
     dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * )
     work, real, dimension( * ) rwork, integer info)

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

💅 Purpose 💅

Okay so CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric 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. So yes, not just “here's your answer,” but “here's your answer and a vibe check on how much you should trust it.” 💀

🌈 Description 🌈

The following steps are performed:

  1. If FACT = 'N', the diagonal pivoting method is used to factor A as
    A = U * D * U**T,  if UPLO = 'U', or
    A = L * D * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

  1. If some D(i,i)=0, so that D is exactly singular, 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. Warning behavior, bestie — not a full stop.
  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.

🦋 Parameters 🦋

  • FACT — CHARACTER*1 🌸 Specifies whether or not the factored form of A has been supplied on entry.
    • = 'F': On entry, AFP and IPIV contain the factored form of A. AP, AFP and IPIV will not be modified.
    • = 'N': The matrix A will be copied to AFP 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. N >= 0.
  • NRHS — INTEGER 🎀 The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • AP — COMPLEX array, dimension (N*(N+1)/2) 🦋 The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

See below for further details.

  • AFP — COMPLEX array, dimension (N*(N+1)/2) 💜

If FACT = 'F', then AFP is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A.

If FACT = 'N', then AFP is an output argument and on exit contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A.

  • IPIV — INTEGER array, dimension (N) 👑

If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by CSPTRF.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.

If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CSPTRF.

  • B — COMPLEX array, dimension (LDB,NRHS) 🌈 The N-by-NRHS right hand side matrix B.
  • LDB — INTEGER ⭐ The leading dimension of the array B. LDB >= max(1,N).
  • X — COMPLEX array, dimension (LDX,NRHS) 💫 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
  • LDX — INTEGER 🫶 The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — REAL 🔥 The estimate of the reciprocal condition number of the matrix A. 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 — REAL 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 — REAL 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 — COMPLEX array, dimension (2*N) ✨
  • RWORK — REAL 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: D(i,i) is exactly zero. The factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
      • = N+1: D 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 🧚

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44

Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 275 of file cspsvx.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

Okay bestie, that's the whole page. Footer vibes: LAPACK Version 3.12.0, source SRC/cspsvx.f(3) — and the footer's date slot is not filled in, so no made-up date from me 💖✨

📦 where this page comes from