DSPSVX(3)

💖 DSPSVX(3) 💖

okay so bestie, this is the DSPSVX(3) page from the Library Functions Manual, and we're gonna read it together like a group chat. No cap, it's giving numerical linear algebra with main character energy. 🫶

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🌸 NAME 🌸

The NAME section is literally just this file path, bestie:

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

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

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

DSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices. (Yes, OTHER is the word on the page — iykyk. 💅)

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🛠️ Function/Subroutine Documentation 🛠️

subroutine DSPSVX (character fact, character uplo, integer n, integer nrhs,
    double precision, dimension( * ) ap, double precision, dimension( * )
    afp, integer, dimension( * ) ipiv, 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)

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

💖 Purpose 💖

okay so here's the tea: DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real 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. Slay. ✨

🌈 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.

2. 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.
3. The system of equations is solved for X using the factored form of A.
4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

🎀 Parameters 🎀

FACT

FACT is CHARACTER*1. It 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

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 DOUBLE PRECISION 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

AFP is DOUBLE PRECISION 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 DSPTRF, 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 DSPTRF, stored as a packed triangular matrix in the same storage format as A.

IPIV

IPIV is 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 DSPTRF. 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 DSPTRF.

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 or INFO = N+1, the N-by-NRHS solution matrix 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. 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 DOUBLE PRECISION array, dimension (3*N).

IWORK

IWORK is INTEGER 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: 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 274 of file dspsvx.f.

📝 Author (generated) 📝

Generated automatically by Doxygen for LAPACK from the source code.

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Cute sign-off time: this page is from LAPACK, Version 3.12.0, and the footer points back to /home/abui...RC/dspsvx.f(3) — that's the source/version/date info as printed here, bestie. 💅💖✨