SRC/dspsvx.f(3)

Header: SRC/dspsvx.f(3) · Library Functions Manual — bestie, this is the whole vibe for this page 💅

💖 NAME

SRC/dspsvx.f — that's the page, and it's literally the home of the dspsvx subroutine. That's the NAME section, no cap ✨

🦋 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

✨ Function/Subroutine Documentation

Full signature, exactly as the page has it:

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)

And the one-liner is literally: DSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices 🫶

🎀 Purpose

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. Iconic behavior, honestly 💖

🌈 Description

Here's the exact step-by-step, bestie:

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.

Singularity check: step 2 is the one where D(i,i)=0 means D is exactly singular, and the routine bails with INFO = i. The INFO = N+1 case is a warning, not a full stop — it still solves and computes error bounds 😭

💅 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). It's 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 automatically by Doxygen for LAPACK from the source code.

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