Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 3/22/2024
Public

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?spsvx

Uses the diagonal pivoting factorization to compute the solution to the system of linear equations with a real or complex symmetric coefficient matrix A stored in packed format, and provides error bounds on the solution.

Syntax

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

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

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

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

call spsvx( ap, b, x [,uplo] [,afp] [,ipiv] [,fact] [,ferr] [,berr] [,rcond] [,info] )

Include Files

  • mkl.fi, lapack.f90

Description

The routine uses the diagonal pivoting factorization to compute the solution to a real or complex system of linear equations A*X = B, where A is a n-by-n symmetric matrix stored in packed format, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

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

The routine ?spsvx performs the following steps:

  1. If fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the factorization is A = U*D*UT orA = L*D*LT, 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 di,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.

Input Parameters

fact

CHARACTER*1. Must be 'F' or 'N'.

Specifies whether or not the factored form of the matrix A has been supplied on entry.

If fact = 'F': on entry, afp and ipiv contain the factored form of A. Arrays ap, afp, and ipiv are not modified.

If fact = 'N', the matrix A is copied to afp and factored.

uplo

CHARACTER*1. Must be 'U' or 'L'.

Indicates whether the upper or lower triangular part of A is stored and how A is factored:

If uplo = 'U', the array ap stores the upper triangular part of the symmetric matrix A, and A is factored as U*D*UT.

If uplo = 'L', the array ap stores the lower triangular part of the symmetric matrix A; A is factored as L*D*LT.

n

INTEGER. The order of matrix A; n 0.

nrhs

INTEGER. The number of right-hand sides, the number of columns in B; nrhs 0.

ap, afp, b, work

REAL for sspsvx

DOUBLE PRECISION for dspsvx

COMPLEX for cspsvx

DOUBLE COMPLEX for zspsvx.

Arrays: ap(size *), afp(size *), b(size ldb by *), work(*).

The array ap contains the upper or lower triangle of the symmetric matrix A in packed storage (see Matrix Storage Schemes).

The array afp is an input argument if fact = 'F'. It contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*UT or A = L*D*LT as computed by ?sptrf, in the same storage format as A.

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

work(*) is a workspace array.

The dimension of arrays ap and afp must be at least max(1, n(n+1)/2); the second dimension of b must be at least max(1,nrhs); the dimension of work must be at least max(1,3*n) for real flavors and max(1,2*n) for complex flavors.

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

ipiv

INTEGER.

Array, size at least max(1, n). The array ipiv is an input argument if fact = 'F'. It contains details of the interchanges and the block structure of D, as determined by ?sptrf.

If ipiv(i) = k > 0, then dii is a 1-by-1 block, and the i-th row and column of A was interchanged with the k-th row and column.

If uplo = 'U'and ipiv(i) =ipiv(i-1) = -m < 0, then D has a 2-by-2 block in rows/columns i and i-1, and (i-1)-th row and column of A was interchanged with the m-th row and column.

If uplo = 'L'and ipiv(i) =ipiv(i+1) = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and (i+1)-th row and column of A was interchanged with the m-th row and column.

ldx

INTEGER. The leading dimension of the output array x; ldx max(1, n).

iwork

INTEGER. Workspace array, size at least max(1, n); used in real flavors only.

rwork

REAL for cspsvx

DOUBLE PRECISION for zspsvx.

Workspace array, size at least max(1, n); used in complex flavors only.

Output Parameters

x

REAL for sspsvx

DOUBLE PRECISION for dspsvx

COMPLEX for cspsvx

DOUBLE COMPLEX for zspsvx.

Array, size ldx by *.

If info = 0 or info = n+1, the array x contains the solution matrix X to the system of equations. The second dimension of x must be at least max(1,nrhs).

afp, ipiv

These arrays are output arguments if fact = 'N'. See the description of afp, ipiv in Input Arguments section.

rcond

REAL for single precision flavors.

DOUBLE PRECISION for double precision flavors.

An 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, berr

REAL for single precision flavors

DOUBLE PRECISION for double precision flavors.

Arrays, size at least max(1, nrhs). Contain the component-wise forward and relative backward errors, respectively, for each solution vector.

info

INTEGER. If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = i, and in, then dii is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.

If info = i, and i = n + 1, then 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.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine spsvx interface are as follows:

ap

Holds the array A of size (n*(n+1)/2).

b

Holds the matrix B of size (n,nrhs).

x

Holds the matrix X of size (n,nrhs).

afp

Holds the array AF of size (n*(n+1)/2).

ipiv

Holds the vector with the number of elements n.

ferr

Holds the vector with the number of elements nrhs.

berr

Holds the vector with the number of elements nrhs.

uplo

Must be 'U' or 'L'. The default value is 'U'.

fact

Must be 'N' or 'F'. The default value is 'N'. If fact = 'F', then both arguments af and ipiv must be present; otherwise, an error is returned.