?spsvx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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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:

If fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the factorization is A = U*D*U^T orA = L*D*L^T, 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.
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.
The system of equations is solved for X using the factored form of A.
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	CHARACTER1. 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 `UDU^T`. If `uplo = 'L'`, the array `ap` stores the lower triangular part of the symmetric matrix `A`; `A` is factored as `LD*L^T`.
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 = UDU^T` or `A = LDL^T` 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(1,3n)` 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 `d_ii` 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 `i≤n`, then `d_ii` 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.

Parent topic: LAPACK Linear Equation Driver Routines

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