?sprfs

Refines the solution of a system of linear equations with a packed symmetric matrix and estimates the solution error.

Syntax

FORTRAN 77:

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

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

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

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

FORTRAN 95:

call sprfs( ap, afp, ipiv, b, x [,uplo] [,ferr] [,berr] [,info] )

lapack_int LAPACKE_ssprfs( int matrix_order, char uplo, lapack_int n, lapack_int nrhs, const float* ap, const float* afp, const lapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dsprfs( int matrix_order, char uplo, lapack_int n, lapack_int nrhs, const double* ap, const double* afp, const lapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_csprfs( int matrix_order, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_float* ap, const lapack_complex_float* afp, const lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zsprfs( int matrix_order, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_double* ap, const lapack_complex_double* afp, const lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a packed symmetric matrix A, with multiple right-hand sides. For each computed solution vector x, the routine computes the component-wise backward error β. This error is the smallest relative perturbation in elements of A and b such that x is the exact solution of the perturbed system:

|δa_ij| ≤ β|a_ij|, |δb_i| ≤ β|b_i| such that (A + δA)x = (b + δb).

Finally, the routine estimates the component-wise forward error in the computed solution ||x - x_e||_∞/||x||_∞ (here x_e is the exact solution).

Before calling this routine:

call the factorization routine ?sptrf
call the solver routine ?sptrs.

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

uplo	CHARACTER*1. Must be 'U' or 'L'. If `uplo = 'U'`, the upper triangle of `A` is stored. If `uplo = 'L'`, the lower triangle of `A` is stored.
n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs` ≥ 0.
ap,afp,b,x,work	REAL for ssprfs DOUBLE PRECISION for dsprfs COMPLEX for csprfs DOUBLE COMPLEX for zsprfs. Arrays: `ap()` contains the original packed matrix `A`, as supplied to ?sptrf. `afp()` contains the factored packed matrix `A`, as returned by ?sptrf. `b(ldb,)` contains the right-hand side matrix `B`. `x(ldx,)` contains the solution matrix `X`. `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` and `x` 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)`.
ldx	INTEGER. The leading dimension of `x`; `ldx ≥ max(1, n)`.
ipiv	INTEGER. Array, DIMENSION at least `max(1, n)`. The `ipiv` array, as returned by ?sptrf.
iwork	INTEGER. Workspace array, DIMENSION at least `max(1, n)`.
rwork	REAL for csprfs DOUBLE PRECISION for zsprfs. Workspace array, DIMENSION at least `max(1, n)`.

Output Parameters

The refined solution matrix X.

ferr, berr

REAL for single precision flavors.

DOUBLE PRECISION for double precision flavors.

Arrays, DIMENSION at least max(1, nrhs). Contain the component-wise forward and 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.

Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine sprfs interface are as follows:

ap	Holds the array `A` of size (`n*(n+1)/2`).
afp	Holds the array `AF` of size (`n*(n+1)/2`).
ipiv	Holds the vector of length `n`.
b	Holds the matrix `B` of size (`n,nrhs`).
x	Holds the matrix `X` of size (`n,nrhs`).
ferr	Holds the vector of length (`nrhs`).
berr	Holds the vector of length (`nrhs`).
uplo	Must be 'U' or 'L'. The default value is 'U'.

Application Notes

The bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual error.

For each right-hand side, computation of the backward error involves a minimum of 4n² floating-point operations (for real flavors) or 16n² operations (for complex flavors). In addition, each step of iterative refinement involves 6n² operations (for real flavors) or 24n² operations (for complex flavors); the number of iterations may range from 1 to 5.

Estimating the forward error involves solving a number of systems of linear equations A*x = b; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately 2n² floating-point operations for real flavors or 8n² for complex flavors.