?pprfs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 6/24/2024

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?pprfs

Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite coefficient matrix stored in a packed format and estimates its error.

Syntax

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

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

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

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

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

Include Files

mkl.fi, lapack.f90

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a symmetric (Hermitian) positive definite matrix A, with multiple right-hand sides. For each computed solution vector x, the routine computes the component-wise backward errorβ. 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||_∞

where x_e is the exact solution.

Before calling this routine:

call the factorization routine ?pptrf
call the solver routine ?pptrs.

Input Parameters

uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates how the input matrix `A` has been factored: 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 spprfs DOUBLE PRECISION for dpprfs COMPLEX for cpprfs DOUBLE COMPLEX for zpprfs. Arrays: `ap()` contains the original matrix `A` in a packed format, as supplied to ?pptrf. `afp()` contains the factored matrix `A` in a packed format, as returned by ?pptrf. `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)`.
iwork	INTEGER. Workspace array, size at least `max(1, n)`.
rwork	REAL for cpprfs DOUBLE PRECISION for zpprfs. Workspace array, size 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, size 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.

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 pprfs 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).
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.

Parent topic: Refining the Solution and Estimating Its Error: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?pprfs

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes

See Also