# ?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( a, af, ipiv, b, x [,uplo] [,ferr] [,berr] [,info] )

## Description

This 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:

`|δaij|/|aij| ≤ β|aij|, |δbi|/|bi| ≤ β|bi|` such that `(A + δA)x = (b + δb)`.

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

Before calling this routine:

- call the factorization routine ?sptrf

- call the solver routine ?sptrs.

## Input Parameters

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, 3*n)` for real flavors and `max(1, 2*n)` for complex flavors.

ldb

  INTEGER. The first dimension of b; `ldb ≥ max(1, n)`.

ldx

  INTEGER. The first 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

x

  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:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size (`n*(n+1)/2`).

af

  Stands for argument afp in Fortan 77 interface. 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.

Parent topic: Refining the Solution and Estimating Its Error