# ?tprfs

Estimates the error in the solution of a system of linear equations with a packed triangular matrix.

## Syntax

FORTRAN 77:

call stprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call dtprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call ctprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call ztprfs( uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, rwork, info )

Fortran 95:

call tprfs( a, b, x [,uplo] [,trans] [,diag] [,ferr] [,berr] [,info] )

## Description

This routine estimates the errors in the solution to a system of linear equations `A*X = B` or `AT*X = B` or AH*X = B with a packed triangular 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)`.

The routine also 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 solver routine ?tptrs.

## Input Parameters

uplo

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

  Indicates whether A is upper or lower triangular:

  If `uplo = 'U'`, then A is upper triangular.

  If `uplo = 'L'`, then A is lower triangular.

trans

  CHARACTER*1. Must be 'N' or 'T' or 'C'.

  Indicates the form of the equations:

  If `trans = 'N'`, the system has the form `A*X = B`.

  If `trans = 'T'`, the system has the form AT*X = B.

  If `trans = 'C'`, the system has the form AH*X = B.

diag

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

  If `diag = 'N'`, A is not a unit triangular matrix.

  If `diag = 'U'`, A is unit triangular: diagonal elements of A are assumed to be 1 and not referenced in the array ap.

n

  INTEGER. The order of the matrix A; n ≥ 0.

nrhs

  INTEGER. The number of right-hand sides; nrhs ≥ 0.

ap, b, x, work

  REAL for stprfs

  DOUBLE PRECISION for dtprfs

  COMPLEX for ctprfs

  DOUBLE COMPLEX for ztprfs.

  Arrays:

  `ap(*)` contains the upper or lower triangular matrix A, as specified by uplo.

  `b(ldb,*)` contains the right-hand side matrix B.

  `x(ldx,*)` contains the solution matrix X.

  `work(*)` is a workspace array.

  The dimension of ap 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)`.

iwork

  INTEGER. Workspace array, DIMENSION at least `max(1, n)`.

rwork

  REAL for ctprfs

  DOUBLE PRECISION for ztprfs.

  Workspace array, DIMENSION at least `max(1, n)`.

## Output Parameters

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 tprfs interface are as follows:

a

  Stands for argument ap in Fortan 77 interface. 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`).

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'.

trans

  Must be 'N', 'C', or 'T'. The default value is 'N'.

diag

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

## Application Notes

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

A call to this routine involves, for each right-hand side, 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 `n2` floating-point operations for real flavors or `4n2` for complex flavors.

Parent topic: Refining the Solution and Estimating Its Error