# ?ptrfs

Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite tridiagonal matrix and estimates its error.

## Syntax

FORTRAN 77:

call sptrfs( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info )

call dptrfs( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info )

call cptrfs( uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call zptrfs( uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info )

Fortran 95:

call ptrfs( d, df, e, ef, b, x [,ferr] [,berr] [,info] )

call ptrfs( d, df, e, ef, 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 symmetric (Hermitian) positive definite tridiagonal 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 ?pttrf

- call the solver routine ?pttrs.

## Input Parameters

uplo

  CHARACTER*1. Used for complex flavors only. Must be 'U' or 'L'.

  Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix A is stored and how A is factored:

  If `uplo = 'U'`, the array e stores the superdiagonal of A, and A is factored as `UH*D*U`.

  If `uplo = 'L'`, the array e stores the subdiagonal of A, and A is factored as `L*D*LH`.

n

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

nrhs

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

d, df, rwork

  REAL for single precision flavors DOUBLE PRECISION for double precision flavors

  Arrays: `d(n)`, `df(n)`, `rwork(n)`.

  The array d contains the n diagonal elements of the tridiagonal matrix A.

  The array df contains the n diagonal elements of the diagonal matrix D from the factorization of A as computed by ?pttrf.

  The array rwork is a workspace array used for complex flavors only.

e,ef,b,x,work

  REAL for sptrfs

  DOUBLE PRECISION for dptrfs

  COMPLEX for cptrfs

  DOUBLE COMPLEX for zptrfs.

  Arrays: `e(n -1)`, `ef(n -1)`, `b(ldb,nrhs)`, `x(ldx,nrhs)`, `work(*)`.

  The array e contains the (n - 1) off-diagonal elements of the tridiagonal matrix A (see uplo).

  The array ef contains the (n - 1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by ?pttrf (see uplo).

  The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

  The array x contains the solution matrix X as computed by ?pttrs.

  The array work is a workspace array. The dimension of work must be at least `2*n` for real flavors, and at least 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)`.

## 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 ptrfs interface are as follows:

d

  Holds the vector of length (n).

df

  Holds the vector of length (n).

e

  Holds the vector of length (n-1).

ef

  Holds the vector of length (n-1).

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

  Used in complex flavors only. Must be 'U' or 'L'. The default value is 'U'.

Parent topic: Refining the Solution and Estimating Its Error