Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 3/31/2023
Public

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?gtrfs

Refines the solution of a system of linear equations with a tridiagonal coefficient matrix and estimates its error.

Syntax

call sgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call dgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info )

call cgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call zgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )

call gtrfs( dl, d, du, dlf, df, duf, du2, ipiv, b, x [,trans] [,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 or AT*X = B or AH*X = B with a tridiagonal 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:

|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 ?gttrf

  • call the solver routine ?gttrs.

Input Parameters

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.

n

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

nrhs

INTEGER. The number of right-hand sides, that is, the number of columns of the matrix B; nrhs 0.

dl,d,du,dlf,
df,duf,du2,
b,x,work

REAL for sgtrfs

DOUBLE PRECISION for dgtrfs

COMPLEX for cgtrfs

DOUBLE COMPLEX for zgtrfs.

Arrays:

dl, dimension (n -1), contains the subdiagonal elements of A.

d, dimension (n), contains the diagonal elements of A.

du, dimension (n -1), contains the superdiagonal elements of A.

dlf, dimension (n -1), contains the (n - 1) multipliers that define the matrix L from the LU factorization of A as computed by ?gttrf.

df, dimension (n), contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.

duf, dimension (n -1), contains the (n - 1) elements of the first superdiagonal of U.

du2, dimension (n -2), contains the (n - 2) elements of the second superdiagonal of U.

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

x(ldx,nrhs) contains the solution matrix X, as computed by ?gttrs.

work(*) is a workspace array; 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 leading dimension of b; ldb max(1, n).

ldx

INTEGER. The leading dimension of x; ldx max(1, n).

ipiv

INTEGER.

Array, size at least max(1, n). The ipiv array, as returned by ?gttrf.

iwork

INTEGER. Workspace array, size (n). Used for real flavors only.

rwork

REAL for cgtrfs

DOUBLE PRECISION for zgtrfs.

Workspace array, size (n). Used for complex flavors only.

Output Parameters

x

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 gtrfs interface are as follows:

dl

Holds the vector of length (n-1).

d

Holds the vector of length n.

du

Holds the vector of length (n-1).

dlf

Holds the vector of length (n-1).

df

Holds the vector of length n.

duf

Holds the vector of length (n-1).

du2

Holds the vector of length (n-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).

trans

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