?gtrfs

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

Syntax

FORTRAN 77:

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 )

Fortran 95:

call gtrfs( dl, d, du, dlf, df, duf, du2, ipiv, b, x [,trans] [,ferr] [,berr] [,info] )

lapack_int LAPACKE_sgtrfs( int matrix_order, char trans, lapack_int n, lapack_int nrhs, const float* dl, const float* d, const float* du, const float* dlf, const float* df, const float* duf, const float* du2, const lapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_dgtrfs( int matrix_order, char trans, lapack_int n, lapack_int nrhs, const double* dl, const double* d, const double* du, const double* dlf, const double* df, const double* duf, const double* du2, const lapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );

lapack_int LAPACKE_cgtrfs( int matrix_order, char trans, lapack_int n, lapack_int nrhs, const lapack_complex_float* dl, const lapack_complex_float* d, const lapack_complex_float* du, const lapack_complex_float* dlf, const lapack_complex_float* df, const lapack_complex_float* duf, const lapack_complex_float* du2, const lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );

lapack_int LAPACKE_zgtrfs( int matrix_order, char trans, lapack_int n, lapack_int nrhs, const lapack_complex_double* dl, const lapack_complex_double* d, const lapack_complex_double* du, const lapack_complex_double* dlf, const lapack_complex_double* df, const lapack_complex_double* duf, const lapack_complex_double* du2, const lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );

Include Files

The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.

Description

The routine performs an iterative refinement of the solution to a system of linear equations A*X = B or A^T*X = B or A^H*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 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:

|δa_ij|/|a_ij| ≤ β|a_ij|, |δb_i|/|b_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||_∞ (here x_e is the exact solution).

Before calling this routine:

call the factorization routine ?gttrf
call the solver routine ?gttrs.

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

trans	CHARACTER1. Must be 'N' or 'T' or 'C'. Indicates the form of the equations: If `trans = 'N'`, the system has the form `AX = B`. If `trans = 'T'`, the system has the form `A`^T`X` = `B`. If `trans = 'C'`, the system has the form `A`^H`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, 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)`.
ipiv	INTEGER. Array, DIMENSION at least `max(1, n)`. The `ipiv` array, as returned by ?gttrf.
iwork	INTEGER. Workspace array, DIMENSION (`n`). Used for real flavors only.
rwork	REAL for cgtrfs DOUBLE PRECISION for zgtrfs. Workspace array, DIMENSION (`n`). Used for complex flavors only.

Output Parameters

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 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'.