Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 3/22/2024
Public

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?gbtrs

Solves a system of linear equations with an LU-factored band coefficient matrix, with multiple right-hand sides.

Syntax

call sgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

call dgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

call cgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

call zgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

call gbtrs( ab, b, ipiv, [, kl] [, trans] [, info] )

Include Files

  • mkl.fi, lapack.f90

Description

The routine solves for X the following systems of linear equations:

A*X = B

if trans='N',

AT*X = B

if trans='T',

AH*X = B

if trans='C' (for complex matrices only).

Here A is an LU-factored general band matrix of order n with kl non-zero subdiagonals and ku nonzero superdiagonals. Before calling this routine, call ?gbtrf to compute the LU factorization of A.

Input Parameters

trans

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

n

INTEGER. The order of A; the number of rows in B; n 0.

kl

INTEGER. The number of subdiagonals within the band of A; kl 0.

ku

INTEGER. The number of superdiagonals within the band of A; ku 0.

nrhs

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

ab, b

REAL for sgbtrs

DOUBLE PRECISION for dgbtrs

COMPLEX for cgbtrs

DOUBLE COMPLEX for zgbtrs.

Arrays: ab(ldab,*), b(ldb,*).

The array ab contains elements of the LU factors of the matrix A as returned by gbtrf. The second dimension of ab must be at least max(1, n).

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. The second dimension of b at least max(1,nrhs).

ldab

INTEGER. The leading dimension of the array ab; ldab 2*kl + ku +1.

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

ipiv

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

Output Parameters

b

Overwritten by the solution matrix X.

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 gbtrs interface are as follows:

ab

Holds the array A of size (2*kl+ku+1,n).

b

Holds the matrix B of size (n, nrhs).

ipiv

Holds the vector of length min(m, n).

kl

If omitted, assumed kl = ku.

ku

Restored as lda-2*kl-1.

trans

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

Application Notes

For each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A + E)x = b, where

|E|  c(kl + ku + 1)ε P|L||U|

c(k) is a modest linear function of k, and ε is the machine precision.

If x0 is the true solution, the computed solution x satisfies this error bound:


Equation

where cond(A,x)= || |A-1||A| |x| || / ||x|| ||A-1|| ||A|| = κ(A).

Note that cond(A,x) can be much smaller than κ(A); the condition number of AT and AH might or might not be equal to κ(A).

The approximate number of floating-point operations for one right-hand side vector is 2n(ku + 2kl) for real flavors. The number of operations for complex flavors is 4 times greater. All these estimates assume that kl and ku are much less than min(m,n).

To estimate the condition number κ(A), call ?gbcon.

To refine the solution and estimate the error, call ?gbrfs.