?gbtrs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?gbtrs

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

Syntax

lapack_int LAPACKE_sgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const float * ab , lapack_int ldab , const lapack_int * ipiv , float * b , lapack_int ldb );

lapack_int LAPACKE_dgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const double * ab , lapack_int ldab , const lapack_int * ipiv , double * b , lapack_int ldb );

lapack_int LAPACKE_cgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const lapack_complex_float * ab , lapack_int ldab , const lapack_int * ipiv , lapack_complex_float * b , lapack_int ldb );

lapack_int LAPACKE_zgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const lapack_complex_double * ab , lapack_int ldab , const lapack_int * ipiv , lapack_complex_double * b , lapack_int ldb );

Include Files

mkl.h

Description

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

A*X = B	if `trans`='N',
`A^T*X = B`	if `trans`='T',
`A^H*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

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
trans	Must be 'N' or 'T' or 'C'.
n	The order of `A`; the number of rows in `B`; `n`≥ 0.
kl	The number of subdiagonals within the band of `A`; `kl`≥ 0.
ku	The number of superdiagonals within the band of `A`; `ku`≥ 0.
nrhs	The number of right-hand sides; `nrhs`≥ 0.
ab	Array `ab` size max(1, ldab*n) The array `ab` contains elements of the LU factors of the matrix `A` as returned by gbtrf.
b	Array b size max(1, ldbnrhs) for column major layout and max(1, ldbn) for row major layout. The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations.
ldab	The leading dimension of the array `ab`; `ldab`≥ 2*`kl` + `ku` +1.
ldb	The leading dimension of `b`; `ldb`≥ max(1, n) for column major layout and `ldb`≥nrhs for row major layout.
ipiv	Array, size at least `max(1, n)`. The `ipiv` array, as returned by ?gbtrf.

Output Parameters

b	Overwritten by the solution matrix `X`.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, parameter i had an illegal value.

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:

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 A^T and A^H 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.

Parent topic: Solving Systems of Linear Equations: LAPACK Computational Routines

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