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 );

• mkl.h

## 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

 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, ldb*nrhs) for column major layout and max(1, ldb*n) 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 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.

## See Also

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