Computes the solution to the system of linear equations with a symmetric or Hermitian positive-definite band coefficient matrix A and multiple right-hand sides.

## Syntax

lapack_int LAPACKE_spbsv (int matrix_layout , char uplo , lapack_int n , lapack_int kd , lapack_int nrhs , float * ab , lapack_int ldab , float * b , lapack_int ldb );

lapack_int LAPACKE_dpbsv (int matrix_layout , char uplo , lapack_int n , lapack_int kd , lapack_int nrhs , double * ab , lapack_int ldab , double * b , lapack_int ldb );

lapack_int LAPACKE_cpbsv (int matrix_layout , char uplo , lapack_int n , lapack_int kd , lapack_int nrhs , lapack_complex_float * ab , lapack_int ldab , lapack_complex_float * b , lapack_int ldb );

lapack_int LAPACKE_zpbsv (int matrix_layout , char uplo , lapack_int n , lapack_int kd , lapack_int nrhs , lapack_complex_double * ab , lapack_int ldab , lapack_complex_double * b , lapack_int ldb );

• mkl.h

## Description

The routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n symmetric/Hermitian positive definite band matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

The Cholesky decomposition is used to factor A as

A = UT*U (real flavors) and A = UH*U (complex flavors), if uplo = 'U'

or A = L*LT (real flavors) and A = L*LH (complex flavors), if uplo = 'L',

where U is an upper triangular band matrix and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A*X = B.

## Input Parameters

 matrix_layout Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR). uplo Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of A is stored: If uplo = 'U', the upper triangle of A is stored. If uplo = 'L', the lower triangle of A is stored. n The order of matrix A; n≥ 0. kd The number of superdiagonals of the matrix A if uplo = 'U', or the number of subdiagonals if uplo = 'L';kd≥ 0. nrhs The number of right-hand sides, the number of columns in B; nrhs≥ 0. ab, b Arrays: ab(size max(1, ldab*n)), bof size max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout. The array ab contains the upper or the lower triangular part of the matrix A (as specified by uplo) in band storage (see Matrix Storage Schemes). 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≥kd +1. ldb The leading dimension of b; ldb≥ max(1, n) for column major layout and ldb≥nrhs for row major layout.

## Output Parameters

 ab The upper or lower triangular part of A (in band storage) is overwritten by the Cholesky factor U or L, as specified by uplo, in the same storage format as A. 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.

If info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, so the factorization could not be completed, and the solution has not been computed.