Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?pbtrs

Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite band coefficient matrix.

Syntax

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

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

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

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

Include Files

  • mkl.h

Description

The routine solves for real data a system of linear equations A*X = B with a symmetric positive-definite or, for complex data, Hermitian positive-definite band matrix A, given the Cholesky factorization of A:

A = UT*U for real data, A = UH*U for complex data if uplo='U'
A = L*LT for real data, A = L*LH for complex data if uplo='L'

where L is a lower triangular matrix and U is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix B.

Before calling this routine, you must call ?pbtrf to compute the Cholesky factorization of A in the band storage form.

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 how the input matrix A has been factored:

If uplo = 'U', U is stored in ab, where A = UT*U for real matrices and A = UH*U for complex matrices.

If uplo = 'L', L is stored in ab, where A = L*LT for real matrices and A = L*LH for complex matrices.

n

The order of matrix A; n 0.

kd

The number of superdiagonals or subdiagonals in the matrix A; kd 0.

nrhs

The number of right-hand sides; nrhs 0.

ab

Array ab is of size max (1, ldab*n).

The array ab contains the Cholesky factor, as returned by the factorization routine, in band storage form.

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

b

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

The size of b is at least max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout.

ldab

The leading dimension of the array ab; ldabkd +1.

ldb

The leading dimension of b; ldb max(1, n) for column major layout and ldbnrhs for row major layout.

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(kd + 1)ε P|UH||U| or |E|  c(kd + 1)ε P|LH||L|

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 approximate number of floating-point operations for one right-hand side vector is 4n*kd for real flavors and 16n*kd for complex flavors.

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

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