?pbtrs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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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 = U^TU` for real data, `A = U^HU` for complex data	if `uplo='U'`
`A = LL^T` for real data, `A = LL^H` 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` = `U`^T`U` for real matrices and `A` = `U`^H`U` for complex matrices. If `uplo = 'L'`, `L` is stored in `ab`, where `A` = `L``L`^T for real matrices and `A` = `L``L`^H 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, ldbnrhs) for column major layout and max(1, ldbn) for row major layout.
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

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|U^H||U| or |E| ≤ c(kd + 1)ε P|L^H||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:

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.

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

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