Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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Document Table of Contents

?tbtrs

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

Syntax

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

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

lapack_int LAPACKE_ctbtrs (int matrix_layout , char uplo , char trans , char diag , 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_ztbtrs (int matrix_layout , char uplo , char trans , char diag , 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 X the following systems of linear equations with a band triangular matrix A, with multiple right-hand sides stored in B:

A*X = B

if trans='N',

AT*X = B

if trans='T',

AH*X = B

if trans='C' (for complex matrices only).

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 A is upper or lower triangular:

If uplo = 'U', then A is upper triangular.

If uplo = 'L', then A is lower triangular.

trans

Must be 'N' or 'T' or 'C'.

If trans = 'N', then A*X = B is solved for X.

If trans = 'T', then AT*X = B is solved for X.

If trans = 'C', then AH*X = B is solved for X.

diag

Must be 'N' or 'U'.

If diag = 'N', then A is not a unit triangular matrix.

If diag = 'U', then A is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array ab.

n

The order of A; the number of rows in B; 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

The array ab contains the matrix A in band storage form.

The size of ab must be max(1, ldab*n)

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

ldab

The leading dimension of 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(n)ε|A|

c(n) is a modest linear function of n, 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 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 b is 2n*kd for real flavors and 8n*kd for complex flavors.

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

To estimate the error in the solution, call ?tbrfs.