?tbtrs

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?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',
`A^T*X = B`	if `trans`='T',
A^H*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 `AX = B` is solved for `X`. If `trans = 'T'`, then `A`^T`X` = `B` is solved for `X`. If `trans = 'C'`, then `A`^H`*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, ldbnrhs) for column major layout and max(1, ldbn) for row major layout.
ldab	The leading dimension of `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(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:

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 A^T and A^H 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.

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

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