?tbtrs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?tbtrs

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

Syntax

call stbtrs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info )

call dtbtrs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info )

call ctbtrs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info )

call ztbtrs( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info )

call tbtrs( ab, b [,uplo] [, trans] [,diag] [,info] )

Include Files

mkl.fi, lapack.f90

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

uplo	CHARACTER*1. 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	CHARACTER1. 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	CHARACTER*1. 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	INTEGER. The order of `A`; the number of rows in `B`; `n`≥ 0.
kd	INTEGER. The number of superdiagonals or subdiagonals in the matrix `A`; `kd`≥ 0.
nrhs	INTEGER. The number of right-hand sides; `nrhs`≥ 0.
ab	REAL for stbtrs DOUBLE PRECISION for dtbtrs COMPLEX for ctbtrs DOUBLE COMPLEX for ztbtrs. The array `ab(ldab,*)` contains the matrix `A` in band storage form. The second dimension of `ab` must be at least `max(1, n)`.
b	REAL for stbtrs DOUBLE PRECISION for dtbtrs COMPLEX for ctbtrs DOUBLE COMPLEX for ztbtrs. The array `b(ldb,*)` contains the matrix `B` whose columns are the right-hand sides for the systems of equations. The second dimension of `b` at least `max(1,nrhs)`.
ldab	INTEGER. The leading dimension of `ab`; `ldab`≥`kd` + 1.
ldb	INTEGER. The leading dimension of `b`; `ldb≥ max(1, n)`.

Output Parameters

Overwritten by the solution matrix X.

info

INTEGER. If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine tbtrs interface are as follows:

ab	Holds the array `A` of size `(kd+1,n)`
b	Holds the matrix `B` of size (`n`, `nrhs`).
uplo	Must be 'U' or 'L'. The default value is 'U'.
trans	Must be 'N', 'C', or 'T'. The default value is 'N'.
diag	Must be 'N' or 'U'. The default value is 'N'.

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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