Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 6/24/2024
Public

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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',

AT*X = B

if trans='T',

AH*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

CHARACTER*1. 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

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; ldabkd + 1.

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

Output Parameters

b

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:


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.