# ?tbtrs

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

## Syntax

FORTRAN 77:

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 )

Fortran 95:

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

## Description

This 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, b

  REAL for stbtrs

  DOUBLE PRECISION for dtbtrs

  COMPLEX for ctbtrs

  DOUBLE COMPLEX for ztbtrs.

  Arrays: `ab(ldab,*)`,`b(ldb,*)`.

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

  The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

  The second dimension of ab must be at least `max(1, n)`, the second dimension of b at least `max(1,nrhs)`.

ldab

  INTEGER. The first dimension of ab; ldab ≥ kd + 1.

ldb

  INTEGER. The first 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.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine tbtrs interface are as follows:

a

  Stands for the argument ab in Fortan 77 interface. 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 `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.

Parent topic: Routines for Solving Systems of Linear Equations