# ?gbtrs

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

## Syntax

FORTRAN 77:

call sgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

call dgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

call cgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

call zgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )

Fortran 95:

call gbtrs( a, b, ipiv, [, kl] [, trans] [, info] )

## Description

This routine solves for X the following systems of linear equations:

A*X = B

  if trans='N',

`AT*X = B`

  if trans='T',

AH*X = B

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

Here A is an LU-factored general band matrix of order n with kl non-zero subdiagonals and ku nonzero superdiagonals. Before calling this routine, call ?gbtrf to compute the LU factorization of A.

## Input Parameters

trans

  CHARACTER*1. Must be 'N' or 'T' or 'C'.

n

  INTEGER. The order of A; the number of rows in B; n ≥ 0.

kl

  INTEGER. The number of subdiagonals within the band of A; kl ≥ 0.

ku

  INTEGER. The number of superdiagonals within the band of A; ku ≥ 0.

nrhs

  INTEGER. The number of right-hand sides; nrhs ≥ 0.

ab, b

  REAL for sgbtrs

  DOUBLE PRECISION for dgbtrs

  COMPLEX for cgbtrs

  DOUBLE COMPLEX for zgbtrs.

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

  The array ab contains the matrix A in band storage (see Matrix Storage Schemes).

  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)`, and the second dimension of b at least `max(1,nrhs)`.

ldab

  INTEGER. The leading dimension of the array ab; ldab ≥ 2*kl + ku +1.

ldb

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

ipiv

  INTEGER. Array, DIMENSION at least `max(1, n)`. The ipiv array, as returned by ?gbtrf.

## 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 gbtrs interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. Holds the array A of size (`2*kl+ku+1,n`).

b

  Holds the matrix B of size (n, nrhs).

ipiv

  Holds the vector of length min(m, n).

kl

  If omitted, assumed `kl = ku`.

ku

  Restored as `lda-2*kl-1`.

trans

  Must be 'N', 'C', or 'T'. 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(kl + ku + 1)ε P|L||U|`

`c(k)` is a modest linear function of k, 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 is 2n(ku + 2kl) for real flavors. The number of operations for complex flavors is 4 times greater. All these estimates assume that kl and ku are much less than min(m,n).

To estimate the condition number `κ∞(A)`, call ?gbcon.

To refine the solution and estimate the error, call ?gbrfs.

Parent topic: Routines for Solving Systems of Linear Equations