# ?gttrs

Solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by ?gttrf.

## Syntax

FORTRAN 77:

call sgttrs( trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info )

call dgttrs( trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info )

call cgttrs( trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info )

call zgttrs( trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info )

Fortran 95:

call gttrs( dl, d, du, du2, b, ipiv [, trans] [,info] )

## Description

This routine solves for X the following systems of linear equations with multiple right hand sides:

A*X = B

  if trans='N',

`AT*X = B`

  if trans='T',

AH*X = B

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

Before calling this routine, you must call ?gttrf to compute the LU factorization of A.

## Input Parameters

trans

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

  Indicates the form of the equations:

  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.

n

  INTEGER. The order of A; n ≥ 0.

nrhs

  INTEGER. The number of right-hand sides, that is, the number of columns in `B; nrhs ≥ 0`.

dl,d,du,du2,b

  REAL for sgttrs

  DOUBLE PRECISION for dgttrs

  COMPLEX for cgttrs

  DOUBLE COMPLEX for zgttrs.

  Arrays: `dl(n -1)`, `d(n)`, `du(n -1)`, `du2(n -2)`, `b(ldb,nrhs)`.

  The array dl contains the `(n - 1)` multipliers that define the matrix L from the LU factorization of A.

  The array d contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.

  The array du contains the (n - 1) elements of the first superdiagonal of U.

  The array du2 contains the (n - 2) elements of the second superdiagonal of U.

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

ldb

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

ipiv

  INTEGER. Array, DIMENSION (n). The ipiv array, as returned by ?gttrf.

## 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 gttrs interface are as follows:

dl

  Holds the vector of length (n-1).

d

  Holds the vector of length (n).

du

  Holds the vector of length (n-1).

du2

  Holds the vector of length (n-2).

b

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

ipiv

  Holds the vector of length (n).

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(n)ε P|L||U|`

`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 `7n` (including n divisions) for real flavors and `34n` (including `2n` divisions) for complex flavors.

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

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

Parent topic: Routines for Solving Systems of Linear Equations