# ?gttrf

Computes the LU factorization of a tridiagonal matrix.

## Syntax

FORTRAN 77:

call sgttrf( n, dl, d, du, du2, ipiv, info )

call dgttrf( n, dl, d, du, du2, ipiv, info )

call cgttrf( n, dl, d, du, du2, ipiv, info )

call zgttrf( n, dl, d, du, du2, ipiv, info )

Fortran 95:

call gttrf( dl, d, du, du2 [, ipiv] [,info] )

## Description

The routine computes the LU factorization of a real or complex tridiagonal matrix A in the form
`A = P*L*U,`

where P is a permutation matrix; L is lower bidiagonal with unit diagonal elements; and U is an upper triangular matrix with nonzeroes in only the main diagonal and first two superdiagonals. The routine uses elimination with partial pivoting and row interchanges.

## Input Parameters

n

  INTEGER. The order of the matrix A; n ≥ 0.

dl, d, du

  REAL for sgttrf

  DOUBLE PRECISION for dgttrf

  COMPLEX for cgttrf

  DOUBLE COMPLEX for zgttrf.

  Arrays containing elements of A.

  The array dl of dimension `(n - 1)` contains the subdiagonal elements of A.

  The array d of dimension n contains the diagonal elements of A.

  The array du of dimension `(n - 1)` contains the superdiagonal elements of A.

## Output Parameters

dl

  Overwritten by the `(n-1)` multipliers that define the matrix L from the LU factorization of A.

d

  Overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A.

du

  Overwritten by the `(n-1)` elements of the first superdiagonal of U.

du2

  REAL for sgttrf

  DOUBLE PRECISION for dgttrf

  COMPLEX for cgttrf

  DOUBLE COMPLEX for zgttrf.

  Array, dimension `(n-2)`. On exit, du2 contains `(n-2)` elements of the second superdiagonal of U.

ipiv

  INTEGER.

  Array, dimension (n). The pivot indices: row i was interchanged with row ipiv(i).

info

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

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

  If `info = i, uii` is 0. The factorization has been completed, but U is exactly singular. Division by zero will occur if you use the factor U for solving a system of linear equations.

## 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 gttrf interface are as follows:

dl

  Holds the vector of length (n-1).

d

  Holds the vectror of length (n).

du

  Holds the vector of length (n-1).

du2

  Holds the vector of length (n-2).

ipiv

  Holds the vector of length (n).

## Application Notes

?gbtrs

  to solve `A*X = B` or `AT*X = B` or `AH*X = B`

?gbcon

  to estimate the condition number of A.

Parent topic: Routines for Matrix Factorization