# NAG Library Routine Document

## 1Purpose

f07cdf (dgttrf) computes the $LU$ factorization of a real $n$ by $n$ tridiagonal matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f07cdf ( n, dl, d, du, du2, ipiv, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: ipiv(n), info Real (Kind=nag_wp), Intent (Inout) :: dl(*), d(*), du(*) Real (Kind=nag_wp), Intent (Out) :: du2(n-2)
C Header Interface
#include nagmk26.h
 void f07cdf_ (const Integer *n, double dl[], double d[], double du[], double du2[], Integer ipiv[], Integer *info)
The routine may be called by its LAPACK name dgttrf.

## 3Description

f07cdf (dgttrf) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals.
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{dl}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
On exit: is overwritten by the $\left(n-1\right)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.
3:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the matrix $A$.
On exit: is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
4:     $\mathbf{du}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$.
On exit: is overwritten by the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
5:     $\mathbf{du2}\left({\mathbf{n}}-2\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: contains the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
6:     $\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer arrayOutput
On exit: contains the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$. ${\mathbf{ipiv}}\left(i\right)$ will always be either $i$ or $\left(i+1\right)$, ${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
7:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

## 7Accuracy

The computed factorization satisfies an equation of the form
 $A+E=PLU ,$
where
 $E∞=OεA∞$
and $\epsilon$ is the machine precision.
Following the use of this routine, f07cef (dgttrs) can be used to solve systems of equations $AX=B$ or ${A}^{\mathrm{T}}X=B$, and f07cgf (dgtcon) can be used to estimate the condition number of $A$.

## 8Parallelism and Performance

f07cdf (dgttrf) is not threaded in any implementation.

## 9Further Comments

The total number of floating-point operations required to factorize the matrix $A$ is proportional to $n$.
The complex analogue of this routine is f07crf (zgttrf).

## 10Example

This example factorizes the tridiagonal matrix $A$ given by
 $A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 .$

### 10.1Program Text

Program Text (f07cdfe.f90)

### 10.2Program Data

Program Data (f07cdfe.d)

### 10.3Program Results

Program Results (f07cdfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017