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.

3

Description

f07cdf (dgttrf) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

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.

4

References

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

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $dl (*)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the dimension of the array dl must be at least $\max (1, n - 1)$ .

On entry: must contain the $(n - 1)$ subdiagonal elements of the matrix $A$ .

On exit: is overwritten by the $(n - 1)$ multipliers that define the matrix $L$ of the $L U$ factorization of $A$ .
3: $d (*)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the dimension of the array d must be at least $\max (1, n)$ .

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 $L U$ factorization of $A$ .
4: $du (*)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the dimension of the array du must be at least $\max (1, n - 1)$ .

On entry: must contain the $(n - 1)$ superdiagonal elements of the matrix $A$ .

On exit: is overwritten by the $(n - 1)$ elements of the first superdiagonal of $U$ .
5: $du2 (n - 2)$ – Real (Kind=nag_wp) arrayOutput: On exit: contains the $(n - 2)$ elements of the second superdiagonal of $U$ .
6: $ipiv (n)$ – 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 $ipiv (i)$ . $ipiv (i)$ will always be either $i$ or $(i + 1)$ , $ipiv (i) = i$ indicating that a row interchange was not performed.
7: $info$ – IntegerOutput: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: Element $〈value〉$ 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.

7

Accuracy

The computed factorization satisfies an equation of the form

A + E = P L U,

where

{‖E‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

is the machine precision.

Following the use of this routine, f07cef (dgttrs) can be used to solve systems of equations

A X = B

A^{T} X = B

, and f07cgf (dgtcon) can be used to estimate the condition number of

A

8

Parallelism and Performance

f07cdf (dgttrf) is not threaded in any implementation.

9

Further 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).

10

Example

This example factorizes the tridiagonal matrix

A

given by

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) .

NAG Library Routine Document

f07cdf (dgttrf)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification