NAG Library Routine Document

f07cdf (dgttrf)

1
Purpose

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

2
Specification

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.

3
Description

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.

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: n0.
2:     dl* – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array dl must be at least max1,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 LU factorization of A.
3:     d* – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array d must be at least max1,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 LU factorization of A.
4:     du* – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array du must be at least max1,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:     du2n-2 – Real (Kind=nag_wp) arrayOutput
On exit: contains the n-2 elements of the second superdiagonal of U.
6:     ipivn – Integer arrayOutput
On exit: contains the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row ipivi. ipivi will always be either i or i+1, ipivi=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=PLU ,  
where
E=OεA  
and ε  is the machine precision.
Following the use of this routine, f07cef (dgttrs) can be used to solve systems of equations AX=B  or ATX=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 = 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 .  

10.1
Program Text

Program Text (f07cdfe.f90)

10.2
Program Data

Program Data (f07cdfe.d)

10.3
Program Results

Program Results (f07cdfe.r)