nag_dgttrf (f07cdc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dgttrf (f07cdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgttrf (f07cdc) computes the LU  factorization of a real n  by n  tridiagonal matrix A .

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dgttrf (Integer n, double dl[], double d[], double du[], double du2[], Integer ipiv[], NagError *fail)

3  Description

nag_dgttrf (f07cdc) 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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     dl[dim]doubleInput/Output
Note: the dimension, dim, 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[dim]doubleInput/Output
Note: the dimension, dim, 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[dim]doubleInput/Output
Note: the dimension, dim, 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:     du2[n-2]doubleOutput
On exit: contains the n-2 elements of the second superdiagonal of U.
6:     ipiv[n]IntegerOutput
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 ipiv[i-1]. ipiv[i-1] will always be either i or i+1, ipiv[i-1]=i indicating that a row interchange was not performed.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR
Uvalue,value 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 function, nag_dgttrs (f07cec) can be used to solve systems of equations AX=B  or ATX=B , and nag_dgtcon (f07cgc) can be used to estimate the condition number of A .

8  Further Comments

The total number of floating point operations required to factorize the matrix A  is proportional to n .
The complex analogue of this function is nag_zgttrf (f07crc).

9  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 .

9.1  Program Text

Program Text (f07cdce.c)

9.2  Program Data

Program Data (f07cdce.d)

9.3  Program Results

Program Results (f07cdce.r)


nag_dgttrf (f07cdc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012