NAG FL Interface
f07cdf (dgttrf)

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1 Purpose

f07cdf computes the LU factorization of a real n × 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 <nag.h>
void  f07cdf_ (const Integer *n, double dl[], double d[], double du[], double du2[], Integer ipiv[], Integer *info)
The routine may be called by the names f07cdf, nagf_lapacklin_dgttrf or its LAPACK name dgttrf.

3 Description

f07cdf 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 https://www.netlib.org/lapack/lug

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: dl(*) Real (Kind=nag_wp) array Input/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 LU factorization of A.
3: d(*) Real (Kind=nag_wp) array Input/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 LU factorization of A.
4: du(*) Real (Kind=nag_wp) array Input/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) array Output
On exit: contains the (n-2) elements of the second superdiagonal of U.
6: ipiv(n) Integer array Output
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). ipiv(i) will always be either i or (i+1), ipiv(i)=i indicating that a row interchange was not performed.
7: info Integer Output
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=O(ε)A  
and ε is the machine precision.
Following the use of this routine, f07cef can be used to solve systems of equations AX=B or ATX=B , and f07cgf can be used to estimate the condition number of A .

8 Parallelism and Performance

f07cdf 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.

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)