F07CRF (ZGTTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07CRF (ZGTTRF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07CRF (ZGTTRF) computes the LU  factorization of a complex n  by n  tridiagonal matrix A .

2  Specification

SUBROUTINE F07CRF ( N, DL, D, DU, DU2, IPIV, INFO)
INTEGER  N, IPIV(N), INFO
COMPLEX (KIND=nag_wp)  DL(*), D(*), DU(*), DU2(N-2)
The routine may be called by its LAPACK name zgttrf.

3  Description

F07CRF (ZGTTRF) 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  Parameters

1:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
2:     DL(*) – COMPLEX (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(*) – COMPLEX (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(*) – COMPLEX (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:     DU2(N-2) – COMPLEX (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 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

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
If INFO=i, Ui,i 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, F07CSF (ZGTTRS) can be used to solve systems of equations AX=B  or ATX=B  or AHX=B , and F07CUF (ZGTCON) 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 real analogue of this routine is F07CDF (DGTTRF).

9  Example

This example factorizes the tridiagonal matrix A  given by
A = -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i .

9.1  Program Text

Program Text (f07crfe.f90)

9.2  Program Data

Program Data (f07crfe.d)

9.3  Program Results

Program Results (f07crfe.r)


F07CRF (ZGTTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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