NAG FL Interface
f07crf (zgttrf)

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1 Purpose

f07crf computes the LU factorization of a complex n × n tridiagonal matrix A .

2 Specification

Fortran Interface
Subroutine f07crf ( n, dl, d, du, du2, ipiv, info)
Integer, Intent (In) :: n
Integer, Intent (Out) :: ipiv(n), info
Complex (Kind=nag_wp), Intent (Inout) :: dl(*), d(*), du(*)
Complex (Kind=nag_wp), Intent (Out) :: du2(n-2)
C Header Interface
#include <nag.h>
void  f07crf_ (const Integer *n, Complex dl[], Complex d[], Complex du[], Complex du2[], Integer ipiv[], Integer *info)
The routine may be called by the names f07crf, nagf_lapacklin_zgttrf or its LAPACK name zgttrf.

3 Description

f07crf 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(*) Complex (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(*) Complex (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(*) Complex (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) Complex (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, f07csf can be used to solve systems of equations AX=B or ATX=B or AHX=B , and f07cuf can be used to estimate the condition number of A .

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07crf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 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.

10 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 ) .  

10.1 Program Text

Program Text (f07crfe.f90)

10.2 Program Data

Program Data (f07crfe.d)

10.3 Program Results

Program Results (f07crfe.r)