NAG CL Interface
f07crc (zgttrf)

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1 Purpose

f07crc computes the LU factorization of a complex n × n tridiagonal matrix A .

2 Specification

#include <nag.h>
void  f07crc (Integer n, Complex dl[], Complex d[], Complex du[], Complex du2[], Integer ipiv[], NagError *fail)
The function may be called by the names: f07crc, nag_lapacklin_zgttrf or nag_zgttrf.

3 Description

f07crc 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[dim] Complex Input/Output
Note: the dimension, dim, 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[dim] Complex Input/Output
Note: the dimension, dim, 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[dim] Complex Input/Output
Note: the dimension, dim, 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 Output
On exit: contains the (n-2) elements of the second superdiagonal of U.
6: ipiv[n] Integer 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-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: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
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 function, f07csc can be used to solve systems of equations AX=B or ATX=B or AHX=B , and f07cuc 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.
f07crc 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 function. 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 function is f07cdc.

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 (f07crce.c)

10.2 Program Data

Program Data (f07crce.d)

10.3 Program Results

Program Results (f07crce.r)