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NAG Toolbox: nag_lapack_zgttrf (f07cr)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgttrf (f07cr) computes the LU  factorization of a complex n  by n  tridiagonal matrix A .

Syntax

[dl, d, du, du2, ipiv, info] = f07cr(dl, d, du, 'n', n)
[dl, d, du, du2, ipiv, info] = nag_lapack_zgttrf(dl, d, du, 'n', n)

Description

nag_lapack_zgttrf (f07cr) 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.

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

Parameters

Compulsory Input Parameters

1:     dl: – complex array
The dimension of the array dl must be at least max1,n-1
Must contain the n-1 subdiagonal elements of the matrix A.
2:     d: – complex array
The dimension of the array d must be at least max1,n
Must contain the n diagonal elements of the matrix A.
3:     du: – complex array
The dimension of the array du must be at least max1,n-1
Must contain the n-1 superdiagonal elements of the matrix A.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array d.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     dl: – complex array
The dimension of the array dl will be max1,n-1
Stores the n-1 multipliers that define the matrix L of the LU factorization of A.
2:     d: – complex array
The dimension of the array d will be max1,n
Stores the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
3:     du: – complex array
The dimension of the array du will be max1,n-1
Stores the n-1 elements of the first superdiagonal of U.
4:     du2n-2 – complex array
Contains the n-2 elements of the second superdiagonal of U.
5:     ipivn int64int32nag_int array
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.
6:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ 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.

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_lapack_zgttrs (f07cs) can be used to solve systems of equations AX=B  or ATX=B  or AHX=B , and nag_lapack_zgtcon (f07cu) can be used to estimate the condition number of A .

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 nag_lapack_dgttrf (f07cd).

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 .  
function f07cr_example


fprintf('f07cr example results\n\n');

% Tridiagonal matrix stored by diagonals
du = [              2   - 1i     2   + 1i    -1   + 1i     1   - 1i  ];
d  = [-1.3 + 1.3i  -1.3 + 1.3i  -1.3 + 3.3i  -0.3 + 4.3i  -3.3 + 1.3i];
dl = [ 1   - 2i     1   + 1i     2   - 3i     1   + 1i               ];

% Factorize.
[dlf, df, duf, du2, ipiv, info] = ...
  f07cr(dl, d, du);

disp('Details of factorization');
fprintf('\n');
disp(' Second super-diagonal of U');
disp(du2');
disp(' First super-diagonal of U');
disp(duf);
disp(' Main diagonal of U');
disp(df(1:4));
disp(df(5:end));
disp(' Multipliers');
disp(dlf);
disp(' Vector of interchanges');
disp(double(ipiv)');


f07cr example results

Details of factorization

 Second super-diagonal of U
   2.0000 - 1.0000i  -1.0000 - 1.0000i   1.0000 + 1.0000i

 First super-diagonal of U
  -1.3000 + 1.3000i  -1.3000 + 3.3000i  -0.3000 + 4.3000i  -3.3000 + 1.3000i

 Main diagonal of U
   1.0000 - 2.0000i   1.0000 + 1.0000i   2.0000 - 3.0000i   1.0000 + 1.0000i

  -1.3399 + 0.2875i

 Multipliers
  -0.7800 - 0.2600i   0.1620 - 0.4860i  -0.0452 - 0.0010i  -0.3979 - 0.0562i

 Vector of interchanges
     2     3     4     5     5


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Chapter Introduction
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