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NAG Toolbox: nag_lapack_dgetrf (f07ad)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgetrf (f07ad) computes the LU factorization of a real m by n matrix.

Syntax

[a, ipiv, info] = f07ad(a, 'm', m, 'n', n)
[a, ipiv, info] = nag_lapack_dgetrf(a, 'm', m, 'n', n)

Description

nag_lapack_dgetrf (f07ad) forms the LU factorization of a real m by n matrix A as A=PLU, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m>n) and U is upper triangular (upper trapezoidal if m<n). Usually A is square m=n, and both L and U are triangular. The function uses partial pivoting, with row interchanges.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     alda: – double array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
The m by n matrix A.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array a.
m, the number of rows of the matrix A.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,n.
The factors L and U from the factorization A=PLU; the unit diagonal elements of L are not stored.
2:     ipivminm,n int64int32nag_int array
The pivot indices that define the permutation matrix. At the ith step, if ipivi>i then row i of the matrix A was interchanged with row ipivi, for i=1,2,,minm,n. ipivii indicates that, at the ith step, a row interchange was not required.
3:     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 factors L and U are the exact factors of a perturbed matrix A+E, where
E c minm,n ε P L U ,  
cn is a modest linear function of n, and ε is the machine precision.

Further Comments

The total number of floating-point operations is approximately 23n3 if m=n (the usual case), 13n23m-n if m>n and 13m23n-m if m<n.
A call to this function with m=n may be followed by calls to the functions:
The complex analogue of this function is nag_lapack_zgetrf (f07ar).

Example

This example computes the LU factorization of the matrix A, where
A= 1.80 2.88 2.05 -0.89 5.25 -2.95 -0.95 -3.80 1.58 -2.69 -2.90 -1.04 -1.11 -0.66 -0.59 0.80 .  
function f07ad_example


fprintf('f07ad example results\n\n');

a = [ 1.80,  2.88,  2.05, -0.89;
      5.25, -2.95, -0.95, -3.80;
      1.58, -2.69, -2.90, -1.04;
     -1.11, -0.66, -0.59,  0.80];

% LU Factorize
[LU, ipiv, info] = f07ad(a);

disp('Details of factorization');
disp(LU);
disp('Pivot indices');
disp(double(ipiv'));


f07ad example results

Details of factorization
    5.2500   -2.9500   -0.9500   -3.8000
    0.3429    3.8914    2.3757    0.4129
    0.3010   -0.4631   -1.5139    0.2948
   -0.2114   -0.3299    0.0047    0.1314

Pivot indices
     2     2     3     4


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Chapter Contents
Chapter Introduction
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