1: $a (lda, n2)$ – double array: lda, the first dimension of the array, must satisfy the constraint $lda \geq m$ .
Columns n1 to n2 contain the vectors to be orthogonalized. The vectors are stored by columns in elements $1$ to $m$ .
2: $n1$ – int64int32nag_int scalar: The indices of the first and last columns of $A$ to be orthogonalized.

Constraint: $n1 \leq n2$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$m$ , the number of elements in each vector.
2: $n2$ – int64int32nag_int scalar: Default: For n2, the second dimension of the array a.
The indices of the first and last columns of $A$ to be orthogonalized.

Constraint: $n1 \leq n2$ .

Output Parameters

1: $a (lda, n2)$ – double array: These vectors store the orthonormal vectors.
2: $cc$ – double scalar: Is used to indicate linear dependence of the original vectors. The nearer cc is to $1.0$ , the more likely vector icol is dependent on vectors n1 to $icol - 1$ . See Further Comments.
3: $icol$ – int64int32nag_int scalar: The column number corresponding to cc. See Further Comments.
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

n1 > n2

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Innerproducts are accumulated using additional precision arithmetic and full machine accuracy should be obtained except when

cc > 0.99999

. (See Further Comments.)

Further Comments

The time taken by nag_orthog_real_gram_schmidt (f05aa) is approximately proportional to

n m^{2}

, where

n = n2 - n1 + 1

Arguments cc and icol have been included to give some indication of whether or not the vectors are nearly linearly independent, and their values should always be tested on exit from the function. cc will be in the range

[0.0, 1.0]

and the closer cc is to

1.0

, the more likely the vector icol is to be linearly dependent on vectors n1 to

icol - 1

. Theoretically, when the vectors are linearly dependent, cc should be exactly

1.0

. In practice, because of rounding errors, it may be difficult to decide whether or not a value of cc close to

1.0

indicates linear dependence. As a general guide a value of

cc > 0.99999

usually indicates linear dependence, but examples exist which give

cc > 0.99999

for linearly independent vectors. If one of the original vectors is zero or if, possibly due to rounding errors, an exactly zero vector is produced by the Gram–Schmidt process, then cc is set exactly to

1.0

and the vector is not, of course, normalized. If more than one such vector occurs then icol references the last of these vectors.

If you are concerned about testing for near linear dependence in a set of vectors you may wish to consider using function nag_lapack_dgesvd (f08kb).

Example

This example orthonormalizes columns 2, 3 and 4 of the matrix:

(\begin{array}{r} 1 & - 2 & 3 & 1 \\ - 2 & 1 & - 2 & - 1 \\ 3 & - 2 & 1 & 5 \\ 4 & 1 & 5 & 3 \end{array}) .

Open in the MATLAB editor: f05aa_example

function f05aa_example


fprintf('f05aa example results\n\n');

a = [ 1, -2,  3,  1;
     -2,  1, -2, -1;
      3, -2,  1,  5;
      4,  1,  5,  3];

% Orthonormalize all but first column of A
n1 = int64(2);
[a, cc, icol, ifail] = f05aa(a, n1);

fprintf('Linear dependence measure for column %1d = %6.4f\n', icol, cc);
fprintf('\nFinal orthonormalized columns\n');
disp(a(:,n1:end));

f05aa example results

Linear dependence measure for column 4 = 0.5822

Final orthonormalized columns
   -0.6325    0.3310   -0.5404
    0.3162   -0.2483    0.2119
   -0.6325   -0.0000    0.7735
    0.3162    0.9104    0.2543

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox