NAG FL Interface
f05aaf (real_​gram_​schmidt)

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1 Purpose

f05aaf applies the Schmidt orthogonalization process to n vectors in m-dimensional space, nm.

2 Specification

Fortran Interface
Subroutine f05aaf ( a, lda, m, n1, n2, s, cc, icol, ifail)
Integer, Intent (In) :: lda, m, n1, n2
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: icol
Real (Kind=nag_wp), Intent (Inout) :: a(lda,n2)
Real (Kind=nag_wp), Intent (Out) :: s(n2), cc
C Header Interface
#include <nag.h>
void  f05aaf_ (double a[], const Integer *lda, const Integer *m, const Integer *n1, const Integer *n2, double s[], double *cc, Integer *icol, Integer *ifail)
The routine may be called by the names f05aaf or nagf_orthog_real_gram_schmidt.

3 Description

f05aaf applies the Schmidt orthogonalization process to n linearly independent vectors in m-dimensional space, nm. The effect of this process is to replace the original n vectors by n orthonormal vectors which have the property that the rth vector is linearly dependent on the first r of the original vectors, and that the sum of squares of the elements of the rth vector is equal to 1, for r=1,2,,n. Inner-products are accumulated using additional precision.

4 References


5 Arguments

1: a(lda,n2) Real (Kind=nag_wp) array Input/Output
On entry: columns n1 to n2 contain the vectors to be orthogonalized. The vectors are stored by columns in elements 1 to m.
On exit: these vectors are overwritten by the orthonormal vectors.
2: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f05aaf is called.
Constraint: ldam.
3: m Integer Input
On entry: m, the number of elements in each vector.
4: n1 Integer Input
5: n2 Integer Input
On entry: the indices of the first and last columns of A to be orthogonalized.
Constraint: n1n2.
6: s(n2) Real (Kind=nag_wp) array Workspace
7: cc Real (Kind=nag_wp) Output
On exit: 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 Section 9.
8: icol Integer Output
On exit: the column number corresponding to cc. See Section 9.
9: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, lda=value, m=value.
Constraint: ldam.
On entry, n1=value and n2=value.
Constraint: n1n2.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Innerproducts are accumulated using additional precision arithmetic and full machine accuracy should be obtained except when cc>0.99999. (See Section 9.)

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f05aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 time taken by f05aaf is approximately proportional to nm2, 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 routine. 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 routine f08kbf.

10 Example

This example orthonormalizes columns 2, 3 and 4 of the matrix:
( 1 −2 3 1 −2 1 −2 −1 3 −2 1 5 4 1 5 3 ) .  

10.1 Program Text

Program Text (f05aafe.f90)

10.2 Program Data

Program Data (f05aafe.d)

10.3 Program Results

Program Results (f05aafe.r)