Language Reference: GSORTH Call

GSORTH Call

CALL GSORTH( P, T, lindep, A ) ;

The GSORTH subroutine computes the Gram-Schmidt orthonormal factorization of the $\text{[math]}$ matrix $\text{[math]}$ , where $\text{[math]}$ is greater than or equal to $\text{[math]}$ . The GSORTH subroutine implements an algorithm described by Golub (1969).

The GSORTH subroutine has a single input argument:

A: is an input $\text{[math]}$ matrix.

The output arguments to the GSORTH subroutine are as follows:

P: is an $\text{[math]}$ column-orthonormal output matrix.
T: is an upper triangular $\text{[math]}$ output matrix.
lindep: is a flag with a value of 0 if columns of A are independent and a value of 1 if they are dependent. The lindep argument is an output scalar.

Specifically, the GSORTH subroutine computes the column-orthonormal $\text{[math]}$ matrix $\text{[math]}$ and the upper triangular $\text{[math]}$ matrix $\text{[math]}$ such that

$\text{[math]}$

If the columns of $\text{[math]}$ are linearly independent (that is, rank $\text{[math]}$ ), then $\text{[math]}$ is full-rank column-orthonormal: $\text{[math]}$ , $\text{[math]}$ is nonsingular, and the value of lindep (a scalar) is set to 0. If the columns of $\text{[math]}$ are linearly dependent (say, rank $\text{[math]}$ ) then $\text{[math]}$ columns of $\text{[math]}$ are set to 0, the corresponding rows of $\text{[math]}$ are set to 0 ( $\text{[math]}$ is singular), and lindep is set to 1. The pattern of zero columns in $\text{[math]}$ corresponds to the pattern of linear dependencies of the columns of $\text{[math]}$ when columns are considered in left-to-right order.

The following statements call the GSORTH subroutine and print the output parameters to the call:

x = {1 1 3 1 2,
     1 0 1 2 3,
     1 1 3 3 4,
     1 0 1 4 5,
     1 1 3 5 6,
     1 0 1 6 7};
call gsorth(P, T, lindep, x);
reset fuzz;
print P, T, lindep;

Figure 23.127 Results of a Gram-Schmidt Orthonormalization

P
0.4082483	0.4082483	-0.5
0.4082483	-0.408248	-0.5
0.4082483	0.4082483	0
0.4082483	-0.408248	0
0.4082483	0.4082483	0.5
0.4082483	-0.408248	0.5

T
2.4494897	1.2247449	4.8989795	8.5732141	11.022704
0	1.2247449	2.4494897	-1.224745	-1.224745
0	0	0	0	0
0	0	0	4	4
0	0	0	0	0

lindep
1

If lindep is 1, you can permute the columns of $\text{[math]}$ and rows of $\text{[math]}$ so that the zero columns of $\text{[math]}$ are rightmost—that is, $\text{[math]}$ , where $\text{[math]}$ is the column rank of $\text{[math]}$ and the equality $\text{[math]}$ is preserved. The following statements show a permutation of columns:

d = loc(vecdiag(T)^=0) || loc(vecdiag(T)=0);
temp = P;
P[,d] = temp;
temp = T;
T[,d] = temp;
print d, P, T;

Figure 23.128 Rearranging Columns

d
1	2	4	3	5

P
0.4082483	0.4082483	-0.5
0.4082483	-0.408248	-0.5
0.4082483	0.4082483	0
0.4082483	-0.408248	0
0.4082483	0.4082483	0.5
0.4082483	-0.408248	0.5

T
2.4494897	1.2247449	8.5732141	4.8989795	11.022704
0	1.2247449	-1.224745	2.4494897	-1.224745
0	0	0	0	0
0	0	4	0	4
0	0	0	0	0

The GSORTH subroutine is not recommended for the construction of matrices of values of orthogonal polynomials; the ORPOL function should be used for that purpose.