Language Reference

GSORTH Call

computes the Gram-Schmidt orthonormalization

CALL GSORTH( , , lindep, );

The inputs to the GSORTH subroutine are as follows:

is an column-orthonormal output matrix.

is an upper triangular output matrix.

lindep

is a flag with a value of 0 if columns of are independent and a value of 1 if they are dependent. The lindep argument is an output scalar.

is an input matrix.

The GSORTH subroutine computes the Gram-Schmidt orthonormal factorization of the matrix , where is greater than or equal to ; that is, the GSORTH subroutine computes the column-orthonormal matrix and the upper triangular matrix such that

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

The GSORTH subroutines implements an algorithm described by Golub (1969).

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.

If lindep is 1, you can rearrange the columns of and rows of so that the zero columns of are right-most - that is, , where is the column rank of and is preserved. The following statements make this rearrangement:

  
    d=rank((ncol(t)-(1:ncol(t))`)#(vecdiag(t)=0)); 
    temp=p; 
    p[,d]=temp; 
    temp=t; 
    t[,d]=temp;
 

An example of a valid GSORTH call follows:

  
    x={1 1 1, 1 2 4, 1 3 9}; 
    xpx=x`*x; 
    call gsorth(p, t, l, xpx);
 

These statements produce the following output matrices:

  
                 P             3 rows      3 cols    (numeric) 
  
                          0.193247 -0.753259 0.6286946 
                          0.386494 -0.530521 -0.754434 
                         0.9018193 0.3887787 0.1886084 
  
  
                 T             3 rows      3 cols    (numeric) 
  
                         15.524175 39.035892 104.99753 
                                 0 2.0491877 8.4559365 
                                 0         0 0.1257389 
  
  
                 L             1 row       1 col     (numeric) 
  
                                           0