generates orthogonal polynomials
- ORPOL( vector<, maxdegree<, weights>>)
The inputs to the ORPOL function are as follows:
The ORPOL matrix function generates orthogonal polynomials.
The result is a column-orthonormal matrix P with the same
number of rows as the vector and with maxdegree+1 columns:
- is an n ×1 (or 1 ×n) vector of
values over which the polynomials are to be defined.
- specifies the maximum degree polynomial to be computed.
Note that the number of columns in the computed
result is 1+maxdegree, whether maxdegree
is specified or the default value is used.
If maxdegree is omitted, IML
uses the default value min(n,19).
If weights is specified,
maxdegree must also be specified.
- specifies an n ×1 (or 1 ×n) vector of
nonnegative weights to be used in defining orthogonality:
If you specify weights, you
must also specify maxdegree.
If maxdegree is not specified or is specified
incorrectly, the default weights (all weights are 1) are used.
P'* diag( weights)P = I .
The result is computed such that P[i,j]
is the value of a polynomial of degree j-1
evaluated at the ith element of the vector.
P' diag( weights)P = I .
The maximum number of nonzero orthogonal polynomials
(r) that can be computed from the vector and the
weights is the number of distinct values in the vector,
ignoring any value associated with a zero weight.
The polynomial of maximum degree has degree of r-1.
If the value of maxdegree exceeds r-1, then columns r+1,
r+2,..., maxdegree+1 of the result are set to 0.
In this case,
The statement below results in a matrix with 3 orthogonal columns:
ORPOL 5 rows 3 cols (numeric)
0.4472136 -0.632456 0.5345225
0.4472136 -0.316228 -0.267261
0.4472136 0 -0.534522
0.4472136 0.3162278 -0.267261
0.4472136 0.6324555 0.5345225
See "Acknowledgments" in the front of
this book for authorship of the ORPOL function.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.