constructpolynomialmatrix

 function f = constructpolynomialmatrix(n,degrees)

 <n> is the number of points
 <degrees> is a vector of polynomial degrees

 return a matrix of dimensions <n> x length(<degrees>)
 with polynomials in the columns.  each column is orthogonalized
 with respect to all of the earlier columns and then made 
 unit-length.  please see the code for the exact details.
 beware of numerical precision issues for high degrees...

 history:
 - 2014/07/31 - now, we orthogonalize and make unit length.
                this changes previous behavior!

 example:
 X = constructpolynomialmatrix(100,0:3);
 figure; subplot(1,2,1); imagesc(X); subplot(1,2,2); imagesc(X'*X);

0001 function f = constructpolynomialmatrix(n,degrees)
0002 
0003 % function f = constructpolynomialmatrix(n,degrees)
0004 %
0005 % <n> is the number of points
0006 % <degrees> is a vector of polynomial degrees
0007 %
0008 % return a matrix of dimensions <n> x length(<degrees>)
0009 % with polynomials in the columns.  each column is orthogonalized
0010 % with respect to all of the earlier columns and then made
0011 % unit-length.  please see the code for the exact details.
0012 % beware of numerical precision issues for high degrees...
0013 %
0014 % history:
0015 % - 2014/07/31 - now, we orthogonalize and make unit length.
0016 %                this changes previous behavior!
0017 %
0018 % example:
0019 % X = constructpolynomialmatrix(100,0:3);
0020 % figure; subplot(1,2,1); imagesc(X); subplot(1,2,2); imagesc(X'*X);
0021 
0022 % do it
0023 f = [];
0024 temp = linspace(-1,1,n)';
0025 for p=1:length(degrees)
0026 
0027   % construct polynomial
0028   polyvector = temp .^ degrees(p);
0029   
0030   % orthogonalize with respect to earlier polynomials and make unit length
0031   polyvector = unitlength(projectionmatrix(f)*polyvector);
0032 
0033   % record
0034   f = cat(2,f,polyvector);
0035   
0036 end