updates docs and cleans up files for differentiation and integration

of Legendre series. this fixes the behavior of intpol. Previously,
intpol set the constant in a way that assumed you were resampling on
the legendre nodes and cancelled the highest degree term. Now, this behvavior
is only used for generating spectral differentiation matrices.
Otherwise, the default is to give the coefficients of the true definite
integral polynomial

chunkie/+lege/dermat.m

@@ -0,0 +1,23 @@
+function mat = dermat(k,u,v)
+%LEGE.DERMAT returns the spectral differentiation matrix on Legendre nodes
+% of order k
+% 
+% input:
+%   k - integer, number of Legendre nodes
+%
+% optional inputs:
+%   u - k x k matrix mapping values at legendre nodes to legendre series
+%      coefficients
+%   v - k x k matrix mapping legendre series coefficients to values at 
+%      legendre nodes
+%
+% output:
+%   mat - spectral differentiation matrix on Legendre nodes
+%
+%see also LEGE.EXPS, LEGE.DERPOL
+
+if nargin < 3
+    [~,~,u,v] = lege.exps(k);
+end
+
+mat = v(:,1:end-1)*lege.derpol(u);

chunkie/+lege/intmat.m

@@ -1,15 +1,24 @@
-function [aint,x,w] = intmat(n)
+function [aint,u,v] = intmat(n,u,v)
 %INTMAT returns the spectral integration matrix on n Gaussian nodes
 % a transcription of part of the Rokhlin routine legeinmt
 %
 % input: 
 %   n - the number of Gaussian nodes 
+%
+% optional inputs:
+%   u - k x k matrix mapping values at legendre nodes to legendre series
+%      coefficients
+%   v - k x k matrix mapping legendre series coefficients to values at 
+%      legendre nodes
 % 
+% output: 
+%   
 
-[x,w,u,v] = lege.exps(n);
+if nargin < 3
+    [~,~,u,v] = lege.exps(n);
+end
 
-coeffs = eye(n);
-polints = lege.intpol(coeffs);
-aint = v*polints(1:end-1,:)*u;
+tmp = lege.intpol(u,'original');
+aint = v*tmp(1:end-1,:);
 
 end

chunkie/+lege/intpol.m

@@ -1,12 +1,42 @@
-function intcoeffs = intpol(coeffs)
+function intcoeffs = intpol(coeffs,const_option)
 %LEGE.INTPOL compute the coefficients of the indefinite 
 % integral of the polynomial with the coefficients given on 
 % input
+%
+% input:
+%  coeffs - array of Legendre series coefficients. if matrix, each column
+%    contains the Legendres series coefficients of a separate function
+% 
+% optional input:
+%  const_option - string. default ('true'). If 'true', then the constant on
+%    output is such that the polynomial(s) described by intcoeffs on output
+%    take the value zero at -1. If 'original', then the constant on output
+%    is such that the polynomial(s) described by intcoeffs(1:end-1,:) on
+%    output take the value zero at -1. This is used for a spectral
+%    differentiation matrix taking point values to point values on a grid
+%    of the same order.
+%
+% output:
+%  intcoeffs - array of Legendre series coefficients for a definte integral
+%    of the input Legendre series 
+%
+
+if nargin < 2
+    const_option = 'true';
+end
 
 sz = size(coeffs);
 szint = sz; szint(1) = szint(1)+1;
 intcoeffs = zeros(szint);
 
+if strcmpi(const_option,'true')
+    ncc = szint(1);
+elseif strcmpi(const_option,'original')
+    ncc = szint(1)-1;
+else
+    error('LEGE.INTPOL: unknown option for constant');
+end
+
 for i = 2:sz(1)
     j = i-1;
     intcoeffs(i+1,:) = coeffs(i,:)/(2*j+1);
@@ -17,7 +47,7 @@
 
 sss=-1;
 dd = zeros(size(intcoeffs(end,:)));
-for k = 2:(szint(1)-1)
+for k = 2:ncc
     dd=dd+intcoeffs(k,:)*sss;
     sss=-sss;
 end

devtools/test/legeexpsunitTest.m

@@ -0,0 +1,32 @@
+%
+
+addpaths_loc();
+
+k = 19;
+
+[x,w,u,v] = lege.exps(k);
+
+dmat = lege.dermat(k,u,v);
+imat = lege.intmat(k,u,v);
+
+pv = sin(x);
+dpv_true = cos(x);
+ipv_true = -cos(x)+cos(-1);
+
+dpv = dmat*pv;
+ipv = imat*pv;
+
+assert(norm(dpv-dpv_true) < 1e-12)
+assert(norm(ipv-ipv_true) < 1e-14)
+
+cfs = randn(k,1);
+cfsint1 = lege.intpol(cfs);
+cfsintold = lege.intpol(cfs,'original');
+fun = @(t) lege.exev(t,cfs);
+
+for j = 1:k
+    itrue = integral(fun,-1,x(j));
+    icoefs = lege.exev(x(j),cfsint1);
+    assert(abs(itrue-icoefs)< 1e-14);
+end
+