Optimized helper functions for divergence

altPascal is now calculated in a single loop. Both divergence helper and extensionOperatorDivergence now create their outputs without inserting into a sparse matrix(which is slow).

src/matlab/alt_pascal.m

@@ -1,17 +1,13 @@
 function [ A ] = alt_pascal( n )
-%   Calculates the alternating pascal triangle.
-%   Will give exact values for n<56
-    if(n>1030); error('Argument should be less than 1031'); end
+%   Calculates the alternating pascal triangle up to n rows.
+%   This is the change of basis matrix from the basis {i in {0,1,...,n-1}:(e^(h*D)-1)^i} to 
+%   the standard basis. {i in {0,1,...,n-1}:(e^(i*h*D))}
     A=eye(n);
-    A(:,1)=1;
+    A(:,1)=2*rem(1:n,2) - 1; %Alternating vector [1,-1,1,-1,...]
     for i=2:n
-        A(i,2:end)=A(i-1,1:end-1)+A(i-1,2:end);
-    end
-    for i=1:n
-        for j=1:i
-            A(i,j)=A(i,j)*(-1)^(i+j);
-        end
+        A(i,2:end)=A(i-1,1:end-1)-A(i-1,2:end);
     end
+
 
 
 end

src/matlab/calculateInteriorRow.m

@@ -9,7 +9,7 @@
     % Our answer. 
     r=idivide(k,int32(2));
     output(r)=-1.0;
-    output(r+1)=1.0;
+    output(r+1)=1.0; %our starting value for this series is (e^(h/2*d/dx)-e^(-h/2*d/dx))
     if (k>2)
         nextTerm=zeros([1,k]);% NextTerm in the series.
         %Initially set to (e^(h/2*d/dx)-e^(-h/2*d/dx))^(3)

src/matlab/divergencehelper.m

@@ -1,14 +1,25 @@
 function D=divergencehelper(k,m)
 %Calculates the mimetic divergence operator of order k,
 %   barring dividing by dx. 
-%   k=order of accuracy
-%
-    r=calculateInteriorRow(k);
-    A=sparse(m+2,m+k-1); %This matrix is just using the interior scheme.Since we've extended our vector field
-    %beyond the boundaries it'd normally have, we don't need to use
-    %seperate power series expansions for the stencils near the boundary. 
+%   k :order of accuracy.
+%   m :number of cells. 
+    interior=calculateInteriorRow(k); %The interior row of our matrix 
+    numNonZeros=m*k; % The number of nonZeroElements in the interior stencil matrix
+    rowList=zeros([1,numNonZeros]);
+    columnList=zeros([1,numNonZeros]);
+    valueList=zeros([1,numNonZeros]);
+    elementsInserted=1;
     for i=2:(m+1)
-        A(i,(i-1):(i+k-2))=r;
+        for j=1:(k)
+            rowList(elementsInserted)=i;
+            columnList(elementsInserted)=i+j-2;
+            valueList(elementsInserted)=interior(j);
+            elementsInserted=elementsInserted+1;
+        end
     end
-    D=A*extensionOperatorDivergence(k,m);
+    interiorScheme=sparse(rowList,columnList,valueList,m+2,m+k-1);
+    % This matrix is just using the interior scheme.
+    % Since we've extended our vector field beyond the boundaries it'd normally have, we don't need to use
+    % seperate power series expansions for the stencils near the boundary. 
+    D=interiorScheme*extensionOperatorDivergence(k,m);
 end

src/matlab/extensionOperatorDivergence.m

@@ -1,22 +1,24 @@
 function E=extensionOperatorDivergence(k,m)
 % Returns a m+k-1 by m+1 one matrix. Which pads an m+1` point vector field
 % With k/2-1 additional points on each side. These points are
-% approximations of what happens if you extrapolate 
+% approximations of
+% f(-(k/2-1)h),f(-(k/2-2)h),....,f(-h),f(1+h),f(1+2h),...,f(1+(k/2-1)h),
+% with O(h^(k+1)) error on each of the approximations. 
 % 
 % 
 % Parameters:
 %                k : Order of accuracy
 %                m : Number of cells
-
-    E=sparse(m+k-1,m+1);
+    numOfRows=m+k-1;
+    numOfColumns=m+1;
+    numNonZeros=(k-2)*(k+1)+m+1;%The number of nonzero elements in E.
+    rowList=zeros([1,numNonZeros]);%Row indices for non zero elements
+    columnList=zeros([1,numNonZeros]); %Column indices for non zero elements
+    valueList=zeros([1,numNonZeros]); %Values of non zero elements
+    elementsInserted=1; %Number of elements inserted into rowList,ColumnList
+    %and ValueList
     r=idivide(k,int32(2));
-    for i=r:m+r
-        E(i,i+1-r)=1;
-    end
-    R=zeros(r-1,k+1);%calculates the matrix R_{i,j}=binom(i-k/2,j) 
-    %Used in the fact that e^(-jh)=∑_{i=0}^(k)binom(-j,i)(exp(h*d/dx)-I)^i+O(h^(k+1)).
-    % This is how you approximate  f(-jh) as a linear combo of
-    %f(0),f(h),...,f(kh) with O(h^(k+1)).  
+    R=zeros(r-1,k+1);% Calculates the matrix R_{i,j}=binom(i-k/2,j-1) 
     R(:,1)=ones([r-1,1]);
     currentTerm=(-1);
     for i=2:(k+1)
@@ -28,18 +30,24 @@
             R(i,j)=R(i+1,j)-R(i,j-1);
         end
     end
-
-    % A=[(-1)^(i+j)*binomial(i,j) for i=0:k,j=0:k]
-    R=R*alt_pascal(k+1)
-    for i=1:(r-1)
+    R=R*alt_pascal(k+1); %Converts R into the standard basis 
+    for i=1:(r-1) %
         for j=1:(k+1)
-            E(i,j)=R(i,j);
-            E(m+k-i,m+2-j)=R(i,j);
+            rowList(elementsInserted)=i;
+            columnList(elementsInserted)=j;
+            valueList(elementsInserted)=R(i,j);
+            rowList(elementsInserted+1)=m+k-i;
+            columnList(elementsInserted+1)=m+2-j;
+            valueList(elementsInserted+1)=R(i,j);
+            elementsInserted=elementsInserted+2;
         end
     end
-
-
-
-
+    for i=r:m+r
+        rowList(elementsInserted)=i;
+        columnList(elementsInserted)=i+1-r;
+        valueList(elementsInserted)=1;
+        elementsInserted=elementsInserted+1;
+    end
+    E=sparse(rowList,columnList,valueList,numOfRows,numOfColumns);
 
 end