MATLAB: Vectorization the Legendre Polynomial Without loop

for looplegendre polynomialsoptimizationvectorization

Hi All,
I would like to speed up this Legendre polynomial code without using loop. It has a significant delay in my associated code
I would like to alwasy keep L as option which can be changed 
t=[1/60:1/60:3];
L=221;
P=zeros(L+1,length(t));
P(1,1:length(t)) = 1;    
P(2,1:length(t)) = t; 
for i = 2:L 
    P(i+1,:) = -(i-1)./i.*P(i-1)+(2.*i-1)./i.*t.*P(i);
end
The equation for the Legendre polynomial is
where and
PS: I corected the mathematical equation, sorry for the inconvenience
Many thanks in advance

Best Answer

Are you sure the the code is working? You calculate the vectors P(i, :) but use the first element only: P(i):
    P(i+1,:) = -(i-1)./i.*P(i-1)+(2.*i-1)./i.*t.*P(i);
    % I assume you want:
    P(i+1,:) = -(i-1)./i.*P(i-1, :)+(2.*i-1)./i.*t.*P(i, :);
    %                            ^                       ^
I do not think that this can be vectorized, because the values of P depend on former values. cumsum etc. is not a solution also. So a small simplification only:
t = 1/60:1/60:3;  % No square brackets needs
L = 221;
P = zeros(L+1, length(t));
P(1, :) = 1;    
P(2, :) = t; 
for n = 2:L 
    P(n+1, :) = (1 - n) / n * P(n - 1, :) + (2 * n - 1) / n * t .* P(n, :);
end
A columnwise processing is faster in general - in my measurements maybe 10%:
t = (1/60:1/60:3).';
L = 221;
P = zeros(length(t), L+1);
P(:, 1) = 1;    
P(:, 2) = t; 
for n = 2:L 
    P(:, n+1) = (1 - n) / n * P(:, n - 1) + (2 * n - 1) / n * t .* P(:, n);
end
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