How to get diff from legendreP faster?

I am trying to get differential from LegendreP, but it took me long time, for some data more that 12 hours !!!

syms mn
for n=1:1025
expr(n)=legendreP(n-1,cos(mn));
end
for n=1:1025
mmm(n)=diff(expr(n),mn,1);
end
teta=0:0.01:pi;
L=zeros(1025,315);
for i=1:length(teta)
      L(:,i)=subs(mmm,teta(i));
end

LegendreP(n,cos(teta)) and first diff were considered. How can I have this one faster???? Thanks

[Walter Roberson]

I do not know at the moment why it might be slow. For whatever it is worth, mmm(n) should be

-n*(legendreP(n-1, cos(mn))*cos(mn) - legendreP(n, cos(mn)))/sin(mn)

This already takes into account that expr(n) is defined in terms of legendreP(n-1,cos(mn))

Answer 1

[Walter Roberson]

I suggest trying

syms mn N

LPD = diff( legendreP(N, cos(mn)), mn );

teta = 0:0.01:pi;

L = zeros(1025,length(teta));

for n = 1:1025
  mmm = matlabFunction( subs(LPD, N, n-1), mn );
  L(n, :) = mmm(teta);
end

[Walter Roberson]

You could try

syms mn N

LPD = diff( legendreP(N, cos(mn)), mn );

teta = 0:0.01:pi;

L = zeros(1025,length(teta));

for n = 1:1025
  mmm = subs(LPD, N, n-1);
  L(n, :) = double(subs(mmm, mn, teta));
end

[Elvis Alexander Agüero Vera]

In Matlab2020a, your last try throws

Error using symengine
Unable to convert expression into double array.

Error in sym/double (line 698)
        Xstr = mupadmex('symobj::double', S.s, 0);

[Elvis Alexander Agüero Vera]

And the main comment throws

Error using symengine
Unable to generate code for partial derivative 'D([2], orthpoly::legendre)(12,
cos(mn))'.

Error in sym/matlabFunction>mup2mat (line 404)
res = mupadmex('symobj::generateMATLAB',r.s,ano,spa,0);

Error in sym/matlabFunction>mup2matcell (line 382)
    c = cellfun(@mup2mat,c,anonymousArgs,sparseArgs,'UniformOutput',false);

Error in sym/matlabFunction (line 188)
    body = mup2matcell(funs, opts.Sparse);

[Walter Roberson]

@Elvis Alexander Agüero Vera

Interesting, I am now seeing the same results that you post.

At the moment I do not recall whether I tested back in 2015; I will check if I have time.