symbolic calculation error in code

syms x n 
syms t 
% syms m
% m=0.7;
U=zeros(1,2,'sym');
A=zeros(1,2,'sym');
B=zeros(1,2,'sym');
 U(1)=exp((sqrt(2)*x)/4)/(exp((sqrt(2)*x)/4)+exp(-(sqrt(2)*x)/4));
for k=1:10
    A(1)=0;
    B(1)=0;
     for i=1:k
        A(1)=A(1)+U(i)*U(k-i+1) ;
     end
     for j=1:k
         for r=1:j
             B(1)=U(r)*U(j-r+1)*U(k-r+1);
         end
     end
       U(k+1)=((diff(U(k),x,2)-B(1)+2*A(1)-U(k)))/k;
end
 disp (U)
 for k=1:10
      series(x,t)=series(x,t)+U(k)*(power(t,k-1));
 end
  series
  C=zeros(5,5);
for x=1:5
     e=(x-1)/1;
     for t=1:5
         f=(t-1)/10;
         C(x,t)=series(e,f)
     end   
 end
 vpa(C,15)

This code is not showing the answer.BUSY is also shown in the workspace.

Answer 1

[Walter Roberson]

tic

syms x n 
syms t 
% syms m
% m = 0.7;
U = zeros(1,2,'sym');
A = zeros(1,2,'sym');
B = zeros(1,2,'sym');
 U(1) = simplify(exp((sqrt(2)*x)/4)/(exp((sqrt(2)*x)/4)+exp(-(sqrt(2)*x)/4)));
for k = 1:10
    A(1) = 0;
    B(1) = 0;
     for i = 1:k
        A(1) = simplify(A(1)+U(i)*U(k-i+1)) ;
     end
     for j = 1:k
         for r = 1:j
             B(1) = simplify(U(r)*U(j-r+1)*U(k-r+1));
         end
     end
       U(k+1) = simplify(((diff(U(k),x,2)-B(1)+2*A(1)-U(k)))/k);
       %disp(U(k+1))
end
 disp (U)

 for k = 1:10
      series(x,t) = simplify(series(x,t)+U(k)*(power(t,k-1)));
 end
  series

series(x, t) = 

  C = zeros(5,5);
  
for x = 1:5
     e = (x-1)/1;
     for t = 1:5
         f = (t-1)/10;
         C(x,t) = series(e,f);
     end   
 end
 vpa(C,15)

ans = 

 toc

Elapsed time is 47.994421 seconds.

[YOGESHWARI PATEL]

One more query you have display the value of sima 1, sigma 2.How this values are diaplayed .I think tic and toc give me the run time.

[Walter Roberson]

The and so on were displayed automatically by Live Script in order to make the expression easier to read. If you are using the regular editor instead of Live Script then you can use pretty() to get something similar.

[YOGESHWARI PATEL]

Thank you .Now its working.

[YOGESHWARI PATEL]

There is a problem with the code in this part

C = zeros(5,5);
  
for x = 1:5
     e = (x-1)/1;
     for t = 1:5
         f = (t-1)/10;
         C(x,t) = series(e,f);
     end   
 end
 vpa(C,15)

when i change the value of k in the upper loop its always showing busy

for k = 1:10
    A(1) = 0;
    B(1) = 0;
     for i = 1:k
        A(1) = simplify(A(1)+U(i)*U(k-i+1)) ;
     end
     for j = 1:k
         for r = 1:j
             B(1) = simplify(U(r)*U(j-r+1)*U(k-r+1));
         end
     end
       U(k+1) = simplify(((diff(U(k),x,2)-B(1)+2*A(1)-U(k)))/k);
       %disp(U(k+1))
end
 disp (U)
for k = 1:10
      series(x,t) = simplify(series(x,t)+U(k)*(power(t,k-1)));
 end

[Walter Roberson]

Your expression gets more and more complicated as k increases, and it takes longer and longer to simplify.

However, simplification is still called for because it reduces the rate at which A and B and U get more complicated.

15 took about 201 seconds on my system. The time required appears to increase with approximately the square of k.

tic

maxk = 15;

syms x n 
syms t 
% syms m
% m = 0.7;
U = zeros(maxk+1,1,'sym');
A = zeros(1,1,'sym');
B = zeros(1,1,'sym');

wb = waitbar(0, 'generating U');
cleanMe = onCleanup(@() delete(wb));

U(1) = simplify(exp((sqrt(2)*x)/4)/(exp((sqrt(2)*x)/4)+exp(-(sqrt(2)*x)/4)));

Utimes = zeros(1,maxk);

for k = 1:maxk
    Utic = tic;
    waitbar(k/maxk, wb);

    A(1) = 0;
    B(1) = 0;
     for i = 1:k
        A(1) = simplify(A(1)+U(i)*U(k-i+1)) ;
     end
     for j = 1:k
         for r = 1:j
             B(1) = simplify(U(r)*U(j-r+1)*U(k-r+1));
         end
     end
     U(k+1) = simplify(((diff(U(k),x,2)-B(1)+2*A(1)-U(k)))/k);
     Utoc = toc(Utic);
     Utimes(k) = Utoc;
end
disp (U)

k = 1:maxk;
plot(k, Utimes, 'displayname', 'U generation times');
xlabel('k'); ylabel('time (s)')
p = polyfit(log(k), log(Utimes), 1);
a = exp(p(2)); b = p(1);
fprintf('Utime is approximately %.5g * k^%.5g\n', a, b);

waitbar(0, wb, 'generating series');

for k = 1:maxk
      waitbar(k/maxk, wb);
      series(x,t) = simplify(series(x,t)+U(k)*(power(t,k-1)));
end

series

C = zeros(5,5);

waitbar(0, wb, 'evaluating numeric')
  
for x = 1:5
     waitbar(x/5, wb);
     e = (x-1)/1;
     for t = 1:5
         f = (t-1)/10;
         C(x,t) = series(e,f);
     end   
 end
 vpa(C,15)

 delete(cleanMe);
 
 toc

[Walter Roberson]

k = 20 took about 600 seconds for me; it looks like the time increase might be cubic in k (which does not surprise me.)

So if you were planning to take this to k = 1000 like in your earlier question, you should expect on the order of 295 days to execute, if your system has enough memory.

You could attempt to rewrite the whole calculation as numeric work, in hope that would speed it up -- which would be entirely possible. However, beyond k = 245 or so, the denominator of the expression would overflow to infinity, so that is the approximate upper limit on numeric work before the calculation becomes completely useles. The calculation is likely to become mostly useless for numeric work long before that.

[YOGESHWARI PATEL]

Thank you