How to convert to a parallel program?

n=41; 
    h=1/(n-1);
    nt=40;   
    x(1)=0; y(1)=0;  

     for i=2:n 
       x(i)=x(i-1)+h;
       y(i)=y(i-1)+h;  
     end  
     
      for j=1:n
       for i=2:n 
        p0(i,j)=1;    
       end   
      end
      
      tic
       tau=0.01;       
   for jt=1:nt    % Vaqt bo`yicha tsikl}          
      t(jt)=jt*tau;    
              
      for j=1:n
      for i=1:n 
       pt(i,j)=exp(-2*x(i)*y(j)*t(jt));
       end      
      end   
   
      % k+0.s cloy
     for j=2:21    % Tenglamalar koeffitsientini hisoblash 
      for i=2:n-1  
       a(i)=1;
       c(i)=1;
       b(i)=2+2*h*h/tau;
       ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));   
       d(i)=2*h*h/tau*p0(i,j)+(p0(i,j-1)-2*p0(i,j)+p0(i,j+1))+ff;   
      end
       %Pragonka
        aa1(1)=0; bb1(1)=1;    
       for i=2:n-1
         aa1(i)=c(i)/(b(i)-aa1(i-1)*a(i));
         bb1(i)=(d(i)+bb1(i-1)*a(i))/(b(i)-aa1(i-1)*a(i));
       end
         
        p(n,j)=exp(-2*y(j)*t(jt));  
       for i=n-1:-1:1 
         p(i,j)=aa1(i)*p(i+1,j)+bb1(i);
       end                   
     end %j    
     
     for j=40:-1:22    % Tenglamalar koeffitsientini hisoblash 
      for i=2:n-1  
       a(i)=1;
       c(i)=1;
       b(i)=2+2*h*h/tau;
       ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));   
       d(i)=2*h*h/tau*p0(i,j)+(p0(i,j-1)-2*p0(i,j)+p0(i,j+1))+ff;   
      end   
       %Pragonka
        aa1(1)=0; bb1(1)=1;    
        for i=2:n-1
         aa1(i)=c(i)/(b(i)-aa1(i-1)*a(i));
         bb1(i)=(d(i)+bb1(i-1)*a(i))/(b(i)-aa1(i-1)*a(i));
        end         
        p(n,j)=exp(-2*y(j)*t(jt));  
        for i=n-1:-1:1 
         p(i,j)=aa1(i)*p(i+1,j)+bb1(i);
        end                          
     end %j        
     
      for i=1:n
        p(i,n)=(4.0*p(i,n-1)-p(i,n-2))/3;
        p(i,1)=(4.0*p(i,2)-p(i,3))/3;
      end      
        
       for j=1:n
        for i=1:n      
          p0(i,j)=p(i,j);  
        end  
       end            
        
    
    % k+1 sloy
    for i=2:21    % Tenglamalar koeffitsientini hisoblash }
     for j=2:n-1  
       a(j)=1;
       c(j)=1;
       b(j)=2+2*h*h/tau;
       ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));   
       d(j)=2*h*h/tau*p0(i,j)+(p0(i-1,j)-2*p0(i,j)+p0(i+1,j))+ff;   
     end
      %Pragonka
       aa1(1)=0; bb1(1)=1;    
       for j=2:n-1
         aa1(j)=c(j)/(b(j)-aa1(j-1)*a(j));
         bb1(j)=(d(j)+bb1(j-1)*a(j))/(b(j)-aa1(j-1)*a(j));
       end        
        p(i,n)=exp(-2*x(i)*t(jt));  
       for j=n-1:-1:1 
         p(i,j)=aa1(j)*p(i,j+1)+bb1(j);
       end    
    end  %i          
       
    for i=40:-1:22   % Tenglamalar koeffitsientini hisoblash }
     for j=2:n-1  
       a(j)=1;
       c(j)=1;
       b(j)=2+2*h*h/tau;
       ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));   
       d(j)=2*h*h/tau*p0(i,j)+(p0(i-1,j)-2*p0(i,j)+p0(i+1,j))+ff;   
     end    
      %Pragonka
       aa1(1)=0; bb1(1)=1;    
       for j=2:n-1
        aa1(j)=c(j)/(b(j)-aa1(j-1)*a(j));
        bb1(j)=(d(j)+bb1(j-1)*a(j))/(b(j)-aa1(j-1)*a(j));
       end
        p(i,n)=exp(-2*x(i)*t(jt));  
       for j=n-1:-1:1 
         p(i,j)=aa1(j)*p(i,j+1)+bb1(j);
       end                          
    end %i
    
      for j=1:n
       p(n,j)=(4.0*p(n-1,j)-p(n-2,j))/3;  
       p(1,j)=(4.0*p(2,j)-p(3,j))/3;          
      end        
                   
       for i=1:n
        for j=1:n      
          p0(i,j)=p(i,j);  
        end  
       end   
    
    
   end % jt
   toc
   
    for i=1:n
        for j=1:n      
          b(j)=(p(i,j)-pt(i,j));  
          max=b(j);
        end  
       end  
        for j=1:n
          px(j)=p(21,j); 
          pxt(j)=pt(21,j);
        end  
              
        t
    p
    pt
    b
    max




      
      
      

[Jan]

What does "parallel program" mean? Do you want to vectrorize the code to increase the speed?

We cannot run the code due to the missing variables nt, n, x, y.

The outer loop is useless, because the contents of pt is overwritten in each iteration. Then it is much cheaper to omit the outer loop and run the code for jt=nt only.

Do you pre-allocate pt before the loops? Iteratively growing arrays need an exponentially growing amount of RAM, even if the final output is not such large.

The current version of your code does not need loops:

pt = exp(-2 * x(:) .* y(:).' * (nt * tau));
%              ^^^     ^^^  maybe this can be omitted depending on
%                           what x and y are

[Shixnazar Ismailov]

Thank you! This is the full program code! I want to use it faster!

[Jan]

Please start with improving the readability: ctrl-a ctrl-i. This automatic identation needs two keyboard activities only, but the code is much clearer.

[Shixnazar Ismailov]

n=41;

h=1/(n-1);

nt=40;

x(1)=0; y(1)=0;

for i=2:n

x(i)=x(i-1)+h;

y(i)=y(i-1)+h;

end

for j=1:n

for i=2:n

p0(i,j)=1;

end

end

tic

tau=0.01;

for jt=1:nt % Vaqt bo`yicha tsikl}

t(jt)=jt*tau;

for j=1:n

for i=1:n

pt(i,j)=exp(-2*x(i)*y(j)*t(jt));

end

end

% k+0.s cloy

for j=2:21 % Tenglamalar koeffitsientini hisoblash

for i=2:n-1

a(i)=1;

c(i)=1;

b(i)=2+2*h*h/tau;

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d(i)=2*h*h/tau*p0(i,j)+(p0(i,j-1)-2*p0(i,j)+p0(i,j+1))+ff;

end

%Pragonka

aa1(1)=0; bb1(1)=1;

for i=2:n-1

aa1(i)=c(i)/(b(i)-aa1(i-1)*a(i));

bb1(i)=(d(i)+bb1(i-1)*a(i))/(b(i)-aa1(i-1)*a(i));

end

p(n,j)=exp(-2*y(j)*t(jt));

for i=n-1:-1:1

p(i,j)=aa1(i)*p(i+1,j)+bb1(i);

end

end %j

for j=40:-1:22 % Tenglamalar koeffitsientini hisoblash

for i=2:n-1

a(i)=1;

c(i)=1;

b(i)=2+2*h*h/tau;

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d(i)=2*h*h/tau*p0(i,j)+(p0(i,j-1)-2*p0(i,j)+p0(i,j+1))+ff;

end

%Pragonka

aa1(1)=0; bb1(1)=1;

for i=2:n-1

aa1(i)=c(i)/(b(i)-aa1(i-1)*a(i));

bb1(i)=(d(i)+bb1(i-1)*a(i))/(b(i)-aa1(i-1)*a(i));

end

p(n,j)=exp(-2*y(j)*t(jt));

for i=n-1:-1:1

p(i,j)=aa1(i)*p(i+1,j)+bb1(i);

end

end %j

for i=1:n

p(i,n)=(4.0*p(i,n-1)-p(i,n-2))/3;

p(i,1)=(4.0*p(i,2)-p(i,3))/3;

end

for j=1:n

for i=1:n

p0(i,j)=p(i,j);

end

end

% k+1 sloy

for i=2:21 % Tenglamalar koeffitsientini hisoblash }

for j=2:n-1

a(j)=1;

c(j)=1;

b(j)=2+2*h*h/tau;

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d(j)=2*h*h/tau*p0(i,j)+(p0(i-1,j)-2*p0(i,j)+p0(i+1,j))+ff;

end

%Pragonka

aa1(1)=0; bb1(1)=1;

for j=2:n-1

aa1(j)=c(j)/(b(j)-aa1(j-1)*a(j));

bb1(j)=(d(j)+bb1(j-1)*a(j))/(b(j)-aa1(j-1)*a(j));

end

p(i,n)=exp(-2*x(i)*t(jt));

for j=n-1:-1:1

p(i,j)=aa1(j)*p(i,j+1)+bb1(j);

end

end %i

for i=40:-1:22 % Tenglamalar koeffitsientini hisoblash }

for j=2:n-1

a(j)=1;

c(j)=1;

b(j)=2+2*h*h/tau;

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d(j)=2*h*h/tau*p0(i,j)+(p0(i-1,j)-2*p0(i,j)+p0(i+1,j))+ff;

end

%Pragonka

aa1(1)=0; bb1(1)=1;

for j=2:n-1

aa1(j)=c(j)/(b(j)-aa1(j-1)*a(j));

bb1(j)=(d(j)+bb1(j-1)*a(j))/(b(j)-aa1(j-1)*a(j));

end

p(i,n)=exp(-2*x(i)*t(jt));

for j=n-1:-1:1

p(i,j)=aa1(j)*p(i,j+1)+bb1(j);

end

end %i

for j=1:n

p(n,j)=(4.0*p(n-1,j)-p(n-2,j))/3;

p(1,j)=(4.0*p(2,j)-p(3,j))/3;

end

for i=1:n

for j=1:n

p0(i,j)=p(i,j);

end

end

end % jt

toc

for i=1:n

for j=1:n

b(j)=(p(i,j)-pt(i,j));

max=b(j);

end

end

for j=1:n

px(j)=p(21,j);

pxt(j)=pt(21,j);

end

t

p

pt

b

max

Answer 1

[Jan]

Pre-allocation is essential. Do not let array grow iteratively, but create them with the final size:

n    = 41; 
h    = 1 / (n-1);
nt   = 40;

% x(1) = 0;
% y(1) = 0;  

% Better:
x = zeros(1, n);  % Pre-allocation
y = zeros(1, n);
for i = 2:n 
   x(i) = x(i - 1) + h;
   y(i) = y(i - 1) + h;  
end

In this case you do not even need a loop:

x = (0:n - 1) * h;
y = x;

Replace:

for j = 1:n
   for i = 2:n 
      p0(i,j)=1;    
   end   
end

by

p0 = ones(n, n);

This is faster and nicer. Pre-allocate t also:

tau = 0.01;
t = (1:nt) * tau;
for jt = 1:nt    % Vaqt bo`yicha tsikl}
   % pt without loops:
   pt = exp(-2 * t(jt) * x.' .* y);

   % Moved out of the loops:
   a = ones(1, n-1);
   c = ones(1, n-1);
   b = repmat(2 + 2 * h * h / tau, 1, n-1);
         
   % k+0.s cloy
   for j = 2:21    % Tenglamalar koeffitsientini hisoblash 
      for i = 2:n-1
         % Don't do this repeatedly inside the loops - see above:
         %   a(i) = 1;
         %   c(i) = 1;
         %   b(i) = 2 + 2 * h * h / tau;
         
         % You have evaluated exp(-2*x(i)*y(j)*t(jt)) already above.
         % Don't do it again.
         ff   = h * h * (-2 * (2 * t(jt)*t(jt) * (x(i)*x(i) + y(j)*y(j)) + x(i)*y(j)) * pt(i, j));
         d(i) = 2 * h * h / tau * p0(i,j) + (p0(i,j-1) - 2*p0(i,j) + p0(i,j+1)) + ff;   
      end
      %Pragonka
      aa1 = zeros(1, n-1);  % Pre-allocate!
      bb1 = zeros(1, n-1);  % Pre-allocate!
      bb1(1) = 1;
      for i = 2:n-1
         aa1(i) = c(i) / (b(i) - aa1(i-1) * a(i));
         bb1(i) = (d(i) + bb1(i-1) * a(i)) / (b(i) - aa1(i-1) * a(i));
      end

      p(n,j) = exp(-2 * y(j) * t(jt));  
      for i = n-1:-1:1 
          p(i,j) = aa1(i) * p(i+1,j) + bb1(i);
      end
   end 

Replace:

for j=1:n
   for i=1:n      
      p0(i,j) = p(i,j);  
   end  
end            

by

p0 = p;

Please try to implement the shown methods in your code.

Use spaces around operators and a proper indentation.

for i = 1:n
   for j = 1:n      
      b(j) = p(i,j) - pt(i,j);  
      max  = b(j);   % ???
   end
end

What is the purpose of this code? b(j) is overwritten in each iteration of i. Finally the value of max is p(n,n) - pt(n,n) and does not depend on the loop in any way. It is not worth to improve the speed of a code, when it does not compute anything useful.

[Shikhnazar Ismailov]

function []=diagonal()

n=41; ns=(n+1)/2; h=1/(n-1); nt=360;

x = zeros(1, n);

y = zeros(1, n);

x = (0:n - 1) * h;

y = x;

p0 = ones(n);

tau=0.01;

for jt=1:nt

t(jt)=jt*tau;

pt = exp(-2 * t(jt) * x.' .* y);

end

tic

for jt=1:nt % Vaqt bo`yicha tsikl}

a = ones(1, n-1);

c = ones(1, n-1);

b = repmat(2 + 2 * h * h / tau, 1, n-1);

aa1 = zeros(1, n-1);

bb1 = zeros(1, n-1);

bb1(1) = 1;

for j=2:ns

parfor i=2:n-1

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d=2*h*h/tau*p0(i,j)+(p0(i,j-1)-2*p0(i,j)+p0(i,j+1))+ff;

aa1(i)=c/(b-aa1(i-1)*a);

bb1(i)=(d+bb1(i-1)*a)/(b-aa1(i-1)*a);

end

p(n,j)=exp(-2*y(j)*t(jt));

for i=n-1:-1:1

p(i,j)=aa1(i)*p(i+1,j)+bb1(i);

end

end %j

for j=n-1:ns-1

parfor i=2:n-1

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d=2*h*h/tau*p0(i,j)+(p0(i,j-1)-2*p0(i,j)+p0(i,j+1))+ff;

aa1(i)=c/(b-aa1(i-1)*a);

bb1(i)=(d+bb1(i-1)*a)/(b-aa1(i-1)*a);

end

p(n,j)=exp(-2*y(j)*t(jt));

for i=n-1:-1:1

p(i,j)=aa1(i)*p(i+1,j)+bb1(i);

end

end %j

for i=1:n

p(i,n)=(4.0*p(i,n-1)-p(i,n-2))/3;

p(i,1)=(4.0*p(i,2)-p(i,3))/3;

end

p0=p;

for i=2:ns

parfor j=2:n-1

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d=2*h*h/tau*p0(i,j)+(p0(i-1,j)-2*p0(i,j)+p0(i+1,j))+ff;

aa1(j)=c/(b-aa1(j-1)*a);

bb1(j)=(d+bb1(j-1)*a)/(b-aa1(j-1)*a);

end

p(i,n)=exp(-2*x(i)*t(jt));

for j=n-1:-1:1

p(i,j)=aa1(j)*p(i,j+1)+bb1(j);

end

end %i

for i=n-1:ns-1

parfor j=2:n-1

ff=h*h*(-2*(2*t(jt)*t(jt)*(x(i)*x(i)+y(j)*y(j))+x(i)*y(j))*exp(-2*x(i)*y(j)*t(jt)));

d=2*h*h/tau*p0(i,j)+(p0(i-1,j)-2*p0(i,j)+p0(i+1,j))+ff;

aa1(j)=c/(b-aa1(j-1)*a);

bb1(j)=(d+bb1(j-1)*a)/(b-aa1(j-1)*a);

end

p(i,n)=exp(-2*x(i)*t(jt));

for j=n-1:-1:1

p(i,j)=aa1(j)*p(i,j+1)+bb1(j);

end

end %i

for j=1:n

p(n,j)=(4.0*p(n-1,j)-p(n-2,j))/3;

p(1,j)=(4.0*p(2,j)-p(3,j))/3;

end

p0=p;

end % jt

toc

px = p0(21,:);

pxt = pt(21,:);

x=0:h:1;

y=0:h:1;

figure ('Position',[600 60 700 600]);

subplot(3,2,1);

meshc(x,y,pt) %Uch o'lchovli 3D grafika

title('АНИК ЕЧИМ 3D ГРАФИГИ')

subplot(3,2,2);

meshc(x,y,p) %Uch o'lchovli 3D grafika

title('CОНЛИ ЕЧИМ 3D ГРАФИГИ')

subplot(3,2,3);

contour(x,y,pt,'ShowText','on','LineWidth',2); %kontur chizish

title('АНИК ЕЧИМ КОНТУР ГРАФИГИ')

subplot(3,2,4);

contour(x,y,p,'ShowText','on','LineWidth',2); %kontur chizish

title('CОНЛИ ЕЧИМ КОНТУР ГРАФИГИ')

subplot(3,2,5);

plot(x,px,x,pxt); %Ikki o'lchovli grafika

title('КЕСИМДА АНИК ВА CОНЛИ ЕЧИМ ЎЗГАРИШИ X БЎЙИЧА')

end

The variable 'aa1' in a parfor cannot be classified.