MATLAB: How can calcuate the lines of code efficiently

MATLABperformance

Hello. This code calculates the Laplacian. I am looking for reducing the time running of it. My main aim is improving the performance of loop running time.
Thanks
%%%code
nx=512;
ny=nx;
dx=1;
dy=dx;
format long;
nxny=nx*ny;
r=zeros(1,nx);
r(1:2)=[2,-1];
T=toeplitz(r);
E=speye(nx);
grad=-(kron(T,E)+kron(E,T));%Tensor
%-- for periodic boundaries
for i=1:nx
    ii=(i-1)*nx+1;
    jj=ii+nx-1;
    grad(ii,jj)=1.0;
    grad(jj,ii)=1.0;
    kk=nxny-nx+i;
    grad(i,kk)=1.0;
    grad(kk,i)=1.0;
end%for
grad = grad /(dx*dy);

Best Answer

Yes, you can if you avoid using a loop and vectorize your code - see here:
tic
%%%code

nx=512;
ny=nx;
dx=1;
dy=dx;
format long;
nxny=nx*ny;
r=zeros(1,nx);
r(1:2)=[2,-1];
T=toeplitz(r);
E=speye(nx);
grad=-(kron(T,E)+kron(E,T));%Tensor


%-- for periodic boundaries


ii=(0:nx-1).*nx+1;
jj = ii+nx-1;
grad(ii,jj)=1.0;
grad(jj,ii)=1.0;
kk=(nxny-nx+1):nxny;
grad(1:nx,kk)=1.0;
grad(kk,1:nx)=1.0;
grad = grad /(dx*dy);
toc
Elapsed time is 0.145899 seconds.
Your code:
tic
%%%code
nx=512;
ny=nx;
dx=1;
dy=dx;
format long;
nxny=nx*ny;
r=zeros(1,nx);
r(1:2)=[2,-1];
T=toeplitz(r);
E=speye(nx);
grad=-(kron(T,E)+kron(E,T));%Tensor
%-- for periodic boundaries
for i=1:nx
    ii=(0:nx-1).*nx+1;
    jj=ii+nx-1;
    grad(ii,jj)=1.0;
    grad(jj,ii)=1.0;
    kk=nxny-nx+i;
    grad(i,kk)=1.0;
    grad(kk,i)=1.0;
end 
toc
Elapsed time is 11.333450 seconds.
Are the results correct? - to check this we call one of the results grad1 and subtract grad1 from grad:
nx=512;
ny=nx;
dx=1;
dy=dx;
format long;
nxny=nx*ny;
r=zeros(1,nx);
r(1:2)=[2,-1];
T=toeplitz(r);
E=speye(nx);
grad=-(kron(T,E)+kron(E,T));%Tensor
%-- for periodic boundaries
for i=1:nx
    ii=(0:nx-1).*nx+1;
    jj=ii+nx-1;
    grad(ii,jj)=1.0;
    grad(jj,ii)=1.0;
    kk=nxny-nx+i;
    grad(i,kk)=1.0;
    grad(kk,i)=1.0;
end 
ii=(0:nx-1).*nx+1;
jj = ii+nx-1;
grad(ii,jj)=1.0;
grad(jj,ii)=1.0;
kk=(nxny-nx+1):nxny;
grad(1:nx,kk)=1.0;
grad(kk,1:nx)=1.0;
grad1 = grad /(dx*dy);
expect_all_zeros = grad1-grad
expect_all_zeros =
   All zero sparse: 262144×262144
Appears to work fine...
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