How do I replace trapz with sum?

Hi My Matlab is R2016b version, and below is a part of code I am struggling with.

I want to change trapz parts into sum to speed up the calculation.

Can you help me?

 elseif fit_method == 2
        tic;

        N=length(xy(:,2));
        N=N-1;
        dt = 10^-8;

        SY = 0;
        t_SY = 0;
        Y_SY = 0;
        SY_SY = 0;
        Y_t = 0;

        for M = 2:N
            SY_comp = trapz(xy(1:M,1),xy(1:M,2));
            SY = SY+SY_comp;
            t_SY_comp = M*trapz(xy(1:M,1),xy(1:M,2));
            t_SY = t_SY + t_SY_comp;
            Y_SY_comp = xy(M,2)*trapz(xy(1:M,1),xy(1:M,2));
            Y_SY = Y_SY + Y_SY_comp;
            SY_SY_comp = (trapz(xy(1:M,1),xy(1:M,2)))^2;
            SY_SY = SY_SY + SY_SY_comp;
            Y_t_comp = xy(M,2)*M;
            Y_t = Y_t + Y_t_comp;
        end;

        Y = sum(xy(1:N,2));
        t = N*(N+1)/2;
        t_t = N*(N+1)*(2*N+1)/6;

        A = double([N, SY, t; SY, SY_SY, Y_SY; t, t_SY, t_t]);
        B = double([Y;Y_SY;Y_t]);
        X = inv(A)*B;
        tau = dt/log(1-X(2,1)*dt);
        fit_success = 0;

        toc;

[Deo]

I just tried to replace "for" with "parfor", and there was 50 % time dedutction. I may still need to change trapz with sum.

Answer 1

[Roger Stafford]

In place of “SY_comp=trapz(xy(1:M,1),xy(1:M,2))” you could write:

SY_comp=1/2*sum(diff(xy(:,1)).*(xy(1:M-1,2)+xy*(2:M,2)));

However, I seriously doubt that this will speed up the calculation. There is no particular reason to assume that Mathworks’ ‘sum’ function is any more efficient than the ‘trapz’ function.

[Deo]

Because of pre-defined delta value, I used like below. 1/2*sum(dt*(xy(1:M-1,2)+xy(2:M,2))) I just tried to do this because few papers I am refering said that way....and the result was 50 % decrease in process time. Thanks for your help. But I am still desiring to see better idea to reduce processing time. Thanks

[Roger Stafford]

If dt is a single scalar value - that is, if all the intervals dt are of the same value - you can use the form

   dt*trapz(Y)

instead of

   trapz(X,Y)

and that should save time. The equivalent formula using ‘sum’ would be:

   dt*((Y(1)+Y(end))/2+sum(Y(2:end-1)));

It is still questionable whether this is faster than trapz(Y).