How to optimize this code for running time?

The first for loop in this code runs really slow, is there a way to optimize it? Thanks for any help.

clear all

M = 500;
fs = 1e6;


f = logspace(0,6,fs);
z = exp(j*2*pi*f/fs);

HcoI3 = 0;
HcoI3_DC = 0;

for l = 1:M
    HcoI3 = HcoI3 + (M-l+2)*(M-l+1)/2*z.^(-(l-1));
end

for l = 1:M
    HcoI3_DC = HcoI3_DC + (M-l+2)*(M-l+1)/2*1^(-(l-1));
end


HcoI3 = 1/HcoI3_DC * HcoI3;
HcoI3_mag = 20*log10(abs(HcoI3));

plot(f, HcoI3_mag)
xlim([1 10e3])
ylim([-60 0])

[Walter Roberson]

Step 1:

Remove the clear all .

Answer 1

[Walter Roberson]

    HcoI3_DC = HcoI3_DC + (M-l+2)*(M-l+1)/2*1^(-(l-1));

When l is finite, then 1^(-(l-1)) is going to be 1, even if -(l-1) is negative or positive or fractional or complex or 0. The only time it would not be 1 is if... huh, after testing even 1^-inf is 1, so the only time you would get something other than 1 is if l is nan.

When you have a calculation whose value is fully predictable, you can increase performance by substituting the constant value for the calculation.

[Walter Roberson]

If you have more than 4 gigabytes of memory (would probably take 8 gigabytes in practice),

l = (1 : M) .';
HcoI3 = sum((M-l+2) .* (M-l+1)/2 .* z.^(-(l-1)), 1);

without a loop.

Here z is a row vector and l is a column vector, and you take advantage of implicit expansion.

If you do not have enough memory, then you can proceed in batches, such as quarters of l at a time, and add the parts together.