Summing up functions using 'sum' leads to wrong values in quadgk

Hello, I have a question concerning the following little example:

h  = @(x) x.^2 + x.^4;
h2 = @(x) sum( [x.^2, x.^4]);

f  = @(x) h(x) .* exp(-abs(x));
f2 = @(x) h2(x) .* exp(-abs(x));

f(3)       % same values here 
f2(3)      %

m = quadgk( f , -inf, inf)      % different values here 
m2 = quadgk( f2 , -inf, inf)    %

The pointwise evaluation of f and f2 yields the same values. Integrating the (pointwise equal) functions f and f2 with quadgk over the real line gives vastly different results. Isn't this strage?! Can some one give me a little hint how this can be? Thanks !

With best regards, Oscar

Answer 1

[Jan]

@quadgk calls the function to be integrated with a row vector as input for a faster vectorized processing. Then sum() operates on a longer row vector, because the dimension to operate on was not specified. Relying on the smart automatic detection of the dimension is a source of bugs. Never do this is serious programs.

Solution:

h2 = @(x) sum([x.^2; x.^4], 1);  % Semicolon, not a comma!

If you use functions instead of anonymous functions, the debugger can reveal what's going on.

[Oscar]

Hi Jan, thank you for this comprehensive answer! I should really have a look at some literature concerning vectorized quadrature in matlab :)