integration of a piecewise function

Can anyone help me understand why

integral(@(mu) (1.0+0.0*mu).*(mu>0)+(1-exp(1.0./mu)).*(mu<0),0,1)

gives a "NaN", while

integral(@(mu) (1.0+0.0*mu).*(mu>0)+(1-(1.0./mu)).*(mu<0),0,1)

gives the correct answer "1.0"?

It's bugging me for so long.

Background, I started with a two variate function

psi_MMS =@(x,mu) (1.0+0.0*x+0.0*mu).*(mu>0) + (1-exp((Tau-x)./(mu+1e-16))).*(mu<0);

And I wanted to integrate it over mu to get a function of x:

phi0_MMS =@(x) integral(@(mu) psi_MMS(x,mu), -1,1);

Thanks!

Answer 1

[Sarah Mohamed]

It looks like NaN values are generated in the first function when you try to multiply -INF and logical 0’s.

For ease of demonstration, let me go ahead and create a handle for the first function as follows:

f1 = @(mu) (1.0+0.0*mu).*(mu>0)+(1-exp(1.0./mu)).*(mu<0);

Let me also go ahead and create a range of values for “mu”, 0.1, 0.01, 0.001, etc, that are close to 0:

mu =  10.^(-1*[1:10]);

If I plug these values into f1, I get the following:

>> f1(mu)

ans =

       1     1   NaN   NaN   NaN   NaN   NaN   NaN   NaN   NaN

Let me try to localize the problem slightly more:

>> (1-exp(1.0./mu))

ans =

     1.0e+43 *

     -0.0000   -2.6881      -Inf      -Inf      -Inf      -Inf      -Inf      -Inf      -Inf      -Inf

At mu = 0.001 and onwards, we find that "exp(1/mu)" evaluates to INF.

Let me now add in the piecewise part of the expression. “mu<0” should give a vector of logical 0’s:

>> (1-exp(1.0./mu)).*(mu<0)

ans =

       0     0   NaN   NaN   NaN   NaN   NaN   NaN   NaN   NaN

In summary, multiplying INF or -INF by a logical 0 (or just zero, for that matter) evaluates to NaN. As a quick check, we can confirm this quickly via:

>> inf*logical(0)

ans =

     NaN

And as a final sanity check, let's try the original integration, but setting the lower limit to 0.01:

>> integral(f1, 0.01, 1)

ans =

      0.9900