Why Does MATLAB Have the Function hypot?

Sometimes I am asked questions like "why did you introduce function xyz into MATLAB? And sometimes I have a compelling answer, even for something that looks simple on the surface. Consider the function hypot.

What Does hypot Compute?

hypot essentially computes the square root of the sum of the squares of 2 inputs. So what's the big deal, right? For "reasonable" input values, there is no big issue.

fhypot = @(a,b) sqrt(abs(a).^2+abs(b).^2);

Other Ways to Compute hypot

Let's create a set of values and compute hypot 3 ways for these: with hypot, fhypot, and sqrt(2)*x. These should all give the same answers, provide a is real and positive.

a = logspace(0,5,6)
format long
aHypot = hypot(a,a)
aFhypot = fhypot(a,a)
aSqrt2 = sqrt(2)*a
a =
1          10         100        1000       10000      100000
aHypot =
1.0e+005 *
Columns 1 through 3
0.000014142135624   0.000141421356237   0.001414213562373
Columns 4 through 6
0.014142135623731   0.141421356237310   1.414213562373095
aFhypot =
1.0e+005 *
Columns 1 through 3
0.000014142135624   0.000141421356237   0.001414213562373
Columns 4 through 6
0.014142135623731   0.141421356237310   1.414213562373095
aSqrt2 =
1.0e+005 *
Columns 1 through 3
0.000014142135624   0.000141421356237   0.001414213562373
Columns 4 through 6
0.014142135623731   0.141421356237310   1.414213562373095


Results

The results are the same, to within round-off, for the 3 methods here. But what happens if the magnitude of a is larger, near realmax perhaps?

realmax
ans =
1.797693134862316e+308


For my computer, a 32-bit Windows machine, realmax for doubles is on the order of 10^308. Let's see what happens if a value nearly that large is used with the different versions of hypot.

a = 1e308
aHypot = hypot(a,a)
aFhypot = fhypot(a,a)
aSqrt2 = sqrt(2)*a
a =
1.000000000000000e+308
aHypot =
1.414213562373095e+308
aFhypot =
Inf
aSqrt2 =
1.414213562373095e+308


What you can see is that the straight-forward method returns Inf instead of a finite answer. And that's why hypot was added to MATLAB, to compute the hypotenuse robustly, avoiding both underflow and overflow.

Published with MATLAB® 7.5

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