# Why Does MATLAB Have the Function hypot?7

Posted by Loren Shure,

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.

Get the MATLAB code

Published with MATLAB® 7.5

Quan replied on : 1 of 7

That was a pretty interesting post Loren. I’m not sure if it will affect the way I do things around here, but it’s something that could come in useful in the future. Thanks for the tip!

Duane Hanselman replied on : 2 of 7

Before hypot existed, I used a=abs(complex(a,b)); In doing so, I was able to tap the robustness of the abs() function.

Loren replied on : 3 of 7

Duane-

Thanks for pointing out that abs is also implemented robustly.

–Loren

Hal K replied on : 4 of 7

Loren,
Does hypot(a,a) any different from norm([a a])?

Hal

Loren replied on : 5 of 7

Hal-

norm is also careful to scale results, like abs and hypot.

–Loren

Shahn replied on : 6 of 7

Hi Loren

I have a binary image and I want to find the distance between every object. I have already computed the centroid for every object. Can Hypot be used for many objects ???

Loren replied on : 7 of 7

Shahn-

You can use hypot if that is what makes sense. Here’s the help page. And here’s a few details from it:

c = hypot(a,b) returns the element-wise result of the following equation, computed to avoid underflow and overflow:

c = sqrt(abs(a).^2 + abs(b).^2)

Inputs a and b must follow these rules:

* Both a and b must be single- or double-precision, floating-point arrays.

* The sizes of the a and b arrays must either be equal, or one a scalar and the other nonscalar. In the latter case, hypot expands the scalar input to match the size of the nonscalar input.

Best wishes,
Loren

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