For this tutorial you may need to read the Wikipedia page about Pythagorean triple (for more information on Pythagorean triple visit http://en.wikipedia.org/wiki/Pythagorean_triple). Pythagorean triples are closely related to the Pythagorean Theorem, which you probably have learned about in high school geometry.
Pythagorean triples represent the three sides of a right triangle, and therefore, obey the Pythagorean Theorem. Let's find the Pythagorean triple that has a components sum of 1000. We will do this using Euclid's formula:
In this example we will see some universal functions in action.
The Euclid's formula defines indices m and n.
- Create
mandnarrays.We will create arrays to hold these indices:
m = numpy.arange(33) n = numpy.arange(33)
- Calculate a, b, and c of the Pythagorean triple.
The second step is to calculate
a,b, andcof the Pythagorean triple using Euclid's formula. Use theouterfunction to get the Cartesian products, difference, and sums we require:a = numpy.subtract.outer(m ** 2, n ** 2) b = 2 * numpy.multiply.outer(m, n) c = numpy.add.outer(m ** 2, n ** 2)
- Find the index.
At this point, we have a number of arrays containing a, b and c values. However, we still need to find the values that conform to the problem's condition. Find the index of those values with the NumPy
wherefunction:idx = numpy.where((a + b + c) == 1000)
- Check the solution.
Check the solution with the
numpy.testingmodule:numpy.testing.assert_equal(a[idx]**2 + b[idx]**2, c[idx]**2)
The following is the complete code:
import numpy import numpy.testing #A Pythagorean triplet is a set of three natural numbers, a < b < c, for which, #a ** 2 + b ** 2 = c ** 2 # #For example, 32 + 42 = 9 + 16 = 25 = 52. # #There exists exactly one Pythagorean triplet for which a + b + c = 1000. #Find the product abc. #1. Create m and n arrays m = numpy.arange(33) n = numpy.arange(33) #2. Calculate a, b and c a = numpy.subtract.outer(m ** 2, n ** 2) b = 2 * numpy.multiply.outer(m, n) c = numpy.add.outer(m ** 2, n ** 2) #3. Find the index idx = numpy.where((a + b + c) == 1000) #4. Check solution numpy.testing.assert_equal(a[idx]**2 + b[idx]**2, c[idx]**2) print a[idx] * b[idx] * c[idx]
Universal functions are not really functions, but objects representing functions. Ufuncs have the
outer method, which we saw in action just now. Many of the NumPy standard universal functions are implemented in C, and are, therefore, faster than regular Python code. Ufuncs support element-by-element processing and type casting, which in practice means less loops.