Rantings from a guy with way too much free time

Recursion Revisited

Recursion, from more than one point-of-view.

A common programming idiom in computer science is solving a problem by self-reference, also known as recursion. In this post, we look at two different implementations of the same problem.

Solve a recursive problem in two different programming language paradigms

Let's look at the solution to a simple problem, compute $f(x)=e^{x}$.

We will illustrate two separate solutions - one in a procedural language (python), and the other in a functional language (elixir).

Let's start off with the functional language. Were does recursion come into play?

We define the function $f(x)=e^x$ as the infinite sum, $\text{ }f(x) = \sum_{n=0}^\infty{\frac{x^n}{n!}}$

In our solution below, we define two separate recursive functions, exp/2 and fac/1. What's interesting to note here is how each of these functions has two separate definitions. This is an aspect of programming in elixir that elegantly uses pattern-matching to return different results depending upon the input. Our two functions nicely dovetail into the base-case and recursive case of a recursive algorithm.

For example, looking at exp/2, the first function definition returns 1 for any value of x (as indicated by the _ preceding the variable) and returns 1. This is the mathematical equivalent of $x^0=1\text{ for any }x$.

The second definition of exp/2 is the recursive case. for any value of $n\gt0$. Moreover, we define exp(x, n) as $\frac{e^x}{n!}$ + exp(x, n-1).

Similarly for the definition of fac/1 we see two definitions; one for $n=0$ and another for all values of $n\gt0$.

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Visualizing the News: Grab your PILlow

2017-11-12 python programming
A Picture's Worth a Gazillion Bits newsapi This weekend, I tripped over a neat news REST-ful api called newsapi. Grab an API key and you're off to the races. There are tons of live headlines - News API can provide headlines from 70 worldwide sources. There are basically two api endpoints: GET GET Register for an account and generate an api-key and let's get started. Accessing the API with Python newsapi can easily be accessed using a browser since the REST-ful method used is a GET method. Continue reading

Making Heads or Tails out of Recursion and Combinatorial Math

2017-11-09 programming math python
GNU — GNU's Not Unix I thought I'd take a look at two topics at once and make some fun by mashing them together. Today's topic is recursion. In computer science (and in math, although we call the equation a recurrence relation) recursion is an often misunderstood concept causing lots of panic and anxiety. In reality - it's pretty easy once you wrap you head around the idiom. What is recursion? Continue reading

Getting Visual With It

Time to Get Vizzy With It! Like Will says: Gettin vizzy wit it Na na na na na na na nana Na na na na nana Gettin vizzy wit it… Plot thickens Time to turn on the lights and see what our data objects look like when we turn them into visualized plots. Sure, we can use the print function to see the numbers within our numpy objects. Better yet, let's turn our objects into pretty graphical images. Continue reading

Do the Numpy

2017-11-07 programming python

Do the Numpy Dance, Is your Chance to do the Nump!

Sir Numpy says:

The Numpy Dance is your chance to do the nump Do the Numpy Nump, come on and do the Numpty Nump Do the Numpy Nump, just watch me do the Numpty Nump Do ya know what I'm doin’, doin’ the Numpty Nump Do the Numpy Nump, do the Numpty Nump

Start at the beginning: NumPy

The first library that we will investigate is numpy. Simply put, numpy allows us to represent mathematical objects like arrays and matricies of different datatypes and performs operations on those objects thereby easing the burden of writing the tedious code within your applications to do the mundane representations and operations.

Diving right in, let's look at some examples of types of objects we can create and how we can operate on them using the library.

creation of arrays

The most basic thing we can do is create a one-dimensional array (also known as a vector):

import numpy as np

v = np.array([1,2,3,4])

Here we've created a row vector containing for elements. Numpy objects are typed and implicitly, we've created an int64 type by passing in the integers. We can verify the type of our array:

>>> v.dtype

We can also verify the dimensions of our array (i.e. the shape):

>>> v.shape

Rather than implicitly typing our array object, we can explicitly specify it when we create or array:

v = np.array([1,2,3,4], dtype=float)

We can also turn a list into an array by shaping it from an array range:

>>> v = np.arange(16).reshape(4,4)
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11],
       [12, 13, 14, 15]])

arange works semantically like our built-in range function, creating an array ranging over the given value. So long as the reshaping integers, n and m are factors of the len(arange(a,b,c)), then we can reshape our range into an n by m array. For example:

>>> v = np.arange(0,32,2).reshape(4,4)
array([[ 0,  2,  4,  6],
       [ 8, 10, 12, 14],
       [16, 18, 20, 22],
       [24, 26, 28, 30]])

We can even create complex-valued arrays:

 >>> c = np.array( [ [1+2.j,2+1.j], [3+5.j,4+1.j] ], dtype=complex )
 array([[ 1.+2.j,  2.+1.j],
       [ 3.+5.j,  4.+1.j]])

operations on arrays

So you created an array, now what? We can perform operations on them that are familair, like addition subtraction, multiplication, and division resulting in the expected semantics for arrays. For example:

# given two arrays, a and b, and vector c
a = np.array([[1,2],[3,4]])
b = np.array([[4,3],[2,1]])
c = np.array([0.5,2])

# add two arrays
>>> a + b
array([[5, 5],
       [5, 5]])
# multiply two arrays
>>> a * b
array([[4, 6],
       [6, 4]])

# inverse
>>> 1/a
array([[ 1.        ,  0.5       ],
       [ 0.33333333,  0.25      ]])

# sin of all values in array a
>>> np.sin(a)
array([[ 0.84147098,  0.90929743],
       [ 0.14112001, -0.7568025 ]])

There are nearly 600 functions in the numpy library - this is a brief sample of only the basic ones. Check out the documentation to learn more. To give you a sense of how extensive the library is, here are the top-level categories of functions:

  • Array creation routines
  • Array manipulation routines
  • Binary operations
  • String operations
  • C-Types Foreign Function Interface (numpy.ctypeslib)
  • Datetime Support Functions
  • Data type routines
  • Optionally Scipy-accelerated routines (numpy.dual)
  • Mathematical functions with automatic domain (numpy.emath)
  • Discrete Fourier Transform (numpy.fft)
  • Financial functions
  • Functional programming
  • NumPy-specific help functions
  • Indexing routines
  • Input and output
  • Linear algebra (numpy.linalg)
  • Logic functions
  • Masked array operations
  • Mathematical functions
  • Matrix library (numpy.matlib)
  • Miscellaneous routines
  • Random sampling (numpy.random)
  • Set routines
  • Sorting, searching, and counting
  • Statistics

Go experiment with the library and get more familiar with function of interest to you. Now that we have a basic undertanding, let's move on to visualizing data representations and transformations within numpy objects.