Software tutorial/Vectors and arrays

In this section we will focus on containers for your numbers: vectors and arrays. For the purposes of this section you should use this terminology:

Vector

A one-dimensional list of numbers

Array

A multi-dimensional arrangement of numbers

Matrix

A two-dimensional array

In this course we will mainly use vectors and matrices, however, arrays are extremely prevalent in process modelling.

Creating vectors

We will create these vectors:

• $a=[4.0,5,6,-2,3,\text{NaN},\mathrm{\infty }]$
• $b=[0,0,\dots ,0]$ with 300 columns of zeros, one row
• $c=[1,1,\dots ,1{]}^T$ with 300 rows of ones in a single column
• $d=[2.6,2.6,\dots ,2.6{]}^T$ with 300 entries of 2.6 in one column
• $e=[4.5,4.6,4.7,\dots ,10.5$, equi-spaced entries
• $f=$ 26 equi-spaced entries starting from 3.0, going down -4.0

MATLAB

Python

a = [4, 5, 6, -2, 3, NaN, inf]; b = zeros(1, 300); c = ones(300, 1); d = ones(300, 1) .* 2.6; e = 4.5:0.1:10.5; f = linspace(3.0, -4.0, 26); >> size(a) ans = 1 7 >> size(c) ans = 300 1 >> size(f) ans = 1 26 In MATLAB, everything is an array, even a scalar is just a $1\times 1$ array, and vectors are either a $1\times n$ or $n\times 1$ array.

import numpy as np a = np.array([4, 5, 6, -2, 3, np.nan, np.inf]) b = np.zeros((1, 300)) # note the extra brackets! c = np.ones((300, 1)) d = np.ones((300, 1)) * 2.6 e = np.arange(4.5, 10.5001, 0.1); # type help(np.arange) to understand why f = np.linspace(3.0, -4.0, 26) >>> a.shape (7,) >>> c.shape (300, 1) >>> f.shape (26,) The NumPy library in Python supports two different data containers: arrays and matrices. We will not use NumPy matrices in this course, because they cannot deal with multiple dimensions (e.g. 3 dimensional arrays). And even though we will probably not use multi-dimensional arrays in this course, it is not worth learning about NumPy matrices, because they are fairly limited. So we will focus on NumPy arrays, which while a bit harder to get used to, are much more powerful than NumPy matrices. A NumPy array is roughly equivalent to a MATLAB array. One difference though with MATLAB, is that a Numpy array with one dimension, in other words a vector, has a shape property of (n,): indicating the vector has n-elements. It is not like MATLAB where one of the dimensions is a 1 (one). If you require the vector to be either a row vector or a column vector, then you must do so explicitly. The following code demonstrates this: np.zeros(300).shape # (300,) a general vector np.zeros((300, 1)).shape # (300, 1) a column vector np.zeros((1, 300)).shape # (1, 300) a row vector

Creating arrays

We will create these arrays:

• $a=\left(\begin{array}{cccc}1& 1& 1& 2\\ 4& -1& -2& 3\end{array}\right)$

• $b=\left(\begin{array}{cccc}0& 0& \dots & 0\\ ⋮& & & ⋮\\ 0& 0& \dots & 0\end{array}\right)$, a $5\times 8$ array of zeros

• $c=\left(\begin{array}{cccc}1& 2& \dots & 10\\ 11& 12& \dots & 20\\ ⋮& ⋮& & ⋮\\ 61& 62& \dots & 70\end{array}\right)$, a $7\times 10$ array of sequence integers

MATLAB

Python

a = [1, 1, 1, 2; 4, -1, -2, 3]; b = zeros(5,8); c = reshape(1:70, 10, 7)' The last line requires a bit more explanation. It first creates a vector with 70 integers, starting from 1 and ending at 70. Then it reshapes that vector into a matrix, with 10 rows and 7 columns. Then it transposes the array into the desired $7\times 10$ result. What would have been the result if you wrote: reshape(1:70, 7, 10): can you explain why?

a = np.array([[1, 1, 1, 2], [4, -1, -2, 3]]) b = np.zeros((5, 8)) c = np.reshape(np.arange(1,71),(7, 10))