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:

• \(a = [4.0, \, 5, \,6, \,-2,\, 3, \text{NaN}, \infty]\)
• \(b = [0, 0, \ldots, 0]\) with 300 columns of zeros, one row
• \(c = [1, 1, \ldots, 1]^T\) with 300 rows of ones in a single column
• \(d = [2.6, 2.6, \ldots, 2.6]^T\) with 300 entries of 2.6 in one column
• \(e = [4.5, 4.6, 4.7, \ldots, 10.5 \), 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:

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

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

• \(c = \left(\begin{array}{cccc} 1 & 2 & \dots & 10 \\ 11 & 12 & \ldots & 20 \\ \vdots & \vdots & & \vdots \\ 61 & 62 & \dots & 70 \\\end{array} \right)\), 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))