Introduction to Computingchitrabhanu/intro_to_computing/...Introduction to Computing Author Plotting Functions with Python (Lecture 8) Created Date 10/11/2019 10:22:55 AM

Introduction to Computing

Plotting Functions with Python (Lecture 8)

9 October, 2019

Numpy

We shall take a detour in this class to learn plotting functions usingPython.

For this we shall need to know a bit about NumPy arrays whichare much faster than the built-in Python lists.

NumPy is the fundamental package for scientific computing inPython.

It (NumPy) is a Python library that provides a multidimensionalarray object, (like a built-in data type), and various derived objects.

To use the NumPy library we have to import it just like any otherPython Library.

>>> import numpy as np

Numpy







Numpy







Numpy







Numpy







Numpy Array

NumPy has a built-in datatype called array.

The easiest way to create a NumPy array is from a Python list usingthe array function in NumPy library.

>>> import numpy as np>>> b = np.array([6, 7, 8])>>> barray([6, 7, 8])>>> type(b)<class 'numpy.ndarray'>

Numpy Array




Numpy Array




Numpy Array

Another way to create an array is using the arange function inNumPy, which is just like the range function in Python.

>>> import numpy as np>>> c = np.arange(0, 2*np.pi, np.pi/4)>>> carray([ 0. , 0.78539816, 1.57079633,

2.35619449, 3.14159265, 3.92699082,4.71238898, 5.49778714])

The function arange takes 3 arguments, start, stop, step,these could be integers or floats. It returns a NumPy array withstart as the first element, start + step as the second element,start + step + step as the third element and so on. The lastelement is the largest such number strictly smaller than stop.

Numpy Array



2.35619449, 3.14159265, 3.92699082,4.71238898, 5.49778714])


Numpy Array



2.35619449, 3.14159265, 3.92699082,4.71238898, 5.49778714])


More examples using arange

The arguments start and step are optional.

>>> np.arange(3)array([0, 1, 2])

>>> np.arange(3.0)array([ 0., 1., 2.])

>>> np.arange(3,7)array([3, 4, 5, 6])

>>> np.arange(3,7,2)array([3, 5])

NumPy LinspaceIf you want to create an array by partitioning an interval into certainnumber of equal parts, then the linspace function in NumPy ismore convenient.

>>> np.linspace(0, 2*np.pi, 10)array([ 0. , 0.6981317 , 1.3962634 ,

2.0943951 , 2.7925268 , 3.4906585 ,4.1887902 , 4.88692191, 5.58505361,6.28318531])

numpy.linspace(start, stop, num) returns num evenly spacedsamples, calculated over the interval [start, stop]. The endpointof the interval is by default included.

>>> np.linspace(2.0, 3.0, num=5)array([ 2. , 2.25, 2.5 , 2.75, 3. ])



2.0943951 , 2.7925268 , 3.4906585 ,4.1887902 , 4.88692191, 5.58505361,6.28318531])





2.0943951 , 2.7925268 , 3.4906585 ,4.1887902 , 4.88692191, 5.58505361,6.28318531])



Multi dimensional arrays

So far we have only seen one dimenstional arrays. But arrays can bemulti dimenstional.

I A one dimensional array is like a list of elements (numbers).I A two dimensional array is like a list of lists or a matrix of

numbers.I A three dimensional array is like a list of list of lists, and so on

We shall only need one and two dimensional arrays so we restrictour discussion to them.











Multidimensional arrays

Here ar esome examples of two dimensional arrays and how tocreate them.

>>> a = np.arange(15)>>> b = a.reshape((3,5))>>> aarray([ 0, 1, 2, 3, 4, 5, 6, 7,

8, 9, 10, 11, 12, 13, 14])>>> barray([[ 0, 1, 2, 3, 4],

[ 5, 6, 7, 8, 9],[10, 11, 12, 13, 14]])


Here ar esome examples of two dimensional arrays and how tocreate them.

>>> a = np.arange(15)>>> b = a.reshape((3,5))>>> aarray([ 0, 1, 2, 3, 4, 5, 6, 7,

8, 9, 10, 11, 12, 13, 14])>>> barray([[ 0, 1, 2, 3, 4],

[ 5, 6, 7, 8, 9],[10, 11, 12, 13, 14]])


Here are some more examples:

>>> a.reshape((5,3))array([[ 0, 1, 2],

[ 3, 4, 5],[ 6, 7, 8],[ 9, 10, 11],[12, 13, 14]])

>>> c = np.array([[1, 4], [9, 16]])>>> carray([[ 1, 4],

[ 9, 16]])

Mathematical operations on NumPy arraysArithmetic operators on arrays apply elementwise. A new array iscreated and filled with the result.

>>> a = np.arange(10)>>> b = a.reshape((2,5))

>>> 2 * aarray([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18])

>>> b ** 2array([[ 0, 1, 4, 9, 16],

[25, 36, 49, 64, 81]])

>>> a + 3array([ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])

>>> b - 4array([[-4, -3, -2, -1, 0],

[ 1, 2, 3, 4, 5]])

Mathematical operations on NumPy arrays

Two arrays can be added, subtracted, multiplied or divided only ifthey have the same dimensions. Again in that case the operation isdone element wise.

>>> a + bTraceback (most recent call last):

File "<stdin>", line 1, in <module>ValueError: operands could not be broadcasttogether with shapes (10,) (2,5)

Examples:

>>> a = np.arange(1,5)>>> b = a.reshape((2,2))

Mathematical operations on NumPy arrays

Two arrays can be added, subtracted, multiplied or divided only ifthey have the same dimensions. Again in that case the operation isdone element wise.

>>> a + bTraceback (most recent call last):

File "<stdin>", line 1, in <module>ValueError: operands could not be broadcasttogether with shapes (10,) (2,5)

Examples:

>>> a = np.arange(1,5)>>> b = a.reshape((2,2))

Examples

>>> barray([[1, 2],

[3, 4]])

>>> b**2 + barray([[ 2, 6],

[12, 20]])

>>> b**2 - barray([[ 0, 2],

[ 6, 12]])

>>> c = np.linspace(0,1,4).reshape((2,2))>>> carray([[ 0. , 0.33333333],

[ 0.66666667, 1. ]])

Examples

>>> c*barray([[ 0. , 0.66666667],

[ 2. , 4. ]])

>>> c/barray([[ 0. , 0.16666667],

[ 0.22222222, 0.25 ]])

>>> c*(b + b**2)array([[ 0., 2.],

[ 8., 20.]])

Applying functions to arraysNumPy has some built-in functions that can be applied to arrays.These functions act elementwise.

I Trigonometric functions: numpy.sin, numpy.cos,numpy.tan etc.

I Exponential and logarithm: numpy.exp, numpy.log,numpy.log10.

I Inverse trigonometric: numpy.arcsin, numpy.arccos,numpy.arctan.

I Roots: numpy.sqrt, numpy.cbrt.

I Round off functions: numpy.floor, numpy.ceil,numpy.round

I Absolute value: numpy.fabs.











































Examples

>>> a = np.linspace(0, 2*np.pi, 13)>>> np.rad2deg(a)array([ 0., 30., 60., 90., 120.,

150., 180., 210., 240.,270., 300., 330., 360.])

>>> b = np.sin(a)>>> np.round(b,3)array([ 0. , 0.5 , 0.866, 1. , 0.866, 0.5 ,

0. , -0.5 , -0.866, -1. , -0.866, -0.5 ,-0. ])

>>> np.fabs(np.round(b,5))array([ 0. , 0.5 , 0.86603, 1. , 0.86603,

0.5 , 0. , 0.5 , 0.86603, 1. ,0.86603, 0.5 , 0. ])

Plotting functions

Now we shall be able to plot some function in python. For this weshall use the powerful Python library matplotlib.

>>> import matplotlib.pyplot as plt>>> import numpy as np>>> x = np.arange(1,5)>>> plt.plot(x, x**2)>>> plt.show()

The Plot

Figure 1: Squares

Matplotlib and Pyplot

Pyplot is a library inside matplotlib.

The function plot in pyplot is used as follows:

If x and y are numpy arrays of the same length then to plot xversus y simply use plt.plot(x,y).

The function show in pyplot displays a figure. When running inipython with its pylab mode, it displays all figures and returns to theipython prompt.

In non-interactive mode, it displays all figures and blocks until thefigures have been closed.













The plot function

The plot function plots the data point from the arrays x and y andjoins those points by a straight line.

It takes a third argument which is a format string that specifies

I the color of the graph: ‘r’ = red, ‘g’ = green, ‘b’ = blue, ‘k’ =black, ‘y’ = yellow;

I the marker for the points: ‘.’ = dot, ‘o’ =circle, ‘x’ =cross.I line style: ‘-’ = solid line, ‘–’ = dashed line, ‘:’= dotted line.

So the string "r.-" means the points in the graph will be plottedwith red dot markers joined by a solid red line.

The plot function






The plot function






Some better examples

>>> import matplotlib.pyplot as plt>>> import numpy as np

>>> x = np.linspace(0,1,100)

>>> plt.plot(x, x**2, 'b-')>>> plt.plot(x, np.sqrt(x), 'b:')

>>> plt.plot(x, x**3, 'r-')>>> plt.plot(x, np.cbrt(x), 'r:')

>>> plt.show()

Graph

Figure 2: Powers

Damped Oscilation


>>> x = np.linspace(0, 10*np.pi, 1000)>>> y = (1/(x+1))*np.cos(x)

>>> plt.plot(x, y)

>>> plt.plot(x, 1/(x+1), 'y:')>>> plt.plot(x, -1/(x+1), 'y:')

>>> plt.show()

Graph

Figure 3: Damped Oscilation

Graph of sin(1x )


>>> x = np.linspace(0,1, 500)>>> x = 1/(10*np.pi) + x

>>> plt.plot(x, np.sin(1/x))

>>> plt.show()

Graph

Figure 4: Graph of sin(1/x)

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Introduction to Computingchitrabhanu/intro_to_computing/...Introduction to Computing Author Plotting Functions with Python (Lecture 8) Created Date 10/11/2019 10:22:55 AM