NUMERICAL METHODS
(MCSC-202)
USING PYTHON
By
Samir Shrestha
Department of Mathematics
Kathmandu University, Dhulikhel
Lecture-4
Python Packages
Python Packages
numpy: Multi-dimensional Array Operations
matplotlib: Plotting Library
scipy: Scientific Computing
pandas: DataFrame (data analysis & manipulate)
seaborn: Statistical Data visualization
Importing Packages
Python Tips
Python Packages should be imported with their commonly
used names:
import numpy as np
import matplotlib.pyplot as plt
import scipy as sp
import pandas as pd
import seaborn as sns
References
1. H. Bhasin, Python Basics A Self-Teaching Introduction, 2019
2. H. P. Langtangen, A Primer on Scientific Programming with Python,
2016
3. Jaan Kiusalaas, Numerical Methods in Engineering with Python 3, 2013
4. Robert Johansson, Numerical Python: Scientific Computing and Data
Science Applications with Numpy, SciPy and Matplotlib, 2019
5. https://www.javatpoint.com/python-tutorial
NumPy
Outlines
Introduction to NumPy
Creating NumPy 1-D, 2-D arrays
Methods on NumPy arrays
Arithmetic operation on arrays
NumPy Linear Algebra
Random numbers in NumPy
Introducing NumPy
NumPy
NumPy stands for numeric python which is a python package for
the computation and processing of the multidimensional and single
dimensional array elements
NumPy performs vectorized computing and hence efficiently
implements the multidimensional arrays
It is capable of manipulation and reshaping the data stored in
multidimensional arrays
NumPy provides the in-built mathematical functions , functions for
linear algebra and random number generation
NumPy doesn't come bundled with Python. It has to be installed
it separately:
$ pip install numpy
In Ubuntu: $ sudo apt-get install python-numpy
In Anaconda : NumPy comes pre-installed
If not, $ conda install numpy or
NumPy
NumPy Math Functions
NumPy
Most commonly used numpy Math function
Trigonometic and Hyperbolic Functions
sin(x), cos(x), tan(x) Trigonometric functions, element-
wise.
arcsin(x), arccos(x), arctan(x)
Inverse trigonometric , element-wise.
Convert angles from radians to
degrees(x)
degrees.
Convert angles from degrees to
radians(x)
radians.
Convert angles from degrees to
deg2rad(x)
radians.
Convert angles from radians to
rad2deg(x)
degrees.
sinh(x), cosh(x), tanh(x) Hyperbolic functions, element-wise.
arcsinh(x), arccosh(x), arctanh(x) Inverse hyperbolic, element-wise
To use NumPy functions, we have to import numpy package
NumPy
Most commonly used numpy Math function continue...
Rounding, Exponential and Log functions
round_(a[, decimals]) Round an array to the given number of decimals
rint(x) Round elements of the array to the nearest integer
fix(x) Round to nearest integer towards zero
floor(x) Return the floor of the input, element-wise
ceil(x) Return the ceiling of the input, element-wise
Calculate the exponential of all elements in the input
exp(x)
array.
log(x) Natural logarithm, element-wise.
Return the base 10 logarithm of the input array,
log10(x)
element-wise.
log2(x) Base-2 logarithm of x.
To use NumPy functions, we have to import numpy package
NumPy
Most commonly used numpy Math function continue...
Other Functions
prod(a[, axis]) Return the product of array elements over a given axis.
sum(a[, axis]) Sum of array elements over a given axis.
Return the cumulative product of elements along a given
cumprod(a[, axis])
axis.
Return the cumulative sum of the elements along a given
cumsum(a[, axis])
axis.
Return the non-negative square-root of an array, element-
sqrt(x)
wise.
cbrt(x) Return the cube-root of an array, element-wise.
fabs(x) Compute the absolute values element-wise.
Returns an element-wise indication of the sign of a
sign(x)
number.
pi Returns the value of 𝜋
To use NumPy functions, we have to import numpy package
NumPy
Creating NumPy Array
NumPy
Creating Numpy Array:
Multi-dimensiona array and matrices can be created using
NumPy module
Creating an Array: Arrays can be created in several ways.
One of them is to use the array function to turn a list into
an array after importing numpy module:
Syntax:
numpy.array(list)
NumPy
Examples: Creating 1-D and 2-D array
1-dimensional array 2-dimensional array
import numpy as np import numpy as np
a = np.array([3, -1, 0, 2.1]) b = np.array([[3, -1, 0],[1, 3, -6]])
print(a) print(b)
print(a.dtype) print(b.dtype)
print(b.shape)
Numpy in-built functions creating 1-D and 2-D array
Syntax: Examples:
numpy.arange(start, stop, step) import numpy as np
numpy.linspace(start,stop,num) np.arange(0, 10, 0.5) # 1-D array
numpy.zeros((dim1, dim2)) np.linspace(-1,1,10) # 1-D array
numpy.ones((dim1, dim2)) np.zeros((3, 2)) # 3x2 zero matrix
numpy.eye(dim) np.ones((2, 3)) # 2x3 matrix of 1
np.eye(3) # 3x3 identity matrix
NumPy
Methods on NumPy Array
Methods on Array
NumPy
import numpy as np
a = np.array([3, -2, 1, 0, 5, -4, 6, 8])
b = np.array([[1, 0, 3, -2], [2, 1, 0, 7], [2, -3, 1, 4]])
Description Method Example
Dimensions of the Array ndim a.ndim
Data type of the array dtype a.dtype
The shape and size of the array shape, size a.shape, a.size
Reshaping the array reshape(m,n) a.reshape(2,4)
maximum, minimum, and sum of the 1- max, min, b.max(), b.min(),b.sum()
D array sum
maximum, minimum, and sum of the 2- max(axis=1), a.max(axis=0),
D array min(axis=0), a.min(axis=1), a.sum(axis
sum(axis=1) =0)
axis = 0 represents column, axis = 1 represents row and default is axis = 0
NumPy
Method and Function on Array Continue ...
import numpy as np
a = np.array([3, -2, 1, 0, 5, -4, 6, 8])
b = np.array([[1, 0, 3, -2], [2, 1, 0, 7], [2, -3, 1, 4]])
Description Method Example
mean, standard deviation, mean, std, var a.mean(),a.std(),a.var(),
variance of an array b.mean(axis=0),b.var(axis=1)
axis = 0 represents column and axis = 1 represents row
NumPy
Indexing and Slicing
NumPy Array
NumPy
Indexing and Slicing 1-D Array
Let us take a 1-D array: Forward index
0 1 2 3 4
import numpy as np
a = np.array([3, -2, 5, 0, 7]) 3 -2 5 0 7
-5 -4 -3 -2 -1 Forward index
Indexing:
Slicing:
a[0] gives 3
a[:] gives array([3, -2, 5, 0, 7]) Updating:
a[1] gives -2
a[0:] gives array( [3, -2, 5, 0, 7]) a[0] = 4 gives a =array( [4, -2, 5, 0, 7])
a[2] gives 5
a[2:4] gives array( [ 5, 0]) a[-1] = 1 gives a =array( [3, -2, 5, 0, 1])
a[-1] gives 7
a[0:5] gives array( [3, -2, 5, 0, 7]) a[1:3]=[-1, 4] gives a = array( [3, -1, 4, 0, 7])
a[-3] gives 5
a[0:-1] gives array( [3, -2, 5, 0])
a[2: ] gives array( [ 0, 7])
NumPy
Indexing and Slicing 2-D array
Let us take a 2-D array: 0 1 2 3
import numpy as np 0 1 0 3 -2
a=np.array([[1, 0, 3, -2], [2, 1, 0, 7], [2, -3, 1, 4]])
1 2 1 0 7
Indexing: 2 2 -3 1 4
Slicing:
a[0,0] gives 1
a[1,:] gives array([2,1,0,7])
a[1,2] gives 0 Slicing:
a[-1,-1] gives 4 0 1 2 3
a[1:3,2:4] gives
a[-2,1] gives 1 0 1 0 3 -2 array([[0,7],[1,4]])
1 2 1 0 7 0 1 2 3
Slicing:
2 2 -3 1 4 0 1 0 3 -2
a[: ,0] gives array([1,2,2])
1 2 1 0 7
0 1 2 3
2 2 -3 1 4
0 1 0 3 -2
1 2 1 0 7
2 2 -3 1 4
NumPy
Arithmetic Operations on
NumPy Array
NumPy
Arithmetic Operations (+,-,*,/,**) on Arrays
Performs elementwise operations
Let us take a 2-D arrays:
Let us take a 1-D arrays: import numpy as np
import numpy as np a2=np.array([[1, 0, 3],[2,1,4],[2,1,5]])
a1=np.array([1, 0, 3, -2]) b2=np.array([[-2, 1, 2],[5,3,-1],[0,3,2]])
b1=np.array([-2, 1, 2, 3]) c2=a2+b2
c1=a1+b1 d2=a2*b2
d1=a1*b1 e2=a2/b2
e1=a1/b1 f2=a2%b2
f1=a1%b1
1 0 3 -2 1 2
a1= 1 0 3 -2
a2= 2 1 4 b2= 5 3 -1
b1= -2 1 2 3
2 1 5 0 3 2
a1*b1= -2 0 6 -6
-2 0 6
a2*b2 = 10 3 -4
0 3 10
NumPy
Arithmetic Operations (+,-,*,/,**) on Arrays continue ...
Let us take a 1-D and 2-D arrays:
import numpy as np
a3=np.array([1, -4, 3])
b3=np.array([[-2, 1, 2],[5,3,-1],[0,3,2]])
c3=a3+b3
d3=a3*b3
e3=a3/b3
f2=a3/b3 -2 1 2
a3= 2 1 4 b3= 5 3 -1
0 3 2
2 1 4 -2 1 2 0 2 6
a3+b3 = 2 1 4 + 5 3 -1 = 7 4 3
2 1 4 0 3 2 2 4 6
a3 expand to 2-D array
NumPy
NumPY Linear Algebra
NumPy
NumPy Linear Algebra
NumPy provides various functions to perform operation on NumPy
arrays, here are a few:
Functions Defintions
dot() Calculates the dot product of two arrays (equivalent to
matix multiplication A*B)
vdot() Calculates the dot product of two vectors
cross() Calculates the vector product of arrays
inner() Inner product of two arrays
outer() Outer product of two vectors
matmul Calculates the matrix multiplication of two arrays
Example1: Example2:
# Scalar product # Vector product
import numpy as np import numpy as np
a = np.array([3,1,2]) a = np.array([3,1,2])
b = np.array([-2,3,-2]) b = np.array([-2,3,-2])
c = np.dot(a,b) c = np.cross(a,b)
NumPy
NumPy Linear Algebra Continue...
Example4:
Example3: # Matrix multiplication changing
# Matrix multiplication array to matrix by using NumPy
import numpy as np function mat
A = np.array([[3,1,2],[-1,2,5]]) import numpy as np
B = np.array([[-2,3,- A = np.array([[3,1,2],[-1,2,5]])
2],[0,1,3],[2,1,-4]]) B = np.array([[-2,3,-
C= np.matmul(A,B) 2],[0,1,3],[2,1,-4]])
A = np.mat(A)
B = np.mat(B)
C= A*B
NumPy
NumPy Linear Algebra linalg Module
NumPy has linalg module to perform operation on NumPy arrays, here
are a few:
Functions Defintions
linalg.det() Calculates the determinant of a matrix
linalg.inv() Caculate the inverse of the matrix
linalg.solve() Solves the linear system of equations
linalg.eig() Eigenvalues and eigenvectors of a square array
linalg.norm() Compute norm of vector or matrix
linalg.cond() Compute condition number of a matrix
Example5: Example6:
# Determinant # Inverse of a matrix
import numpy as np import numpy as np
A = np.array([[-2,3,-2],[0,1,3],[2,1,-4]]) A = np.array([[-2,3,-2][0,1,3],[2,1,-4]])
Det = np.linalg.det(A) A_inv = np.linalg.inv(A)
NumPy
NumPy Linear Algebra Continue...
Example7: Example8:
# System of linear equations # Norm of a 1-D array
import numpy as np import numpy as np
A = np.array([[2,4,2],[-2,-3,1],[2,2,-3]]) v = np.array([[2,4,2,-3]])
b = np.array([16,-5,-3]) norm = np.linalg.norm(v)
X = np.linalg.solve(A,b)
Example9: Example10:
# Norm of a 2-D array # Eigen-value & Eigen-Vector
import numpy as np import numpy as np
v=np.array([[2,4,2],[-2,-3,1],[2,2,-3]]) v=np.array([[2,4,2],[-2,-3,1],[2,2,-3]])
norm = np.linalg.norm(v) [EinVal, EinVec],= np.linalg.eig(v)
NumPy
Matrix Library: NumPY module numpy.matlib
The NumPy numpy.matlib module contains all the function in the
numpy name space that returns matrices instead of ndarray:
Functions Defintions
matlib.zeros() Returns zero matix
matlib.ones() Returns matrix of ones
matlib.eye() Returns matrix with 1s in diagonal and elsewhere 0s
matlib.identity() Returns square identiy matrix
matlib.repmat(a,m,n) Repeates 0-D to 2-D matrix of size mxn
matlib.rand() Returns a matrix of uniformally distributed random
values with given shape
matlib.randn() Returns a matrix of standard normally distributed
random values with given shape
NumPy
Random Numebrs in
NumPY
NumPy
Random Numbers in NumPy
Numpy offers the random module to work with random numbers
Functions Defintions
ranom.rand(m,n) Uniformaly distributed random number on [0,1] of
size mxn
ranom.randn(m,n) Standard normal ly distributed random number of
size mxn
ranom.randint(L,H,(m,n)) Random intergers from low L to high H (exclusive)
of given size mxn
Numpy also offers the random module to generate the random
numbers from various types of distribution. See www.numpy.org
documentation
End of Lecture-4
Lecture-5
Matplotlib
34
