Numerica

My own experimental implementations of numerical methods as homework.
Use documentation to see how to use, and check test.py for real examples.

Table of Contents

• Usage
  • Importing
  • Function Definition
  • Matrix Definition
• Documentation
  • 1- Solving Nonlinear Equations
    • Root Bracketing Methods
      • Graph
      • Bisection
      • Regula-Falsi
    • Iterative Methods
      • Fixed-Point Iteration
      • Newton-Raphson
      • Secant
  • 2- Matrix Operations
    • Basic Operations
      • Definition
      • Creating an Identity Matrix by n
      • Getting Dimensions of a Matrix
      • Transpose of a Matrix
    • Finding Inverse of a Matrix
      • Gauss-Jordan Method
    • Matrix Utils
      • Concat Matrices by Row (Horizontal)
      • Concat Matrices by Column (Vertical)
      • Map a Row of Matrix
      • Map all Matrix Cells
      • Is Matrix Check
      • Slice Matrix Vertically
  • 3- Solving Systems of Linear Equations
    • Gauss Elimination
    • Jacobi
    • Gauss-Seidel
  • 4- Solving Systems of Nonlinear Equations
  • 5- Numerical Integration
    • Trapezoidal
    • Simpson
  • 6- Numerical Differentiation
    • Euler Methods
      • Backward
      • Forward
      • Midpoint
  • 7- Finite Differences
    • Determine Degree of a Polynomial
  • 8- Interpolation
    • Lagrange
  • 9- Regression
    • Least Squares
• Resources
• Testing Package
• Uploading to PyPI

Usage

python >= 3.8 is required

Importing

import numerica as n
from numerica import f // function definition
from numerica import m // matrix definition

Function Definition

f('expression')

fx = f('3x^2 + 2x + 3')
fx(2)

Matrix Definition

m(
    a11, a12, a13;
    a21, a22, a23;
    a31, a32, a33
)

matrix = m('1,2,3; 4,5,6; 7,8,9');

Documentation

1- Solving Nonlinear Equations

Root Bracketing Methods

Graph

n.nl_graph(fx, dx, epsilon, x)

Bisection

n.nl_bisection(fx, epsilon, a, b)

Regula-Falsi

n.nl_regulafalsi(fx, epsilon, a, b)

Iterative Methods

Fixed-Point Iteration

n.nl_fixedpoint(hx, epsilon, x)

Newton-Raphson

n.nl_newtonraphson(fx, epsilon, x)

Secant

n.nl_secant(fx, epsilon, x0, x1)

2- Matrix Operations

Basic Operations

Matrix Definition

m(
    a11, a12, a13;
    a21, a22, a23;
    a31, a32, a33
)

Identity Matrix

n.m_id(n)

Size of Matrix

(m, n) = n.m_size(A)

Transpose of a Matrix

n.m_transpose(A)

Finding Inverse of a Matrix

Gauss-Jordan Method

n.mi_gaussjordan(A)

Matrix Utils

Concat Matrices by Row (Horizontal)

n.m_rowconcat(A, B)

Concat Matrices by Column (Vertical)

n.m_colconcat(A, B)

Map a Row of Matrix

n.m_rowmap(A, i, iteratee)

Map all Matrix Cells

n.m_cellmap(A, iteratee)

Is Matrix Check

n.is_matrix(A)

Slice Matrix Vertically

n.m_rowslice(A, start, stop, step)

3- Solving Systems of Linear Equations

Gauss Elimination

n.ls_gauss(A, C)

Jacobi

n.ls_jacobi(A, C, X, epsilon=0.001)

Gauss-Seidel

n.ls_gaussseidel(A, C, X, epsilon=0.001)

4- Solving Systems of Nonlinear Equations

5- Numerical Integration

Trapezoidal

n.itg_trapezoidal(fx, x0, xn, n)

Simpson

n.itg_simpson(fx, x0, xn, n)

6- Numerical Differentiation

Euler Methods

Backward

n.diff_backward(fx, x)

Forward

n.diff_forward(fx, x)

Midpoint

n.diff_midpoint(fx, x)

7- Finite Differences

Determine Degree of a Polynomial

n.fd_degree(pair_tuples)
n.fd_degree([(x0,y0), (x1,y1), (x2,y3), ...])

8- Interpolation

Lagrange

n.itp_lagrange(pair_tuples)
n.itp_lagrange([(x0,y0), (x1,y1), (x2,y3), ...], x)

9- Regression

Least Squares

n.reg_leastsquares(pair_tuples, degree) // returns polynomial
n.reg_leastsquares_solve(pair_tuples, x, degree) // solves polynomial 
n.reg_leastsquares_solve([(x0,y0), (x1,y1), (x2,y3), ...], x, deg)

Resources

• YTU Numerical Analysis Lecture Notes
• https://mat.iitm.ac.in/home/sryedida/public_html/caimna/index1.html

Testing Package

Test Directly as Script

python3.8 -m numerica

or Install Package Locally (from repo root dir)

pip3.8 install .

and Test It from REPL

import numerica as n
# ...

or Use test.py Interactively

python3.8 -i test.py
# ...

or Just Test and Exit

python3.8 test.py

Uploading to PyPI

Install Twine

pip3.8 install twine

Build

rm -rf build & rm -rf dist & rm -rf numerica.egg-info
python3.8 setup.py sdist bdist_wheel

Upload

twine upload dist/*