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Swayam

Computational Mathematics with SageMath

Institute of Chemical Technology (ICT) and NPTEL via Swayam

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Overview

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Computational Mathematics with SageMath: This twelve week course aims to use SageMath, a Python based free and open source computer algebra system (CAS) to explore concepts in Calculus, Applied Linear Algebra, Numerical Methods and basic Linear Programming Problems. The course will begin with the introduction of basic Python programming language. Next, we shall provide a quick introduction to SageMath software along with plotting 2D and 3D objects. The focus will be on using SageMath to explore topics in Calculus, Applied Linear Algebra, Numerical Method and Linear Programming Problems along with several applications.INTENDED AUDIENCE :UG and PG students of Mathematics and BE Students, Teachers teaching in UG/PG Level.PREREQUISITES : Basic knowledge of Calculus, Linear Algebra and ODE and Numerical Methods.INDUSTRIES SUPPORT :Any industry dealing with Data Science and Numerical Computations.

Syllabus

Week 1:
  • Installation of Python
  • Getting Started with Python
  • Python as an advanced calculator
  • Lists in Python
  • Tuple, Sets and Dictionaries in Python
  • Functions and Branching
Week 2:
  • For loop in Python
  • While loop in Python
  • Creating Modules and Introduction to NumPy
  • Use of NumPy module
  • Python Graphics using MatplotLib
Week 3:
  • Use of SciPy and SymPy in Python
  • Classes in Python - Part 01
  • Classes in Python - Part 02
  • Introduction and Installation of SageMath
  • Exploring integers in SageMath
Week 4:
  • Solving Equations in SageMath
  • 2d Plotting with SageMath
  • 3d Plotting with SageMath
  • Calculus of one variable with SageMath - Part 1
  • Calculus of one variable with SageMath - Part 2
  • Applications of derivatives
Week 5:
  • Integration with SageMath
  • Improper Integral using SageMath
  • Application of integration using SageMath
  • Limit and Continuity of real valued functions
  • Partial Derivative with SageMath
  • Local Maximum and Minimum
Week 6:
  • Application of local maximum and local minimum
  • Constrained optimization using Lagrange multipliers
  • Working with vectors in SageMath
  • Solving system of linear Equations in SageMath
  • Vector Spaces in SageMath
  • Basis and dimensions of vector spaces in SageMath
Week 7:
  • Matrix Spaces with SageMath
  • Linear Transformations Part 1 with SageMath
  • Linear Transformations Part 2 with SageMath
  • Eigenvalues and Eigenvectors Part 1 with SageMath
  • Eigenvalues and Eigenvectors Part 2 with SageMath
Week 8:
  • Inner Product Part 1 with SageMath
  • Inner Product Part 2 with SageMath
  • Orthogonal Decomposition with SageMath
  • Least Square Solution with SageMath
  • Singular Value Decomposition (SVD) with SageMath
  • Application of SVD to image processing
  • Solving System of linear ODE using Eigenvalues and Eigenvectors
Week 9:
  • Google Page Rank Algorithm using SageMath
  • Finding Roots of algebraic and transcendental equations in SageMath
  • Numerical Solutions of System of linear equations in SageMath
  • Interpolations in SageMath
  • Numerical Integration in SageMath
  • Numerical Eigenvalues
Week 10:
  • Solving 1st and 2nd order ODE with SageMath
  • Euler's Method to solve 1st order ODE with SageMath
  • Fourth Order Runge-Kutta Method
  • RK4 method for System of ODE and Applications
  • Solving ODE using Laplace Transforms in SageMath
Week 11:
  • Introduction to Linear Programming Problems (LPP)
  • Solving Linear Programming Problmes using Graphical Methods
  • Basics Definitions and Results in LPP
  • Theory of Simplex Method
  • Simplex Methods in SageMath - Part 01
  • Simplex Methods in SageMath - Part 02
  • Simplex Methods in Matrix Form
Week 12:
  • Revised Simplex Method in SageMath
  • Two Phase Simplex Method in SageMath
  • Big-M Method in SageMath
  • Duality of Linear Program
  • Dual Simplex Method in SageMath
  • Review and What next in SageMath?

Taught by

Prof. Ajit Kumar

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