Course127106019109 lectures
Numerical Methods for Engineers
Prof. Niket Kaisare · IIT Madras
About
The development of fast, efficient and inexpensive computers has significantly increased the range of engineering problems that can be solved reliably. Numerical Methods use computers to solve problems by step-wise, repeated and iterative solution methods, which would otherwise be tedious or unsolvable by hand-calculations. This course is designed to give an overview of numerical methods of interest to scientists and engineers. However, the focus being on the techniques themselves, rather than specific applications, the contents should be relevant to varied fields such as engineering, management, economics, etc.
INTENDED AUDIENCE: First/Second Year UG students in any branch of engineering (or science)
PREREQUISITES: 12th standard Math background
Lectures
Week 1 - Computation and Error Analysis
- Errors and Approximations
- Truncation and Round-Off Errors
- Binary Numbers: Introduction
- Floating Point: Real numbers in decimal system
- Floating Point in Binary system
- Tutorial Example: Iterative Method
- Tutorial Example: Direct Method
- Tutorial Example: Sequential Method
Week 2 - Linear Systems and Equations
- Linear Algebra: Basics
- Introduction to Linear Equations
- Rank Condition for Solving Linear Equations
- Motivating Gauss Elimination
- Gauss Elimination
- Tutorial Recap: Gauss Elimination
- Back Substitution to find solution
- Gauss Jordan and LU Decomposition
Week 3 - Linear Equations - 2
- Partial Pivoting in Gauss Elimination
- Analysis of Gauss Elimination
- Tri-Diagonal Systems: Practical Relevance
- Thomas Algorithm for Tri-Diagonal Systems
- Gauss Siedel Method
- Analysis of Gauss Siedel Method
- Gauss Siedel vs. Jacobi Methods
- Bonus: Example using MS Excel
- Summary: Linear Equations
Week 4: Nonlinear Equations in Single Variable
- Introduction to Nonlinear Equations
- Bisection Method
- Analysis of Bisection Method
- Bonus: Excel Solution for Bisection Method
- Regula-Falsi Method
- Bonus: Excel Solution for Regula-Falsi Method
- Regula-Falsi vs. Secant Method
- Bonus: Excel Solution for Secant Method
- Some special cases
Week 5: Nonlinear equations in Single and Multiple Variables
- Fixed-Point Iteration
- Newton-Raphson Method
- Analysis of Fixed-Point Iteration
- Analysis of Newton-Raphson
- Problems with Newton-Raphson
- Multi-Variable Fixed-Point Iteration
- Multi-Variable Newton-Raphson
- Out of Syllabus: Improvements to NR Methods
- Out of Syllabus: Roots of a polynomial
- Summary
Week 6: Regression (Curve Fitting)
- Introduction: Regression and Interpolation
- Linear Regression in One Variable
- Recap: Formula for Linear Regression
- Bonus: Linear Regression using MS-Excel
- Linear Regression in Multiple Variables
- Matrix Method for Multi-Linear Regression
- Polynomial Regression
- Functional Regression
- Bonus: X-Y versus Y-X data (Using MS Excel)
Week 7: Interpolation
- Interpolation: Introduction & A Naïve Extension
- Bonus: MS-Excel for Naïve Interpolation
- Lagrange Interpolating Polynomials
- Newton's Forward Difference Polynomial
- Newton's Divided Differences: Derivation
- Interpolation Examples
- Bonus: MS-Excel for Newton's Polynomial
- Summary: Regression and Interpolation
Week 8: Numerical Differentiation
- Numerical Differentiation: Introduction
- Numerical Differentiation Formula and Analysis
- Derivation using "Method of undetermined coefficients"
- Three-point differentiation formulae
- Bonus: Differentiation using MS-Excel
- Numerical Differentiation Examples
- Truncation vs. Round-Off Errors
- Summary of Numerical Differentiation
Week 9: Numerical Integration
- Numerical Integration: Introduction
- Trapezoidal rule and Derivation
- Simpson's Rules for Integration
- Bonus: MS-Excel for Numerical Integration
- Error Analysis for Simpson's Rules
- Numerical Integration Examples
- Bonus: Integration using MS-Excel
- Summary of "Newton Cotes" Formulae
- Richardson's Extrapolation
- Gauss Quadrature
- Summary of Numerical Integration
Week 10: Ordinary Differential Equations – Initial Value Problems (ODE-IVP)
- Introduction to ODE-IVP
- Motivation using an Example (Bonus)
- Euler's Methods and Second-Order Methods
- Second-Order Runge-Kutta Methods
- Summary of RK-2
- Higher order RK Methods
- Bonus: ODE-IVP using MS-Excel
- Bonus: RK-2 and RK-4 Methods using MS-Excel
- Summary and Recap
Week 11: ODE-IVP (Part-2)
- Introduction to Predictor-Corrector Methods
- Stability of Implicit Methods: Overview
- Stability Analysis of Euler's Methods
- Extension to multiple variables
- Local vs. Global Truncation Errors
- Richardson's Extrapolation
- Stiff System of ODEs: Introduction
- Adaptive Step-sizing
- Adaptive step-sizing and Embedded Methods
- Bonus: Errors and Extrapolation using MS-Excel
- Summary and Recap (Weeks 10 and 11)
Week 12: ODE - Boundary Value Problems
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Madras
- Department
- Multidisciplinary
- Teacher
- Prof. Niket Kaisare
- Level
- Undergraduate
- Language
- English
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
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