Learn · Numerical Methods for Engineers
Lecture 39 of 109Week 5: Nonlinear equations in Single and Multiple Variables
Problems with Newton-Raphson
Prof. Niket Kaisare · IIT Madras
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All the lectures
- 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
- 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
- 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
- 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
- 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
- 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)
- 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
- 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
- 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
- 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
- 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)
- Introduction to ODE-BVP
- Shooting Method: An Overview
- Finite Difference Method: An Overview
- Solution using Shooting Method
- Algorithm for Shooting Method
- Problems with Shooting Method
- Solving ODE-BVP using Finite Difference Method
- Microsoft Excel based Solution
- Recap of Week-12 (ODE-BVP)