Back to Courses

Numerical Methods

Comprehensive study of numerical techniques for solving mathematical problems using computational approaches

14 Sessions
0% Complete
Course Progress0%
14
Total Sessions
42
Materials
28 hrs
Estimated Time

Course Sessions

01
Introduction to Numerical Analysis and Error Analysis
Fundamental concepts of numerical analysis including sources of errors, floating-point arithmetic, machine epsilon, and error propagation. Introduction to Taylor series and its role in numerical methods.
3 hours
7 Materials

Learning Objectives:

  • Understand the fundamental concepts and scope of numerical analysis
  • Analyze different types of errors: round-off, truncation, and discretization errors
  • Master IEEE 754 floating-point representation and machine epsilon calculations
  • Apply Taylor series expansion for error analysis and algorithm development
  • Compute absolute and relative errors, and understand error propagation
  • Evaluate condition numbers and numerical stability concepts

Available Materials:

Complete Lecture Notes (45 pages)
IEEE 754 Standard Reference Guide
Error Analysis Workbook with 25 Problems
MATLAB/Python Code Examples
Interactive Error Visualization Tools
Problem Set 1 (10 problems)
Solution Manual Chapter 1
02
Root Finding Methods: Bracketing Techniques
Comprehensive study of bracketing methods for finding zeros of nonlinear equations including bisection method, false position method, and their convergence analysis.
3 hours
7 Materials

Learning Objectives:

  • Implement and analyze the bisection method with convergence proofs
  • Master the false position (regula falsi) method and its variants
  • Understand bracketing principles and intermediate value theorem applications
  • Analyze convergence rates and stopping criteria for bracketing methods
  • Handle multiple roots and numerical challenges in root finding
  • Compare bracketing methods in terms of robustness and efficiency

Available Materials:

Detailed Method Derivations (30 pages)
Convergence Analysis Proofs
Algorithm Flowcharts and Pseudocode
Programming Assignments (Python/MATLAB)
Test Functions Library (20 examples)
Comparative Analysis Report Template
Video Demonstrations of Algorithm Execution
03
Root Finding Methods: Open Methods
Advanced root finding techniques including Newton-Raphson method, secant method, and modified Newton methods. Analysis of convergence, multiple roots, and computational complexity.
3.5 hours
7 Materials

Learning Objectives:

  • Derive and implement Newton-Raphson method with geometric interpretation
  • Master the secant method and understand its superlinear convergence
  • Handle multiple roots using modified Newton methods
  • Analyze quadratic convergence of Newton method and linear convergence of secant method
  • Implement deflation techniques for polynomial root finding
  • Compare open methods with bracketing methods for various function types

Available Materials:

Mathematical Derivations and Proofs (35 pages)
Newton Method Convergence Analysis
Secant Method Implementation Guide
Modified Newton Methods for Multiple Roots
Comparative Performance Study
Programming Project: Complete Root Finder Package
Real-world Application Examples
04
Direct Methods for Linear Systems: Gaussian Elimination
Comprehensive treatment of direct methods for solving linear systems including Gaussian elimination with partial and complete pivoting, LU decomposition, and computational complexity analysis.
4 hours
8 Materials

Learning Objectives:

  • Master Gaussian elimination with partial pivoting for numerical stability
  • Understand and implement complete pivoting strategies
  • Derive and implement LU decomposition using Doolittle and Crout methods
  • Analyze computational complexity: O(n³) operations for elimination
  • Handle special matrix structures: tridiagonal, symmetric, and banded matrices
  • Understand forward and backward substitution algorithms
  • Evaluate numerical stability and condition number effects

Available Materials:

Comprehensive Linear Algebra Review (40 pages)
Pivoting Strategy Analysis and Implementation
LU Decomposition Algorithms with Examples
Computational Complexity Analysis
Special Matrix Solvers (Tridiagonal, Symmetric)
Numerical Stability Case Studies
Programming Assignment: Linear System Solver Package
Performance Benchmarking Tools
05
Iterative Methods for Linear Systems
Detailed study of iterative methods including Jacobi, Gauss-Seidel, and SOR methods. Convergence analysis, spectral radius, and preconditioning techniques for large sparse systems.
3.5 hours
8 Materials

Learning Objectives:

  • Implement Jacobi and Gauss-Seidel methods with matrix splitting theory
  • Master Successive Over-Relaxation (SOR) method and optimal relaxation parameter
  • Analyze convergence conditions using spectral radius and diagonal dominance
  • Apply iterative methods to large sparse systems from engineering applications
  • Understand preconditioning techniques for accelerating convergence
  • Compare direct vs iterative methods for different problem types and sizes

Available Materials:

Matrix Splitting Theory and Convergence Proofs (38 pages)
SOR Method Optimization Analysis
Sparse Matrix Applications in Engineering
Preconditioning Techniques Guide
Convergence Criteria and Stopping Rules
Large-Scale Problem Examples
Performance Comparison Studies
Programming Project: Iterative Solver Suite
06
Polynomial Interpolation and Approximation
Comprehensive study of polynomial interpolation including Lagrange interpolation, Newton divided differences, Hermite interpolation, and error analysis. Chebyshev polynomials and optimal approximation.
4 hours
8 Materials

Learning Objectives:

  • Master Lagrange interpolation formula and its computational implementation
  • Understand Newton divided differences and forward/backward difference formulas
  • Implement Hermite interpolation for functions with derivative information
  • Analyze interpolation error using Weierstrass approximation theorem
  • Apply Chebyshev polynomials for optimal node placement and minimax approximation
  • Understand Runge phenomenon and strategies to avoid oscillations

Available Materials:

Complete Interpolation Theory (50 pages)
Lagrange and Newton Method Implementations
Hermite Interpolation with Applications
Chebyshev Polynomials and Optimal Approximation
Error Analysis and Runge Phenomenon Study
Polynomial Approximation Project
Interactive Interpolation Visualization Tools
Real Data Approximation Examples
07
Spline Interpolation and Curve Fitting
Advanced interpolation techniques using cubic splines, B-splines, and least squares curve fitting. Natural and clamped boundary conditions, spline properties, and applications in computer graphics.
3.5 hours
8 Materials

Learning Objectives:

  • Derive and implement natural cubic spline interpolation
  • Understand clamped and not-a-knot boundary conditions for splines
  • Master the construction of cubic spline systems and their solution
  • Apply B-spline basis functions for flexible curve representation
  • Implement least squares polynomial fitting for overdetermined systems
  • Compare spline interpolation with polynomial interpolation advantages

Available Materials:

Spline Theory and Mathematical Foundations (42 pages)
Cubic Spline Implementation Algorithms
Boundary Condition Analysis and Applications
B-spline Basis Functions and Properties
Least Squares Curve Fitting Methods
Computer Graphics Applications
Spline Construction Programming Project
Comparative Analysis of Interpolation Methods
08
Numerical Differentiation and Richardson Extrapolation
Finite difference approximations for derivatives, error analysis, and Richardson extrapolation for improved accuracy. Forward, backward, and central difference formulas with applications.
3 hours
8 Materials

Learning Objectives:

  • Derive finite difference formulas using Taylor series expansion
  • Implement forward, backward, and central difference approximations
  • Analyze truncation errors and optimal step size selection
  • Master Richardson extrapolation for higher-order accuracy
  • Apply numerical differentiation to partial differential equations
  • Understand the trade-off between truncation and round-off errors

Available Materials:

Finite Difference Theory and Derivations (35 pages)
Error Analysis and Optimal Step Size Studies
Richardson Extrapolation Implementation
Partial Derivative Approximation Methods
Step Size Selection Guidelines
PDE Application Examples
Programming Assignment: Derivative Calculator
Accuracy vs Efficiency Trade-off Analysis
09
Numerical Integration: Newton-Cotes Formulas
Comprehensive study of numerical quadrature including trapezoidal rule, Simpson's rules, and higher-order Newton-Cotes formulas. Composite rules, error analysis, and adaptive integration.
4 hours
8 Materials

Learning Objectives:

  • Derive trapezoidal and Simpson's rules using polynomial interpolation
  • Implement composite trapezoidal and Simpson's rules for improved accuracy
  • Understand Newton-Cotes formulas of various orders and their stability
  • Analyze quadrature errors and convergence rates
  • Apply Romberg integration for systematic error reduction
  • Implement adaptive quadrature methods with automatic error control

Available Materials:

Quadrature Theory and Newton-Cotes Derivations (45 pages)
Composite Rule Implementations and Error Analysis
Romberg Integration Algorithm and Examples
Adaptive Integration Strategies
Stability Analysis of High-Order Formulas
Integration Project: Comprehensive Quadrature Package
Performance Comparison Studies
Engineering Application Examples
10
Gaussian Quadrature and Special Integration Methods
Advanced integration techniques including Gauss-Legendre, Gauss-Laguerre, and Gauss-Hermite quadrature. Orthogonal polynomials, weight functions, and applications to improper integrals.
3.5 hours
8 Materials

Learning Objectives:

  • Understand orthogonal polynomial theory and Gaussian quadrature principles
  • Implement Gauss-Legendre quadrature for standard intervals
  • Apply Gauss-Laguerre quadrature for semi-infinite intervals
  • Master Gauss-Hermite quadrature for infinite intervals with exponential weights
  • Transform integrals to utilize appropriate Gaussian quadrature methods
  • Compare Gaussian quadrature efficiency with Newton-Cotes methods

Available Materials:

Orthogonal Polynomial Theory (40 pages)
Gaussian Quadrature Node and Weight Tables
Gauss-Legendre Implementation Guide
Special Weight Function Applications
Integral Transformation Techniques
High-Precision Integration Examples
Programming Project: Gaussian Quadrature Suite
Comparison with Monte Carlo Methods
11
Ordinary Differential Equations: Initial Value Problems
Comprehensive treatment of numerical methods for solving initial value problems including Euler methods, Runge-Kutta methods, multistep methods, and stability analysis.
4.5 hours
8 Materials

Learning Objectives:

  • Implement Euler method and improved Euler method with error analysis
  • Master Runge-Kutta methods of orders 2, 3, and 4 with derivations
  • Understand adaptive step size control and embedded RK methods
  • Analyze stability regions and stiff differential equations
  • Implement multistep methods: Adams-Bashforth and Adams-Moulton
  • Apply predictor-corrector methods for enhanced accuracy and stability

Available Materials:

ODE Theory and Numerical Method Derivations (55 pages)
Runge-Kutta Method Family Implementation
Adaptive Step Size Control Algorithms
Stability Analysis and Stiff Equation Treatment
Multistep Method Implementations
Comprehensive ODE Solver Project
Engineering and Physics Application Examples
Performance and Accuracy Comparison Studies
12
Boundary Value Problems and Shooting Methods
Numerical methods for boundary value problems including shooting method, finite difference methods, and collocation techniques. Linear and nonlinear boundary conditions.
3.5 hours
8 Materials

Learning Objectives:

  • Understand boundary value problem formulation and solution approaches
  • Implement shooting method for linear and nonlinear boundary value problems
  • Master finite difference methods for BVP discretization
  • Apply collocation methods using orthogonal polynomials
  • Handle various boundary condition types: Dirichlet, Neumann, and mixed
  • Analyze convergence and stability of boundary value problem methods

Available Materials:

Boundary Value Problem Theory (38 pages)
Shooting Method Implementation and Convergence
Finite Difference BVP Schemes
Collocation Method Applications
Boundary Condition Implementation Guide
Linear and Nonlinear BVP Examples
Programming Assignment: BVP Solver Package
Engineering Applications in Heat Transfer and Mechanics
13
Eigenvalue Problems and Matrix Computations
Numerical methods for computing eigenvalues and eigenvectors including power method, QR algorithm, and Jacobi method. Applications to vibration analysis and principal component analysis.
4 hours
8 Materials

Learning Objectives:

  • Implement power method and inverse power method for dominant eigenvalues
  • Understand deflation techniques for multiple eigenvalue computation
  • Master QR algorithm with shifts for complete eigenvalue computation
  • Apply Jacobi method for symmetric matrix eigenproblems
  • Understand Gershgorin circle theorem for eigenvalue localization
  • Apply eigenvalue methods to engineering problems and data analysis

Available Materials:

Eigenvalue Theory and Matrix Analysis (48 pages)
Power Method Implementation and Convergence Analysis
QR Algorithm with Shift Strategies
Jacobi Method for Symmetric Matrices
Gershgorin Circles and Eigenvalue Bounds
Vibration Analysis Application Examples
Principal Component Analysis Implementation
Comprehensive Eigenvalue Solver Project
14
Partial Differential Equations: Finite Difference Methods
Numerical solution of partial differential equations using finite difference methods. Elliptic, parabolic, and hyperbolic PDEs with stability and convergence analysis.
5 hours
8 Materials

Learning Objectives:

  • Discretize elliptic PDEs using finite difference schemes
  • Implement explicit and implicit methods for parabolic PDEs
  • Master stability analysis using von Neumann method
  • Apply finite difference methods to hyperbolic PDEs
  • Understand CFL condition and numerical stability requirements
  • Solve heat equation, wave equation, and Laplace equation numerically

Available Materials:

PDE Theory and Classification (50 pages)
Finite Difference Scheme Derivations
Stability Analysis Methods and Criteria
Heat Equation Implementation (Explicit/Implicit)
Wave Equation Finite Difference Solutions
Elliptic PDE Solver Development
CFL Condition Analysis and Applications
Engineering PDE Applications Project