Syllabus of Numerical Methods

Unit 1: Introduction to Numerical Methods

• Introduction to Numerical Methods

  • Definition and importance of numerical methods.
  • Real-world applications of numerical methods in engineering, science, and finance.
  • Error analysis: types of errors (absolute, relative), propagation of errors.

• Basic Concepts

  • Floating-point arithmetic and round-off errors.
  • Stability and convergence of numerical algorithms.
  • Precision and accuracy in numerical computations.

Unit 2: Solution of Nonlinear Equations

• Bisection Method

  • Concept and procedure.
  • Convergence analysis.
  • Advantages and limitations.

• Newton-Raphson Method

  • Derivation and procedure.
  • Convergence criteria.
  • Application to real-world problems.

• Secant Method

  • Concept and comparison with Newton-Raphson.
  • Implementation and convergence.
  • Advantages and disadvantages.

• Fixed-Point Iteration

  • Methodology and geometric interpretation.
  • Convergence conditions.
  • Practical applications.

Unit 3: Interpolation and Polynomial Approximation

• Introduction to Interpolation

  • Definition and need for interpolation.
  • Interpolation vs. extrapolation.

• Lagrange Interpolation

  • Concept and formulation.
  • Construction of Lagrange polynomials.
  • Error in interpolation.

• Newton’s Divided Difference Interpolation

  • Divided difference table.
  • Newton’s forward and backward interpolation formulas.
  • Application to evenly and unevenly spaced data.

• Spline Interpolation

  • Cubic splines: definition and properties.
  • Construction of spline curves.
  • Comparison with polynomial interpolation.

Unit 4: Numerical Differentiation and Integration

• Numerical Differentiation

  • Finite difference approximations.
  • Forward, backward, and central difference formulas.
  • Error analysis in numerical differentiation.

• Numerical Integration

  • Trapezoidal rule: derivation and error analysis.
  • Simpson’s 1/3 and 3/8 rules: derivation and error analysis.
  • Application of numerical integration to area and volume calculations.

• Romberg Integration

  • Concept of Romberg integration.
  • Implementation and error reduction.
  • Comparison with other integration methods.

Unit 5: Solution of Systems of Linear Equations

• Direct Methods

  • Gaussian Elimination: procedure and pivoting strategies.
  • LU Decomposition: method and applications.
  • Matrix inversion using LU decomposition.

• Iterative Methods

  • Jacobi Method: concept and convergence criteria.
  • Gauss-Seidel Method: implementation and comparison with Jacobi method.
  • Successive Over-Relaxation (SOR): acceleration of convergence.

• Error Analysis

  • Condition number and its significance.
  • Numerical stability of solutions.
  • Practical applications of solving linear systems.

Unit 6: Eigenvalues and Eigenvectors

• Power Method

  • Concept and implementation.
  • Convergence and practical considerations.
  • Application to finding the largest eigenvalue and corresponding eigenvector.

• Jacobi Method for Eigenvalues

  • Procedure for symmetric matrices.
  • Convergence analysis.
  • Advantages and limitations.

• QR Algorithm

  • Concept and steps of the QR algorithm.
  • Application to find all eigenvalues of a matrix.
  • Error analysis and efficiency considerations.

Unit 7: Numerical Solutions of Ordinary Differential Equations (ODEs)

• Initial Value Problems

  • Euler’s method: derivation and error analysis.
  • Improved Euler’s method and Heun’s method.
  • Runge-Kutta methods: derivation and applications (2nd, 3rd, 4th order).

• Boundary Value Problems

  • Finite difference method for BVPs.
  • Shooting method: concept and implementation.
  • Application to physical and engineering problems.

Unit 8: Numerical Solutions of Partial Differential Equations (PDEs)

• Classification of PDEs

  • Parabolic, hyperbolic, and elliptic PDEs.
  • Examples and physical significance.

• Finite Difference Methods for PDEs

  • Explicit and implicit methods for solving PDEs.
  • Crank-Nicolson method for parabolic PDEs.
  • Stability and convergence analysis.

• Numerical Methods for Elliptic PDEs

  • Solution of Laplace and Poisson equations.
  • Iterative methods: Jacobi, Gauss-Seidel, SOR.
  • Application to heat conduction, fluid flow, and electrostatics problems.

Unit 9: Curve Fitting and Approximation

• Least Squares Method

  • Fitting linear and nonlinear curves.
  • Error minimization in curve fitting.
  • Application to data analysis and regression.

• Polynomial Regression

  • Fitting polynomial curves to data.
  • Determination of polynomial coefficients.
  • Comparison with linear regression.

• Approximation Theory

  • Chebyshev polynomials.
  • Minimax approximation.
  • Application to function approximation.

Unit 10: Numerical Optimization

• Unconstrained Optimization

  • Gradient descent method: concept and implementation.
  • Newton’s method for optimization.
  • Convergence analysis and practical applications.

• Constrained Optimization

  • Lagrange multipliers and the Karush-Kuhn-Tucker (KKT) conditions.
  • Simplex method for linear programming.
  • Application to resource allocation and optimization problems in engineering and economics.