# Numerical Methods and Optimization for SPPU 15 Course (TE - II - Mech. - 302047) (OLD EDITION)

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Unit – I : Roots of Equation and Error Approximations Roots of Equation Bisection Method, Newton Raphson method and Successive approximation method. Error Approximations Types of Errors: Absolute, Relative, Algorithmic, Truncation, Round off Error, Error Propagation, Concept of convergence-relevance to numerical methods. (Chapters - 1, 2) Unit – II : Simultaneous Equations Gauss Elimination Method with Partial pivoting, Gauss-Seidal method and Thomas algorithm for Tri-diagonal Matrix, Jacob iteration method. (Chapter - 3) Unit – III : Optimization Introduction to optimization, Classification, Constrained optimization (maximum two constrains): Graphical and Simplex method, One Dimensional unconstrained optimization: Newton’s Method. Modern Optimization Techniques: Genetic Algorithm (GA), Simulated Annealing (SA). (Chapter - 4) Unit – IV : Numerical Solutions of Differential Equations Ordinary Differential Equations [ODE] Taylor series method, Euler Method, Runge-Kutta fourth order, Simultaneous equations using RungeKutta order method. Partial Differential Equations [PDE] : Finite Difference methods Introduction to finite difference method, Simple Laplace method, PDEs- Parabolic explicit solution, Elliptic-explicit solution. (Chapters - 5 and 6) Unit – V : Curve Fitting and Regression Analysis Curve Fitting Least square technique - Straight line, Power equation, Exponential equation and Quadratic equation. Regression Analysis Introduction to multi regression analysis, Lagrange’s Interpolation, Newton’s Forward interpolation, Inverse interpolation (Lagrange’s method only). (Chapters - 7, 8) Unit – VI : Numerical Integration Numerical Integration (1D only) Trapezoidal rule, Simpson’s 1/3rd Rule, Simpson’s 3/8th Rule, Gauss Quadrature 2 point and 3 point method. Double Integration Trapezoidal rule, Simpson’s 1/3rd Rule. (Chater - 9)

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