Floating - point representation and errors-Normalized floating-point forms, Errors in representing numbers, Floating point machine number and machine epsilon, Loss of significance and its avoidance (Chapter 2 in Cheney and Kincaid).
Roots of equations - locating roots of f(x)=0 Bisection method and convergence analysis, Newton’s method and convergence analysis, failure of Newton’s method due to bad starting points, modification of Newton’s method for multiple roots, Newton’s method for System of Non-linear equations, Secant method and convergence analysis, Golden ratio.
Interpolation and numerical differentiation - polynomial interpolation and its existence Lagrange and Newton form of interpolating Polynomial, Divided difference and recursive property, Inverse interpolation, Error in Polynomial interpolation, First and Second derivative formulae via interpolation Polynomials.
Numerical integration - Trapezoidal, Simpson’s and adaptive Simpson rules and Error analysis.
System of linear equations - Gausssian elimination and back substitution –partial and complete pivoting, Tridiagonal and pentadiagonal banded systems, Thomas algorithm, Doolittle, Cholesky and Crout LU decomposition methods, Jacobi and Gauss – Seidel iterative methods and convergence theorems. Power (and inverse power) method of obtaining largest (smallest) eigenvalue and corresponding
eigenvector.
Ordinary differential equations - initial value problem, Picard’s, Taylor series, Runge-Kutta first, second and fourth order methods, adaptive Runge-Kutta method of fifth order (derivation of only Runge-Kutta first and second order methods), boundary value problems-shooting methods for linear differential equations.
Linear programming - first Primal form, Graphical solution method, Transforming problems into first primal form, dual problem, Theorem on primal and dual problems, Second Primal form. Simplex method, Approximate solution of inconsistent linear systems.
NUMERICAL ANALYSIS and LINEAR PROGRAMMING - Lab Programs
Roots of equations - locating roots of f(x)=0 Bisection method and convergence analysis, Newton’s method and convergence analysis, failure of Newton’s method due to bad starting points, modification of Newton’s method for multiple roots, Newton’s method for System of Non-linear equations, Secant method and convergence analysis, Golden ratio.
Interpolation and numerical differentiation - polynomial interpolation and its existence Lagrange and Newton form of interpolating Polynomial, Divided difference and recursive property, Inverse interpolation, Error in Polynomial interpolation, First and Second derivative formulae via interpolation Polynomials.
Numerical integration - Trapezoidal, Simpson’s and adaptive Simpson rules and Error analysis.
System of linear equations - Gausssian elimination and back substitution –partial and complete pivoting, Tridiagonal and pentadiagonal banded systems, Thomas algorithm, Doolittle, Cholesky and Crout LU decomposition methods, Jacobi and Gauss – Seidel iterative methods and convergence theorems. Power (and inverse power) method of obtaining largest (smallest) eigenvalue and corresponding
eigenvector.
Ordinary differential equations - initial value problem, Picard’s, Taylor series, Runge-Kutta first, second and fourth order methods, adaptive Runge-Kutta method of fifth order (derivation of only Runge-Kutta first and second order methods), boundary value problems-shooting methods for linear differential equations.
Linear programming - first Primal form, Graphical solution method, Transforming problems into first primal form, dual problem, Theorem on primal and dual problems, Second Primal form. Simplex method, Approximate solution of inconsistent linear systems.
NUMERICAL ANALYSIS and LINEAR PROGRAMMING - Lab Programs
- Write a program to find the roots of an equation f (x) = 0 using Bisection method.
- Write a program to find the simple/multiple roots of f (x) = 0 using Newton – Raphson method.
- Write a program to find the roots of f(x) = 0 using Secant method.
- Write a program to find the integral of a function using Trapezoidal rule.
- Write a program to find the integral of a function using Simpson’s 1/3rd and 3/8th rule using switch case.
- Write a program to find the integral of a function using adaptive Simpson method
- Write a program to solve the system of equations Ax = b in tridiagonal form using Thomas Algorithm.
- Write a program to solve the system of equations Ax = b using Gauss elimination method.
- Write a program to solve the system of equations Ax = b using Jacobi Iteration method.
- Write a program to solve the system of equations Ax = b using Gauss-Seidel method
- Write a program to find the largest (or smallest) Eigen value and corresponding eigen vector of a square matrix using power (or inverse power) method.
- Write a program to solve first and second order ordinary differential equations (initial value problem) using Runge-Kutta fourth order method.
- Write a program to solve first order ordinary differential equations (initial value problem) using adaptive Runge-Kutta method .
- Write a program to solve second order ordinary differential equations (boundary value problem) using shooting method based on adaptive Runge-Kutta method and Newton-Raphson method.
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