Computational Methods

UNIT I

Numerical solution to Linear algebraic & transcendental equations- Numerical algorithms and their complexities, Computer implementation and efficiency, Root finding- bracketing methods: Bracketing Methods, graphical methods, Bisection method, False Position (Regula Falsi), Root finding -Open Methods: Simple Fixed-Point Iteration, Newton-Raphson method, Secant methods, Brent's method

UNIT II

Numerical linear algebra- Gauss elimination, Pivoting, Tridiagonal systems, LU factorization, Gauss elimination as LU factorization, Cholesky factorization, Matrix inverse and condition, Error analysis and system condition. Iterative Methods: Gauss-Seidel method, Nonlinear Systems. Eigenvalues: The Power Method, Interpolations, Lagrange's, piecewise/splines

UNIT III

Numerical Differentiation- High-Accuracy differentiation formulas, Richardson Extrapolation, Derivatives of unequally spaced data, Partial Derivatives. Numerical Integration: Newton- Cotes Formulas, The trapezoidal rule, Simpson’s Rules, Higher-Order Newton-Cotes formulas, Integration with unequal segments, Numerical Integration of Functions, Romberg integration, Gauss quadrature, Adaptive quadrature

UNIT IV:

Ordinary differential equations- Euler's Method, Runge-Kutta Methods, Adaptive methods, finite difference methods, Initial value problems, Boundary value problems, Partial differential equations

Text Books:

1. Erwin Kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons, 2006.
2. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 35th Edition, 2000.

Reference Books:

1. Numerical Methods in Engineering & Science (with Programs in C,C++ & MATLAB), B. S.Grewal, Khanna Publishers.
2. Numerical Methods for Engineers, Steven Chapra, Raymond Canale, McGraw-Hill Higher Education, 2010