Numerical Methods Lecture Notes For Engineering And Science Students

NUMERICAL-ANALYSISConcepts, Methods, and ApplicationsGATE/NET-JRFRa j kumar sharmaM.sc in mathematics and scientific ComputingNational Institute of Technology ,Manipur Prepared for GATE / NET Mathematics AspirantsContents / TopicsSystems of Linear Equations: Direct methods (Gaussian elimination, LU decomposition,Cholesky factorization), Iterative methods (Jacobi, Gauss-Seidel), Convergence for diagonally dom-inant matrices Numerical Solutions of Nonlinear Equations: Bisection method, Secant method, Newton-Raphson method, Fixed point iteration Numerical Solution of Initial Value Problems: Euler method, Modified Euler method,Runge-Kutta method of order 2 Interpolating polynomail Problems: ; Interpolation: Lagrange and Newton forms of interpo-lating polynomial, Error in polynomial interpolation of a function; Numerical differentiation anderror, Numerical integration: Trapezoidal and Simpson rules, Newton-Cotes integration formulas,composite rules, mathematical errors involved in numerical integration formulaeSystems of Linear Equations: Direct MethodsWe consider solving a system of linear equations of the form: Ax = b,where ARn n, x Rn, b Rn.Gaussian EliminationGaussian elimination reduces the matrix Ato anupper triangular matrix Uusingelementary rowoperations. Then, the solution is obtained by back-substitution:Ux = c.Gaussian Elimination MethodDefinition:Gaussian Elimination is a systematic method for solving a system of linear equations. It reduces thecoefficient matrix Ato an upper triangular matrix Uusing elementary row operations , andthen the solution is obtained by back-substitutionfromU x=c, where cis the transformed right-handside vector.Method / Steps:1.