Quantitative Methods Answers

using finite difference, show that the data f(-3)=13, f(-2)=7,f(-1)=3,f(0)=1,f(1)=1,f(2)=3,f(3)=7. represents a second degree polynomial. obtain this polynomial using interpolation and find f(2.5).

a table of values is to be constructed for the function f(x) =1 /1+x in the interval [1,4] with equal step length. determine the spacing h such that quadratic interpolation gives result with accuracy 1×10 - 6

use modified euler 's method to find the approximate solution of IVP y' =2xy, y(1)=1, at x=1.5 with h=0.1. if the exact solution is y(x) =ex2-1, find the error.

For h=0.5,. 25,.125 solve integration from 0 to 1, dx/1+x2 and improve the accuracy by romberg integration. Compare your value with the real value.

derive a suitable numerical differentiation formula of 0(h2) to find f''(2. 4) with h =0.1 given the table f(0.1)=3.41, f(1.2) =2.68, f(2.4)=1. 37, f(3.9) =-1. 48.

the position of f(x) of a particle moving in a line at various times xk is given in the following table. estimate the velocity and acceleration of the particle at x =1.2. f(1.0)=2.72, f(1.2)=3.32, f(1.4)=4.06, f(1.6)=4.96, f(1.8)=6.05, f(2. 0)=7.39, f(2.2)=9.02.

determine the constants a, b, c in the differentiation formula y '' (x0) =ay(x0-h) +by(x0) +cy(x0 +h). so that the method of the highest possible order and the error term of the method.

using the classical R-K method of O(h4) calculate approximate solution of the IVP y'=1-x+4y, y(0) =1 at x=0.6, taking h=0.1 and 0.2. use extrapolation technique to improve the accuracy.

for the linear system of equations [1 2 - 2,1 1 1, 2 2 1][x y z] =[1 3 5] set up the gauss - jacobi and gauss - seidel iteration schemes in matrix form. also check the convergence of the two schemes.

find the dominant eigenvalue and the corresponding eigenvector for the matrix A =[-4 14 0, - 5 13 0, - 1 0 2]using five itrations of the power method and taking y(0)=[1 1 1] as the initial vector.