Numerical Method
Q21.
Given X: 0 10 16 Y: 6 16 28 The interpolated value X=4 using piecewise linear interpolation isQ22.
A root \alpha of equation f(x)=0 can be computed to any degree of accuracy if a 'good' initial approximation x_0 is chosen for whichQ23.
The formula P_k = y_0 + k \triangledown y_0+ \frac{k(k+1)}{2} \triangledown ^2 y_0 + \dots + \frac{k \dots (k+n-1)}{n!} \triangledown ^n y_0 isQ24.
With respect to the numerical evaluation of the definite integral,K=\int_{a}^{b}x^{2}dx , where a and b are given, which of the following statements is/are TRUE? (I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral. (II) The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.Q25.
Function f is known at the following points: The value of \int_{0}^{3}f(x)dx computed using the trapezoidal rule isQ26.
The secant method is used to find the root of an equation f(x)=0. It is started from two distinct estimates x_{a} and x_{b} for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if f(x_{b}) is very small and then x_{b} is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by ?. Which is the suitable expression that is to be put in place of ? so that it follows all steps of the secant method?Q27.
x+y/2=9 3x+y=10 What can be said about the Gauss-Siedel iterative method for solving the above set of linear equations?Q28.
Consider the series x_{n+1}=\frac{x_{n}}{2}+\frac{9}{8x_{n}},x_{0}=0.5 obtained from the Newton-Raphson method. The series converges toQ29.
The trapezoidal method is used to evaluate the numerical value of \int_{0}^{1}e^x dx. Consider the following values for the step size h. i. 10^{-2} ii. 10^{-3} iii. 10^{-4} iv. 10^{-5} For which of these values of the step size h, is the computed value guaranteed to be correct to seven decimal places. Assume that there are no round-off errors in the computation.Q30.
If the trapezoidal method is used to evaluate the integral obtained \int_{0}^{1} x^2dx, then the value obtained