Numerical Method
Q21.
Match the following iterative methods for solving algebraic equations and their orders of convergence.\begin{array}{|l|l|l|l|} \hline & \text { Method } & & \text { Order of Convergence } \\ \hline \text { 1. } & \text { Bisection } & \text { P. } & \text { 2 or more } \\ \hline \text { 2. } & \text { Newton-Raphson } & \text { Q. } & 1.62 \\ \hline \text { 3. } & \text { Secant } & \text { R. } & 1 \\ \hline \text { 4. } & \text { Regula falsi } & \text { S. } & 1 \text { bit per iteration } \\ \hline \end{array}Q22.
Which of the following statements applies to the bisection method used for finding roots of functions:Q23.
X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the following is true?Q24.
If f(l) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3) using Lagrange's interpolation formula?Q25.
The Newton-Raphson iteration X_{n+1}=(X_{n}/2)+(3/(2X_{n})) can be used to solve the equationQ26.
The Newton-Raphson method is to be used to find the root of the equation f(x)=0 where x_o is the initial approximation and f' is the derivative of f. The method convergesQ27.
The trapezoidal rule for integration gives exact result when the integrand is a polynomial of degreeQ28.
The trapezoidal method to numerically obtain \int_a^b f(x) dx has an error E bounded by \frac{b-a}{12} h^2 \max f''(x), x \in [a, b] where h is the width of the trapezoids. The minimum number of trapezoids guaranteed to ensure E \leq 10^{-4} in computing \ln 7 using f=\frac{1}{x} isQ29.
Consider the following iterative root finding methods and convergence properties: The correct matching of the methods and properties isQ30.
A piecewise linear function f (x) is plotted using thick solid lines in the figure below (the plot is drawn to scale). If we use the Newton-Raphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be