Numerical Method
Q1.
The Guass-Seidal iterative method can be used to solve which of the following sets?Q2.
What is the sum to infinity of the series,3+6 x^{2}+9 x^{4}+12 x^{6}+\ldots \text { given }|x| \lt 1Q3.
Consider the function f(x) = x^{2} - 2x - 1. Suppose an execution of the Newton-Raphson method to find a zero of f(x) starts with an approximation x_{0} = 2 of x. What is the value of x_{2}, the approximation of x that algorithm produces after two iterations, rounded to three decimal places?Q4.
The velocity v (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time t (in minutes) as follows: The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson's 1/3^{rd} rule is _______________.Q5.
The bisection method is applied to compute a zero of the function f(x) = x^{4}-x^{3}-x^{2}-4 in the interval [1,9]. The method converges to a solution after _____iterations.Q6.
The Newton-Raphson iteration x_{n+1}=\frac{1}{2}(x_{n}+\frac{R}{x_{n}}) can be used to compute theQ7.
Newton-Raphson method is used to compute a root of the equation x^{2} -13=0 with 3.5 as the initial value. The approximation after one iteration isQ9.
In the Newton-Raphson method, an initial guess of x_{0}=2 is made and the sequence x_{0},x_{1},x_{2}... is obtained for the function 0.75x^{3}-2x^{2}-2x+4=0 Consider the statements (I) x_{3}=0. (II) The method converges to a solution in a finite number of iterations. Which of the following is TRUE?Q10.
Consider the polynomial p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 , where a_i \neq 0, \forall i. The minimum number of multiplications needed to evaluate p on an input x is: