GATE CS 2015 Set 3 Q50 — Numerical Methods (Historical)

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 …

In the Newton-Raphson method, an initial guess of $x_0= 2$ is made and the sequence $x_0,x_1,x_2\:\dots$ is obtained for the function $$0.75x^3-2x^2-2x+4=0$$ Consider the statements $x_3\:=\:0$ The method converges to a solution in a finite number of iterations. Which of the following is TRUE?

With respect to the numerical evaluation of the definite integral, $K = \int \limits_a^b \:x^2 \:dx$, where $a$ and $b$ are given, which of the following statements is/are TRUE? The value of $K$ obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral. The value of $K$ obtained using the …

Function $f$ is known at the following points: $x$ 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 $f(x)$ 0 0.09 0.36 0.81 1.44 2.25 3.24 4.41 5.76 7.29 9.00 The value of $\int_{0}^{3} f(x) \text{d}x$ computed using the trapezoidal rule is

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.

The Newton-Raphson iteration $x_{n+1} = \frac{1}{2}\left(x_n+\frac{R}{x_n}\right)$ can be used to compute the