GATE CSE 2015 | Set 2 | Question: 39

Question

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?

$\textbf{Secant}$

Initialize $x_a, x_b, \varepsilon, N$ // $\varepsilon$ = convergence integer
// N = maximum no. iterations
$f_b=f(x_b)$
i=0
while ( i < N and $|f_b| > \varepsilon)$ do
     i = i+1 // update counter
     $x_t$ =? //missing expression for intermediate value
     $x_a =x_b$ //reset $x_a$
     $x_b = x_t$  // reset $x_b$
     $f_b = f(x_b)$  //function value at new $x_b$
end while
if $|f_b| > \varepsilon$ then // loop is terminated with i=N
     write "Non-convergence"
else
     write "return $x_b$"
end if

• A. $x_b - (f_b-f(x_a)) f_b / (x_b-x_a)$
• B. $x_a - (f_a-f(x_a)) f_a / (x_b-x_a)$
• C. $x_b - (x_b-x_a) f_b / (f_b-f(x_a)) $
• D. $x_a - (x_b-x_a) f_a / (f_b-f(x_a)) $

Answer 1

C. 

Also, $x_b - (x_b-x_a) f_b / (f_b-f(x_a)) $ = $(x_a f_b - x_b f(x_a)) / (f_b-f(x_a)) $

Ref: https://www.math.ohiou.edu/courses/math3600/lecture6.pdf

Comments

Ok @babe :) ,

… wait that sounds weird!

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This is a question from numerical methods and in no possible ways is going to come in the examination. So, do not worry.

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lol

Answer 2

I think it can either c or d according to the algorithm