Question

Determine the root of f(x)=sinx+cos(1+x^2)-1 where x is in radians. Use 4 iterations of the secant method with initial guesses of xi-1=1.5 and xi=2.5

Answer 1

Initialization. Two initial guesses $x_0$ and $x_1$ are chosen.

Iteration. For $n=1,2,3,...$

$$x_{n+1}=x_n-f(x_n)\cdot\dfrac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})}$$

Given $f(x)=\sin(x)+\cos(1+x^2)-1$

$x_0=1.5, x_1=2.5$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} Step & x_0 & x_1 & x_2 & f(x_2) \\ \hline 1 & 1.500000 & 2.500000 & 2.356929 & 0.669 842 \\ \hdashline 2 & 2.500000 & 2.356929 & 2.547287 & -0.082828 \\ \hdashline 3 & 2.356929 & 2.547287 & 2.526339 & 0.031471 \\ \hdashline 4 & 2.547287 & 2.526339 & 2.532107 & 0.000570 \\ \hdashline 5 & 2.526339 & 2.532107 & 2.532213 & -0.000004 \end{array}$$

Root is : $x=2.532213$