A hybrid root$-$finding method. Write a python code implementing the following "hybrid" method to evaluate numerically roots of a function f$($x$)$

:

Set a tolerance $\backslash$epsi

and start with an initial interval $[$x$0,$x$1]$

that contains a root of f$($x$)$

$($how to grant the presence of a root in the interval?$)$;

Apply a step of the secant method to f$($x$)$

using as points x$0$

and x$1$

and denote the number obtained by c

$.$

If c

is inside $[$x$0,$x$1]$

$,$ set x$2=$c

; otherwise set x$2=($x$0+$x$1)/2$

$.$

Repeat the steps above until the tolerance is reached.

Test the method evaluating the root of x$2=2$

with an accuracy $\backslash$epsi $=1013$

$.$

Comparing methods. Find the root of sinx$=0\mathrm{.}5625$

with an accuracy $\backslash$epsi $=1012$

using the following methods: bisection, secant, Newton and the hybrid method above. In case of the bisection and hybrid methods, start from the interval $[0,\backslash$pi $/2]$

$.$ In case of the secant and bisection methods, start from the point $0$

$.$ For each method, print the number of steps that were needed to get the desired precision.

Conditioning. Find the root of the polynomial p$($x$)=$x$58$x$4+25\mathrm{.}6$x$340\mathrm{.}96$x$2+32\mathrm{.}768$x$10\mathrm{.}48576$

between $0$ and $3$ using Newton's method. Assume that, in the range $[0,3]$

$,$ p$($x$)$

is known with an accuracy of $\backslash$Delta p$1012$

and evaluate the absolute error on the numerical value of the root due to the conditioning of p$($x$)$

at that value.

Remark: feel free to re$-$use any code or part of code from the textbook.A hybrid root$-$finding method. Write a python code implementing the following "hybrid"

method to evaluate numerically roots of a function $f(x)$ :

Set a tolerance $$ and start with an initial interval $x_0,x_1$ that contains a root of $f(x)($how

to grant the presence of a root in the interval?$)$;

Apply a step of the secant method to $f(x)$ using as points $x_0$ and $x_1$ and denote the

number obtained by $c.$

If $c$ is inside $x_0,x_1,$ set $x_2=c$; otherwise set $x_2=\frac{x_0+x_1}{2}.$

Repeat the steps above until the tolerance is reached.

Test the method evaluating the root of $x^2=2$ with an accuracy $={10}^{-13}.$

Comparing methods. Find the root of $sinx=0\mathrm{.}5625$ with an accuracy $={10}^{-12}$ using the

following methods: bisection, secant, Newton and the hybrid method above. In case of the

bisection and hybrid methods, start from the interval $0,\frac{}{2}.$ In case of the secant and

bisection methods, start from the point $0.$ For each method, print the number of steps that were

needed to get the desired precision.

Conditioning. Find the root of the polynomial

$p(x)=x^5-8x^4+25\mathrm{.}6x^3-40\mathrm{.}96x^2+32\mathrm{.}768x-10\mathrm{.}48576$ between $0$ and $3$ using Newton's

method. Assume that, in the range $[0,3],p(x)$ is known with an accuracy of $p{10}^{-12}$ and

evaluate the absolute error on the numerical value of the root due to the conditioning of $p(x)$ at

that value.