Programming Exercise $1$

In this programming exercise we will use PYTHON to implement the root finding algorithms taught in class.

Write a PYTHON program to implement the false position method to find the roots of equation

$f(x)=2xcos(2x)-(x-2{)}^2,$

accurate up to four significant digits.

a$)$ First, define a PYTHON function in the following manner:

def falsepos$($func$,$ a$,$ b$,$ es$)$:

where func represents $f(x),a,b$ the bracket and es is the stopping criteria. Thereafter, call this function

for different brackets to find the roots. Report their values.

b$)$ Make a plot of $f(x)$ vs$.x$ and identify these roots via markers.

c$)$ Verify your solutions using the in$-$built PYTHON function brentq for Brent's method. Note that you need

to import scipy to access this function.

Now write a PYTHON function to implement the secant method:

def falsepos$($func$,$ x$0,$ x$1,$ es$)$:

where $x0,x1$ are the two starting points.

a$)$ Use the function to find roots of $f(x)=x^4+65x^3+86x+128,$ accurate up to $4$ significant digits. Report

their values.

b$)$ Again make a plot of the above function to and mark all the roots.

c$)$ Verify your solutions using the in$-$built PYTHON function roots, which is available in numpy.

Utilizing the above functions, find the shortest distance between the curve $y=x^2+1$ and the point $(1,0).$

Report the shortest distance. Which method converges faster, the false position or the secant method? $($report

the number of iterations in both cases$).$ In this programming exercise we will use PYTHON to implement the root finding algorithms taught in class.

Write a PYTHON program to implement the false position method to find the roots of equation

$f(x)=2xcos(2x)-(x-2{)}^2,$

accurate up to four significant digits.

a$)$ First, define a PYTHON function in the following manner:

def falsepos$($func$,$ a$,$ b$,$ es$)$:

where func represents $f(x),a,b$ the bracket and $es$ is the stopping criteria. Thereafter, call this function

for different brackets to find the roots. Report their values.

b$)$ Make a plot of $f(x)$ vs$.x$ and identify these roots via markers.

c$)$ Verify your solutions using the in$-$built PYTHON function brentq for Brent's method. Note that you need

to import scipy to access this function.

Now write a PYTHON function to implement the secant method:

def falsepos$($func$,$ x$0,$ x$1,$ es$)$:

where $x0,x1$ are the two starting points.

a$)$ Use the function to find roots of $f(x)=x^4+65x^3+86x+128,$ accurate up to $4$ significant digits. Report

their values.

b$)$ Again make a plot of the above function to and mark all the roots.

c$)$ Verify your solutions using the in$-$built PYTHON function roots, which is available in numpy.

Utilizing the above functions, find the shortest distance between the curve $y=x^2+1$ and the point $(1,0).$

Report the shortest distance. Which method converges faster, the false position or the secant method $?($report

the number of iterations in both cases$).$