$1.$ Prove the following:

$($a$)$ If a$,$ b$,$ c$,$ d are machine numbers show that

a $$ b and c $$ d implies fl$($a $+$ c$)$ fl$($b $+$ d$)$

$($b$)$ If x is a machine number then show fl$(2$x$)=2$x$,$ assuming that there is no overflow. The same is also true for the division. $($Hint: consider the binary representation of x$).$

$($c$)$ Construct a counter$-$example to show that the associative property of the addition does not hold for floating point numbers.

$2.$ Implement the following algorithm in Python. In each iteration print x with $15$ decimal digits, and run your code for n $=30.$ In $$exact$($infinite precision$)$ arithmetic the result should always be x$.$ Try your code with the following values m $=10,9,8.$ Can you explain the complete loss of accuracy after few iterations for m $=10,9?$ Why the last choice m $=8$ is not suffering from the same catastrophic behaviour? $($Recall that in Python the standard representation of numbers is with double precision$)$

Choose m and n

Set x :$=$ m$+1$ m

for k $=1,...,$ n do

print$($x$)$

x :$=($x $1)($m $+1)$

end

$3.$ Consider the positive solution of the equation $^2=$ $+1,$ which is the golden ratio $=(1+5)/2.$ To approximate $$ we consider the following three, mathematically equivalent, iterative algorithms

Algr $1$: $0=2,$ k$+1=$ sqrt$($ k $+1),$ k $=1,...$

Algr $2$: $0=2,$ k$+1=1+1/$k $,$ k $=1,...$

Algr $3$: $0=2,$ k$+1=$ k$^21,$ k $=1,...1$

Implement each algorithm in Python and compute the absolute and relative errors Ek $=|$k $$ $|,$ Rk $=$ Ek$/|$$|$ respectively, for k $=1,....$ Derive for each algorithm an approximate formula for the ratio Ek$+1/$Ek and verify your findings numerically. Which of the algorithms are appropriate, in terms of stability, for computing $?$

$4.$ The Bessel functions of order $0$ satisfy the recurrence relation

Jn$1($x$)+$ Jn$+1($x$)=(2$n $/$x $)$ Jn$($x$),$ n $=1,2,....$

Assume that we want to compute the sequence J$2(1),$ J$3(1),...,$ J$8(1),$ given the values of J$0(1)=0\mathrm{.}765198$ and J$1(1)=0\mathrm{.}440051.$ Compute these values with a Python program which uses the recurrence above and compare with the exact values computed by the Python function jv in scipy.special. Are the values computed by the recurrence formula accurate? What are the errors? What do you think is to blame?

$5.$ A loan of A euros is paid back in n equal monthly instalments of M euros each. If r is the monthly interest rate then Ar $=$ M $[11/(1+$ r$)^$n$].$ Take A $=10000,$ M $=250,$ n $=60$ and propose a method for finding the interest rate r$.$