Problem $3.$ Iterations and Convergence

$($a$)$ Padovan Sequence

Like the Fibonacci sequence we learned in the class, the Padovan sequence is a sequence of integers $P(n)$ defined by the initial values

$P(1)=P(2)=P(3)=1,$

and the recurrence relation

$P(j)=P(j-2)+P(j-3).$

We are to calculate the first $100$ Padovan numbers. To do so$,$ initialize a $1001$ column vector $p$ using matrix. The first three entities are the initial values defined above. Use a for loop to run $j$ from $4$ to $100,$ calculate $P(j)$ and store them in $p[j].$

Save $p$ in $A15.$ dat using R$\text{'}$s command

write.table$(...,$ col.names$=$FALSE, row.names$=$FALSE$).$

$($b$)$ Iterative solution for $\sqrt[\phantom{2}]{2}$

In Classwork $5,$ we discussed the Newton$-$Raphson method for finding the numerical value of $\sqrt[\phantom{2}]{2}.$ The Newton$-$Raphson method converges very quickly $($known as $2^{nd}$ order convergence$).$ Below we try a $($much$)$ slower method $(1^{st}$ order convergence$).$

The iterative equation to be used in this problem is given by

$x_n=\frac{0\mathrm{.}1x_{n-1}+2}{x_{n-1}+0\mathrm{.}1}$

$x_n$ will approach $\sqrt[\phantom{2}]{2}$ when $n$ approaches $.$

Start from $x_1=1.$ Write an R program that finds $\sqrt[\phantom{2}]{2},$ with a tolerance of ${10}^{-6}.$

Keep track of how many iterations, $n\_$iter, that is needed to achieve the required tolerance. Save $n_i$ ter and the corresponding estimated value of $\sqrt[\phantom{2}]{2}$ as a $12$ row vector in A$16.$ dat using the R$\text{'}$s command write.table$(...,$ col.names$=$FALSE, row.names$=$FALSE$).$