Given data

$(x_1,y_1),(x_2,y_2),$cdots,$(x_n,y_n)$

we know that it is always possible to find the unique $($at most$)n-1$ degree polynomial

$P(x)$ that interpolates the data, i$.$e$.,P(x_i)=y_i,i=1,2,$dots,$n.$

In class we learned that by assuming $P$ has the form

$P(x)=a_0+a_{1}x+a_2{x}^2+$cdots$+a_{n-1}{x}^{n-1}$

we can solve a system of equations for the unknowns $a_0,a_1,$dots,$a_{n-1}$ that is of the form

$VA=Y.$ Here $V$ is a Vandermonde matrix that depends on $x_1,x_2$dots,$x_n,$ and $Y$ is

the vector with components $y_1,y_2$dots,$y_n.$

In the method of divided differences we assume that $P$ has the form

$P(x)=a_0+a_1(x-x_1)+a_2(x-x_1)(x-x_2)+$cdots$+a_{n-1}(x-x_1)(x-x_2)cdots(x-x_{n-1}).$

We are justified in doing this by the result of problem $(3).$

In this case we solve a system of equations for the unknowns $a_0,a_1,$dots,$a_{n-1}$ that is of

the form $DA=Y,D$ is a matrix that depends on $x_1,x_2$dots,$x_n,$ and $Y$ is the vector

with components $y_1,y_2$dots,$y_n.$

$($a$)$ Write a MATLAB function that generates this matrix $D.$ Your function has as

input $x=(x_1,x_2$dots,$x_n)$ and has as output the $nn$ matrix $D.$ Let the prototype

be $D=$ dividedDifference $(x).$

$($b$)$ Write a MATLAB function which finds the polynomial interpolating the given

data. Use your function from part $(4$a$)$ to help you. Your function will have

input $x$ and $Y,$ the $x_j$ and $y_j$ variables respectively, and will output the coeffi$-$

cients, $P=(a_0,a_1,$dots,$a_{n-1}),$ of the desired polynomial. The prototype is $P=$

divDiffPoly $(x,Y).$

$($c$)$ Show that the $nn$ matrix $D$ has rank $n.$