(Solved): The aim of this lab is to write a function to compute the coefficients for a Hermite polynomial ...

The aim of this lab is to write a function to compute the coefficients for a Hermite polynomial interpolating function using divided differences and then to use the coefficients to compute the Hermite interpolating polynomial. The algorithm for the divided difference formulas for the Hermite polynomial coefficients is: INPUT $$X=\left(x_0,\dots ,x_n\right),Y=\left(f\left(x_0\right),\dots ,f\left(x_n\right)\right),YP=\left(f^{?}\left(x_0\right),\dots ,f^{?}\left(x_n\right)\right)$$ OUTPUT $$Q_{i,j}$$, the appropriate divided differences, where the Hermite interpolating polynomial is $$H_{2n+1}(x)=Q_{0,0}+\underset{i=1}{\overset{2n+1}{?}}{Q}_{i,i}\underset{j=0}{\overset{i?1}{?}}\left(z?z_j\right)$$ and $$z_{2i}=x_i,z_{2i+1}=x_i$$ for $$i=0,\dots ,n$$ STEP 1. For $$i=0,1,\dots n$$ $$\begin{array}{l}{z}_{2i}=x_i\\ z_{2i+1}=x_i\\ Q_{2i0}=f\left(x_i\right)\\ Q_{2i+1,0}=f\left(x_i\right)\\ Q_{2i+1}=f^{?}\left(x_i\right)\end{array}$$ If $$i?=0$$ then $$Q_{2,1}=\frac{Q_{2i,0}?Q_{2i?10}}{z_{21}?z_{2j?1}}$$ End If End STEP 2. For $$i=2,3,\dots ,2n+1$$ For $$j=2,3,\dots ,i$$ $$Q_{i,j}=\frac{Q_{i,?1}?Q_{i?1,?1}}{z_i?z_{i?j}}$$ End End STEP 3. OUTPUT $$Q_{i,j}$$ STOP (1). Write a matlab routine called main.m that calls a function that you write to compute the generalized divided differences given above. (2). Use a Hermite polynomial to interpolate the function $$f(x)=e^{0.1x^2}$$ given below in tabular form, at the point $$x=1.25$$. Use $$H_5(1.25)$$ with nodes $$x_0,x_1,x_2$$ and $$H_3(1.25)$$, with nodes $$x_0,x_1$$. (3). What is your estimate of the errors? Justify your answer. Find error bounds for these approximations.