Exercise 1. Find the polynomial of least degree that interpolates the data $𝒟=\{(x_i,y_i),i=0,1,\dots ,n\}$$=\{(3,10),(7,146),(1,2),(2,1)\}$.

Exercise 2. Find the polynomial of least degree that interpolates the data $𝒟=\{(x_i,y_i,y_i^{\text{'}}),i=0,1,\dots ,n\}=$$𝒟=\{(3,10,14),(1,2,-6)\}$.

Exercise 3. Find the polynomial of least degree that interpolates the data $𝒟=\{(x_i,y_i,y_i^{\text{'}},y_i^{\text{'}\text{'}}),i=0,1,\dots ,n\}=$$𝒟=\left\{(1,2,-6,10)\right\}$.

Exercise 4. Consider $ℱ=\{f|f:[a,b]\to ℝ$}, ${𝒫}_n=\{p|p.\left(t\right)=a_0+a_{1}t+\cdots +a_n{t}^n\}$, $𝒙\in {ℝ}^{n+1}$, $x_i\in [a,b]$, $i\ne j\Rightarrow x_i\ne x_j$. Define the operators $S:ℱ\to {ℝ}^{(n+1)\times 2}$, $I:{ℝ}^{(n+1)\times 2}\to {𝒫}_n\subset ℱ$,

For each of the mappings, $S$, $I$, $P=I\circ S$, determine if the mapping is linear, and if so write its matrix representation. Also determine if the mappings are surjective, bijective, one-to-one.

Exercise 5. Recall the eigenvalue problem for finite-dimensional linear operators, $𝑨:{ℝ}^m\to {ℝ}^m$, $𝑨𝑿=𝑿𝚲$, $𝑿\in {ℝ}^{m\times m}$, $𝚲\in {ℂ}^{m\times m}$, $𝚲=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_m)$. The generalization of the finite-dimensional problem to linear operators on vector space $𝒰$, $L:𝒰\to 𝒰$ is immediate, $Lu=\lambda u$, $u\in 𝒰$, $\lambda \in ℂ$. What is the appropriate generalization of $C\left(𝑿\right)$ for the operator $P:ℱ\to ℱ$?

Exercise 6. Provide a relative error bound for the polynomial interpolation of $f:[-1,1]\to ℝ$, $f\left(t\right)=\cosh \left(t\right)$ by a polynomial $p_{22}\left(t\right)$ of degree $n=22$.

Exercise 7. Prove that the above bound $\epsilon ⩽5\times {10}^{-16}$, irrespective of the choice of distinct interpolation nodes.

Exercise 8. In the limit $x_1\to x_0$ the divided difference

has limit $f[x_0,x_1]\to f^{\text{'}}\left(x_0\right)$. Write and establish the validity of the finite difference form of the product rule ${(fg)}^{\text{'}}=f^{\text{'}}g+f{g}^{\text{'}}$.

Exercise 9. Repeat the above for second order finite differences and ${(fg)}^{\text{'}\text{'}}=f^{\text{'}\text{'}}g+2f^{\text{'}}{g}^{\text{'}}+f{g}^{\text{'}\text{'}}$.

Exercise 10. A natural cubic spline has zero curvature at the end points. Prove that of all cubic spline interpolations of data $𝒟=\{(x_i,y_i=f\left(x_i\right)),i=0,1,\dots ,n\}$, the natural spline $S\left(t\right)$ curvature two-norm is bounded by the function curvature two-norm

Exercise 11. Recall the behavior of global polynomial interpolation for the Runge function, and interpret the significance of the above inequality for spline interpolation. In particular study the bound for the curvature of the global polynomial interpolant

Exercise 12. Prove that best inf-norm approximant of $f:[-1,1]\to ℝ$, $f\left(t\right)=\cosh \left(t\right)$ by a quadratic polynomial has form $p_2\left(t\right)=a+b{t}^2$, with $b=\cosh 1-1$. Compute $a$.

Exercise 13. Compare the error bound for $p_2\left(t\right)$ from Ex 12 with that for $p_{22}\left(t\right)$ from Ex. 6.

Exercise 14. Prove that

is exact for $f\in {𝒫}_4$, (polynomials of degree up to 4).

Exercise 15. Determine the quadrature formula

that is exact for all functions of form $f\left(t\right)=a{e}^t+b\cos (\pi t/2)$.

Exercise 16. Determine the quadrature formula

that is exact for all functions of form $f\left(t\right)=a+b\cos \left(t\right)$.

Exercise 17. Verify that the above formula is exact for

Exercise 18. Can the quadrature

be exact for all quadratic polynomials?

Exercise 19. Let the quadrature

be exact for all polynomials of degree $n=2k$, $k\in \mathbb{N}$. Show that if the quadrature nodes $x_i$ are symmetric around the origin the formula is also exact for polynomials of degree $n+1$.

Exercise 20. Determine the minimum number of subintervals needed to approximate

to absolute accuracy $e<0.5\times {10}^{-7}$ using the composite trapezoid rule.

Exercise 21. Repeat above for the composite Simpson rule.

Exercise 22. Consider the equation $f\left(x\right)=0$, $f:ℝ\to ℝ$, $f\in C^{\left(2\right)}\left(ℝ\right)$. Generate an approximating sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$, $x_n\to r$ as $n\to \infty$, $f\left(r\right)=0$, based upon constructing the linear interpolant of data $𝒟=\{(x_{n-2},f_{n-2}),(x_{n-1},f_{n-1})\}$ to obtain $x_n$ ($f_k=f\left(x_k\right)$).

Exercise 23. Consider the equation $f\left(x\right)=0$, $f:ℝ\to ℝ$, $f\in C^{\left(2\right)}\left(ℝ\right)$. Generate an approximating sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$, $x_n\to r$ as $n\to \infty$, $f\left(r\right)=0$, based upon constructing the linear interpolant of data $𝒟=\left\{(x_{n-1},f_{n-1},f_{n-1}^{\text{'}})\right\}$ to obtain $x_n$ ($f_k=f\left(x_k\right),f_k^{\text{'}}=f\left(x_k\right)$).

Exercise 24. Consider the equation $f\left(x\right)=0$, $f:ℝ\to ℝ$, $f\in C^{\left(2\right)}\left(ℝ\right)$. Generate an approximating sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$, $x_n\to r$ as $n\to \infty$, $f\left(r\right)=0$, based upon constructing the linear interpolant of data $𝒟=\{(x_{n-3},f_{n-3}),(x_{n-2},f_{n-2}),(x_{n-1},f_{n-1})\}$ to obtain $x_n$ ($f_k=f\left(x_k\right)$).

Exercise 25. Consider the equation $f\left(x\right)=0$, $f:ℝ\to ℝ$, $f\in C^{\left(2\right)}\left(ℝ\right)$. Generate an approximating sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$, $x_n\to r$ as $n\to \infty$, $f\left(r\right)=0$, based upon constructing the linear interpolant of data $𝒟=\left\{(x_{n-1},f_{n-1},f_{n-1}^{\text{'}},f_{n-1}^{\text{'}\text{'}})\right\}$ to obtain $x_n$ ($f_k=f\left(x_k\right),f_k^{\text{'}}=f\left(x_k\right)$, $f_k^{\text{'}\text{'}}=f^{\text{'}\text{'}}\left(x_k\right)$).