These exercises focus on midterm examination preparation.

1Track 1

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)\}$.

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)\}$.

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\}$.

4. Find $a,b,c,d$ such that

   ${\displaystyle S\left(x\right)=\{\begin{array}{ll}1-2x& x\in (-\infty ,-3]\\ a+bx+c{x}^2+d{x}^3& x\in [-3,4]\\ 157-32x& x\in [4,\infty )\end{array}.,}$

   is a natural spline on the $[-3,4]$ interval.

5. Apply the Gram-Schmidt algorithm to orthonormalize the function set $\{1,t,t^2\}$ with respect to the scalar product

   ${\displaystyle (f,g)=\underset{-1}{\overset{+1}{\int }}f\left(t\right)g\left(t\right)dt.}$

6. Let $\{{\phi }_0\left(t\right),{\phi }_1\left(t\right),{\phi }_2\left(t\right)\}$ denote the orthonormalized set found above. Find the best $2$-norm approximant $g\left(t\right)=c_0{\phi }_0\left(t\right)+c_1{\phi }_1\left(t\right)+c_2{\phi }_2\left(t\right)$ of $f\left(t\right)=\sin (\pi t/2)$ on the interval $[-1,1]$.

7. As above, find the best $2$-norm approximant of $f\left(t\right)=\cos (\pi t/2)$ on the interval $[-1,1]$.

8. Find the best approximant $g\left(t\right)=\lambda t$ of $f\left(t\right)=\sin t$ on the interval $[0,\pi /2]$ in the $\infty$-norm.

2Track 2

1. In the limit $x_1\to x_0$ the divided difference

   ${\displaystyle f[x_0,x_1]=[y_1,y_0]=\frac{y_1-y_0}{x_1-x_0},y_i=f\left(x_i\right),i=0,1,}$

   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 ${(fg)}^{\text{'}}=f^{\text{'}}g+f{g}^{\text{'}}$.

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

3. 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

   ${\displaystyle \underset{x_0}{\overset{x_n}{\int }}{\left[S^{\text{'}\text{'}}\left(t\right)\right]}^{2}dt⩽\underset{x_0}{\overset{x_n}{\int }}{\left[f^{\text{'}\text{'}}\left(t\right)\right]}^{2}dt.}$

4. Find $a,b,c,d$ such that

   ${\displaystyle \left|e\left(1\right)\right|=\left|e\left(0\right)\right|\Rightarrow |\cosh 1-a-b|=|1-a|\Rightarrow }$ ${\displaystyle S\left(x\right)=\{\begin{array}{ll}1-2x& x\in (-\infty ,-3]\\ a+bx+c{x}^2+d{x}^3& x\in [-3,4]\\ 157-32x& x\in [4,\infty )\end{array}.,}$

   is a natural spline on the $[-3,4]$ interval.

5. Present an analysis of the conditioning of quadratic spline interpolation.

6. Apply the Gram-Schmidt algorithm to orthonormalize the function set $\{1,t,t^2\}$ with respect to the scalar product

   ${\displaystyle (f,g)=\underset{-1}{\overset{+1}{\int }}\frac{f\left(t\right)g\left(t\right)}{\sqrt{1-t^2}}dt.}$

7. Find the best approximant $g\left(t\right)=a+bt$ of $f\left(t\right)=\sin t$ on the interval $[0,\pi /2]$ in the 2-norm and the $\infty$-norm.

8. 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 $a$.