Problem sheet 2

Please submit the question marked * to your tutor.

E2.1

Use elementary calculus to prove that $$ \max_{x\in [x_0,x_1]} (x-x_0)(x_1-x) = \frac14 (x_0-x_1)^2 $$ and hence complete the proof of Corollary 2.2.

E2.2 *

Consider $f(x) = x^3+2x+3$.

(a) Write down the linear polynomial $p_1$ that interpolates $f$.

(b) Use Corollary 2.2 to bound $|e(x)|$ over $x\in [0,1]$ where $e:= f-p_1$. Compare your bound with the actual error, which can be found analytically in this case.

(c) Find the quadratic polynomial $p_2$ that interpolates $f$ at $x_0$, $x_1$, and the additional point $x_2 = 2$.

(d) Next find the cubic polynomial $p_3$ which interpolates $f$ at $x_0, x_1, x_2$ and the additional point $x_3 = −1$. Comment on the relation between $f$ and $p_3$.

E2.3

The following function interpolates a given function $f $ between the two points $x_0$ and $x_1$ (input arguments $x_0$ and $x_1$). p1 is a vector containing all the values of the function $p_1(x)$ at 100 equally spaced points in $[x_0,x_1]$.

import numpy as np 
def f(x):
    return np.sqrt(x)
    
def linear_interp(x0,x1,mesh):   
    vecone = np.ones(100,)
    p1 =  f(x0) + ((f(x1) - f(x0))/(x1-x0))*(mesh - x0)    
    return p1

mesh = np.linspace(0,1,100)
p1 = linear_interp(0,1,mesh)

Write code to plot $p_1(x)$ and $f(x)$ (in different colours) on one graph, and then the error $e(x) := f(x) − p_1(x)$ on another graph.

import matplotlib.pyplot as plt
# fill in

E2.4*

In this exercise, numerically investigate $$ e_h:= \max_{x\in [-h,h]} |f(x) - p_1(x)| $$ for $h>0$, where the maximum is taken over 100 equally spaced points in $[-h,h]$.

Run your program for the case $f(x) = \exp(x)$ and draw up a table of $e_h$ against $h$, for $h = 1/8, 1/16, 1/32, \cdots, 1/256$. Investigate the convergence rate as $h\to 0$ experimentally, by making the conjecture $e_h = C h^\alpha$ where $C$ and $\alpha$ are constants and then finding approximations to $\alpha$.

Note that as in lectures, computing $\log_2(e_{2h}/e_h)$ will give approximations to $\alpha$. This can be done either by editing the program or using a calculator.

Repeat the exercise for $f(x) = \sin x$. Explain your observations by appealing to the theory from lectures.

def f(x):
    return np.exp(x)
    
#takes as input h and returns the maximum error eh as defined above
def max_err_linear_interp(h):   
    #fill in
    return eh