Chapter 9

Make circle a subclass of an ellipse. Chapter \(7.2 .3\) presents class Circle. Make a similar class Ellipse for representing an ellipse. Then create a new class Circle that is a subclass of Ellipse. Name of program file: Ellipse_Circle.py.

Make super- and subclass for a point. A point \((x, y)\) in the plane can be represented by a class: We can extend the Point class to also contain the representation of the point in polar coordinates. To this end, create a subclass PolarPoint whose constructor takes the polar representation of a point, \((r, \theta)\), as arguments. Store \(r\) and \(\theta\) as attributes and call the superclass constructor with the corresponding \(x\) and \(y\) values (recall the relations \(x=r \cos \theta\) and \(y=r \sin \theta\) between Cartesian and polar coordinates). Add a _-str__- method in class PolarPoint which prints out \(r, \theta, x\), and \(y\). Verify the implementation by initializing three points and printing these points. Name of program file: PolarPoint.py. 0

Represent people by a class hierarchy. Classes are often used to model objects in the real world. We may represent the data about a person in a program by a class Person, containing the person's name, address, phone number, date of birth, and nationality. A method __str__ may print the person's data. Implement such a class Person. A worker is a person with a job. In a program, a worker is naturally represented as class Worker derived from class Person, because a worker is a person, i.e., we have an is-a relationship. Class Worker extends class Person with additional data, say name of company, company address, and job phone number. The print functionality must be modified accordingly. Implement this Worker class. A scientist is a special kind of a worker. Class Scientist may therefore be derived from class Worker. Add data about the scientific discipline (physics, chemistry, mathematics, computer science,...). One may also add the type of scientist: theoretical, experimental, or computational. The value of such a type attribute should not be restricted to just one category, since a scientist may be classified as, e.g., both experimental and computational (i.e., you can represent the value as a list or tuple). Implement class Scientist. Researcher, postdoc, and professor are special cases of a scientist. One can either create classes for these job positions, or one may add an attribute (position) for this information in class Scientist. We adopt the former strategy. When, e.g., a researcher is represented by a class Researcher, no extra data or methods are needed. In Python we can create such an "empty" class by writing pass (the empty statement) as the class body: It is a continuous debate in computer science whether multiple inheritance is a good idea or not. One obvious problem \(^{11}\) in the present example is that class Professor inherits two names, one via Teacher and one via Scientist (both these classes inherit from Person). Neither of the two widely used languages Java and C# allow multiple inheritance. Nor in this book will we persue the idea of multiple inheritance further. Name of program file: Person.py.

Compute convergence rates of numerical integration methods. Most numerical methods have a discretization parameter, call it \(n\), such that if \(n\) increases (or decreases), the method performs better. Often, the relation between the error in the numerical approximation (compared with the exact analytical result) can be written as and $$ E_{i}=C n_{i}^{r} $$ Divide the first equation by the second to eliminate \(C\), and then take the logarithm to solve for \(r\) : $$ r=\frac{\ln \left(E_{i-1} / E_{i}\right)}{\ln \left(n_{i-1} / n_{i}\right)} $$ We can compute \(r\) for all pairs of two successive experiments. Usually, the "last \(r^{n}\), corresponding to \(i=N\) in the formula above, is the "best" \(r\) value \(^{12}\). Knowing \(r\), we can compute \(C\) as \(E_{N} n_{N}^{-r}\). Having stored the \(n_{i}\) and \(E_{i}\) values in two lists \(n\) and \(E\), the following code snippet computes \(r\) and \(C\) : from scitools. convergencerate import convergence_rate \(\mathrm{C}, \mathrm{r}=\) convergence_rate \((\mathrm{n}, \mathrm{E})\) Construct a test problem for integration where you know the analytical result of the integral. Run different numerical methods (the Midpoint method, the Trapezoidal method, Simpson's method, Monte Carlo integration) with the number of evaluation points \(n=2^{k}+1\) for \(k=2, \ldots, 11\), compute corresponding errors, and use the code snippet above to compute the \(r\) value for the different methods in questions. The higher the absolute error of \(r\) is, the faster the method converges to the exact result as \(n\) increases, and the better the method is. Which is the best and which is the worst method? Let the program file import methods from the integrate module and the module with the Monte Carlo integration method from Exercise 9.14. Name of program file: integrators_convergence.py. $$ E=C n^{r} $$ where \(E\) is the error, and \(C\) and \(r\) are constants. Suppose you have performed an experiment with a numerical method using discretization parameters \(n_{0}, n_{1}, \ldots, n_{N}\). You have computed the corresponding errors \(E_{0}, E_{1}, \ldots, E_{N}\) in a test problem with an analytical solution. One way to estimate \(r\) goes as follows. For two successive experiments we have

Make a calculus calculator class. Given a function \(f(x)\) defined on a domain \([a, b]\), the purpose of many mathematical exercises is to sketch the function curve \(y=f(x)\) compute the derivative \(f^{\prime}(x)\), find local and global extreme points, and compute the integral \(\int_{a}^{b} f(x) d x .\) Make a class CalculusCalculator which can perform all these actions for any function \(f(x)\) using numerical differentiation and integration, and the method explained in Exercise \(7.36\) or \(7.37\) for finding extrema. Here is an interactive session with the class where we analyze \(f(x)=\) \(x^{2} e^{-0.2 x} \sin (2 \pi x)\) on \([0,6]\) with a grid (set of \(x\) coordinates) of 700 points: Design the class such that the above session can be carried out. Hint: Use classes from the Diff and Integrator hierarchies (Chapters \(9.2\) and \(9.3\) ) for numerical differentiation and integration (with, e.g., Central2 and Trapezoidal as default methods for differentiation and integration). The method set_differentiation_method takes a subclass name in the Diff hierarchy as argument, and makes an attribute df that holds a subclass instance for computing derivatives. With set_integration_method we can similarily set the integration method as a subclass name in the Integrator hierarchy, and then compute the integral \(\int_{a}^{b} f(x) d x\) and store the value in the attribute integral. The extreme_points method performs a print on a MinMax instance, which is stored as an attribute in the calculator class. Name of program file: CalculusCalculator.py.