School of Mathematics and Statistics MAST30028 Numerical Methods & Scientific Computing Week 2
Drag and drop the folder Week2 from L: \MAST30028 to C:\…\MATLAB and include it in the path, as described in Week 1. Now MATLAB knows how to find the files in Week2.
1 Further MATLAB
This relates to material presented in Lecture 3 (Further MATLAB)
Exercise Set 1
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a. b.
Define the function x !→ x2 cos(πx)e−x2 using an anonymous function. Evaluate it at x = 4 and plot it on the interval [−5, 5] using ezplot.
c.
d. e.
Many root-finders , trying to solve
take the form of a fixed point iteration
until some convergence criterion is satisfied.
Let x1, x2 be the previous and current iterate. Write anonymous functions that take x1, x2 and a tolerance
tol and return true if the condition given below is satisfied and false otherwise. • the absolute difference between x1 and x2 is less than the tolerance.
• the relative difference between x1 and x2 is less than the tolerance.
Hint: these are very simple functions!
Write a function that looks for the fixed point of a function by iterating it. The inputs are : a function handle to the function you want to iterate i.e. the function g, a function handle to the criterion you want to apply (from the previous exercise), the initial guess and the tolerance. The iterator should stop if the condition is not satisfied in 10,000 iterations.
Test your function from above by feeding it the Newton iteration to find √5.
Modify your newtonFuncWhileLoop so that it finds the square root of the input a from the initial guess
x0, with default value for a of 5 if no value is given. Test your new function. Introduction to NumPy/SciPy
f(x) = 0
xn+1 = g(xn) n = 0,1,2,···
This relates to material presented in Lab 2.
While MATLAB has a large installed base and is popular in numerical analysis and engineering circles, an
increasing fraction of scientific computing is being done on open source platforms, including R for statistics and machine learning and Octave, for scientific computing. However, the fastest growing segment appears to be packages which extend the general purpose object-oriented language Python to create overall a MATLAB-like environment, sometimes called the ‘SciPy stack’.
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Python is a popular introductory programming language with a rich set of data types (lists, tuples, sets and dictionaries) for general-purpose programming. But it doesn’t have arrays, especially numeric arrays which are the dominant datatype in MATLAB and commonly used in scientific programming.
IPython is a Python interpreter, allowing interactive evaluation of Python commands and programs.
NumPy is a module that extends Python by adding the ndarray datatype to Python, which are arrays filled with items of the same numeric type — called numeric arrays. With numeric arrays now defined, it supports vectorization of the common mathematical functions over arrays, linear algebra operations, random numbers and more functions such as FFT.
SciPy is a module that builds on NumPy to add functions for root-finding, numerical integration, solving differential equations, interpolation, optimization , special functions and the like.
Matplotlib is a module for graphics. The submodule PyPlot is enough to do simple plots of the kind we meet in the next lecture.
Spyder is an Integrated Development Environment IDE that includes a text editor with debugger, an IPython console, and more development tools. Like the MATLAB Desktop environment.
Anaconda is a distribution that allows a simple download of all of the above plus more modules plus a package manager. Other distributions are Enthought Canopy and Python(x,y)[Windows only]. You can download Anaconda on to your own machine: just Google ‘anaconda downloads’.
Python comes in 2 branches: Python2 (actually Python 2.7) and Python3 (currently up to Python 3.6) which is NOT backwards-compatible with Python2! Since the distribution in the labs is a couple of years old, it uses Python 2.7. But Python 2 will no longer be supported after 2020 so the general advice is for new learners to use Python3. You can choose which version you want to install when you go to Anaconda downloads.
Exercise Set 2
Run each code section in lab2.py that uses NumPy, see what it does by using the Variable Explorer, Object Inspector etc. then write MATLAB code that does the same thing, if possible.
3 Simple plotting Exercise Set 3
This relates to material presented in Lecture 4 (Plotting in MATLAB).
Plotting in MATLAB
Write MATLAB commands to create the following kinds of plots:
a. Plot sin θ versus θ for 60 points in the interval 0 ≤ θ ≤ 2π. Connect the points with a dashed line and label the points with open circles.
b. Create a plot that will reveal whether the data {yn} decay geometrically with n e.g. yn = 4.3(0.5)n
c. Create a plot that will reveal whether the data {yn} decay algebraically with n e.g. yn = 7.2n−2
d. Plot y = x2;y = sin(x2), and y = sin(x) on the same graph. Label them with arrows pointing to the respective curve (for example, have an arrow from the label ’y = x2’ to the curve y = x2). I suggest you do this interactively with Plot Tools.
e. Use a parametric plot to plot the ellipse
Plotting with Matplotlib
x2 y2
22 + 32 = 1
By following the examples in lec4.py, write Python commands to create the same plots as the Exercises above. It will be sufficient to use the PyPlot submodule of Matplotlib.
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