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MAT2043: Numerical and Computational Methods
Worksheet 2: Mathematical Programming
M-files
When writing programs, it is a good idea to write your code in a file. This can be done using any editor or by using the built-in editor. The files should always be saved with the extension .m File names cannot begin with numbers or include hyphens. Open the editor by clicking on the icon underneath the File menu (labeled New M-File) and type the following lines in it:
a(1)=1; for n=2: a(n)=a(n-1)+1/n^2; end plot(a,'r*')
Click on the Save and Run button in the menubar of the editor to save the program and to run it in Matlab. Remember to use the extension .m for the saved file. Alternatively, if the file was created with another editor and saved as filename.m, then it can be run in Matlab by typing filename
Programming patterns
Programming patterns are standard segments of code that form parts of larger algorithms. These patterns allow us to represent standard mathematical expressions and tasks e.g. sums in mathematics can be evaluated using for-loops in programming.
Piecewise-functions
Mathematical formula Code Pattern
if condition Y = a; else Y = b; end
The simplest example of a piecewise-function is |x| :
Create a new script and type:
Save this script as absx.m in your working directory. At the Matlab command line type:
ls % check that I am in my working directory absx.m Y = absx(-2) Y = 2 Y = absx(2) Y = 2
Maximum or minimum value of a vector
Mathematical formula Code Pattern
Y = x(1); for i = 2:n if Y < x(i) Y = x(i); end end
Create a new script called maxx.m and type:
At the Matlab command line, type:
x = [1 5 20 -4 500 2] % a (row) vector of numbers x = 1 5 20 -4 500 2
Y = maxx(x) Y = 500
Summation
Mathematical formula Code Pattern
z = 0; for i=1:n z = z + x(i)*y(i); end
Create a new script called sum2.m and type:
Iteration
Mathematical formula Code Pattern
Iterate until:
or
Y = x0; Z = f(Y); i = 1; while abs(Y-Z)>tol && i<=imax Y = Z; Z = f(Y); i = i + 1; end
The first condition is checking for convergence while the second condition stops an endless while-loop!
Create a new script called my_iter.m and type:
At the Matlab command line, type:
Z = my_iter(0.2,0.1,1000) Z = 0.
Recursion
Mathematical formula Code Pattern
Fibonacci sequence function F = Fib(n) if n== F = 0; else if n== F = 1; else F = Fib(n-1) + Fib(n-2); end end
All recursive functions have two parts:
- “Are we done yet?” Check terminating condition (in the code this is the “if n==0” and “if n==1” statements). If not, go to step 2.
- “Simplify problem” and call my-simplified-self. (in the above code this is the second “else” statement)
Create a new script called Fib.m and type:
and at the Matlab command line type:
F = Fib(6) F = 8
Construction of complex algorithms
Combining the above code patterns, we can solve more complex problems e.g. find the infinity norm of a matrix A
To solve this problem, one first finds the vector
and then you can apply the “max” code pattern to find Y. The Matlab script looks like this:
