Lecture Notes 3/4/13
Review of Matlab: Issues that showed up on exam 1
problems with variable names and functions
Variable names have some limitations. These are outlined on page 11 of Palm.
 must begin with a letter
 can only contain letters, numbers and underscore _
 Case Sensitive (A and a are not the same in variable names)
 no more than 63 characters
defining a variable follows a pattern
variable name = expression or value
trying to do things the other way will cause problems.
Functions
programming functions are not the same as mathematical functions. In
programming a function is a piece of code that we want to easily reuse. This is
somewhat similar to a unit operation in Pro/II.
For example a pfd in Pro/II might look like this:
Here we are sending some streams to a unit operation which produces some output,
which goes to another unit operation, which produces some output.
Matlab can be thought of in a similar way:
Here we are sending variables into functions to produce new variables as outputs.
There are both user-defined and built-in functions. User-defined functions are ones
we write. Built-in functions are ones like plot(), polyfit(), rref() in Matlab, or
average() in Excel.
A function in matlab can contain a mathematical functions. I have seen this called a
‘function function’. I suppose if the mathematical function was to calculate friction
factors we would have a ‘friction factor function function’ which frankly just makes
me think of Fox in Socks by Dr. Suess and we should stop before we make up a
tweedle-beatle-puddle-paddle-bottle-battle function.
Solving Non-linear Equations Numerically
built in functions in matlab: (1 variable only)
fzero – finds roots for any function. According to the matlab documentation
this uses a combination of bisection, secant and inverse quadratic interpolation
methods.
roots – finds roots for polynomials. According to the matlab documentations
this ‘simply calculates the eigenvalues of the companion matrix’.
Algorithms for solving non-linear equations:
Bisection, Regula-Falsi, Newton’s (or Newton-Raphson), Secant and FixedPoint Iteration. Newton and Fixed-Point can be extended to systems fairly easily.
What follows is a graphical demonstration of what each method does, and a brief
summary.
Bisection
 Select a range
(points A and B).
 Find the
midpoint.
 Determine
which side the root is
on.
 set the new
range
 repeat until
changes in A and B
are within our desired
tolerance.
In this example, the
first midpoint
becomes our new
value for B. The new
midpoint would become A, etc. This method is really only set up to work
when the line we are working with crosses the x axis. When this is the case,
multiplying f(A) and f(B) will result in a negative number. so multiplying
f(midpoint) by f(a) and f(b) will tell us which half of our range contains the
root.
Regula-Falsi
 Select a range (points a and b)
 Draw a straight line between a and b.
 Where this line crosses the x-axis, set a new range endpoint.
 repeat until changes in A and B are within our desired tolerance.
As with bisection, the function we are evaluating must cross the x axis, and
we can use the same method to determine which endpoint to shift in order to
maintain this condition. In the example I sketched above, point b is shifted.
We can interpolate between points a and b to produce a straight line. our
new point b is at the intercept of this line. We use this new point to generate
a new line, etc.
Newton’s Method (also called Newton-Raphson)




Pick a point (yes just one with this method)
Draw a line tangent to your function (i.e. a line with a slope equal to that of
your function at that point, or a line with a slope equal to the value of the
derivative of your function at that point.)
set a new point where this line intercepts the x axis and f(intercept)
repeat until your x values are changing less than your tolerance.
Secant Method
This is a modification of Newton’s method. It uses 2 points to generate a slope
rather than taking a
derivative.
 Pick 2 points
(x1 and x2)
 Straight line
between x1 and x2
 at intercept of
line get x3
 straight line
between x2 and x3
 repeat until
change is within
tolerance.
Fixed Point Iteration
1. Starting with your equation in the form f(x)=0, rewrite as x=g(x).
There are very likely multiple options for what g(x) will actually look
like. For example:
if abs(g’(x))<1, it should converge, pick a g(x) accordingly.
2. plot y=g(x) and y=x. The intersection occurs at the root we are
searching for.
3. pick a point
4. draw a line in the x direction from the point to y=x, then in the y
direction back to y=g(x), this becomes our new point.
5. repeat step four until the change in x is within tolerance.