Introductory Programming
Fall 2005

Evaluation 10 on Thursday (last one)

Two solutions to Evaluation 9:

a) In testsqrt.m:

function testsqrt(v)
    for x=v
        mine = mysqrt(x);
        theirs = sqrt(x);
        diff = mine-theirs;
        rel = diff / theirs
    end
end


b) In mysqrt.m:

function root = mysqrt(a)
    est = a;

    for i=1:5
        est = est - f(est, a) / fp(est);
    end
    root = est;
end

function y = f(x, b)
    y = x^2 - b;
end

function y = fp(x)
    y = 2*x;
end


Who calls who?

1) testsqrt calls mysqrt and sqrt

   (notice the symmetry)

2) mysqrt calls f and fp

   (notice the encapsulation)

   (why do we have to pass a as an input variable to f?)

3) draw the stack diagrams here...



Nested functions
----------------

One way to avoid passing values around as input variables
is to use nested functions:

function root = mysqrt(a)  % a is defined here...

    function y = f(x)
        y = x^2 - a;       % and used here...
    end

    function y = fp(x)
        y = 2*x;
    end 

    est = a;               % and here.

    for i=1:5
        est = est - f(est) / fp(est);
    end
    root = est;
end


Relative error
--------------

What does relative error tell us about the number of correct digits?

How does the error for the sqrt algorithm depend on x?

How can we iterate until convergence in MATLAB?

What does the following loop do?

i = 0
while i<10
   i = i + 1
end

What about this one?

x = a;
while true
    xnew = (x + a/x) / 2
    if abs(xnew - x) < 1e-6
        break
    end
    x = xnew;
end
r = xnew;