How do I plot convergence in Newton's method to find square root?

I have the following function to find the square root of a number a, in this case it is 2:

function [sqrta,n] = newton() 
a = 2;
tol = 1e-10;
n = 0;    
sqrta = a;
sqrtaminus1 = a;
while abs(sqrta^2 - a) > tol
    sqrta = 0.5*(sqrtaminus1 + (a/sqrtaminus1));
    n = n + 1;
    sqrtaminus1 = sqrta;  
    plot(sqrta,n);
end

I run this by:

>> [sqrta,n] = newton()

I'm making some mistake while plotting for convergence. I know I am making a fatal error without understanding the concept of convergence. It'd be great if someone could help me. Thanks!

Answer 1

[Duncan Lilley]

If you are trying to plot each stage of Newton's method, here are some suggestions:

1. In order for multiple plots to display on the axes, you will need to use the 'hold on' command.
2. Since 'sqrtaminus1' is keeping track of the previous value of 'sqrta', it makes sense to update it before recalculating the value of 'sqrta' inside the 'while' loop.
3. In order to view the plot, either provide a line specification to draw point markers or use the 'sqrtaminus1' value to draw a line from the previous value to the newly calculated value of 'sqrta'.
4. Since the value of 'n' represents the stage within Newton's method, it may make more sense to plot it as the x-value.

Take a look at the following changes:

function [sqrta,n] = newton() 
a = 2;
tol = 1e-15;
n = 0;  
sqrta = a;
sqrtaminus1 = a;

axes
hold on

while abs(sqrta^2 - a) > tol
    sqrtaminus1 = sqrta;
    sqrta = 0.5*(sqrtaminus1 + (a/sqrtaminus1));
    n = n + 1;
    plot([n-1 n], [sqrtaminus1 sqrta], 'r-');
end

hold off

This will plot each stage of Newton's method.

Answer 2

[Yujun Sun]

function [sqrta,n] = newton1()

a = 2;

tol = 1e-15;

n = 0;

sqrta = a;

sqrtaminus1 = a;

axes

%hold on

while abs(sqrta^2 - a) > tol

sqrtaminus1 = sqrta;

sqrta = 0.5*(sqrtaminus1 + (a/sqrtaminus1));

n = n + 1;

plot([n-1 n], [sqrtaminus1 sqrta], 'r-');

pause(2)

end

end