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  Controlling Error and Conditional Statements

Controlling Error and Conditional Statements:

Measuring error and the Residual:

If we are attempting to find a numerical solution of an equation f(x) = 0 then there are a few different ways we can measure the error of our approximation. The most direct way to calculate the error would be as:

{Error at step n} = en= xn− x

Where xn is the n-th approximation alsoxis the true value. Nevertheless we usually do not know thevalue of x, or we wouldn’t be trying to approximate it. This makes it not possible to know the errordirectly as well as so we must be more clever.

One possible strategy that habitually works is to run a program until the approximation xnstopschanging. The trouble with this is that sometimes doesn’t work. Just for the reason that the program stopchanging does not necessarily mean that xn is close to the true solution.

For Newton’s method we have the subsequent principle- At every step the number of significantdigits roughly doubles. While this is asignificant statement about the error (ever since it signifiesNewton’s method converges really quickly) it is to some extent hard to use in a program.

Rather than calculate how close xn is to x* in this and many other situations it is much more practicalto measure how close the equation is to being satisfied in other words how close yn= f(xn) is to0. We will use the quantity rn= f(xn) − 0, called the residual, in many different situations. Mostof the time we merely care about the size of rn, so we look at |rn| = |f(xn)|.

The if ... end statement

If we have a definite tolerance for |rn| = |f(xn)| then we are able to incorporate that into our Newtonmethod program using an if ... end statement-

function x = mynewton(f,f1,x0,n,tol)
% Solves f(x) = 0 by doing n steps of Newton’s method starting at x0.
% Inputs: f -- the function, input as an inline
% f1 -- it’s derivative, input as an inline
% x0 -- starting guess, a number
% tol -- desired tolerance, prints a warning if |f(x)|>tol
% Output: x -- the approximate solution
x = x0; % set x equal to the initial guess x0
for i = 1:n % Do n times
x = x - f(x)/f1(x) % Newton’s formula
end
r = abs(f(x))
if r > tol
warning (’The desired correctness was not attained’)
end

In this program if checks if abs(y) > tol is true or else else not. If it is true then it does the whole thingamong there and end. If not accurate then it omits ahead to end.

In the command window explains the function

> f = inline(’x^3-5’,’x’)

And its derivative

> f1 = inline(’3*x^2’,’x’).

Then utilize the program with n = 5, tol = .01, and x0 = 2. Next, change tol to 10−10 and repeat.
The loop: while ... end

While the preceding program will tell us if it worked or not we still have to input n the number ofsteps to take. Therefore far for a well-behaved problem if we make n too small then the tolerance will notbe attained and we will have to go back as well as increase it or if we make n too big then the programwill take more steps than necessary.

One way to manage the quantity of steps taken is to iterate until the residual |rn| = |f(x)| = |y| issmall enough. In Mat lab this is merely accomplished with a while ... end loop.

function x = mynewtontol(f, f1, x0, tol)

x = x0; % set x equal to the initial guess x0
y = f(x);
while abs(y) > tol % Do until the tolerence is reached.
x = x - y/f1(x) % Newton’s formula
y = f(x)
end

The statement while  end is a loopalike to for  end however instead of going through theloop a fixed number of times it keeps going so long as the statement abs(y) > tol is true.

One clear drawback of the program is that abs(y) might never get slighter than tol. If thishappens the program would carry onrunning over as well as over until we stop it. Try this by setting theforbearance to a really small number:

>tol = 10^(-100)

after that run the program again for f(x) = x3 − 5.

You are able to use Ctrl-c to stop the program when it’s stuck.

One way to evade an infinite loop is add a counter variable say i in addition to a maximum number ofiterations to the programs.

Utilizing the while statement this can be accomplished as:

function x = mynewtontol(f,f1,x0,tol)
x = x0; i=0; % set x equal to the initial guess x0. set counter to zero
y = f(x);
while abs(y) > tol & i < 1000 % Do until the tolerence is reached or max iter.
x = x - y/f1(x) % Newton’s formula
y = f(x)
i = i+1;
end

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