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  Bisection method and Locating Roots

The Bisection Method and Locating Roots:

Bisecting also the if ... else ... end statement

Recall the bisection method also assume that c = f(a) <0 and d = f(b) >0. If f is non stop then obviously it must be zero at some x between a and b. The bisection method subsequently consists of looking half way between a and b for the zero of f that is let x = (a + b)/2 and calculate y = f(x).

If not this is zero then from the signs of c, d and y we are able to decide which new interval to subdivide. Particularly if c and y have the same sign then [x, b] must be the new interval but if c and y have different signs,then [a, x] must be the new interval.

Make a decision to do different things in different situations in a program is called flow control. The mainly common way to do this is the if else end statement which is an extension of the if end statement we have used already.

Bounding the Error:

One good thing about the bisection method is that we donot have with Newton’s method is that we always know that the definite solution x is inside the current interval [a, b] since f(a) and f(b) have different signs. This permits us to be sure about what the maximum error can be. Precisely the error is forever less than half of the length of the current interval [a, b] that is:

{Absolute Error} = |x − x| <(b − a)/2,

where x is the center point among the current a and b.

527_bisectionmethod.jpg

The bisection method

The following function program as well available on the webpage does n iterations of the bisection method and returns not merely the final value but as well the maximum possible error:

function [x e] = mybisect(f,a,b,n)
% function [x e] = mybisect(f,a,b,n)
% Does n iterations of the bisection method for a function f
% Inputs: f -- an inline function
% a,b -- left and right edges of the interval
% n -- the number of bisections to do.
% Outputs: x -- the estimated solution of f(x) = 0
% e -- an upper bound on the error
format long
c = f(a); d = f(b);
if c*d > 0.0
error(’Function has same sign at both endpoints.’)
end
disp(’ x y’)
for i = 1:n
x = (a + b)/2;
y = f(x);
disp([ x y])
if y == 0.0 % solved the equation exactly
e = 0;
break % jumps out of the for loop
end
if c*y < 0
b=x;
else
a=x;
end
end
e = (b-a)/2;

Another significant aspect of bisection is that it always works. We saw that Newton’s method is able tofail to converge to x if x0 isn’t close enough to x. on the contrary the current interval [a, b] in bisection will forever get decreased by a factor of 2 at every step and so it will always eventually shrink down as small as you want it.

Locating a root:

The bisection method as well as Newton’s method is both used to obtain closer and closer approximations of a solution, but both require starting places. The bisection method needs two points a andb that have a root between them and Newton’s method requires one point x0 which is reasonably close to a root. How do you arise with these starting points? It depends. If you are solving an equation once afterwards the best thing to do first is to just graph it. From acorrect graph you can see approximately where the graph crosses zero.

There are other circumstances where you aren’t just solving an equation once however have to solve the same equation many times but with different coefficients. This happens habitually when you are developing software for a specific application. In this circumstance the first thing you want to take benefit of is the natural domain of the problem that is on what interval is a solution physically reasonable. If that is known then it is simple to get close to the root by merely checking the sign of the function at a fixed number of points inside the interval. When the sign changes from one point to the next there is a root between those points. The following program will gaze for the roots of a function f on a specified interval [a0, b0].
function [a,b] = myrootfind(f,a0,b0)
% function [a,b] = myrootfind(f,a0,b0)
% Looks for subintervals where the function changes sign
% Inputs: f -- an inline function
% a0 -- the left edge of the domain
% b0 -- the right edge of the domain
% Outputs: a -- an array, giving the left edges of subintervals
% on which f changes sign
% b -- an array, giving the right edges of the subintervals
n = 1001; % number of test points to use
a = []; % start empty array
b = [];
x = linspace(a0,b0,n);
y = f(x);
for i = 1:(n-1)
if y(i)*y(i+1) < 0 % The sign changed, record it
a = [a x(i)];
b = [b x(i+1)];
end
end
if a == []
warning(’no roots were found’)
end

The final circumstance is writing a program that will look for roots with no given information. This is a difficult problem as well as one that is not often encountered in engineering applications. Formerly a root has been located on an interval [a, b] these a and b can serve as the beginning points for the bisection as well as secant methods. For Newton’s method one would desire to choose x0 between a and b. One clear choice would be to let x0 be the bisector of a andbthat is x0 = (a + b)/2. An even better choice would be to utilize the secant method to choose x0.

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