MATLAB CODING PROBLEM Use either bisection method or Newtons

MATLAB CODING PROBLEM

Use either bisection method or Newton`s method, solve the equation d(x)=0.8(cos^2(x)-sin^2(x)), attach the code.

Solution

Following are the steps to find the approximate solution of given equation using Bisection method:

Let us assume that we have to find out the roots of f(x), whose solution is lies in the range (a,b), which we have to determine. The only condition for bisection method is that f(a) and f(b) should have opposite signs (f(a) negative and f(b) positive). When f(a) and f(b) are of opposite signs at least one real root between ‘a’ and ‘b’ should exist.

For the first approximation we assume that root to be,

x0=(a+b)/2

Then we have to find sign of f(x0).

If f(x0) is negative the root lies between a and x0. If f(x0) is positive the root lies between x0 and b.

Now we have new minimized range, in which our root lies.

The next approximation is given by,

x1 = (a+x0)/2………….if f(x0) is negative.

x1 = (x0+b)/2………….if f(x0) is positive.

In this taking midpoint of range of approximate roots, finally both values of range converges to a single value, which we can take as a approximate root.