Method of Bisection

• If the function is continuous on [a,b] and f(a) and f(b) have different signs , Bisection Method obtains a new interval that is half of the current interval and the sign of the function at the end points of the interval are different.
  
  
• This allows us to repeat the Bisection procedure to further reduce the size of the interval.

Short basic of Bisection Method

• The Bisection method is one of the simplest methods to find a zero of an on linear function.
  
• It is also called interval halving method.
  
• To use the Bisection method, one needs an initial interval that is known to contain a zero of the function.
  
• The method systematically reduces the interval . It does this by dividing the interval into two equal parts , perform a simple test and based on the result of the test half of the interval is thrown away.
  
• The procedure is repeated until the desired interval size is obtained.

Basis of Bisection Method

• Theorem: An equation f(x)=0 , where f(x) is a real continuous function , has at least one root between a and b iff (a)f(b)<0.
  
  
  

Solution Methods

Several ways to solve nonlinear equations are possible.

• Analytical Solutions: possible for special equations only.
  
• Graphical Solutions: Useful for providing initial guesses for other methods.
  
• Numerical Solutions:

1. Bisection Method
2. Newton’s Method
3. Secant Method
4. False position Method
5. Fixed point iterations

Roots of Equations/Zeros of a function

• A number r that satisfies an equation is called a root of the equation.
  
  The equation x^4 - 3x^3 - 7x^2 +15x=-18
  has four roots -2,3,3 and -1
  
• Let f(x) be a real-valued function of a real variable. Any number r for which f(r)=0 is called a zero of the function.
  
• Examples: 2 and 3 are zeros of the function
  f(x) = (x-2)(x-3)

• The real zeros of a function f(x)are the values of x at which the graph of the function crosses (or touches ) the x-axis.

Root finding Problems

• Root-Finding Problem is the problem of finding a root of the equation f(x)=0, where f(x) is a function of a single variable x.
  
• Given a function f(x), find x=r such that f(r)=0.
  
• The equation , will be called algebraic or transcendental according as f(x) is purely a polynomial in x or contains some other functions , such as trigonometric , logarithmic or exponential functions etc. For example
  
          x^3-x+2=0               algebraic equation
  
   sinx +x^2-logx=0           transcendental equation