computational methods chapter two and chapter three, Lecture notes of Computational Methods

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• CHAPTER

Open Methods

• For bracketing methods, the root is located within an interval prescribed by a lower and an upper bound. Repeated application of these methods always result in closer estimates of the true value of the root.
• In contrast, the open methods are based on formulas that require only a single starting value of or two starting values that do not necessarily bracket the root. However, open methods usually do so much more quickly than the bracketing methods.
• it has two types of methods, those are; fixed point iteration and Newton-Raphson Method

Whereas sin could be put in to the form of eqn(1) by adding x to both sides to yield The utility of eqn(1) is that it provides a formula to predict a new value of x as a function of an old value of x. Thus, given an initial guess at the root , eqn(1) can be used to compute a new estimate as expressed by the iterative formula ). The sequence has the pattern =g( ) =g( ) =g( ) The approximate error can be determined using the error estimator: %

Example 1: Use simple fixed point iteration to locate the root of Soln: the function can be separated directly and expressed in the form of eqn(1) as: = starting with an initial guess of this iteration can be applied to compute

Derivation

The Newton-Raphson method is based on the principle that if the initial guess of the root of f (x) = 0 is at x i , then if one draws the tangent to the curve at f (x i ) , the point xi+ 1 where the tangent crosses the x - axis is an improved estimate of the root (Figure 1 ). Figure 1 Geometrical illustration of the Newton-Raphson method f ( x ) f ( xi ) f ( xi+ 1 ) xi+ 2 xi+ 1 xi θ [ xi, f ( xi )]

• Using the definition of the slope of a function, at x = x i f ‘(xi) = tan θ
• Equation ( 1 ) is called the Newton-Raphson formula for solving nonlinear equations of the form f ( x ) =
  1. So starting with an initial guess, xi , one can find the next guess, xi+ 1 , by using Equation ( 1 ). One can repeat this process until one finds the root within a desirable tolerance.

Drawbacks of the Newton-Raphson Method

• The above equation is called the secant method.
• This method now requires two initial guesses, but unlike the bisection method, the two initial guesses do not need to bracket the root of the equation.
• The secant method is an open method and may or may not converge. However, when secant method converges, it will typically converge faster than the bisection method. However, since the derivative is approximated as given by Equation ( 2 ), it typically converges slower than the Newton-Raphson method