Numerical Analysis Lecture 7

Solution of Non-Linear Equations Chapter 2 Solution of Non-Linear Equations

Introduction Bisection Method Regula-Falsi Method Method of iteration Newton - Raphson Method Secant Method Muller’s Method Graeffe’s Root Squaring Method

Secant Method

We choose x0, x1 close to the root ‘a’ of f (x) =0 such that f (x0) f(x1) As a next approximation x2 is obtained as the point of intersection of y = 0 and the chord passing through the points (x0, f(x0 )), (x1 f(x1 )).

Putting y = 0 we get Generally This sequence converges to the root ‘b’ of f (x) = 0 i.e. f( b ) = 0.

Convergence of Secant Method Here the sequence is generated by the rule starting with x0 and x1 as It will converge to ‘a’ ,that is f ( a ) = 0

The Secant method converges faster than linear and slower than Newton’s quadratic.

Example Do three iterations of secant method to find the root of taking

Though X5 is not the true root, yet this is good approximation to the root and convergence is faster than bisection.

Comparison: In secant method, we do not check whether the root lies in between two successive approximates Xn-1, and Xn. This checking was imposed after each iteration, in Regula –Falsi method.

Muller’s Method

In Muller’s method, f (x) = 0 is approximated by a second degree polynomial; that is by a quadratic equation that fits through three points in the vicinity of a root. The roots of this quadratic equation are then approximated to the roots of the equation f (x) = 0.

This method is iterative in nature and does not require the evaluation of derivatives as in Newton-Raphson method. This method can also be used to determine both real and complex roots of f (x) = 0.

Suppose, be any three distinct approximations to a root of f (x) = 0.

Noting that any three distinct points in the (x, y)-plane uniquely, determine a polynomial of second degree. A general polynomial of second degree is given by

Suppose, it passes through the points then the following equations will be satisfied

Fig: Quadratic polynomial.

Eliminating a, b, c, we obtain

which can be written as

The above equation can be written as That was a second degree polynomial. Now, introducing the notation The above equation can be written as

The above equation can be written as

We further define

With these substitution we get a simplified Equation as

Or

To compute set f = 0, we obtain where A direct solution will lead to loss of accuracy and therefore to obtain max accuracy we rewrite as:

so that, or Here, the positive sign must be so chosen that the denominator becomes largest in magnitude.

we can get a better approximation to the root, by using

Example Find the root of the equation x3 – x–1 = 0 using Muller’s method.

Numerical Analysis Lecture 7