03.05.1
Chapter 03.05
Secant Method of Solving Nonlinear Equations
After reading this chapter, you should be able to:
1.
derive the secant method to solve for the roots of a nonlinear equation,
2.
use the secant method to numerically solve a nonlinear equation.
What is the secant method and why would I want to use it instead of the Newton
Raphson method?
The NewtonRaphson method of solving a nonlinear equation
0
)
(
x
f
is given by the
iterative formula
)
(
)
(
1
i
i
i
i
x
f
x
f
= x
x
(1)
One of the drawbacks of the NewtonRaphson method is that you have to evaluate the
derivative of the function.
With availability of symbolic manipulators such as Maple,
MathCAD, MATHEMATICA and MATLAB, this process has become more convenient.
However, it still can be a laborious process, and even intractable if the function is derived as
part of a numerical scheme.
To overcome these drawbacks, the derivative of the function,
)
(
x
f
is approximated as
1
1
)
(
)
(
)
(
i
i
i
i
i
x
x
x
f
x
f
x
f
(2)
Substituting Equation (2) in Equation (1) gives
)
(
)
(
)
)(
(
1
1
1
i
i
i
i
i
i
i
x
f
x
f
x
x
x
f
x
x
(3)
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.
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 Spring '08
 Kaw,A
 Secant method, Rootfinding algorithm, solving nonlinear equations

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