Numerical Methods
for
Engineers and Scientists
Third
Edition
Amos Gilat • Vish Subramaniam
Chapter 3
Solving Nonlinear Equations
To find the roots of the equation, we can use:
1.
2.
3.
4.
5.
Bisection method
Regula Falsi method
Newton’s (Newton-Raphson) method
Secant method
Fixed-Point Iteration method
As the estimated numerical solutions are close to the true solution, it is anticipated
(n)
(n)
( n −1)
that the different xNS
is small compared to the value of xNS
, and the Estimated
− xNS
Relative Error is approximately the same as the True Relative Error.
Bisection Method
Additional notes on the bisection method:
1. The method always converges to an answer, provided a root was trapped in the interval
[a, b] to begin with.
2. The method may fail when the function is tangent to the axis and does not cross the
x-axis at f(x)=0.
3. The method converges slowly relative to other methods.
Regula Falsi Method
Newton’s (Newton-Raphson) Method
1. Started with one initial (guess) point.
2. Finding a numerical solution of an equation of the form f(x)=0 where f(x) is continuous
and differentiable and the equation is known to have a solution near a given point.
Secant Method
1. If the analytical form of f’(x) is difficult to get, secant method can be used to replace
Newton’s method.
2. Started with two initial points.
Fixed-Point Iteration Method
The fixed-point iteration method converges if, in the neighborhood of the fixed point,
the derivative of g(x) has an absolute value that is smaller than 1.
Systems of nonlinear equations (solved by using Newton’s method)
A system of nonlinear equations consists of two, or more nonlinear equations that have
to be solved simultaneously.
1.
2.
3.
4.
Estimate (guess) an initial solution, x1, x2, x3, ….
Calculate the Jacobian
Solve for Δx1, Δx2, Δx3….
Calculate new estimate x1, i+1, x2, i+1……
But, the initial guess for the Newton’s method should be close to the solution.
The partial derivative (the elements of the Jacobian matrix) have to be determined.