Ridder’s Method For Solving Linear and Nonlinear Equations

Ridders’ method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a continuous function $f(x)$ . The method is due to C. Ridders.^[1]^[2]Ridders’ method is simpler than Muller’s method or Brent’s method but with similar performance.^[3] The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall order of convergence of the method is $Sqrt(2)$ . If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval at least halves on each iteration, so convergence is guaranteed.

Example On Using This Code

Input

func =@(x) x^3 + 5*x - 5;             % Equation we interest to solve
xLow = -50;                           % Lower interval bound
xUpp = 100;                           % Upper interval bound
xZero = rootSolve(func,xLow,xUpp)     % Calling Ridder's Function

Output

xZero = 0.868830020341475
func(xZero) = 0.0

Method

Given two values of the independent variable, $x_(0)$ and $x_(2)$ , which are on two different sides of the root being sought, i.e., $f(x_(0)) . f(x_(2)) < 0$ , the method begins by evaluating the function at the midpoint $x_(1) = (x_(0) + x_(2)) / 2$ . One then finds the unique exponential function $e^(ax)$ such that function

$h(x)=f(x)e^{ax}$

satisfies

$h(x_{1})=(h(x_{0})+h(x_{2}))/2$

Specifically, parameter a is determined by

$e^{a(x_{1}-x_{0})}={\frac {f(x_{1})-\operatorname {sign} [f(x_{0})]{\sqrt {f(x_{1})^{2}-f(x_{0})f(x_{2})}}}{f(x_{2})}}.$

The false position method is then applied to the points $(x_(0), h(x_(0)))$ and $(x_(2), h(x_(2)))$ , leading to a new value $x_(3)$ between $x_(0)$ and $x_(2)$ ,

$x_{3}=x_{1}+(x_{1}-x_{0}){\frac {\operatorname {sign} [f(x_{0})]f(x_{1})}{\sqrt {f(x_{1})^{2}-f(x_{0})f(x_{2})}}},$

which will be used as one of the two bracketing values in the next step of the iteration. The other bracketing value is taken to be $x_(1)$ if $f(x_(1)) . f(x_(3)) < 0$ (well-behaved case), or otherwise whichever of $x_(0)$ and $x_(2)$ has function value of opposite sign to $f(x_(3))$ . The procedure can be terminated when a given accuracy is obtained.

References

Ridders, C. (1979). “A new algorithm for computing a single root of a real continuous function”. IEEE Transactions on Circuits and Systems. 26: 979–980. doi:10.1109/TCS.1979.1084580.
Kiusalaas, Jaan (2010). Numerical Methods in Engineering with Python (2nd ed.). Cambridge University Press. pp. 146–150. ISBN 978-0-521-19132-6.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). “Section 9.2.1. Ridders’ Method”. Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.