Ridders' method

In numerical analysis, 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${\displaystyle f(x)}$ . 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 ${\displaystyle {\sqrt {2}}}$ . 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.

Method

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

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

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

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

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 ${\displaystyle x_{1}}$ if ${\displaystyle f(x_{1})f(x_{3})<0}$ (well-behaved case), or otherwise whichever of ${\displaystyle x_{0}}$ and ${\displaystyle x_{2}}$ has function value of opposite sign to ${\displaystyle f(x_{3})}$. The procedure can be terminated when a given accuracy is obtained.

References

1. 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.
2. Kiusalaas, Jaan (2010). Numerical Methods in Engineering with Python (2nd ed.). Cambridge University Press. pp. 146–150. ISBN 978-0-521-19132-6.
3. 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.