Ridders' method

Key takeaways

Ridders' method belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ridders' method to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ridders' method from memory before moving on to harder problems.

Reference excerpt

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 f ( x ) {\displaystyle f(x)} . The method is due to C. Ridders. Ridders' method is simpler than Muller's method or Brent's method but with similar performance. 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 with respect to function evaluations rather than with respect to number of iterates is 2 {\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, x 0 {\displaystyle x_{0}} and x 2 {\displaystyle x_{2}} , which are on two different sides of the root being sought so that f ( x 0 ) f ( x 2 ) < 0 {\displaystyle f(x_{0})f(x_{2})<0} , the method begins by evaluating the function at the midpoint x 1 = ( x 0 + x 2 ) / 2 {\displaystyle x_{1}=(x_{0}+x_{2})/2} . One then finds the unique exponential function e a x {\displaystyle e^{ax}} such that function h ( x ) = f ( x ) e a x {\displaystyle h(x)=f(x)e^{ax}} satisfies h ( x 1 ) = ( h ( x 0 ) + h ( x 2 ) ) / 2 {\displaystyle h(x_{1})=(h(x_{0})+h(x_{2}))/2} . Specifically, parameter a {\displaystyle a} is determined by

e a ( x 1 − x 0 ) = f ( x 1 ) − sign ⁡ [ f ( x 0 ) ] f ( x 1 ) 2 − f ( x 0 ) f ( x 2 ) f ( x 2 ) . {\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 ( x 0 , h ( x 0 ) ) {\displaystyle (x_{0},h(x_{0}))} and ( x 2 , h ( x 2 ) ) {\displaystyle (x_{2},h(x_{2}))} , leading to a new value x 3 {\displaystyle x_{3}} between x 0 {\displaystyle x_{0}} and x 2 {\displaystyle x_{2}} ,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Ridders' method

Start with the simplest possible case. Write down what Ridders' method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ridders' method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ridders' method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ridders' method

In research: Ridders' method appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ridders' method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ridders' method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Applied mathematics stubs, Root-finding algorithms, so understanding it makes those chapters shorter.
In everyday life: Look for Ridders' method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Ridders' method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ridders' method means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ridders' method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ridders' method in simple terms?

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 f ( x ) {\displaystyle f(x)} . The method is due to C.

Why does Ridders' method matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ridders' method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ridders' method.