Secant method

Key takeaways

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

Reference excerpt

In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method, so it is considered a quasi-Newton method. Historically, it is as an evolution of the method of false position, which predates Newton's method by over 3000 years.

The method The secant method is an iterative numerical method for finding a zero of a function f. Given two initial values x0 and x1, the method proceeds according to the recurrence relation

x n = x n − 1 − f ( x n − 1 ) x n − 1 − x n − 2 f ( x n − 1 ) − f ( x n − 2 ) = x n − 2 f ( x n − 1 ) − x n − 1 f ( x n − 2 ) f ( x n − 1 ) − f ( x n − 2 ) . {\displaystyle x_{n}=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}={\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}.}

This is a nonlinear second-order recurrence that is well-defined given f and the two initial values x0 and x1. Ideally, the initial values should be chosen close to the desired zero.

Derivation of the method Starting with initial values x0 and x1, we construct a line through the points (x0, f(x0)) and (x1, f(x1)), as shown in the picture above. In point–point form, the equation of this line is

y = f ( x 1 ) − f ( x 0 ) x 1 − x 0 ( x − x 1 ) + f ( x 1 ) . {\displaystyle y={\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{1})+f(x_{1}).}

The root of this linear function, that is the value of x such that y = 0 is

x = x 1 − f ( x 1 ) x 1 − x 0 f ( x 1 ) − f ( x 0 ) . {\displaystyle x=x_{1}-f(x_{1}){\frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.}

We then use this new value of x as x2 and repeat the process, using x1 and x2 instead of x0 and x1. We continue this process, solving for x3, x4, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between xn and xn−1):

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Secant method

Start with the simplest possible case. Write down what Secant method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Secant 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 Secant 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 Secant method

In research: Secant method appears in science 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 Secant 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: Secant method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quasi-Newton methods, so understanding it makes those chapters shorter.
In everyday life: Look for Secant 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 Secant method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Secant 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 Secant method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Secant method in simple terms?

In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method, so it is considered a quasi-Newton method.

Why does Secant method matter?

Because it connects several science 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 Secant 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 Secant method.