computer science

Muller's method

Muller's method is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Muller's method rather than just read about it. In short: Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0. It was first presented by David E.

Key takeaways

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

Reference excerpt

Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0. It was first presented by David E. Muller in 1956.

Muller's method proceeds according to a third-order recurrence relation similar to the second-order recurrence relation of the secant method. Whereas the secant method proceeds by constructing a line through two points on the graph of f corresponding to the last two iterative approximations and then uses the line's root as the next approximation at every iteration, by contrast, Muller's method uses three points corresponding to the last three iterative approximations, constructs a parabola through these three points, and then uses a root of the parabola as the next approximation at every iteration.

Derivation Muller's method uses three initial approximations of the root, x 0 , x 1 {\displaystyle x_{0},x_{1}} and ⁠ x 2 {\displaystyle x_{2}} ⁠, and determines the next approximation x 3 {\displaystyle x_{3}} by considering the intersection of the x-axis with the parabola through ⁠ ( x 0 , f ( x 0 ) ) {\displaystyle (x_{0},f(x_{0}))} ⁠, ( x 1 , f ( x 1 ) ) {\displaystyle (x_{1},f(x_{1}))} and ⁠ ( x 2 , f ( x 2 ) ) {\displaystyle (x_{2},f(x_{2}))} ⁠. Consider the quadratic polynomial

P ( x ) = a ( x − x 2 ) 2 + b ( x − x 2 ) + c , {\displaystyle P(x)=a(x-x_{2})^{2}+b(x-x_{2})+c,}

that passes through ⁠ ( x 0 , f ( x 0 ) ) {\displaystyle (x_{0},f(x_{0}))} ⁠, ( x 1 , f ( x 1 ) ) {\displaystyle (x_{1},f(x_{1}))} and ⁠ ( x 2 , f ( x 2 ) ) {\displaystyle (x_{2},f(x_{2}))} ⁠. To simplify notation, define the differences

h 0 = x 1 − x 0 , h 1 = x 2 − x 1 {\displaystyle h_{0}=x_{1}-x_{0},\quad h_{1}=x_{2}-x_{1}}

and

δ 0 = f ( x 1 ) − f ( x 0 ) h 0 , δ 1 = f ( x 2 ) − f ( x 1 ) h 1 . {\displaystyle \delta _{0}={\frac {f(x_{1})-f(x_{0})}{h_{0}}},\quad \delta _{1}={\frac {f(x_{2})-f(x_{1})}{h_{1}}}.}

Substituting each of the three points ⁠ ( x 0 , f ( x 0 ) ) {\displaystyle (x_{0},f(x_{0}))} ⁠, ( x 1 , f ( x 1 ) ) {\displaystyle (x_{1},f(x_{1}))} and ( x 2 , f ( x 2 ) ) {\displaystyle (x_{2},f(x_{2}))} into P ( x ) {\displaystyle P(x)} and solving simultaneously for a , b {\displaystyle a,b} and c {\displaystyle c} gives

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Muller's method

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

In research: Muller's method appears in computer 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 Muller's 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: Muller's method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Root-finding algorithms, so understanding it makes those chapters shorter.
In everyday life: Look for Muller's 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.

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How to study Muller's method in 20 minutes

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

Frequently asked questions

What is Muller's method in simple terms?

Muller's method is a root-finding algorithm, a numerical method for solving equations of the form f(x) = 0. It was first presented by David E.

Why does Muller's method matter?

Because it connects several computer 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 Muller's 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 Muller's method.