Simple rational approximation

Key takeaways

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

Reference excerpt

Simple rational approximation (SRA) is a subset of interpolating methods using rational functions. Especially, SRA interpolates a given function with a specific rational function whose poles and zeros are simple, which means that there is no multiplicity in poles and zeros. Sometimes, it only implies simple poles. The main application of SRA lies in finding the zeros of secular functions. A divide-and-conquer algorithm to find the eigenvalues and eigenvectors for various kinds of matrices is well known in numerical analysis. In a strict sense, SRA implies a specific interpolation using simple rational functions as a part of the divide-and-conquer algorithm. Since such secular functions consist of a series of rational functions with simple poles, SRA is the best candidate to interpolate the zeros of the secular function. Moreover, based on previous researches, a simple zero that lies between two adjacent poles can be considerably well interpolated by using a two-dominant-pole rational function as an approximating function.

One-point third-order iterative method: Halley's formula The origin of the interpolation with rational functions can be found in the previous work done by Edmond Halley. Halley's formula is known as one-point third-order iterative method to solve f ( x ) = 0 {\displaystyle \,f(x)=0} by means of approximating a rational function defined by

h ( z ) = a z + b + c . {\displaystyle h(z)={\frac {a}{z+b}}+c.}

We can determine a, b, and c so that

h ( i ) ( x ) = f ( i ) ( x ) , i = 0 , 1 , 2. {\displaystyle h^{(i)}(x)=f^{(i)}(x),\qquad i=0,1,2.}

Then solving h ( z ) = 0 {\displaystyle \,h(z)=0} yields the iteration

x n + 1 = x n − f ( x n ) f ′ ( x n ) ( 1 1 − f ( x n ) f ″ ( x n ) 2 ( f ′ ( x n ) ) 2 ) . {\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}\left({\frac {1}{1-{\frac {f(x_{n})f''(x_{n})}{2(f'(x_{n}))^{2}}}}}\right).}

This is referred to as Halley's formula. This geometrical interpretation h ( z ) {\displaystyle h(z)} was derived by Gander (1978), where the equivalent iteration also was derived by applying Newton's method to

g ( x ) = f ( x ) f ′ ( x ) = 0. {\displaystyle g(x)={\frac {f(x)}{\sqrt {f'(x)}}}=0.}

We call this algebraic interpretation g ( x ) {\displaystyle g(x)} of Halley's formula.

One-point second-order iterative method: Simple rational approximation Similarly, we can derive a variation of Halley's formula based on a one-point second-order iterative method to solve f ( x ) = α ( ≠ 0 ) {\displaystyle \,f(x)=\alpha (\neq 0)} using simple rational approximation by

h ( z ) = a z + b . {\displaystyle h(z)={\frac {a}{z+b}}.}

Then we need to evaluate

h ( i ) ( x ) = f ( i ) ( x ) , i = 0 , 1. {\displaystyle h^{(i)}(x)=f^{(i)}(x),\qquad i=0,1.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Simple rational approximation

Start with the simplest possible case. Write down what Simple rational approximation 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 Simple rational approximation 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 Simple rational approximation 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 Simple rational approximation

In research: Simple rational approximation 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 Simple rational approximation 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: Simple rational approximation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Interpolation, so understanding it makes those chapters shorter.
In everyday life: Look for Simple rational approximation 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 Simple rational approximation in 20 minutes

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

Frequently asked questions

What is Simple rational approximation in simple terms?

Simple rational approximation (SRA) is a subset of interpolating methods using rational functions. Especially, SRA interpolates a given function with a specific rational function whose poles and zeros are simple, which means that there is no multiplicity in poles and zeros.

Why does Simple rational approximation 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 Simple rational approximation?

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 Simple rational approximation.