mathematics

Modified Richardson iteration

Modified Richardson iteration is a mathematics 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 Modified Richardson iteration rather than just read about it. In short: Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in work that was published in 1911.

Key takeaways

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

Reference excerpt

Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in work that was published in 1911. It is similar to the Jacobi and Gauss–Seidel method. We seek the solution to a set of linear equations, expressed in matrix terms as

A x = b . {\displaystyle Ax=b.}

The Richardson iteration is

x ( k + 1 ) = x ( k ) + ω ( b − A x ( k ) ) , {\displaystyle x^{(k+1)}=x^{(k)}+\omega \left(b-Ax^{(k)}\right),}

where ω {\displaystyle \omega } is a scalar parameter that has to be chosen such that the sequence x ( k ) {\displaystyle x^{(k)}} converges. It is easy to see that the method has the correct fixed points, because if it converges, then x ( k + 1 ) ≈ x ( k ) {\displaystyle x^{(k+1)}\approx x^{(k)}} and x ( k ) {\displaystyle x^{(k)}} has to approximate a solution of A x = b {\displaystyle Ax=b} .

Convergence Subtracting the exact solution x {\displaystyle x} , and introducing the notation for the error e ( k ) = x ( k ) − x {\displaystyle e^{(k)}=x^{(k)}-x} , we get the equality for the errors

e ( k + 1 ) = e ( k ) − ω A e ( k ) = ( I − ω A ) e ( k ) . {\displaystyle e^{(k+1)}=e^{(k)}-\omega Ae^{(k)}=\left(I-\omega A\right)e^{(k)}.}

Thus,

‖ e ( k + 1 ) ‖ = ‖ ( I − ω A ) e ( k ) ‖ ≤ ‖ I − ω A ‖ ‖ e ( k ) ‖ , {\displaystyle \left\|e^{(k+1)}\right\|=\left\|\left(I-\omega A\right)e^{(k)}\right\|\leq \left\|I-\omega A\right\|\left\|e^{(k)}\right\|,}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Modified Richardson iteration

Start with the simplest possible case. Write down what Modified Richardson iteration 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 Modified Richardson iteration 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 Modified Richardson iteration 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 Modified Richardson iteration

In research: Modified Richardson iteration 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 Modified Richardson iteration 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: Modified Richardson iteration is common in secondary-school and first-year university syllabi. It links to neighbouring topics Iterative methods, Numerical linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Modified Richardson iteration 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Modified Richardson iteration” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Modified Richardson iteration in 20 minutes

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

Frequently asked questions

What is Modified Richardson iteration in simple terms?

Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in work that was published in 1911.

Why does Modified Richardson iteration 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 Modified Richardson iteration?

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 Modified Richardson iteration.