Inverse iteration

Key takeaways

Inverse 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 Inverse iteration to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Inverse iteration from memory before moving on to harder problems.

Reference excerpt

In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method. It appears to have originally been developed to compute resonance frequencies in the field of structural mechanics. The inverse power iteration algorithm starts with an approximation μ {\displaystyle \mu } for the eigenvalue corresponding to the desired eigenvector and a vector b 0 {\displaystyle b_{0}} , either a randomly selected vector or an approximation to the eigenvector. The method is described by the iteration

b k + 1 = ( A − μ I ) − 1 b k C k , {\displaystyle b_{k+1}={\frac {(A-\mu I)^{-1}b_{k}}{C_{k}}},}

where C k {\displaystyle C_{k}} are some constants usually chosen as C k = ‖ ( A − μ I ) − 1 b k ‖ . {\displaystyle C_{k}=\|(A-\mu I)^{-1}b_{k}\|.} Since eigenvectors are defined up to multiplication by constant, the choice of C k {\displaystyle C_{k}} can be arbitrary in theory; practical aspects of the choice of C k {\displaystyle C_{k}} are discussed below. At every iteration, the vector b k {\displaystyle b_{k}} is multiplied by the matrix ( A − μ I ) − 1 {\displaystyle (A-\mu I)^{-1}} and normalized. It is exactly the same formula as in the power method, except replacing the matrix A {\displaystyle A} by ( A − μ I ) − 1 . {\displaystyle (A-\mu I)^{-1}.}

The closer the approximation μ {\displaystyle \mu } to the eigenvalue is chosen, the faster the algorithm converges; however, incorrect choice of μ {\displaystyle \mu } can lead to slow convergence or to the convergence to an eigenvector other than the one desired. In practice, the method is used when a good approximation for the eigenvalue is known, and hence one needs only few (quite often just one) iterations.

Theory and convergence The basic idea of the power iteration is choosing an initial vector b {\displaystyle b} (either an eigenvector approximation or a random vector) and iteratively calculating A b , A 2 b , A 3 b , . . . {\displaystyle Ab,A^{2}b,A^{3}b,...} . Except for a set of zero measure, for any initial vector, the result will converge to an eigenvector corresponding to the dominant eigenvalue. The inverse iteration does the same for the matrix ( A − μ I ) − 1 {\displaystyle (A-\mu I)^{-1}} , so it converges to the eigenvector corresponding to the dominant eigenvalue of the matrix ( A − μ I ) − 1 {\displaystyle (A-\mu I)^{-1}} . Eigenvalues of this matrix are ( λ 1 − μ ) − 1 , . . . , ( λ n − μ ) − 1 , {\displaystyle (\lambda _{1}-\mu )^{-1},...,(\lambda _{n}-\mu )^{-1},} where λ i {\displaystyle \lambda _{i}} are eigenvalues of A {\displaystyle A} . The largest of these numbers corresponds to the smallest of ( λ 1 − μ ) , . . . , ( λ n − μ ) . {\displaystyle (\lambda _{1}-\mu ),...,(\lambda _{n}-\mu ).} The eigenvectors of A {\displaystyle A} and of ( A − μ I ) − 1 {\displaystyle (A-\mu I)^{-1}} are the same, since

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Inverse iteration

Start with the simplest possible case. Write down what Inverse 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 Inverse 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 Inverse 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 Inverse iteration

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

How to study Inverse iteration in 20 minutes

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

Frequently asked questions

What is Inverse iteration in simple terms?

In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known.

Why does Inverse 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 Inverse 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 Inverse iteration.