Incomplete Cholesky factorization

Key takeaways

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

Reference excerpt

In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method. The Cholesky factorization of a positive definite matrix A of order N is A = LL* where L is a lower triangular matrix. An incomplete Cholesky factorization is given by a lower triangular matrix K that is sparser than L, but in some sense similar to L. The corresponding preconditioner is KK*. There are many ways of constructing incomplete Cholesky factorizations. This article reviews two of them.

Motivation Consider the following matrix as an example:

A = [ 5 − 2 0 − 2 − 2 − 2 5 − 2 0 0 0 − 2 5 − 2 0 − 2 0 − 2 5 − 2 − 2 0 0 − 2 5 ] {\displaystyle \mathbf {A} ={\begin{bmatrix}5&-2&0&-2&-2\\-2&5&-2&0&0\\0&-2&5&-2&0\\-2&0&-2&5&-2\\-2&0&0&-2&5\\\end{bmatrix}}}

If we apply the full regular Cholesky decomposition, it yields:

L = [ 2.24 0 0 0 0 − 0.89 2.05 0 0 0 0 − 0.98 2.02 0 0 − 0.89 − 0.39 − 1.18 1.63 0 − 0.89 − 0.39 − 0.19 − 1.95 0.45 ] {\displaystyle \mathbf {L} ={\begin{bmatrix}2.24&0&0&0&0\\-0.89&2.05&0&0&0\\0&-0.98&2.02&0&0\\-0.89&-0.39&-1.18&1.63&0\\-0.89&-0.39&-0.19&-1.95&0.45\\\end{bmatrix}}}

And, by definition:

A = L L ′ {\displaystyle \mathbf {A} =\mathbf {L} \mathbf {L'} }

However, by applying Cholesky decomposition, we observe that some zero elements in the original matrix end up being non-zero elements in the decomposed matrix, like elements (4,2), (5,2) and (5,3) in this example. These elements are known as "fill-ins". This is not an issue per se, but it is very problematic when working with sparse matrices, since the fill-ins generation is mostly unpredictable and reduces the matrix sparsity, impacting the efficiency of sparse matrix algorithms. Therefore, given the importance of the Cholesky decomposition in matrix calculations, it is extremely relevant to repurpose the regular method, so as to eliminate the fill-ins generation. Incomplete Cholesky factorizations do exactly that. They yield some matrix K {\displaystyle \mathbf {K} } that is sparser than L {\displaystyle \mathbf {L} } and gives an approximation A ≈ K K ′ {\displaystyle \mathbf {A} \approx \mathbf {K} \mathbf {K'} } .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Incomplete Cholesky factorization

Start with the simplest possible case. Write down what Incomplete Cholesky factorization 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 Incomplete Cholesky factorization 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 Incomplete Cholesky factorization 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 Incomplete Cholesky factorization

In research: Incomplete Cholesky factorization 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 Incomplete Cholesky factorization 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: Incomplete Cholesky factorization 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 Incomplete Cholesky factorization 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 Incomplete Cholesky factorization in 20 minutes

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

Frequently asked questions

What is Incomplete Cholesky factorization in simple terms?

In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method.

Why does Incomplete Cholesky factorization 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 Incomplete Cholesky factorization?

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 Incomplete Cholesky factorization.