mathematics

Stone's method

Stone's method 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 Stone's method rather than just read about it. In short: In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations. The method uses an incomplete LU decomposition, which approximates the exact LU decomposition, to get an iterative solution of the problem.

Key takeaways

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

Reference excerpt

In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations. The method uses an incomplete LU decomposition, which approximates the exact LU decomposition, to get an iterative solution of the problem. The method is named after Harold S. Stone, who proposed it in 1968. The LU decomposition is an excellent general-purpose linear equation solver. The biggest disadvantage is that it fails to take advantage of coefficient matrix to be a sparse matrix. The LU decomposition of a sparse matrix is usually not sparse, thus, for a large system of equations, LU decomposition may require a prohibitive amount of memory and number of arithmetical operations. In the preconditioned iterative methods, if the preconditioner matrix M is a good approximation of coefficient matrix A then the convergence is faster. This brings one to idea of using approximate factorization LU of A as the iteration matrix M. A version of incomplete lower-upper decomposition method was proposed by Stone in 1968. This method is designed for equation system arising from discretisation of partial differential equations and was firstly used for a pentadiagonal system of equations obtained while solving an elliptic partial differential equation in a two-dimensional space by a finite difference method. The LU approximate decomposition was looked in the same pentadiagonal form as the original matrix (three diagonals for L and three diagonals for U) as the best match of the seven possible equations for the five unknowns for each row of the matrix.

Algorithm method stone is For the linear system Ax = b calculate incomplete LU factorization of matrix A Ax = (M-N)x = (LU-N)x = b Mx(k+1) = Nx(k)+b , with ||M|| >> ||N|| Mx(k+1) = LUx(k+1) = c(k) LUx(k) = L(Ux(k+1)) = Ly(k) = c(k) set a guess k = 0, x(k) r(k)=b - Ax(k) while ( ||r(k)||2 ≥ ε ) do evaluate new right hand side c(k) = Nx(k) + b solve Ly(k) = c(k) by forward substitution y(k) = L−1c(k) solve Ux(k+1) = y(k) by back substitution x(k+1) = U−1y(k) end while

Footnotes

References Stone, H. L. (1968). "Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations". SIAM Journal on Numerical Analysis. 5 (3): 530–538. Bibcode:1968SJNA....5..530S. doi:10.1137/0705044. hdl:10338.dmlcz/104038. - the original article Ferziger, J.H. and Peric, M. (2001). Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin. ISBN 3-540-42074-6.{{cite book}}: CS1 maint: multiple names: authors list (link) Acosta, J.M. (2001). Numerical Algorithms for Three Dimensional Computational Fluid Dynamic Problems. PhD Thesis. Polytechnic University of Catalonia. This article incorporates text from the article Stone's_method on CFD-Wiki that is under the GFDL license.

Worked examples

Example 1 — a first encounter with Stone's method

Start with the simplest possible case. Write down what Stone's method 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 Stone'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 Stone'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 Stone's method

In research: Stone's method 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 Stone'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: Stone's method 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 Stone'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 Stone's method in 20 minutes

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

Frequently asked questions

What is Stone's method in simple terms?

Why does Stone's method 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 Stone'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 Stone's method.