mathematics

Truncated Newton method

Truncated Newton 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 Truncated Newton method rather than just read about it. In short: The truncated Newton method, originated in a paper by Ron Dembo and Trond Steihaug, also known as Hessian-free optimization, are a family of optimization algorithms designed for optimizing non-linear functions with large numbers of independent variables. A truncated Newton method consists of repeated application of an iterative optimization algorithm to approximately solve Newton's equations, to determine an update…

Key takeaways

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

Reference excerpt

The truncated Newton method, originated in a paper by Ron Dembo and Trond Steihaug, also known as Hessian-free optimization, are a family of optimization algorithms designed for optimizing non-linear functions with large numbers of independent variables. A truncated Newton method consists of repeated application of an iterative optimization algorithm to approximately solve Newton's equations, to determine an update to the function's parameters. The inner solver is truncated, i.e., run for only a limited number of iterations. It follows that, for truncated Newton methods to work, the inner solver needs to produce a good approximation in a finite number of iterations; conjugate gradient has been suggested and evaluated as a candidate inner loop. Another prerequisite is good preconditioning for the inner algorithm.

References

Further reading Grippo, L.; Lampariello, F.; Lucidi, S. (1989). "A Truncated Newton Method with Nonmonotone Line Search for Unconstrained Optimization". J. Optimization Theory and Applications. 60 (3): 401–419. CiteSeerX 10.1.1.455.7495. doi:10.1007/BF00940345. S2CID 18990650. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Nash, Stephen G.; Nocedal, Jorge (1991). "A numerical study of the limited memory BFGS method and the truncated-Newton method for large scale optimization". SIAM J. Optim. 1 (3): 358–372. CiteSeerX 10.1.1.474.3400. doi:10.1137/0801023. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)

Worked examples

Example 1 — a first encounter with Truncated Newton method

Start with the simplest possible case. Write down what Truncated Newton 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 Truncated Newton 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 Truncated Newton 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 Truncated Newton method

In research: Truncated Newton 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 Truncated Newton 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: Truncated Newton method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Applied mathematics stubs, Optimization algorithms and methods, so understanding it makes those chapters shorter.
In everyday life: Look for Truncated Newton 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 Truncated Newton method in 20 minutes

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

Frequently asked questions

What is Truncated Newton method in simple terms?

Why does Truncated Newton 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 Truncated Newton 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 Truncated Newton method.