mathematics

Posynomial

Posynomial 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 Posynomial rather than just read about it. In short: A posynomial, also known as a posinomial in some literature, is a function of the form f ( x 1 , x 2 , … , x n ) = ∑ k = 1 K c k x 1 a 1 k ⋯ x n a n k {\displaystyle f(x_{1},x_{2},\dots ,x_{n})=\sum _{k=1}^{K}c_{k}x_{1}^{a_{1k}}\cdots x_{n}^{a_{nk}}} where all the coordinates x i {\displaystyle x_{i}} and coefficients c k {\displaystyle c_{k}} are positive real numbers, and the exponents a i k {\displaystyle a_{ik}}…

Key takeaways

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

Reference excerpt

A posynomial, also known as a posinomial in some literature, is a function of the form

f ( x 1 , x 2 , … , x n ) = ∑ k = 1 K c k x 1 a 1 k ⋯ x n a n k {\displaystyle f(x_{1},x_{2},\dots ,x_{n})=\sum _{k=1}^{K}c_{k}x_{1}^{a_{1k}}\cdots x_{n}^{a_{nk}}}

where all the coordinates x i {\displaystyle x_{i}} and coefficients c k {\displaystyle c_{k}} are positive real numbers, and the exponents a i k {\displaystyle a_{ik}} are real numbers. Posynomials are closed under addition, multiplication, and nonnegative scaling. For example,

f ( x 1 , x 2 , x 3 ) = 2.7 x 1 2 x 2 − 1 / 3 x 3 0.7 + 2 x 1 − 4 x 3 2 / 5 {\displaystyle f(x_{1},x_{2},x_{3})=2.7x_{1}^{2}x_{2}^{-1/3}x_{3}^{0.7}+2x_{1}^{-4}x_{3}^{2/5}}

is a posynomial. Posynomials are not the same as polynomials in several independent variables. A polynomial's exponents must be non-negative integers, but its independent variables and coefficients can be arbitrary real numbers; on the other hand, a posynomial's exponents can be arbitrary real numbers, but its independent variables and coefficients must be positive real numbers. This terminology was introduced by Richard J. Duffin, Elmor L. Peterson, and Clarence Zener in their seminal book on geometric programming. Posynomials are a special case of signomials, the latter not having the restriction that the c k {\displaystyle c_{k}} be positive.

References Richard J. Duffin; Elmor L. Peterson; Clarence Zener (1967). Geometric Programming. John Wiley and Sons. p. 278. ISBN 0-471-22370-0. Stephen P Boyd; Lieven Vandenberghe (2004). Convex optimization. Cambridge University Press. ISBN 0-521-83378-7. Harvir Singh Kasana; Krishna Dev Kumar (2004). Introductory Operations Research: Theory and Applications. Springer. ISBN 3-540-40138-5. Weinstock, D.; Appelbaum, J. (2004). "Optimal solar field design of stationary collectors". Journal of Solar Energy Engineering. 126 (3): 898–905. doi:10.1115/1.1756137.

External links S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A Tutorial on Geometric Programming

Worked examples

Example 1 — a first encounter with Posynomial

Start with the simplest possible case. Write down what Posynomial 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 Posynomial 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 Posynomial 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 Posynomial

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

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

Frequently asked questions

What is Posynomial in simple terms?

A posynomial, also known as a posinomial in some literature, is a function of the form f ( x 1 , x 2 , … , x n ) = ∑ k = 1 K c k x 1 a 1 k ⋯ x n a n k {\displaystyle f(x_{1},x_{2},\dots ,x_{n})=\sum _{k=1}^{K}c_{k}x_{1}^{a_{1k}}\cdots x_{n}^{a_{nk}}} where all the coordinates x i {\displaystyle x_{…

Why does Posynomial 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 Posynomial?

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 Posynomial.