Hurwitz polynomial

Key takeaways

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

Reference excerpt

In mathematics, a Hurwitz polynomial (named after German mathematician Adolf Hurwitz) is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. Such a polynomial must have coefficients that are positive real numbers. The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the imaginary axis (i.e., a Hurwitz stable polynomial). A polynomial function P(s) of a complex variable s is said to be Hurwitz if the following conditions are satisfied:

P(s) is real when s is real. The roots of P(s) have real parts which are zero or negative. Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the Routh–Hurwitz stability criterion.

Examples A simple example of a Hurwitz polynomial is:

x 2 + 2 x + 1. {\displaystyle x^{2}+2x+1.}

The only real solution is −1, because it factors as

( x + 1 ) 2 . {\displaystyle (x+1)^{2}.}

In general, all quadratic polynomials with positive coefficients are Hurwitz. This follows directly from the quadratic formula:

x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}

where, if the discriminant b2−4ac is less than zero, then the polynomial will have two complex-conjugate solutions with real part −b/2a, which is negative for positive a and b. If the discriminant is equal to zero, there will be two coinciding real solutions at −b/2a. Finally, if the discriminant is greater than zero, there will be two real negative solutions, because b 2 − 4 a c < b {\displaystyle {\sqrt {b^{2}-4ac}}<b} for positive a, b and c.

Properties For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive (except for quadratic polynomials, which also imply sufficiency). A necessary and sufficient condition that a polynomial is Hurwitz is that it passes the Routh–Hurwitz stability criterion. A given polynomial can be efficiently tested to be Hurwitz or not by using the Routh continued fraction expansion technique.

References

Wayne H. Chen (1964) Linear Network Design and Synthesis, page 63, McGraw Hill.

Worked examples

Example 1 — a first encounter with Hurwitz polynomial

Start with the simplest possible case. Write down what Hurwitz polynomial claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Hurwitz polynomial 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 Hurwitz polynomial 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 Hurwitz polynomial

In research: Hurwitz polynomial appears in science 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 Hurwitz polynomial 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: Hurwitz polynomial is common in secondary-school and first-year university syllabi. It links to neighbouring topics Polynomials, so understanding it makes those chapters shorter.
In everyday life: Look for Hurwitz polynomial 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 Hurwitz polynomial in 20 minutes

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

Frequently asked questions

What is Hurwitz polynomial in simple terms?

In mathematics, a Hurwitz polynomial (named after German mathematician Adolf Hurwitz) is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. Such a polynomial must have coefficie…

Why does Hurwitz polynomial matter?

Because it connects several science 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 Hurwitz polynomial?

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 Hurwitz polynomial.