Knuth–Eve algorithm: Difference between revisions

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{{Short description|Algorithm for evaluating polynomials}}

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TheIn [[computer science]], the '''Knuth-EveKnuth–Eve algorithm''' is an [[algorithm]] for [[polynomial evaluation]]. Specifically, it can evaluate an arbitraryIt [[polynomial]]Polynomial evaluation#Evaluation with [[real number|real]] coefficients of one real variable. It [[Polynomial_evaluation#Evaluation_with_preprocessingpreprocessing|preprocesses]] the coefficients of the polynomial to reduce the number of multiplications required at [[Execution_Execution (computing)#Runtime|runtime]].

The key ideasIdeas used in thisthe algorithm were originally proposed by [[Donald Knuth]] in 1962. His procedure opportunistically exploits structure in the polynomial being evaluated.<ref name="knuth1962"/> In 1964, James Eve determined for which polynomials this structure exists, and they gave a simple method of [[#"Preconditioning"|"preconditioning"]] polynomials (explained below) to endow them with that structure.<ref name="eve1964"/>{{refn|group=note|Article published under the name J. Eve, which is associated with the name James Eve by the ACM Digital Library.<ref name=JamesEveACMDL>{{Cite web |title=James Eve (Newcastle University) |url=https://dl.acm.org/do/10.1145/contrib-81100250587/abs/ |access-date=2025-07-30 |website=Author DO Series | publisher=ACM Digital Library |doi=10.1145/contrib-81100250587/abs |doi-broken-date=31 July 2025 |language=en}}</ref>}}

== Algorithm ==

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=== Preliminaries ===

Consider an arbitrary polynomial <math>p \in \mathbb{R}[x]</math> of [[Degree of a polynomial|degree]] <math>n</math>. Assume that <math>n \geq 3</math>. Define <math>m</math> such that: if <math>n</math> is odd then <math>n = 2m+1</math>, and if <math>n</math> is even then <math>n = 2m+2</math>.<ref name="eve1964"/>

Unless otherwise stated, all variables in this article represent either [[real number]]s or univariate [[polynomial]]s with real coefficients.<ref name="knuth1962"/><ref name="eve1964"/> All operations in this article are done over <math>\mathbb{R}</math>.<ref name="eve1964"/>

Unless otherwise specified, all variables represent real numbers.

Again, the goal is to create an algorithm that returns <math>p(x)</math> given any <math>x \in \mathbb{R}</math>. The algorithm is allowed to depend on the polynomial <math>p</math> itself. And ideally, itsince wouldits usecoefficients asare fewknown operationsin {{--}}advance.<ref as few additions and multiplications {{--}} as possible.name="knuth1962"/>

=== Overview ===

==== Key Ideaidea ====

Using [[polynomial long division]], we can write

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<math display="block"> p(x) = q(x) \cdot (x^2 - \alpha) + (\beta x + \gamma), </math>

where <math>x^2 - \alpha</math> is the divisor. Picking a value for <math>\alpha</math> fixes both the quotient <math>q</math> and the coefficients in the remainder <math>\beta</math> and <math>\gamma</math>. Then, theThe key idea is to cleverly choose <math>\alpha</math> such that <math>\beta = 0</math>, so that

<math display="block"> p(x) = q(x) \cdot (x^2 - \alpha) + \gamma. </math>

Finally<ref name="overill1997"/> This way, weno operations are needed to compute the remainder polynomial, since it's just a constant. We apply this procedure [[Recursion (computer science)|recursively]] to <math>q</math>, expressing

<math display="block"> p(x) = \left( \left( q_mq(x) \cdot (x^2 - \alpha_m) + \gamma_m \right) \cdots \right) \cdot (x^2 - \alpha_1) + \gamma_1. </math>

After <math>m</math> recursive calls, the quotient <math>q_mq</math> is either a [[Linear function (calculus)|linear]] or a [[Quadratic function|quadratic]] polynomial. In this base case, the polynomial can be evaluated with (say) [[Horner's method]].<ref name="knuth1962"/><ref name="overill1997"/><ref name="muller2016"/>

==== "Preconditioning" ====

For arbitrary <math>p</math>, it may not be possible to force <math>\beta = 0</math> at every step of the recursion.<ref name="knuth1962"/> Still, considerConsider the polynomials <math>p^e</math> and <math>p^o</math> with coefficients taken from the even and odd terms of <math>p</math> respectively, so that

<math display="block"> p(x) = p^e(x^2) + x \cdot p^o(x^2). </math>

If every [[Zero of a function#Polynomial roots|root]] of <math>p^o</math> is real, then it is possible to write <math>p</math> in the form given above. Each <math>\alpha_i</math> is a different root of <math>p^o</math>, counting [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiple roots]] as distinct.<ref name="overill1997"/> Furthermore, if allat theleast <math>n-1</math> roots of <math>p</math> (except perhaps one) lie in one half of the [[Complex number|complex plane]], then every root of <math>p^o</math> is real.<ref name="eve1964"/>

Ultimately, it may be necessary to "precondition" <math>p</math> by shifting it {{--}} by setting <math>p(x) \gets p(x + t)</math> for some <math>t</math> {{--}} to endow it with the structure that allmost of its roots lie in one half of the complex plane. At runtime, this shift has to be "undone" by first setting <math>x \gets x - t</math>.<ref name="eve1964"/>

=== Preprocessing Stepsstep ===

The following algorithm is run once for a given polynomial <math>p</math>.<ref name="knuth1962"/><ref name="overill1997"/> At this point, the values of <math>x</math> that <math>p</math> will be evaluated on are not known.<ref name="knuth1962"/>

<div style="margin-left: 35px;">

{{framebox|blue}}

* Let <math>r_1, \cdots, r_n \in \mathbb{C}</math> be the complex roots of <math>p</math>, sorted in descending order by real part

* LetChoose any <math display="inline">t = \max_{i = 1}^ngeq \text{Re}(r_ir_2)</math><hr/>

* LetSet <math>q_0(x)p =\gets p(x + t)</math>

<hr/>

* Let <math>q_0^e</math> and <math>q_0^o</math> be the polynomials such that <math>q_0(x) = q_0^e(x^2) + x \cdot q_0^o(x^2)</math>

* Let <math>\alpha_1, \cdots \alpha_m \in \mathbb{R}p^e</math> be all the roots ofand <math>q_0p^o</math>. Allbe ofthe itspolynomials rootssuch willthat be<math>p(x) real.= p^e(x^2) + x \cdot p^o(x^2)<hr/math>

* Let <math>\alpha_1, \cdots \alpha_m \in \mathbb{R}</math> be the roots of <math>p^o</math>. All of its roots will be real.

<hr/>

* Initialize <math>q \gets p</math>

* For <math>i \gets 1, \cdots, m</math>:

** Divide <math>q_{i-1}q</math> by <math>x^2 - \alpha_i</math> to get quotient <math>q_iq^\prime \in \mathbb{R}[x]</math> and remainder <math>\gamma_i \in \mathbb{R}</math>. The remainder will be a constant polynomial {{--}} a number.

** Set <math>q \gets q^\prime</math>
<hr/>
* ''Output:'' The derived values <math>t</math>, <math>\alpha_1, \cdots, \alpha_m</math>, and <math>\gamma_1, \cdots, \gamma_m</math>; as well as the base-case polynomial <math>q_mq</math>

{{frame-footer}}

</div>

==== RuntimeBetter Stepschoice of t ====

While any <math>t \geq \text{Re}(r_2)</math> can work, it is possible to remove one addition during evaluation if <math>t</math> is also chosen such that two roots of <math>p(x + t)</math> are symmetric about the origin. In that case, <math>\alpha_1</math> can be chosen such that the shifted polynomial has a factor of <math>x^2 - \alpha_1</math>, so <math>\gamma_1 = 0</math>. It is always possible to find such a <math>t</math>.<ref name="eve1964"/>

== Notes ==

One possible algorithm for choosing <math>t</math> is:{{cn|date=July 2025}}

<div style="margin-left: 35px;">

{{framebox|blue}}

* If <math>r_1 \in \mathbb{R}</math>:

** If <math>r_2 \in \mathbb{R}</math>: <math display="inline">t = \tfrac{1}{2} (r_1 + r_2)</math>

** Else: <math>t = \text{Re}(r_2)</math>

* Else: <math>t = \text{Re}(r_1)</math>

{{frame-footer}}

</div>

=== Evaluation step ===

The following algorithm evaluates <math>p</math> at some, now known, point <math>x</math>.<ref name="knuth1962"/><ref name="eve1964"/><ref name="overill1997"/><ref name="muller2016"/>

<div style="margin-left: 35px;">

{{framebox|blue}}

* Set <math>x \gets x - t</math>

* Let <math>s = x^2</math>. Compute this once so it can be reused.

<hr/>

* Compute <math>y \gets q(x)</math> using [[Horner's method]]

* For <math>i \gets m, \cdots, 2, 1</math>:

** Let <math>y \gets y \cdot (s - \alpha_i) + \gamma_i</math>

* ''Output:'' <math>y</math>

{{frame-footer}}

</div>

Assuming <math>t</math> is chosen optimally, <math>\gamma_1 = 0</math>. So, the final iteration of the loop can instead run

<math display="block">y \gets y \cdot (s - \alpha_i),</math>

saving an addition.<ref name="eve1964"/>

== NotesAnalysis ==

In total, evaluation using the Knuth–Eve algorithm for a polynomial of degree <math>n</math> requires <math>n</math> additions and <math>\lfloor n/2 \rfloor + 2</math> multiplications, assuming <math>t</math> is chosen optimally.<ref name="eve1964"/>

No algorithm to evaluate a given polynomial of degree <math>n</math> can use fewer than <math>n</math> additions or fewer than <math>\lceil n/2 \rceil</math> multiplications during evaluation. This result assumes only addition and multiplication are allowed during both preprocessing and evaluation.<ref name="erickson2003"/>{{Better source needed|date=July 2025}}

The Knuth–Eve algorithm is not [[Condition number|well-conditioned]].<ref name="mesztenyi1967"/>

== Footnotes ==

{{Reflist|group=note}}

== References ==

{{reflist |refs=

<ref name="knuth1962">{{cite journal |last1=Knuth |first1=Donald |title=Evaluation of polynomials by computer |journal=Communications of the ACM |date=December 1962 |volume=5 |issue=12 |pages=595-599595–599 |doi=10.1145/355580.369074 |url=https://dl.acm.org/doi/10.1145/355580.369074 |access-date=25 July 2025}}</ref>

<ref name="eve1964">{{cite journal |last1=Eve |first1=J.James |title=The evaluation of polynomials |journal=Numerische Mathematik |date=December 1964 |volume=6 |issue=1 |pages=17-2117–21 |doi=10.1007/BF01386049 |url=https://link.springer.com/article/10.1007/BF01386049 |access-date=25 July 2025|url-access=subscription }}</ref>

<ref name="overill1997">{{cite journal |last1=Overill |first1=Richard |title=Data parallel evaluation of univariate polynomials by the Knuth-Eve algorithm |journal=Parallel Computing |date=12 June 1997 |volume=23 |issue=13 |pages=2115-21272115–2127 |doi=10.1016/S0167-8191(97)00096-3 |url=https://www.sciencedirect.com/science/article/pii/S0167819197000963 |access-date=25 July 2025|url-access=subscription }}</ref>

*<ref name="erickson2003">{{cite web |last1=Erickson |first1=Jeff |title=Evaluating polynomials |date=10 March 2003 |url=https://jeffe.cs.illinois.edu/teaching/497/08-polynomials.pdf |website=CS 497: Concrete Models of Computation |publisher=University of Illinois Urbana-Champaign |access-date=25 July 2025}}</ref>

*<ref name="mesztenyi1967">{{cite journal |last1=Mesztenyi |first1=C.Charles |title=Stable evaluation of polynomials |journal=Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics |date=January 1967 |volume=71B |issue=1 |pages=11-1711–17 |doi=10.6028/jres.071B.003 |url=https://nistdigitalarchives.contentdm.oclc.org/digital/collection/p16009coll6/id/8767787671 |access-date=25 July 2025}}</ref>

*<ref name="muller2016">{{cite book |last1=Muller |first1=Jean-Michel |title=Elementary functions: Algorithms and implementation |date=17 November 2016 |publisher=Birkh&auml;user Boston |___location=Boston, MA |isbn=978-1-4899-7983-4 |doi=10.1007/978-1-4899-7983-4_5 |pages=82-8482–84 |url=https://link.springer.com/chapter/10.1007/978-1-4899-7983-4_5#citeas |access-date=25 July 2025}}</ref>

}}

{{DEFAULTSORT:Knuth-Eve algorithm}}

== References ==

[[Category:Algorithms]]

{{refbegin}}

[[Category:Polynomials]]

* {{cite book |last1=Muller |first1=Jean-Michel |title=Elementary functions: Algorithms and implementation |date=17 November 2016 |publisher=Birkh&auml;user Boston |___location=Boston, MA |isbn=978-1-4899-7983-4 |doi=10.1007/978-1-4899-7983-4_5 |pages=82-84 |url=https://link.springer.com/chapter/10.1007/978-1-4899-7983-4_5#citeas |access-date=25 July 2025}}

[[Category:Algorithms and data structures]]

* {{cite journal |last1=Mesztenyi |first1=C. |title=Stable evaluation of polynomials |journal=Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics |date=January 1967 |volume=71B |issue=1 |pages=11-17 |doi=10.6028/jres.071B.003 |url=https://nistdigitalarchives.contentdm.oclc.org/digital/collection/p16009coll6/id/87677 |access-date=25 July 2025}}

* {{cite web |last1=Erickson |first1=Jeff |title=Evaluating polynomials |url=https://jeffe.cs.illinois.edu/teaching/497/08-polynomials.pdf |website=CS 497: Concrete Models of Computation |publisher=University of Illinois Urbana-Champaign |access-date=25 July 2025}}

{{refend}}