Faddeev–LeVerrier algorithm: Difference between revisions

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[[Image:Urbain Le Verrier.jpg|220px|thumb|right|[[Urbain Le Verrier]] (1811&ndash;1877)<br> The discoverer of [[Neptune]].]]

In mathematics ([[linear algebra]]), the '''Faddeev–LeVerrier algorithm''' is a [[Recurrence relation|recursive]] method to calculate the coefficients of the [[characteristic polynomial]] <math>pp_A(\lambda)=\det (\lambda I_n - A)</math> of a square [[Matrix (mathematics)|matrix]], {{mvar|A}}, named after [[Dmitry Konstantinovich Faddeev]] and [[Urbain Le Verrier]]. Calculation of this polynomial yields the [[eigenvalue]]s of {{mvar|A}} as its roots; as a matrix polynomial in the matrix {{mvar|A}} itself, it vanishes by the fundamental [[Cayley–Hamilton theorem]]. Computing determinantthe characteristic polynomial directly from the definition of characteristicthe polynomial, however,determinant is computationally cumbersome insofar as it introduces a new symbolic quantity <math>\lambda</math>; by contrast, the Faddeev-Le Verrier algorithm works directly with coefficients of matrix <math>A</math>.

because <math>\lambda</math> is new symbolic quantity, whereas this algorithm works directly with coefficients of matrix <math>A</math>.

The algorithm has been independently rediscovered several times, in somedifferent form or anotherforms. It was first published in 1840 by [[Urbain Le Verrier]], subsequently redeveloped by P. Horst, [[Jean-Marie Souriau]], in its present form here by Faddeev and Sominsky, and further by J. S. Frame, and others.<ref>[[Urbain Le Verrier]]: ''Sur les variations séculaires des éléments des orbites pour les sept planètes principales'', ''J. de Math.'' (1) '''5''', 230 (1840), [http://gallica.bnf.fr/ark:/12148/bpt6k163849/f228n35.capture# Online]</ref><ref>Paul Horst: ''A method of determining the coefficients of a characteristic equation''. ''Ann. Math. Stat.'' '''6''' 83-84 (1935), {{DOIdoi|10.1214/aoms/1177732612}}</ref><ref>[[Jean-Marie Souriau]], ''Une méthode pour la décomposition spectrale et l'inversion des matrices'', ''Comptes Rend.'' '''227''', 1010-1011 (1948).</ref><ref>D. K. Faddeev, and I. S. Sominsky, ''Sbornik zadatch po vyshej algebra'' ([http://www.isinj.com/aime/Problems%20in%20Higher%20Algebra%20-%20Faddeev,%20Sominskii%20(MIR,1972).pdf Problems in higher algebra] {{Webarchive|url=https://web.archive.org/web/20160309010057/http://www.isinj.com/aime/Problems%20in%20Higher%20Algebra%20-%20Faddeev,%20Sominskii%20(MIR,1972).pdf |date=2016-03-09 }}, Mir publishers, 1972), MoskowMoscow-Leningrad (1949). Problem '''979'''.</ref><ref>J. S. Frame: ''A simple recursion formula for inverting a matrix (abstract)'', ''Bull. Am. Math. Soc.'' '''55''' 1045 (1949), {{DOIdoi|10.1090/S0002-9904-1949-09310-2}}</ref> (For historical points, see Householder.<ref>

{{cite book|first=Alston S.|last=Householder|title=The Theory of Matrices in Numerical Analysis |publisher=Dover Books on Mathematics|year=2006|author-link=Alston Scott Householder | isbn=0486449726}}</ref> An elegant shortcut to the proof, bypassing [[Newton polynomial]]s, was introduced by Hou.<ref>Hou, S. H. (1998). [http://epubs.siam.org/doi/pdf/10.1137/S003614459732076X "Classroom Note: A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm"] ''SIAM review'' '''40(3)''' 706-709, {{doi|10.1137/S003614459732076X}} .</ref> The bulk of the presentation here follows Gantmacher, p.&nbsp;88.<ref>{{cite book| last= Gantmacher|first=F.R. | title=The Theory of Matrices |year=1960| publisher= Chelsea Publishing|___location= NY | isbn = 0-8218-1376-5 }}</ref>)

==The Algorithm==

The objective is to calculate the coefficients {{math|''c_k''}} of the characteristic polynomial of the {{math|''n''×''n''}} matrix {{mvar|A}},

::<math>pp_A(\lambda)\equiv \det(\lambda I_n-A)=\sum_{k=0}^{n} c_k \lambda^k~,</math>

where, evidently, {{math|''c_n''}} = 1 and(characteristic polynomials are [[Monic polynomial|monic polynomials]]) and {{math|''c''}}₀ = (−1)^''n'' det {{mvar|A}}.

The coefficients are{{math|''c_{n determined− recursivelyi}''}} fromare the top down,determined by dintinduction of the auxiliary matriceson {{mvar|Mi}}, using an auxiliary sequence of matrices

:<math> \begin{align}

M_0 &\equiv 0 & c_n &= 1 \qquad &(k=0) \\

Line 26 ⟶ 25:

:<math>M_3= A^2-A\mathrm{tr} A -\frac{1}{2}\Bigl(\mathrm{tr} A^2 -(\mathrm{tr} A)^2\Bigr) I,</math>

::<math>c_{n-3}=- \tfrac{1}{6}\Bigl( (\operatorname{tr}A)^3-3\operatorname{tr}(A^2)(\operatorname{tr}A)+2\operatorname{tr}(A^3)\Bigr)=-\frac{1}{3}(c_n \mathrm{tr} A^3+c_{n-1} \mathrm{tr} A^2 +c_{n-2}\mathrm{tr} A); </math>

etc.,<ref>Zadeh, Lotfi A. and Desoer, Charles A. (1963, 2008). ''Linear System Theory: The State Space Approach'' (Mc Graw-Hill; Dover Civil and Mechanical Engineering) {{ISBN|9780486466637}} , pp 303&ndash;305;

</ref><ref>Abdeljaoued, Jounaidi and Lombardi, Henri (2004). ''Méthodes matricielles - Introduction à la complexité algébrique'',

(Mathématiques et Applications, 42) Springer, {{ISBN|3540202471}} .</ref>

Line 38 ⟶ 37:

The proof relies on the modes of the [[adjugate matrix]], {{math|''B_k ≡ M_n−k''}}, the auxiliary matrices encountered. &nbsp;

This matrix is defined by

::<math>(\lambda I-A) B = I ~ pp_A(\lambda)</math>

and is thus proportional to the [[Resolvent formalism|resolvent]]

:<math>B = (\lambda I-A)^{-1} I ~ pp_A(\lambda) ~. </math>

It is evidently a matrix polynomial in {{math|''λ''}} of degree {{math|''n−1''}}. Thus,

Line 58 ⟶ 57:

This can be easiest achieved through the following auxiliary equation (Hou, 1998),

::<math>\lambda \frac{\partial pp_A(\lambda) }{ \partial \lambda} -n p =\operatorname{tr} AB~.</math>

This is but the trace of the defining equation for {{mvar|B}} by dint of [[Jacobi's formula]],

:<math>\frac{\partial pp_A(\lambda)}{\partial \lambda}= pp_A(\lambda) \sum^\infty _{m=0}\lambda ^{-(m+1)} \operatorname{tr}A^m =

pp_A(\lambda) ~ \operatorname{tr} \frac{I}{\lambda I -A}\equiv\operatorname{tr} B~. </math>

Inserting the polynomial mode forms in this auxiliary equation yields

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:<math> M_{m} =A M_{m-1} - \frac{1}{m-1} (\operatorname{tr}A M_{m-1}) I ~,</math>

and, in comportance with the [[Cayley–Hamilton theorem]],

:<math> \operatorname{adj}(A) =(-1)^{n-1} M_{n}=(-1)^{n-1} (A^{n-1}+c_{n-1}A^{n-2}+ ...+c_2 A+ c_1 I)=(-1)^{n-1} \sum_{k=1}^n c_k A^{k-1}~.</math>

The final solution might be more conveniently expressed in terms of complete exponential [[Bell polynomials]] as

Line 91 ⟶ 88:

==An equivalent but distinct expression==

A compact determinant of an {{mvar|m}}×{{mvar|m}}-matrix solution for the above Jacobi's formula may alternatively determine the coefficients {{mvar|c}},<ref>Brown, Lowell S. (1994). ''Quantum Field Theory'', Cambridge University Press. {{ISBN|978-0-521-46946-3}}, p. 54; Also see, Curtright, T. L., Fairlie, D. B. and Alshal, H. (2012). "A Galileon Primer", arXiv:1212.6972 , section 3.</ref><ref>{{Cite book|title=Methods of Modern Mathematical Physics|last1=Reed|first1=M.|last2=Simon|first2=B.|publisher=ACADEMIC PRESS, INC.|year=1978|isbn=0-12-585004-2|volume=Vol. 4 Analysis of Operators|___location=USA|pages=323-333323–333, 340, 343}}</ref>

:<math>c_{n-m} = \frac{(-1)^m}{m!}

\begin{vmatrix} \operatorname{tr}A & m-1 &0&\cdots&0\\

\operatorname{tr}A^2 &\operatorname{tr}A& m-2 &\cdots&0\\

\vdots & \vdots & & & \vdots \\

\operatorname{tr}A^{m-1} &\operatorname{tr}A^{m-2}& \cdots & \cdots & 1 \\

Line 102 ⟶ 99:

== See also ==

* [[Characteristic polynomial]]

* [[Exterior algebra#Leverrier's%20algorithm|Exterior algebra § Leverrier's algorithm]]

* [[JacobiHorner's formulamethod]]

* [[Fredholm determinant]]

==References==

{{reflist}}

Barbaresco F. (2019) Souriau Exponential Map Algorithm for Machine Learning on Matrix Lie Groups. In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science, vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_10

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