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MATLAB offers a function polyeig for computing polynomial eigenvalues, which appear, for instance in quadratic eigenvalue problems (see here for some applications) such as:

\begin{equation} (M\lambda^2 + R\lambda + K)x =0 \end{equation}

where $M$, $R$ and $K$ are matrices. Any realistic way of accomplishing this in Mathematica? By realistic I mean one that actually employs specialized algorithms.


P.S. If anyone from Wolfram is reading this: any perspectives on whether this may appear in future releases of Mathematica?

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1  
I had not heard that name for this type of problem before. Definitely an interesting subject. –  rcollyer Aug 24 '12 at 14:32
    
@rcollyer, indeed, a very interesting subject; even now, there is still so much research on efficient methods for the quadratic case that exploit the structure of the coefficient matrices. The book by Gohberg, Lancaster, and Rodman is one very nice reference on the subject. –  J. M. Aug 24 '12 at 14:45
    
@J.M. Looks like I need to renew my membership in SIAM. –  rcollyer Aug 24 '12 at 14:50
    
Interesting question. +1. Could you mention an application where this is needed? –  user21 Aug 24 '12 at 14:56
2  
@J.M, thanks, I'll have a look. I added this as a suggestion for future development - but can not say if/when this will be taken up. –  user21 Aug 24 '12 at 15:17

3 Answers 3

up vote 22 down vote accepted

(I've been waiting for somebody to ask this question for months... :D )

Here's the Mathematica implementation of the Frobenius companion matrix approach discussed by Jim Wilkinson in his venerable book (for completeness and complete analogy with built-in functions, I provide these three):

PolynomialEigenvalues[matCof : {__?MatrixQ}] := 
 Module[{p = Length[matCof] - 1, n = Length[First[matCof]]},
   Eigenvalues[{ArrayFlatten[
      Prepend[NestList[RotateRight, PadRight[{IdentityMatrix[n]}, p], 
        p - 2], -Rest[matCof]]], 
     SparseArray[{Band[{1, 1}] -> First[matCof], {k_, k_} -> 1}, {n p, n p}]}]
   ] /; Precision[matCof] < Infinity && SameQ @@ (Dimensions /@ matCof)

PolynomialEigenvectors[matCof : {__?MatrixQ}] := 
 Module[{p = Length[matCof] - 1, n = Length[First[matCof]]},
   Map[Take[#, n] &, 
    Eigenvectors[{ArrayFlatten[
       Prepend[NestList[RotateRight, PadRight[{IdentityMatrix[n]}, p],
          p - 2], -Rest[matCof]]], 
      SparseArray[{Band[{1, 1}] -> First[matCof], {k_, k_} -> 1}, {n p, n p}]}]]
   ] /; Precision[matCof] < Infinity && SameQ @@ (Dimensions /@ matCof)

PolynomialEigensystem[matCof : {__?MatrixQ}] := 
 Module[{p = Length[matCof] - 1, n = Length[First[matCof]]},
   MapAt[Map[Take[#, n] &, #] &, 
    Eigensystem[{ArrayFlatten[
       Prepend[NestList[RotateRight, PadRight[{IdentityMatrix[n]}, p],
          p - 2], -Rest[matCof]]], 
      SparseArray[{Band[{1, 1}] -> First[matCof], {k_, k_} -> 1}, {n p, n p}]}], 2]
   ] /; Precision[matCof] < Infinity && SameQ @@ (Dimensions /@ matCof)

Here's how to verify that they work as expected:

m = (* matrix dimensions *);
n = (* degree of matrix polynomial *);
pcofs = Table[RandomReal[{-9, 9}, {m, m}, WorkingPrecision -> 20], {n + 1}];

(* should return an array of zeros *)
MapThread[Function[{λ, \[ScriptV]}, Chop[Fold[#1 λ + #2 &, 0, pcofs].\[ScriptV]]],
          PolynomialEigensystem[pcofs]]

(* should return an array of zeros *)
Table[Det[Fold[#1 λ + #2 &, 0, pcofs]] // Chop, {λ, PolynomialEigenvalues[pcofs]}]

There are more efficient ways to solve, say, the quadratic eigenvalue problem if the coefficient matrices have a nice structure (see this, for instance), but at least the method here, based on the QZ algorithm, is general.

share|improve this answer
    
This statement in the paper "a system is largely determined by just a few of the eigenvalues nearest to the real axis" makes me think of Green's functions in condensed matter (or any particle physics). The lifetime of the quasi-particles are inversely proportional to their imaginary frequencies. In other words, the system is dominated by the long lived modes. It makes me wonder how this can be applied to Green's functions. I have not gone through the paper nor your solution enough to vote either way, unfortunately. –  rcollyer Aug 24 '12 at 14:49
    
"I have not gone through the paper nor your solution enough to vote either way, unfortunately." - That's fine. As another note, what I called here the "Frobenius method" is referred to as linearization in the book by Gohberg/Lancaster/Rodman. As for Green's functions, well... –  J. M. Aug 24 '12 at 14:53
    
+1 Very nice. It's nice to see with how little code something like this can be added. –  user21 Aug 24 '12 at 15:19
    
I agree with ruebenko. Definitely worth a vote. –  Daniel Lichtblau Aug 24 '12 at 15:36
    
@Daniel, well, the simplicity of the code is due to ArrayFlatten[] now being available, as well as the Eigen*[] functions now supporting matrix pencils (e.g. Eigenvalues[{a, b}]); so, thanks for that, I guess. :) (This is actually one of the first things I wrote when I first found out about ArrayFlatten[].) –  J. M. Aug 24 '12 at 15:40

As I've already mentioned in my previous answer, the method of using the QZ algorithm for matrix pencils on the Frobenius companion linearization of the polynomial eigenproblem is not always the most efficient approach. To illustrate this, I'll outline a general method for solving a hyperbolic quadratic eigenvalue problem, which is known to have all its eigenvalues real. (Encapsulating the strategy as a working Mathematica routine is left as an exercise.)

The method also makes use of a linearization; this linearization, as described in this paper by Higham, Mackey, Mackey, and Tisseur, makes use of block symmetric matrices whose blocks are block Hankel and block antitriangular. A Mathematica routine for constructing the matrix $X_m(P(\lambda))$ (in the notation of section 3.3 of that paper) follows:

BlockSymmetricBasis[k_Integer?NonNegative, matCof : {__?MatrixQ}] :=
 Module[{p = Length[matCof] - 1, lm, um},
   Switch[k,
    0, -ArrayFlatten[Partition[PadRight[Take[matCof, -p], 2 p - 1], p, 1]],
    p, ArrayFlatten[Partition[PadLeft[Take[matCof, p], 2 p - 1], p, 1]],
    _,
    lm = ArrayFlatten[Partition[PadLeft[Take[matCof, k], 2 k - 1], k, 1]]; 
    um = ArrayFlatten[Partition[PadRight[Take[matCof, k - p], 2 (p - k) - 1], p - k, 1]];
    ArrayFlatten[{{lm, 0}, {0, -um}}]]] /;
  SameQ @@ (Dimensions /@ matCof) && k < Length[matCof]

From here, one now has a family of pencils $X_{m-1}(P(\lambda))-\lambda X_m(P(\lambda))$ at disposal; the key is to choose among these pencils such that $X_m$ is positive definite (i.e., we want to pick out which of the $X_{m-1}(P(\lambda))-\lambda X_m(P(\lambda))$ is a symmetric-definite pencil).

To continue further with the discussion, here is a concrete example of a hyperbolic quadratic eigenvalue problem:

polycof = N@{{{3, 2, 1}, {2, 3, 2}, {1, 2, 3}},
             {{-2, -1, -1}, {-1, -3, 2}, {-1, 2, -1}},
             {{-5, 1, -2}, {1, -4, -3}, {-2, -3, -5}}};

We check which of the $X_m$ are positive definite:

Table[PositiveDefiniteMatrixQ[BlockSymmetricBasis[k, polycof]], {k, 0, 2}]
   {False, True, False}

We thus continue further with the pencil $X_0-\lambda X_1$. Now, we check the condition number $\kappa$ of $X_1$:

X0 = BlockSymmetricBasis[0, polycof]; X1 = BlockSymmetricBasis[1, polycof];
LinearAlgebra`MatrixConditionNumber[X1]
   22.2727

The value of the condition number obtained is rather modest in size, so we can continue with one of the usual approaches for symmetric-definite pencils, which uses Cholesky decomposition. First, build the intermediate matrix:

ℳ = CholeskyDecomposition[X1]; lft = LinearSolve[Transpose[ℳ]];
ℋ = lft[Transpose[lft[X0]]];

Here are the eigenvalues:

λ = Eigenvalues[ℋ]
   {6.61035, -1.8856, 1.37725, 1.21165, -1.06445, -0.124207}

Check the eigenvalues:

Table[Det[Fold[#1 λ + #2 &, 0, polycof]] // Chop, {λ, %}]
    {3.47379*10^-10, 0, 0, 0, 0, 0}

Here are the eigenvectors:

lf = LinearSolve[ℳ];
\[ScriptV] = Take[lf[#], 3] & /@ Eigenvectors[ℋ]
   {{-0.46788, 0.903438, -0.600705}, {0.420022, -0.335861, -0.176273},
    {-0.425712, -0.113381, 0.331892}, {-0.0699161, -0.208497, -0.204744},
    {0.196805, -0.0360833, 0.26503}, {0.069413, 0.115626, -0.103772}}

Check eigenvalues and eigenvectors:

MapThread[Function[{λ, \[ScriptV]}, Chop[Fold[#1 λ + #2 &, 0, polycof].\[ScriptV]]],
          {λ, \[ScriptV]}]
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

Had the result of LinearAlgebra`MatrixConditionNumber[X1] been rather large (say, $\approx 10^7$), an alternative route would have been the eigendecomposition of X1, which would have gone like this:

ℳ = #2.#1 & @@ MapThread[#1@#2 &, {{Composition[DiagonalMatrix, Sqrt], Transpose},
     Eigensystem[X1]}];
lf = LinearSolve[ℳ];
ℋ = lf[Transpose[lf[X0]]]

λ = Eigenvalues[ℋ] (* eigenvalues *)

lft = LinearSolve[Transpose[ℳ]];
\[ScriptV] = Take[lft[#], 3] & /@ Eigenvectors[ℋ] (* eigenvectors *)

See this reference for more information on definite matrix polynomials, which is the general class of matrix polynomials with real eigenvalues.


To summarize: the Frobenius linearization + QZ route works generally, but one should be on the lookout for methods that exploit structure if you will be solving a lot of structured problems, since they will usually be more efficient with space, computational effort, or both.

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Great work you are doing by putting all this together - a great +1 –  Vitaliy Kaurov Aug 25 '12 at 7:34
    
Are the lower case script letters not available in the standard unicode? –  rm -rf Aug 25 '12 at 14:37
    
@R.M., to use \[ScriptV] as an example, this says that the corresponding Unicode is F6C7; if I look this up in a Unicode table, it's listed as a "private use" character that renders like this: . If you paste this in Mathematica, it is automagically turned into \[ScriptV], but it looks unseemly to me if I used that character here. –  J. M. Aug 25 '12 at 14:52
    
Ugh... what I expected. I ran into several such characters, where there is a perfectly valid equivalent in Unicode, but Mathematica uses their own in the private range. A couple of other examples that come to mind are \[Function] and \[Rule]... –  rm -rf Aug 25 '12 at 15:10

[Not really a viable answer but too long for a comment]

Here is a method that is not well suited for numerical matrices, at least not in the first steps. But it is conceptually quite simple. The idea is that the values of lambda for which we get nontrivial null vectors are precisely those that make the determinant of the polynomial matrix vanish.

The code below uses an interpolation method to compute this determinant. That is not (conceptually, once again) needed. But it works around the limited heuristics of built-in Det.

interpDet[mat_, lam_, deg_] := Module[
  {vals, mats},
  mats = Table[mat /. lam -> j, {j, Length[mat]*deg + 1}];
  vals = Det /@ mats;
  InterpolatingPolynomial[vals, lam]]

The main body of code is quite short. Here I allow for working with finite precision in the eigenvector step. This is purely for speed purposes.

polyEigensystem[matrices : {__?MatrixQ}, prec_: Infinity] := Module[
  {lam, n = Length[matrices], mat, det, eigvals, eigvecs}, 
  mat = Plus @@ (matrices*lam^Range[n - 1, 0, -1]);
  det = interpDet[mat, lam, n - 1]; (*could use Det[] instead...*)
  eigvals = Union[Solve[det == 0, lam]];
  eigvecs = Map[NullSpace, N[mat /. eigvals, prec]];
  {lam /. eigvals, Flatten[eigvecs, 1]}]

Borrowing the testing code from JM:

m = 12;
n = 3;
SeedRandom[11111];
pcofs = Table[RandomInteger[{-9, 9}, {m, m}], {n + 1}];

Quick test. Though nowhere near as quick as JM's code. But it does give exact eigenvalues.

Timing[e1 = polyEigensystem[pcofs, 20];]

Out[490]= {1.15, Null}

In[500]:= e1[[1]] // N // Sort

(* Out[500]= {-4.90804, -1.22881, -1.08631 - 0.460071 I, -1.08631 + 
  0.460071 I, -1.00005 - 0.536795 I, -1.00005 + 
  0.536795 I, -0.812783 - 1.17189 I, -0.812783 + 
  1.17189 I, -0.613395 - 0.983785 I, -0.613395 + 
  0.983785 I, -0.5483, -0.216034 - 0.114503 I, -0.216034 + 
  0.114503 I, -0.0949477 - 0.755516 I, -0.0949477 + 
  0.755516 I, -0.0475538 - 1.59625 I, -0.0475538 + 1.59625 I, 
 0.134809 - 0.691959 I, 0.134809 + 0.691959 I, 0.266427 - 1.00571 I, 
 0.266427 + 1.00571 I, 0.336446, 0.408772 - 0.625837 I, 
 0.408772 + 0.625837 I, 0.564009 - 0.729785 I, 0.564009 + 0.729785 I, 
 0.605961 - 0.0502088 I, 0.605961 + 0.0502088 I, 0.60609 - 0.527523 I,
  0.60609 + 0.527523 I, 0.722651 - 1.9656 I, 0.722651 + 1.9656 I, 
 0.956883 - 0.0966356 I, 0.956883 + 0.0966356 I, 1.23438 - 1.22286 I, 
 1.23438 + 1.22286 I} *)

Here is the result from JM's code:

Timing[e2 = PolynomialEigensystem[N[pcos, 20]];]

Out[495]= {0.6, Null}

In[501]:= e2[[1]] // N // Sort

(* Out[501]= {-4.90804, -1.22881, -1.08631 - 0.460071 I, -1.08631 + 
  0.460071 I, -1.00005 - 0.536795 I, -1.00005 + 
  0.536795 I, -0.812783 - 1.17189 I, -0.812783 + 
  1.17189 I, -0.613395 - 0.983785 I, -0.613395 + 
  0.983785 I, -0.5483, -0.216034 - 0.114503 I, -0.216034 + 
  0.114503 I, -0.0949477 - 0.755516 I, -0.0949477 + 
  0.755516 I, -0.0475538 - 1.59625 I, -0.0475538 + 1.59625 I, 
 0.134809 - 0.691959 I, 0.134809 + 0.691959 I, 0.266427 - 1.00571 I, 
 0.266427 + 1.00571 I, 0.336446, 0.408772 - 0.625837 I, 
 0.408772 + 0.625837 I, 0.564009 - 0.729785 I, 0.564009 + 0.729785 I, 
 0.605961 - 0.0502088 I, 0.605961 + 0.0502088 I, 0.60609 - 0.527523 I,
  0.60609 + 0.527523 I, 0.722651 - 1.9656 I, 0.722651 + 1.9656 I, 
 0.956883 - 0.0966356 I, 0.956883 + 0.0966356 I, 1.23438 - 1.22286 I, 
 1.23438 + 1.22286 I} *)
share|improve this answer
    
The thing with using Solve[] is that it won't return the infinite eigenvalues (which will happen if the leading coefficient matrix is singular); if you're just computing eigenvalues, it's easy to do a PadRight[eigvals, n p, Infinity], but I don't know a straightforward way to isolate the eigenvectors corresponding to infinite eigenvalues. Also, Union[] is a bad idea if your $\lambda$-matrix genuinely has multiple eigenvalues. –  J. M. Aug 26 '12 at 2:18
    
@J.M. The point of using Union is that you do not want to repeat eigenvectors. The null space might have dimension anywhere between 1 and the multiplicity. The number of elements it gives will be the dimension of that null space. Regarding the other issue you raise, I confess I did not consider infinite eigenvalues. –  Daniel Lichtblau Aug 26 '12 at 3:10

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