A factorization of the inverse of the shifted companion matrix

Transcription

1 Electronic Journal of Linear Algebra Volume 26 Volume 26 (203) Article A factorization of the inverse of the shifted companion matrix Jared L Aurentz jaurentz@mathwsuedu Follow this and additional works at: Recommended Citation Aurentz, Jared L (203), "A factorization of the inverse of the shifted companion matrix", Electronic Journal of Linear Algebra, Volume 26 DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository For more information, please contact scholcom@uwyoedu

2 A FACTORIZATION OF THE INVERSE OF THE SHIFTED COMPANION MATRIX JARED L AURENTZ Abstract A common method for computing the zeros of an n-th degree polynomial is to compute the eigenvalues of its companion matrix C A method for factoring the shifted companion matrix C ρi is presented The factorization is a product of 2n essentially 2 2 matrices and requires O(n) storage Once the factorization is computed it immediately yields factorizations of both (C ρi) and (C ρi) The cost of multiplying a vector by (C ρi) is shown to be O(n), and therefore, any shift and invert eigenvalue method can be implemented efficiently This factorization is generalized to arbitrary forms of the companion matrix Key words Polynomial, Root, Companion matrix, Shift and invert AMS subject classifications 65F8, 5A8 Introduction Finding the roots of an n-th degree polynomial is equivalent to finding the eigenvalues of the corresponding companion matrix It has been shown in [] that the companion matrix can be factored into a product of essentially 2 2 matrices Moreover, these factors can be reordered arbitrarily to produce different forms of the companion matrix Recently, methods have been proposed in [2] that exploit this structure to compute the entire set of roots in O(n 2 ) time For large n computing the entire set of roots may be infeasible One option for computing a subset of the roots is to use a shift and invert strategy We will show that the shifted companion matrix, the inverse of the shifted companion matrix and the adjoints of both these matrices can be written in a factored form that requires O(n) storage and can be applied to a vector in O(n) operations This means that the inverse power method or any subspace iteration method can be used to compute a modest subset of the spectrum in O(n) time In Section 2, we introduce the idea of core transformation and describe the Fielder factorization In Section 3, we state and prove the main factorization theorems In Section 4, we extend the factorization to related operators In Section 5, we give some numerical examples that illustrate the use of the factorization to compute a few roots of a polynomial of high degree Received by the editors on August 23, 202 Accepted for publication on May 4, 203 Handling Editor: Bryan L Shader Department of Mathematics, Washington State University, Pullman, WA , USA (jaurentz@mathwsuedu) 23

3 232 JL Aurentz 2 The factored companion matrix In order to describe the Fiedler factorization we introduce the idea of a core transformation A core transformation is any matrix G i that is the identity matrix everywhere except at the intersection of rows and columns i and i + These matrices are essentially 2 2 and can be viewed as transformations that act on either two adjacent rows or two adjacent columns Graphically we represent a core transformation with an arrowed bracket,, where the arrowheads represent the rows that are transformed It is easy to see that two core transformations G i and G j commute, G i G j = G j G i as long as i j > This can be seen graphically as two brackets that don t overlap, = = If i j, then the factors do not commute in general This appears graphically as overlapping brackets, Fiedler showed that for an n-th degree, monic polynomial p(x) = x n +a x n + +a n x+a n the associated upper-hessenberg companion matrix a a n a n 0 C = 0 can be written as a product of n core transformations, C = C C 2 C n C n For i n, the C i have the form I i C i = a i 0, (2) I n i

4 A Factorization of the Inverse of the Shifted Companion Matrix 233 where I k is the k k identity matrix For i = n, there is a special factor, C n = diag{,,, a n } Fiedler also showed that if the factors {C,C 2,,C n } are multiplied together in an arbitrary order, the resulting product is similar to C and therefore can also be considered a companion matrix for p(x) In any such product, many of the factors commute, so we only need to keep track of the ones that don t To do this we introduce the position vector p This is an (n )-tuple that takes the value p i = 0 if C i is to the left of C i+, and p i = if C i+ is to the left of C i The products of the C i are not unique to the position vector p since C i and C j commute if i j > This means the position vector p represents an equivalence class of companion matrices For example, when n = 4, the product C = C C 3 C 2 C 4 is equivalent to the products C 3 C C 2 C 4, C C 3 C 4 C 2 and C 3 C C 4 C 2 since they are all consistent with the same position vector p = {0,,0} Definition 2 Let {G,G 2,,G n } be a set of core transformations and let p be a position vector that describes the relative order of the G i Any product of these core transformations that is consistent with p will be written as {G,G 2,,G n } p (a) Upper-Hessenberg (b) Lower-Hessenberg (c) CMV-pattern Fig 2: Graphical representation of some factorizations Example 22 (Upper-Hessenberg matrix) For the upper-hessenberg companion matrix the factorization has position vector p = {0,0,,0} This means that C can be written as C = p {C,C 2,,C n } = C C 2 C n This is represented graphically in Figure 2a as a descending sequence of core transformations Example 23 (Lower-Hessenberg matrix) For the lower-hessenberg companion matrix the factorization has position vector p = {,,,} This means that C can be written as C = p {C,C 2,,C n } = C n C n C This is represented graphically in Figure 2b as an ascending sequence of core transformations

5 234 JL Aurentz Example 24 (CMV matrix) For the CMV companion matrix the factorization has position vector p = {0,,0,,} This means that C can be written as C = p {C,C 2,,C n } = (C C 3 )(C 2 C 4 ) This is represented graphically in Figure 2c as a zigzag sequence of core transformations 3 Factorization theorems In this section, we will show that for the upper- Hessenberg companion matrix C there exists a sequence of 2n core transformations whose product equals C ρi We will then extend the result to companion matrices with arbitrary position vector p Before we state and prove the main results we introduce the Horner vector, h(ρ) C n This vector contains all the intermediate steps used when applying Horner s rule to the polynomial p, at the shift ρ The entries are defined recursively, h (ρ) = ρ+a, h i+ (ρ) = ρh i (ρ)+a i+ (3) Notice that h n (ρ) = p(ρ) We also introduce two new types of core transformations: R i = I i ρ 0 I n i (32) for i =,,n, I i H i = h i (ρ) 0 I n i (33) for i =,,n, and H n = diag{i n, h n (ρ)} Here I k is still the k k identity matrix Theorem 3 For an n-th degree, monic polynomial, p(x) = x n + a x n + + a n x + a n, with a n 0, the shifted, upper-hessenberg companion matrix C associated with p(x), can be written as the product of 2n core transformations, C ρi = H H 2 H n 2 H n H n R n R n 2 R 2 R, with H i and R i defined as in (32) and (33), respectively Proof The case n = is trivial, C ρi = [ a ρ] = [ h (ρ)] = H n For n >,

6 A Factorization of the Inverse of the Shifted Companion Matrix 235 assume that after k steps, we have C ρi = I k h k (ρ) a k+ H H k ρ R k R We compute the next factor by first performing an interchange of rows k and k +, I k h k (ρ) a k+ ρ = I k 0 0 I n k I k ρ h k (ρ) a k+ Then we use the correct Gauss transformation to eliminate the (k +,k) entry, I k ρ h k (ρ) a k+ = I k 0 h k (ρ) I n k I k ρ 0 h k+ (ρ) Combining the row interchange with the Gauss transform we get the factor H k, I k I k h k (ρ) a k+ ρ = H ρ k 0 h k+ (ρ)

7 236 JL Aurentz The last step is to eliminate the (k,k +) entry using another Gauss transform, I k ρ 0 h k+ (ρ) = I k 0 0 h k+ (ρ) I k ρ 0 This core transformation is exactly R k, I k I k ρ 0 h k+ (ρ) = 0 0 h k+ (ρ) R k I n k Putting it all together we get I k I k h k (ρ) a k+ ρ = H h k+(ρ) a k+2 k ρ R k, which implies that C ρi = I k h k+(ρ) a k+2 H H k ρ R k R Example 32 (Upper-Hessenberg factors) When n = 3 the shifted, upper- Hessenberg companion matrix admits the factorization C ρi = H H 2 H 3 R 2 R, h (ρ) a 2 a 3 ρ = ρ h (ρ) 0 h 2 (ρ) 0 h 3 (ρ) ρ ρ 0 0

8 A Factorization of the Inverse of the Shifted Companion Matrix 237 It should be noted that this factorization is a natural extension of the Fiedler factorization in the sense that when ρ = 0, H i = C i and R i = I for all i, C 0I = H H n H n R n R = C C n C n In order to extend the factorization to companion matrices with arbitrary position vectors p, we need the following lemma Lemma 33 For an n-th degree, monic polynomial, p(x) = x n +a x n + + a n x+a n, with a n 0, the companion matrix C associated with p(x) and position vector p, has only one non-zero diagonal entry, namely C, = a Proof The case n = and n = 2 are trivial For the induction step it is sufficient to note that the k-th diagonal entry of C is uniquely determined by the factors C k and C k and the (k ) entry of p This is true because core transformations only act locally For p k = 0, the k-th diagonal of C is the k-th diagonal entry of C k C k, I k 2 a k a k C k C k = I n k The k-th diagonalentry is 0 For p k =, the k-th diagonalof C is the k-th diagonal entry of C k C k, I k 2 a k 0 C k C k = a k I n k The importance of Lemma 33 is that every companion matrix formed via the Fiedler factors has the same main diagonal Definition 34 Let P i C n n for i =,,n be the following diagonal matrix 0 P i = 0 I n i

9 238 JL Aurentz When i = 0, P i = I n Corollary 35 For an n-th degree, monic polynomial, p(x) = x n +a x n + +a n x+a n, with a n 0, the shifted, companion matrix C associated with p(x) and position vector p, can be written as {H,C 2,,C n } ρp p Proof By Lemma 33 we know the main diagonal of C ρi, a ρ h (ρ) C ρi = ρ = ρ ρ ρ Now we subtract off ρp and leave the rest of C ρi in the matrix D, C ρi = D ρp where h (ρ) D = 0 0 Since C and D only differ in the (,) entry D has the same off diagonal entries as C The (,) entry of C was determined by the factor C Replacing the (,) entry of C with h (ρ) transforms C into H Therefore,the matrix D can be written as D = p {H,C 2,,C n }, and therefore, C ρi = p {H,C 2,,C n } ρp Theorem 36 For an n-th degree, monic polynomial, p(x) = x n + a x n + +a n x+a n, with a n 0, the shifted, companion matrix C associated with p(x) and position vector p, can be written as the product of 2n core transformations, C ρi = A A 2 A n 2 A n H n B n B n 2 B 2 B,

10 A Factorization of the Inverse of the Shifted Companion Matrix 239 where A i = H i and B i = R i when p i = 0 and A i = R T i and B i = H i when p i = with H i and R i defined as in (32) and (33), respectively Proof The case n = is the same as in Theorem 3 For n >, the induction step has two cases depending on the value of p k For the first case, we start by assuming that p k = 0 and after k steps C ρi has been factored as C ρi = A A k ( p {H k,c k+,,c n } ρp k )B k B, where the A i and B i are chosen exactly as in the statement of the theorem We begin by removing the factor H k exactly as in Theorem 3 This does not affect the rest of the C i for i > k + because non-intersecting core transformations commute, A A k H k ( p {I,C k+,,c n } ρh k P k ) B k B Let s take a closer look at the product ρh k P k, ρh k P k = 0 0 ρ 0 ρh k (ρ) ρ This product can be written as ρh k P k = F k +ρp k+, where F k = 0 0 ρ 0 ρh k (ρ) 0 Since F k is non-zero only at the intersection of rows and columns k and k+, the only C i in the product p {I,C k+,,c n } that F k interacts with is C k+ Computing

11 240 JL Aurentz the sum we get I k ρ 0 C k+ F k = 0 h k+ (ρ) 0 0 I n k 2 Multiplying by the correct Gauss transform on the right to eliminate the (k,k + ) entry we get I k I k 0 0 ρ 0 C k+ F k = 0 h k+ (ρ) which is exactly I n k 2 C k+ F k = H k+ R k Putting this all together we obtain ( ) H k p {I,C k+,,c n } ρh k P k = and since P k+ R k = P k+, we obtain ( ) H k p {I,C k+,,c n } ρh k P k Using the induction hypothesis we get our result, C ρi = H k ( p {H k+,c k+2,,c n }R k ρp k+ ), = H k ( p {H k+,c k+2,,c n } ρp k+ )R k I n k 2 A A k H k ( p {H k+,c k+2,,c n } ρp k+ )R k B k B For the second case we start by assuming that p k = and after k steps C ρi has been factored as C ρi = A A k ( p {H k,c k+,,c n } ρp k )B k B

12 A Factorization of the Inverse of the Shifted Companion Matrix 24 We begin by removing the factor H k but this time we factor on the right since p k = tells us that H k is to the right of C k+, ) A A k ( p {I,C k+,,c n } ρp k H k H k B k B Again taking a closer look at the product ρp k H k, 0 ρp k H 0 0 k = ρ ρh k (ρ) ρ This product can be written as where Fk T = ρp k H k = F T k +ρp k ρ ρh k (ρ) 0 Since Fk T is non-zeroonly at the intersectionofrowsand columns k and k+, the only C i in the product p {I,C k+,,c n } that F k interacts with is C k+ Computing the sum we get I k 0 0 C k+ Fk T = ρ h k+ (ρ) 0 0 I n k 2 Multiplying by the correct Gauss transform on the left to eliminate the (k +,k) entry we get I k I k C k+ Fk T = ρ 0 0 h k+ (ρ) I n k 2 I n k 2

13 242 JL Aurentz which is exactly Putting this all together we get ( p {I,C k+,,c n } ρp k H k C k+ F T k = R T kh k+ ( R T k ) H k = p {H k+,c k+2,,c n } ρp k+ ) H k, and since (Rk T) P k+ = P k+, we obtain ( ) p {I,C k+,,c n } ρp k H k H k = ( ) p {H k+,c k+2,,c n } ρp k+ H k R T k Using the induction hypothesis we get our result, C ρi = A A k R T k ( p {H k+,c k+2,,c n } ρp k+ ) H k B k B It should be noted that Theorem 3 is a special case of Theorem 36 Example 37 (CMV factors) When n = 3 the shifted, CMV companion matrix with p = {,0} admits the factorization C ρi = R T H 2 H 3 R 2 H, h (ρ) 0 a 2 ρ a 3 = ρ 0 ρ h 2 (ρ) 0 h 3 (ρ) ρ 0 h (ρ) 0 Schematically, the factorization looks like (C ρi) = when n = 5 The only difference from one shifted companion matrix to another is which side of the V a particular H i, R i or R T i resides

14 A Factorization of the Inverse of the Shifted Companion Matrix Extensions In this section, we will see that this factorization gives easy access to some other operators Remember what C ρi looks like, C ρi = A A n H n B n B In order to use a shift and invert strategy to compute the roots it is necessary to have access to the inverse of the shifted operator Here we get the factored inverse of the shifted matrix for free, (C ρi) = B Bn H n A n A Example 4 (CMV factors) When n = 3 the inverse of the shifted, CMV companion matrix with p = {, 0} admits the factorization (C ρi) = (H ) (R 2 ) (H 3 ) (H 2 ) (R T ), ρ 2 ρ a 3 h 3 (ρ) ρa 2 +a 3 ρh (ρ) h (ρ)a 3 = ρ h 2 (ρ) 0 h (ρ) ρ 0 h 3 (ρ) 0 0 ρ h 2 (ρ) Some sparse eigenvalue methods, for instance unsymmetric-lanczos, require access to the adjoint We also get the factored adjoint for free, (C ρi) = B B n H na n A The adjoint of the inverse is available as well, ((C ρi) ) = (A ) (An ) (Hn ) (Bn ) (B ) It is interesting to note that all of these operators have a similar sparsity structure More over the cost of applying any of these operators to a vector is the same The cost to compute the Horner vector is roughly 2n Once the Horner vector is computed each of the H i and R i or their inverses, transposes or adjoints must be applied to the vector Since each of these core transformations requires 2 flops, except H n, the flop count sums to roughly 4n So the total flop count for applying the operator is roughly 6n and only 4n if the Horner vector has already been computed

15 244 JL Aurentz 5 Numerical experiments In this section, we give two examples of using a shift and invert strategy to compute subsets of the roots of a polynomial of high degree To compute the roots we used the shifted inverse of the upper-hessenberg companion matrix in conjunction with a version of the unsymmetric-lanczos algorithm as outlined in [3] The scaling coefficients were chosen such that the resulting tridiagonal matrix was complex symmetric Its eigenvalues could then be computed using the complex symmetric eigensolver described in [4] Implicit restarts were performed using a Krylov-Schur-like method described in [3] Forthefirstexample,wecomputedthe0rootsofthepolynomialp(x) = x 0000 i that are closest to Since these roots are known, we can compute the relative error exactly λ λ λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 λ 9 λ 0 κ(λ) λ µ µ Table 5: Error table for computed roots of x 0000 i Table 5 shows the relative errors of the computed root λ against the exact root µ Since we are running unsymmetric-lanczos, we can compute a condition number for the roots If v is the right eigenvector and w is the left eigenvector associated with λ then κ(λ) is defined as κ(λ) = v 2 w 2 w v Figure 5 shows the 0 computed roots of i = closest to against the 20 actual roots of i = closest to In this example, the shift and invert strategy computed the 0 closest roots without leaving any gaps For the second example, we

16 A Factorization of the Inverse of the Shifted Companion Matrix 245 Fig 5: Computed Roots of x 0000 i against Actual Roots compute the 0 roots closest to i = of a millionth degree, complex polynomial with coefficients whose real and imaginary parts are both normally distributed with mean 0 and variance λ i λ i λ i λ i λ i λ i λ i λ i λ i λ i Table 52: Computed roots near i = for a millionth degree random polynomial Table52showstheestimatesofthe0rootsnearesti = thatwerecomputed using our implementation Table 53 shows severalresidualsfor the computed roots The value p(λ) λp (λ) comes

17 246 JL Aurentz λ κ(λ) p(λ) λp (λ) Cv λv C v λ λ λ λ λ λ λ λ λ λ Table 53: Error table for computed roots of a millionth degree random polynomial from the first order Taylor expansion of p(µ), where µ is the exact root, centered at the computed root λ, This implies that If µ λ is small then p(λ) p (λ) simply divide by the magnitude of λ, p(µ) = p(λ)+p (λ)(µ λ)+o ( (µ λ) 2) µ λ = p(λ) p (λ) +O( µ λ 2) infinity norm residual of the Ritz pair (λ,v) is a good error estimate To get a relative error estimate p(λ) λp (λ) The last column of Table 53 is the 6 Conclusions We have shown that arbitrary shifted, Fiedler companion matrices can be factored into 2n essentially 2 2 matrices and therefore require O(n) storage This factorization gives immediate access to the factorization of the inverse of the shifted companion matrix as well We have also shown that the cost of applying either of these factorizationsto a vector is O(n) making it possible to employ any shift and invert eigenvalue method efficiently Acknowledgment I thank the editor and referee for their careful reading of the manuscript and for their questions, criticisms, and suggestions, which lead to significant improvements in the paper

18 A Factorization of the Inverse of the Shifted Companion Matrix 247 REFERENCES [] M Fiedler A note on companion matrices Linear Algebra and its Applications, 372:325 33, 2003 [2] JL Aurentz, R Vandebril, and DS Watkins Fast computation of the zeros of a polynomial via factorization of the companion matrix SIAM Journal on Scientific Computing, 35(): , 203 [3] DS Watkins The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods SIAM, Philadelphia, 2007 [4] JK Cullum and RA Willoughby Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol Theory Birkhauser, Boston, 985