nonlinear matrix equations and structured linear algebra

Power series matrix equationsQuadratic matrix equations

Matrix pth rootDARE-type matrix equations

Nonlinear matrix equations and structured linearalgebra

Beatrice Meini

Dipartimento di Matematica, Universita di Pisa, Italy

11th ILAS Conference, Coimbra, July 19–22, 2004

Beatrice Meini Nonlinear matrix equations and structured linear algebra



Some classes of matrix equations

1 Power series matrix equations: X = A−1 + A0X + A1X2 + · · ·

MotivationTheoretical propertiesAlgorithms

2 Quadratic matrix equations: X = A−1 + A0X + A1X2

Quasi-Birth-Death processes“Non-Markovian” matrix equations

3 Matrix pth root: X p = AMatrix sign functionInverse of a matrix Laurent polynomial

4 DARE-type matrix equations: X = C +∑d

i=1 AiX−1Bi

MotivationAlgorithms












i=1 AiX−1Bi






A simple queueing problem

4 3 2 1

One server which attends to one customer at a time, in orderof their arrivals.

Time is discretized into intervals of fixed length.

A random number of customers joins the system during eachinterval.

Customers are indefinitely patient!






Define:

αn: the number of new arrivals in (n − 1, n);Xn: the number of customers in the system at time n.

Then

Xn+1 =

{Xn + αn+1 − 1 if Xn + αn+1 ≥ 1

0 if Xn + αn+1 = 0.

If {αn} are independent random variables, then {Xn} is aMarkov chain with space state N.

If in addition the αn’s are identically distributed, then{Xn} is homogeneous.






The transition matrix P = (pi ,j)i ,j∈N, such that

pi ,j = P[X1 = j |X0 = i ], for all i , j in N.

is

P =

q0 + q1 q2 q3 . . .

q0 q1 q2. . .

q0 q1. . .

0. . .

. . .

where qi is the probability that i new customers join the queueduring a unit time interval.





M/G/1-type Markov chains

Introduced by M. F. Neuts in the 80’s, model a large varietyof queueing problems.

The transition matrix is

P =

B0 B1 B2 B3 . . .A−1 A0 A1 A2 . . .

A−1 A0 A1. . .

A−1 A0. . .

0. . .

. . .

where Ai−1,Bi ∈ Rm×m, for i ≥ 0, are nonnegative such that∑+∞

i=−1 Ai ,∑+∞

i=0 Bi , are stochastic.P is upper block Hessenberg and is block Toeplitz except forits first block row.





M/G/1-type Markov chains

The computation of performance measures of the queueing model(stationary vector, mean waiting time, etc.) is ultimately reducedto the computation of the minimal component-wise solution G ,among the nonnegative solutions, of

X = A−1 + A0X + A1X2 + · · ·





Some properties of G

Let S(z) = zI −∑+∞

i=−1 z i+1Ai .If the M/G/1-type Markov chain is positive recurrent, then:

G is row stochastic.

det S(z) has exactly m zeros in the closed unit disk.

The eigenvalues of G are the zeros of det S(z) in the closedunit disk.

Therefore G is the minimal solvent (Gohberg, Lancaster, Rodman’82)





Some properties of S(z)

The power series S(z) = zI −∑+∞

i=−1 z i+1Ai belongs to theWiener algebra W , therefore it is analytic for |z | < 1,continuous for |z | = 1.

Under some mild additional assumptions S(z) is analytic for|z | < r , where r > 1, and there exists a smallest modulus zeroξ of det S(z) such that 1 < |ξ| < r .





Some properties of G

The function φ(z) = I −∑+∞

i=−1 z iAi has a weak canonicalfactorization in W

φ(z) =(I −

+∞∑i=0

z iUi

)(I − z−1G ), |z | = 1,

where:

U(z) = I −∑+∞

i=0 z iUi is analytic for |z | < 1, det U(z) 6= 0 for|z | ≤ 1;

L(z) = I − z−1G is analytic for |z | > 1, det L(z) 6= 0 for|z | > 1, det L(1) = 0.





Functional iterations

Natural iteration{Xn+1 =

∑+∞i=−1 AiX

i+1n , n ≥ 0

X0 = 0

History Several variants proposed by Neuts (’81, ’89),Ramaswami (’88), Latouche (’93), Bai (’97).

Convergence Convergence analysis performed by Meini (’97), Guo(’99). Convergence is linear, and for some problemsit may be extremely slow.





Linearization of the matrix equation

I − A0 −A1 −A2 . . .

−A−1 I − A0 −A1. . .

−A−1 I − A0. . .

0. . .

. . .

�� GG 2

G 3

...

=

A−1

00...

.

G can be interpreted by means of the solution of an infinite blockHessenberg, block Toeplitz system





Cyclic reduction: history

Introduced in the late ’60s by Buzbee, Golub and Nielson forsolving block tridiagonal systems in the context of ellipticequations.

Stability and convergence properties: Amodio and Mazzia(’94), Yalamov (’95), Yalamov and Pavlov (’96), etc.

Rediscovered by Latouche and Ramaswami (Logarithmicreduction) in the context of Markov chains (’93);

Extended to infinite block Hessenberg, block Toeplitz systemsby Bini and Meini (starting from ’96).





The cyclic reduction algorithm

Original system:I − A0 −A1 −A2 . . .

−A−1 I − A0 −A1. . .

−A−1 I − A0. . .

0. . .

. . .

GG 2

G 3

...

=

A−1

00...






Block even-odd permutation:

I − A0 −A2 . . . −A−1 −A1 . . .

I − A0

. . . −A−1

. . .

0. . . 0

. . .

−A1 −A3 . . . I − A0 −A2 . . .

−A−1 −A1

. . . I − A0

. . .

0. . .

. . . 0. . .

G2

G4

...GG3

...

=

00...

A−1

0...

In compact form:[I − H1 −H2

−H3 I − H4

] [g−g+

]=

[0b

]






Structure of the matrix:

[I − H1 −H2

−H3 I − H4

]=

Schur complementation:

I − H4 − H3(I − H1)−1H2 = +

=

Upper block Hessenberg matrix, block Toeplitz except for its firstblock row






Resulting system:I − A

(1)0 −A

(1)1 −A

(1)2 . . .

−A(1)−1 I − A

(1)0 −A

(1)1 . . .

−A(1)−1 I − A

(1)0

. . .

0. . .

. . .

GG 3

G 5

...

=

A−1

00...






One more step of the same procedure:I − A

(2)0 −A

(2)1 −A

(2)2 . . .

−A(2)−1 I − A

(2)0 −A

(2)1 . . .

−A(2)−1 I − A

(2)0

. . .

0. . .

. . .

GG 5

G 9

...

=

A−1

00...






At the n-th step:I − A

(n)0 −A

(n)1 −A

(n)2 . . .

−A(n)−1 I − A

(n)0 −A

(n)1 . . .

−A(n)−1 I − A

(n)0

. . .

0. . .

. . .

GG 2n+1

G 2·2n+1

...

=

A−1

00...






At the limit as n →∞:I − A

(∞)0 0

−A(∞)−1 I − A

(∞)0

−A(∞)−1 I − A

(∞)0

0. . .

. . .

GG ∗

G ∗

...

=

A−1

00...

where G ∗ = limn Gn.Therefore G = (I − A

(∞)0 )−1A−1






Functional interpretation

A(n+1)(z) = zA(n)odd(z) + A

(n)even(z)(I − A

(n)odd(z))−1A

(n)even(z)

A(n+1)(z) = A(n)even(z) + A

(n)odd(z)(I − A

(n)odd(z))−1A

(n)even(z)

where

A(n)(z) =+∞∑i=0

z i A(n)i , A(n)(z) =

+∞∑i=−1

z i+1A(n)i





Applicability of CR: the role of Wiener algebra

Theorem (Bini, Latouche, Meini ’04)

For any n ≥ 0 one has:

1 A(n)(z) and A(n)(z) belong to W+.

2 I − A(n)odd(z) is invertible for |z | ≤ 1 and its inverse belongs to

W+.

3 φ(n)(z) = I − z−1A(n)(z) has a weak canonical factorization

φ(n)(z) =(I −

+∞∑i=0

z iU(n)i

)(I − z−1G 2n

), |z | = 1.





Convergence of CR


Let ξ be the zero of smallest modulus of det S(z) such that|ξ| > 1. Then:

1 {A(n)(z)}n −→ A(∞)−1 + zA

(∞)0 uniformly over any compact

subset of {z ∈ C : |z | < ξ}.2 ‖A(n)

i ‖ ≤ γ|ξ|−i ·2nand ‖A(n)

i ‖ ≤ γ|ξ|−i ·2n, for any i ≥ 1,

n ≥ 0.

3 ‖A(n)0 − A

(∞)0 ‖ ≤ γ|ξ|−2n

for any n ≥ 0.

4 ρ(A(∞)0 ) ≤ ρ(A

(∞)0 ) < 1.

5 ‖G − G (n)‖ ≤ γ|ξ|−2n, where G (n) = (I − A

(n)0 )−1A−1.





Computational issues

The matrix power series A(n)(z), A(n)(z) are approximated bymatrix polynomials of degree at most dn.

The computation of such matrix polynomials by means ofevaluation/iterpolation at the roots of unity can be performed in

O(m3dn + m2dn log dn)

arithmetic operations





Doubling method

History Introduced by W.J. Stewart (’95) to solve generalblock Hessenberg systems, applied by Latouche andStewart (’95) for computing G , improved by Bini andMeini (’98) by exploiting the Toeplitz structure ofthe block Hessenberg matrices.

Idea Successively solve finite block Hessenberg systems ofblock size which doubles at each iterative step.





Doubling method

Truncation at block size n of the infinite system :

I − A0 −A1 −A2 . . . −An−1

−A−1 I − A0 −A1. . .

...

−A−1 I − A0. . . −A2

. . .. . . −A1

0 −A−1 I − A0

X

(n)1

X(n)2...

X(n)n

=

A−1

0...0

.





Doubling method: convergence


For any n ≥ 1 one has:

0 ≤ X(n)1 ≤ X

(n+1)1 ≤ G.

X(n)i ≤ G i for i = 1, . . . , n.

For any ε > 0 there exist positive constants γ and σ such that

‖G − X(n)1 ‖∞ ≤ γ(|ξ| − ε)−n,

where ξ is the zero of smallest modulus of det S(z) such that|ξ| > 1.





Doubling method: algorithm

The algorithm consists in successively solving systems of block size2, 4, 8, 16, . . . .

Size doubling at each step =⇒ Quadratic convergence

Use of FFT and Toeplitz structure =⇒ The 2n × 2n blocksystem can be solved in O(m32n + m2n2n) arithmeticoperations.












i=1 AiX−1Bi






Quasi-Birth-Death processes

If Ai = 0 for i > 1 the M/G/1-type Markov chain is called aQuasi-Birth-Death process (QBD).

Problem

Computation of the minimal component-wise solution G , among thenonnegative solutions, of

X = A−1 + A0X + A1X2





Linearization of the matrix equation

I − A0 −A1 0−A−1 I − A0 −A1

−A−1 I − A0. . .

0. . .

. . .

�� GG 2

G 3

...

=

A−1

00...

.

G can be interpreted by means of the solution of an infinite blocktriangular, block Toeplitz system





Cyclic reduction for QBD’s

At the n-th stepI − A

(n)0 −A

(n)1 0

−A(n)−1 I − A

(n)0 −A

(n)1

−A(n)−1 I − A

(n)0

. . .

0. . .

. . .

�� GG 2n+1

G 2·2n+1

...

=

A−1

00...

.





Cyclic reduction for QBD’s

At the limit as n →∞:I − A

(∞)0 0

−A(∞)−1 I − A

(∞)0

−A(∞)−1 I − A

(∞)0

0. . .

. . .

GG ∗

G ∗

...

=

A−1

00...

where G ∗ = limn Gn.Therefore G = (I − A

(∞)0 )−1A−1





Cyclic reduction

Recursive (algebraic) relations

A(n+1)−1 = A

(n)−1K

(n)A(n)−1,

A(n+1)0 = A

(n)0 + A

(n)−1K

(n)A(n)1 + A

(n)1 K (n)A

(n)−1,

A(n+1)1 = A

(n)1 K (n)A

(n)1 ,

A(n+1)0 = A

(n)0 + A

(n)1 K (n)A

(n)−1, n ≥ 0

where K (n) = (I − A(n)0 )−1.





“Non-Markovian” matrix equations

GivenX = A−1 + A0X + A1X

2

such that det S(z) = det(zI − A−1 − zA0 − z2A1) has zeros

|ξ1| ≤ · · · ≤ |ξm| < |ξm+1| ≤ · · · ≤ |ξ2m|

compute the solution G such that λ(G ) = {ξ1, . . . , ξm}.Such G is called the minimal solvent (Gohberg, Lancaster,Rodman ’82)

Applications Quadratic eigenvalue problems (damped vibrationproblems), polynomial factorization, Markov chains,etc.

Available algorithms Functional iterations, Bernoulli’s method,Newton’s iteration (Higham and Kim ’00, ’01)





Functional interpretation

Theorem (Bini, Gemignani, Meini ’03)

If ρ(G ) < 1 and if in addition

X = A−1 + XA0 + X 2A1

has a minimal solvent R, then

φ(z) = I − z−1A−1 − A0 − zA1 is invertible in{z ∈ C : ρ(G ) < |z | < 1/ρ(R)}H(z) = φ(z)−1 =

∑+∞i=−∞ z iHi is such that

Hi =

{G−iH0, i < 0,

H0Ri i > 0.

Therefore, G = H−1H−10 and R = H−1

0 H1.





Linearization

In matrix form:

. . .. . .

. . . 0A−1 A0 A1

A−1 A0 A1

A−1 A0 A1

0. . .

. . .. . .

...

R2

RIGG 2

...

H0 =

...00I00...

where R = H1H

−10 is the minimal solvent of

X 2A1 + XA0 + A−1 = X .




Matrix sign functionInverse of a matrix Laurent polynomial








i=1 AiX−1Bi






Matrix pth root

Assumptions A ∈ Cm×m with no eigenvalues on the closednegative real axis.

Definition The principal matrix pth root of A, A1/p, is theunique matrix X such that:

1 X p = A.2 The eigenvalues of X lie in the segment{ z : −π/p < arg(z) < π/p }.

Applications Computation of the matrix logarithm, computationof the matrix sector function (control theory).

Algorithms Schur decomposition (Smith ’03). Most of theavailable iterative algorithms are useless for theirnumerical instability, unless A is very wellconditioned (Smith ’03)!





Matrix sign function

Let

C =

0 I

. . .. . .. . . I

A 0

∈ Cmp×mp,

be the block companion matrix for zpI − A = 0.

If p = 2 then (Higham ’97)

Sign(C ) =

[0 A−1/2

A1/2 0

].

If p > 2?






Benner, Byers, Mehrmann and Xu (’00) prove that ifU ∈ Cmp×m is such that CU = UY for some nonsingularY ∈ Cp×p, and U1 is nonsingular, then X = U2U

−11 is a pth

root of A.

For an appropriate choice of subspace, X is the principal pthroot.

Does the matrix sign of C give the appropriate subspace? YES






Theorem (Bini, Higham, Meini, ’04)

If p = 2q, where q is odd, then the first block column of Sign(C )is given by

1

p

θ0I

θ1A1/p

θ2A2/p

...

θp−1A(p−1)/p

where θi =

∑p−1j=0 ωij

pσj , for i = 0, . . . , p − 1,

ωp = cos(2πp ) + i sin(2π

p ) and σj = −1 forj = bq/2c+ 1, . . . , bq/2c+ q, σj = 1 otherwise.





Matrix sign function:open issues

Design of a numerically stable algorithm for computing Sign(C ),having a low computational cost: the matrix sign iteration

Cn+1 = (Cn + C−1n )/2, n ≥ 0,

with C0 = C , requires O(m3p3) arithmetic operations at each step!





Inverse of a matrix Laurent polynomial: the square root

Theorem (Meini ’04)

The Laurent matrix polynomial

R(z) = z−1(I − A) + 2(I + A) + z(I − A)

is invertible for any z ∈ C such that r < |z | < 1/r where

r = ρ((A1/2 − I )(A1/2 + I )−1

)< 1.

Moreover, H(z) = R(z)−1 = H0 +∑+∞

i=1 (z i + z−i )Hi is such that

A1/2 = 4AH0.






Sketch of proof.

Consider the Cayley transform w : C \ {−1} → C,

w(x) = (x − 1)/(x + 1),

which maps the right half plane C+ into the open unit disk.

Observe that

W = w(A1/2) = (A1/2 − I )(A1/2 + I )−1

is the minimal solvent of (I −A)+2(I +A)X +(I −A)X 2 = 0.

Deduce thatR(z) =

(z(A1/2 + I )−(A1/2− I )

)(z−1(A1/2 + I )−(A1/2− I )

)Beatrice Meini Nonlinear matrix equations and structured linear algebra





We may compute A1/2 by approximating the constantcoefficient of R(z)−1.

Cyclic reduction applied to

. . .. . .

. . . 0I − A 2(I + A) I − A

I − A 2(I + A) I − AI − A 2(I + A) I − A

0. . .

. . .. . .

converges quadratically to Diag(. . . , 4A1/2, 4A1/2, 4A1/2, . . .).





Inverse of a matrix Laurent polynomial: the pth root

Theorem (Bini, Higham, Meini ’04)

If p = 2q, where q is odd, then the Laurent matrix polynomial

R(z) = z−qp∑

j=0

z j

(p

j

)(A + (−1)j+1I

)is invertible for any z ∈ C such that r < |z | < 1/r , for a suitabler > 1. Moreover, H(z) = R(z)−1 = H0 +

∑+∞i=1 (z i + z−i )Hi is

such that

A1/p = 4p sin(π/p)A

q−1∑j=0

αjHj

where α0 = 12

(p−2q−1

), αj =

( p−2q−j−1

), j = 1, . . . , q − 1.





Inverse of a matrix Laurent polynomial: open issues

Design of a numerically stable algorithm for computing the centralp coefficients of H(z).

Some ideas: use of Graeffe iteration, evaluation/interpolationtechniques












i=1 AiX−1Bi






DARE-type matrix equations

Assumptions B, Ai and Di , i = 1, . . . , d , nonnegative m ×mmatrices, such that B is sub-stochastic andB + Di + A1 + · · ·+ Ad , i = 1, . . . , d , are stochastic.

Problem Computation of the minimal component-wisesolution of the matrix equation

X +d∑

i=1

AiX−1Di = C ,

where C = I − B.Motivation Analysis of certain Markov processes called Tree-Like

processes: single server queues with LIFO servicediscipline, medium access control protocol with anunderlying stack structure, etc. (Latouche,Ramaswami ’99)





Tree-like processes

The generator matrix of a Tree-Like process has the form

Q =

C0 Λ1 Λ2 . . . Λd

V1 W 0 . . . 0

V2 0 W. . .

......

.... . .

. . . 0Vd 0 . . . 0 W

,

where C0 is an m ×m matrix,

Λi =[

Ai 0 0 . . .], Vi =

Di

00...

, for 1 ≤ i ≤ d





Tree-like processes

The infinite matrix W is recursively defined by

W =

C Λ1 Λ2 . . . Λd

V1 W 0 . . . 0

V2 0 W. . .

......

.... . .

. . . 0Vd 0 . . . 0 W

.





Tree-like processes


The matrix W can be factorized as W = UL, where

U =

S Λ1 Λ2 . . . Λd

0 U 0 . . . 0

0 0 U.. .

......

.... . .

. . . 00 0 . . . 0 U

, L =

I 0 0 . . . 0

Y1 L 0 . . . 0

Y2 0 L. . .

......

.... . .

. . . 0Yd 0 . . . 0 L

and S is the minimal solution of X +

∑di=1 AiX

−1Di = C.

Consequence: Once the matrix S is known, the stationaryprobability vector can be computed by using the UL factorizationof W .





Natural fixed point iteration

The sequencesSn = C +

∑1≤i≤d

AiGi ,n,

Gi ,n+1 = (−Sn)−1Di , for 1 ≤ i ≤ d , n ≥ 0,

with G1,0 = . . . = Gd ,0 = 0, monotonically converge to S andGi = (−S)−1Di , i = 1, . . . , d , respectively (Latouche andRamaswami ’99)





Cyclic reduction + fixed point iteration

Multiply

S +d∑

j=1

AjS−1Dj = C

by S−1Di , for i = 1, . . . , d .

Observe that Gi = (−S)−1Di , i = 1, . . . , d , is a solution

Di + (C +∑

1≤j≤dj 6=i

AjGj)X + AiX2 = 0.

We may prove that Gi is the minimal solvent.





Cyclic reduction + fixed point iteration

Set G1,0 = G2,0 = · · · = Gd ,0 = 0

For n = 0, 1, 2, . . .

For i = 1, . . . , d :1 define

Fi,n = C +∑

1≤j≤i−1

AjGj,n +∑

i+1≤j≤d

AjGj,n−1.

2 compute, by means of cyclic reduction, the minimalsolvent Gi,n of

Di + Fi,nX + AiX2 = 0,

The sequences {Gi ,n : n ≥ 0} monotonically converge to Gi , for1 ≤ i ≤ d





Newton’s iteration

Set S0 = C

For n = 0, 1, 2, . . .

1 Compute Ln = Sn − C +∑d

i=1 AiS−1n Di .

2 Compute the solution Yn of

X −d∑

i=1

AiS−1n XS−1

n Di = Ln (1)

3 Set Sn+1 = Sn − Yn

The sequence {Sn}n converges quadratically to S .

Open issues: efficient computation of the solution of (1).


New book

Numerical Methods for Structured Markov Chains

D.A. Bini (University of Pisa)G. Latouche (Universite Libre de Bruxelles)B. Meini (University of Pisa)

Oxford University Press, 2005


Wiener algebra

Definition

The Wiener algebra W is the set of complex m ×m matrix valuedfunctions A(z) =

∑+∞i=−∞ z iAi such that

∑+∞i=−∞ |Ai | is finite.

Definition

The set W+ is the subalgebra of W made up by power series ofthe kind

∑+∞i=0 z iAi .



Definition

Let M be an m ×m matrix, with no pure imaginary eigenvalues.Let M = S(D + N)S−1 be the Jordan decomposition of M, whereD = Diag(λ1, λ2, . . . , λm) and N is nilpotent and commutes withD. Then the matrix sign of M is the matrix

Sign(M) = S Diag(Sign(λ1),Sign(λ2), . . . ,Sign(λm))S−1,


nonlinear matrix equations and structured linear algebra

Documents