Journal of Computational and Applied Mathematics

Contents lists available atScienceDirect

Journal of Computational and Applied Mathematics

journal homepage:www.elsevier.com/locate/cam

Accurate numerical solution for structured M-matrix algebraic Riccati equations

Changli Liu

^a

, Wei-Guo Wang

^b,1

, Jungong Xue

^c,2

, Ren-Cang Li

^d,∗,3

aCollege of Mathematics, Sichuan University, Chengdu 610065, PR China

bSchool of Mathematical Sciences, Ocean University of China, Qingdao, 266100, PR China

cSchool of Mathematical Science, Fudan University, Shanghai 200433, PR China

dDepartment of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, USA

a r t i c l e i n f o

Article history:

Received 15 October 2020 MSC:

15A24 60G17 65F30 65H10 Keywords:

M-matrix algebraic Riccati equation M-matrix

Doubling algorithm

High entrywise relative accuracy

a b s t r a c t

This paper is concerned with aM-matrix algebraic Riccati equation (mare)XDX−AX− XB+C = 0 for whichAis block-diagonal and its defining matrixW =

[ B −D

−C A ]

is a nonsingular or irreducible singularM-matrix. Such anmare can be decomposed into many coupled algebraic Riccati equations (ares) that can be solved by the Jacobi- or Gauss–Seidel-like iteration updating scheme at the outer-loop while by a doubling algorithm in the inner loop for each coupledare, as first proposed by Meini (2013). The goals of this paper are two-fold. One is to resolve a critical technical detail in Meini’s algorithm that was not addressed. It is about whether eacharein the inner loop has a minimal nonnegative solution. It is proved that the defining matrix of each coupled areduring a doubling iteration is indeed a nonsingular or irreducible singularM-matrix and, as a result, they do have minimal nonnegative solutions and a doubling algorithm is an efficient way to compute them. The other goal is to design a highly accurate implementation of the doubling algorithm for the inner loop so that all entries of the minimal nonnegative solution to the originalmareare calculated with high entrywise relative accuracies, regardless of their magnitudes. This is made possible by a novel way of constructing triplet representations for the coupledareduring doubling iterations.

Numerical examples are presented to demonstrate that the resulting algorithm can indeed deliver an entrywise relatively accurate solution.

©2021 Elsevier B.V. All rights reserved.

1. Introduction

Consider the algebraic Riccati equation (are)

XDX

−

AX

−

XB

+

C

=

0

,

^(1.1a)

∗ Corresponding author.

E-mail addresses: [email protected](C. Liu),[email protected](W.-G. Wang),[email protected](J. Xue),[email protected](R.-C. Li).

1 Supported in part by NNSFC, China 11771408 and 11871444, and the Shandong Province Natural Science Foundation, China Grant ZR2017MA027.

2 Supported in part by the NNSFC, China 10971036 and Laboratory of Mathematics for Nonlinear Science, Fudan University, China.

3 Supported in part by NSF DMS-1719620 and DMS-2009689.

https://doi.org/10.1016/j.cam.2021.113614 0377-0427/©2021 Elsevier B.V. All rights reserved.

whereX

∈

_R^n×mis the unknown, and the sizes of the constant coefficient matricesA

,

^B

,

^{C, andD}are determined by the partitioning:

W

=

[

m n

m B

−

D

n

−

C A

]

∈

_R^(m+n)×(m+n)

.

^(1.1b)

We call this matrixW in(1.1b)thedefining matrix ofare(1.1a).

Theares(1.1a)that we are interested in are those whenW is anM-matrix. Following [1], we will callare(1.1a)a M-Matrix algebraic Riccati equation(mare) ifW is anM-matrix. Previously, when the termmarewas first coined in [2], it was required that

defining matrixW is a nonsingular or irreducible singularM-matrix, (1.2) which is often the case forares arising from relevant applications such as applied probability and transportation theory (see [3–9] and the references therein). It is these applications that motivated early studies onmares by the numerical linear algebra communities. It was shown in [3,5,10] that are (1.1a) has a unique minimal nonnegative solutionΦ^, i.e., entrywise

Φ

≤

X for any other nonnegative solutionXofare(1.1a)

under assumption(1.2), making all studies on such anarea natural thing to do. That is whymarewas defined the way it was in [2].

In general,W just being anM-matrix is too broad to allow one to say much about the solution to the associatedare. Guo [10] constructed an example:

m

=

n

=

1

,

^W

=

[ 0 0

−

1 0 ]

,

^(1.3)

which is anM-matrix. The associatedmareis 1

=

0 and thus has no solution, not to mention a minimal nonnegative solution. This simple example shows thatare(1.1)being simply anmare, i.e.,Wbeing anM-matrix, does not necessarily guarantee itself to have a solution. But Guo [10] made a remarkable observation. What really matters, behind the assumption(1.2)that guarantees the existence of the minimal nonnegative solutionΦ, is, beyondW being anM-matrix, that there is a positive vectoru

∈

_R^m+n such thatWu

≥

0. He introduced the notation of regular M-matrix for such anM-matrix. It is well-known that under(1.2), there exists some positive vectorusuch thatWu

≥

0. Consequently, Guo [10] expanded the set of ares that have minimal nonnegative solutions. Later Guo and Lu [11] proved that the doubling algorithms [6,12,13] still converge at least linearly ifW is a regularM-matrix and if rank(W)

≥

m

+

n

−

1.

The latter automatically holds if(1.2)holds. Inspired by these developments, we streamline the nomenclature ofares in connection with anM-matrix as follows.

Definition 1.1. We callare(1.1)anmareifW is anM-matrix [1], aregular mareifW is a regularM-matrix [10,11], astrongly regularmareifW is a regularM-matrix and if rank(W)

≥

m

+

n

−

1, and finally asuper-regularmareif its defining matrixW satisfies(1.2).

Evidently, a super-regularmareis a strongly regularmarewhich is a regularmarewhich is anmare. According to [10,11], any regularmarehas a minimal nonnegative solution, and the doubling algorithms globally converge for strongly regularmares and locally the convergence is linear (with the linear rate 1

/

2) or quadratic, which extended earlier results for a super-regularmare, previously called anmare[2].

A super-regularmarewas really the focus of the study in the past 20 years or so. It is very well understood nowadays both theoretically and numerically. C. Guo and his collaborators completed much of the studies into the existence and basic properties of the unique minimal nonnegative solutionΦ[3,4,14]. The first structure-preserving doubling algorithm (sda) was proposed by X. Guo, Lin and Xu [6] in 2006, and it was immediately clear at that moment thatsdais much superior to Newton’s method in solvingmarefor the unique minimal nonnegative solutionΦ. Soon after,sdawas improved by two other more efficient doubling algorithmssda-ss [12] andadda[13] withaddaprovably being the best. A highly accurate implementation ofadda was discovered first by Nguyen and Poloni [15] for a singular but irreducibleW and then by Xue and Li [16] for nonsingularW as well. An entrywise relative perturbation theory formarewas established earlier in [2,17].

In this paper, we will study super-regular mare (1.1a) coming from multi-type queues with general customer impatience [18,19] and risk processes [20]. Evidently, existing doubling algorithms can be used to calculate its minimal nonnegative solutionΦ, and, if entrywise accuracy ofΦ is what is needed, we can apply accadda [15,16], the highly accurate implementation ofadda. However, for themare,Ais a block-diagonalM-matrix, and as a result themarecan be broken into many coupledares of smaller sizes, a structure that should be taken advantage of for more efficient methods. Rightly, Meini [21] did just that. She proposed an inner–outer iterative method, where the Jacobi- or Gauss–

Seidel-like updating scheme is used as an outer iteration while a doubling algorithm serves as the inner iteration for each smaller-sizedare. The method was analyzed theoretically and demonstrated numerically. However, the theoretical

2

argument there [21] is incomplete in that it did not properly justify that the smaller-sizedares have minimal nonnegative solutions, although it was proved that their defining matrices are indeedM-matrices which, however, are not enough as Guo’s example(1.3)indicates.

We have two primary goals in this paper. One is to provide a rigorous theoretical analysis of the inner–outer iterative method in [21]. It is proved that each smaller-sizedarein the inner doubling iterations are indeed a super-regularmare, and thus it has a unique minimal nonnegative solution and the doubling algorithm is guaranteed convergent and the convergence is at least linear but often quadratic. Our second goal is to devise a highly accurate implementation of Meini’s algorithm based on accadda[15,16]. The key for making that possible is a novel way to construct entrywise accurate triplet representations of all definingM-matrices of the smaller-sizedares during the doubling iterative process, assuming that a triplet representation for the defining matrixW ofmare(1.1a)is knowna priori.

The paper is organized as follows. Section 2 collects necessary preliminaries on M-matrix, its accurate inverse, and super-regular mare that will be needed later. In Section 3, we investigate the structured mare theoretically to lay the foundation for our highly accurate algorithm in Section5. We review the highly accurate doubling algorithm accadda[15,16] with an additional output that was not in the original accadda. In Section5, we first fill in the gap of technical incompleteness we mentioned earlier and then present our highly accurate algorithm. Two numerical examples are presented in Section6to demonstrate that our new highly accurate algorithm can indeed deliver computed minimal nonnegative solution with nearly full entrywise relative accuracy in the working precision. We draw our conclusions in Section7.

Notation.R^m×nis the set of allm

×

nreal matrices,Rⁿ

=

_R^n×1, andR

=

_R¹.I_n(or simplyIif its dimension is clear from the context) is then

×

nidentity matrix. The superscript in

·

^Ttakes transpose. ForX

∈

_R^m×n,X_(i,j)refers to its (i

,

^j)th entry,

|

X

| ∈

_R^m×ntakes entrywise absolute value. InequalityX

≤

YmeansX_(i,j)

≤

Y_(i,j)for all (i

,

j), and similarly forX

<

^Y, X

≥

Y, andX

>

^Y. In particular,X

≥

0 means thatXis entrywise nonnegative. For a square matrixX, denote by

ρ

^(X) its spectral radius and by eig(X) the set of its eigenvalues counted algebraic multiplicities; diag(X) is a diagonal matrix extracting the diagonal part ofX, and offdiag(X)

=

X

−

diag(X).1_n

∈

_Rⁿis then-vector of all ones and1_m×_n

∈

_R^m×n is them

×

nmatrix of all ones. The symboluis the unit machine roundoff.

2. Preliminaries

2.1. M-matrix

A matrixA

∈

_R^n×nis called aZ-matrixifA(i,^j)

≤

0 for alli

̸=

j[22, p. 284]. AnyZ-matrixAcan be written assI

−

Nwith N

≥

0, and it is called anM-matrixifs

≥ ρ

^{(N), a}singular M-matrixifs

= ρ

^{(N), and a}nonsingular M-matrixifs

> ρ

^(N).

The results inTheorem 2.1are either well-known [22] or can be proved straightforwardly. For item (e), the reader is referred to [23, Lemma 2.5].

Theorem 2.1.

(a)If A is a nonsingular M-matrix and B is Z -matrix satisfying B

≥

A, then B is a nonsingular M-matrix.

(b) If A is an irreducible singular M-matrix and B is Z -matrix satisfying B

≥

A, then B is a nonsingular or irreducible singular M-matrix. If also B

̸=

A, then B is a nonsingular M-matrix.

(c) If A is a Z -matrix and if Au

≥

0for someu

>

0, then A is an M-matrix.

(d)If A is a Z -matrix and if Au

>

^{0for someu}

>

0, then A is a nonsingular M-matrix.

(e) Let A

∈

_R^n×nbe a nonsingular or irreducible singular M-matrix, conformally partitioned as A

=

[A11 A12

A₂₁ A₂₂ ]

,

where A₁₁and A₂₂are square matrices. Then A₁₁and A₂₂are nonsingular M-matrices, and their Schur complements A₂₂

−

A₂₁A⁻₁₁¹A₁₂

,

^A11

−

A₁₂A⁻₂₂¹A₂₁

are nonsingular M-matrices if A is a nonsingular M-matrix, or irreducible singular M-matrices if A is an irreducible singular M-matrix.

2.2. Accurate inverses of an M-matrix

The key ingredient in recent work [15,16] to achieve high entrywise relative accuracy is the GTH-like algorithm for inverting a nonsingularM-matrix due to Alfa, Xue, and Ye [24]. They proposed to represent a nonsingularM-matrixA by the so-calledtriplet representationwhich can determineA⁻¹to high entrywise relative accuracy. Specifically, a triplet representation (offdiag(A)

,

^u

,

v^{) of theM-matrixA}

∈

_R^n×nconsists of offdiag(A) which is obtained by simply resetting the diagonal part ofAto 0, 0

<

^u

∈

_Rⁿ, andv

=

Au

≥

0. Often for convenience, we will not distinguishAfrom its triplet representation and write

A

=

(offdiag(A)

,

^u

,

v⁾

.

3

It is proved [25] that if all entries of offdiag(A),u, andvare known to high entrywise relative accuracy, then all entries ofA⁻¹ are determined to a comparable high entrywise relative accuracy, or equivalently the solutionxtoAx

=

b for anyb

≥

0 is determined to a comparable high entrywise relative accuracy. Numerically, the GTH-like algorithm of Alfa, Xue, and Ye [24], using the idea in [26], computes the LU decompositionA

=

LU, via the Gaussian elimination without pivoting and without any cancellation⁴and, consequently,LandUare computed with high entrywise relative accuracy.

Moreover, the diagonal entries ofLare all 1 and its off-diagonal entries are non-positive,Uhas positive diagonal entries and non-positive off-diagonal entries. These properties ofLand U ensure that the solutionxof Ax

=

b

≥

0 can be computed to the claimed accuracy, without any cancellation. For more details, the reader is referred to [1].

2.3. Properties ofmare

We will summarize important results for a super-regularmare(1.1). They are mostly due to [3,5,10] (see also [1]).

SinceW is assumed a nonsingular or irreducible singularM-matrix, there existu₁

∈

_R^mandu₂

∈

_Rⁿsuch that u₁

>

⁰

,

^u2

>

⁰

,

[u

ˆ

₁ u

ˆ

₂ ]

:=

W [u₁

u₂ ]

≥

0

,

^(2.1a)

whereu₁andu₂can be chosen to satisfy [22]

u

ˆ

₁

>

⁰

,

u

ˆ

₂

>

⁰

,

^ifW is a nonsingularM-matrix

;

(2.1b)

u

ˆ

₁

=

0

,

u

ˆ

₂

=

0

,

^ifW is an irreducible singularM-matrix

.

^(2.1c)

It is well-known thatare(1.1)is equivalent to [1,27]

H [I

X ]

=

[I

X ]

M

,

^(2.2a)

whereM

=

B

−

DX and H

=

[I_m

−

I_n ]

W

=

[B

−

D C

−

A ]

.

^(2.2b)

Eq. (2.2a)is an eigenvalue problem, seeking an invariant subspace of H. Denote the set of the eigenvalues, counted algebraic multiplicities, ofHby

eig(H)

= { λ

1

, . . . , λ

m+_n

} ,

^(2.3)

where

λ

ifor 1

≤

i

≤

m

+

nare ordered by their nonincreasing real parts, i.e.,

ℜ

(

λ

j)

≤ ℜ

(

λ

i) fori

<

^j.

Theorem 2.2([3,5,10]).Suppose that(1.1)is a super-regularmare, i.e., W in(1.1b)is a nonsingular or an irreducible singular M-matrix.

(a)

λ

mand

λ

m+₁are real,

ℜ

(

λ

m+₂)

<

⁰

< ℜ

(

λ

m−₁), and

ℜ

(

λ

m+_n)

≤ · · · ≤ ℜ

(

λ

m+₂)

≤ λ

m+₁

≤

0

≤ λ

m

≤ ℜ

(

λ

m−₁)

≤ · · · ≤ ℜ

(

λ

1)

.

^(2.4) In particular, this implies

λ

m+₁

<

⁰

< λ

mif W is nonsingular.

(b) mare(1.1)has a unique minimal nonnegative solutionΦ. Moreover, eig(B

−

DΦ⁾

= { λ

1

, . . . , λ

m

} ,

^eig(A

−

_ΦD)

= {− λ

m+1

, . . . , − λ

m+n

} .

(c) If W isirreducible, thenΦ

>

^{0, and A}

−

_ΦD and B

−

DΦ^areirreducibleM-matrices.

(d)If W isnonsingular, then A

−

_ΦD and B

−

DΦ^arenonsingularM-matrices.

(e) Φ^u1

≤

u₂. Moreover,Φ^u1

<

^u2if W is nonsingular.

(f) H has a unique m-dimensional eigenspace associated with its eigenvalues inC0+

:= {

z

∈

_C

: ℜ

(z)

≥

0

}

, and [I_m

Φ ]

is a basis matrix of the eigenspace.

3. The structuredmare

The type ofmarecoming from multi-type queues with general customer impatience [18,19] and risk processes [20] has an additional block diagonal structure inA. In this section, we will analyze suchmare, inspired by Meini [21]. Specifically, considermare:

XDX

−

AX

−

XB

+

C

=

0

,

^(3.1a)

4 By cancellation we mean any subtraction of a real number from another one of the same sign.

4

whereAis aK

×

K block diagonal matrix:

A

=

⎡

⎢

⎢

⎣

n1 n2 ... ⁿK

n1 A₁

n2 A₂

... ...

nK AK

⎤

⎥

⎥

⎦

∈

_R^n×n

,

ⁿ

=

K

∑

i=₁

n_i

,

^(3.1b)

and, as before,B

∈

_R^m×m,C

∈

_R^n×m, andD

∈

_R^m×n. AssumeK

≥

2 sincemare(3.1)withK

=

1 reduces to the one that has been well-studied.

Correspondingly, we partitionC,Dand the unknownXas

C

=

⎡

⎢

⎢

⎣

m

n1 C₁

n2 C₂

... ...

nK C_K

⎤

⎥

⎥

⎦

,

^D

=

[

n1 n2 ... ⁿK

m D₁ D₂

· · ·

D_K ]

,

^X

=

⎡

⎢

⎢

⎣

m

n1 X₁

n2 X₂

... ...

nK X_K

⎤

⎥

⎥

⎦

.

^(3.2)

The structuredmare(3.1)can be equivalently turned into a system of coupled matrix Riccati equations inX_j:

X_jD_jX_j

−

A_jX_j

−

X_jB_j(X)

+

C_j

=

0 for 1

≤

j

≤

K

,

^(3.3)

where

B_j(X)

=

B

−

∑

i̸=_j

D_iX_i

∈

_R^m×m for 1

≤

j

≤

K

.

^(3.4)

It can be seen thatB_j(X)

−

D_jX_j

=

B

−

DX. Let for 1

≤

j

≤

K W_j(X)

=

[B_j(X)

−

D_j

−

C_j A_j ]

,

^Hj(X)

=

[I_m

−

I_nj ]

W_j(X)

=

[B_j(X)

−

D_j C_j

−

A_j ]

.

^(3.5)

We will still formally denote the defining coefficient matrix ofmare(3.1)by the matrixW of(1.1b):

W

=

[ B

−

D

−

C A ]

.

Formare (3.1)originally arising from the aforementioned applications, W is an irreducible singular M-matrix. As we mentioned before, the argument in Meini [21] is incomplete in the sense that although all involvedW_j(

·

) during the doubling iterations was indeed proven to be anM-matrix, that alone is not enough to guarantee that the associatedmare (3.3)has a minimal nonnegative solution. One of our two goals is to remove this incompleteness. Specifically, we will show that eachW_j(

·

) during the doubling iterations is a nonsingular or irreducible singular M-matrix, and thus each mare(3.3)is super-regular and has a minimal nonnegative solution which can be found efficiently by any of the doubling algorithms formare[1].

For the ease of future reference, we will callmare(3.1)astructuredmareif its definingW is anM-matrix. Our study in this paper is for the caseW is a nonsingular or irreducible singularM-matrix, i.e.,(3.1)is also a super-regularmare.

Then it has a unique minimal nonnegative solutionΦ

∈

_R^n×mwhich is partitioned, similarly toXin(3.2), as

Φ

=

⎡

⎢

⎢

⎣

m

n1 Φ1

n2 Φ2

... ...

nK ΦK

⎤

⎥

⎥

⎦

∈

_R^n×m

.

^(3.6)

Evidently, for each 1

≤

j

≤

K,Φjis a nonnegative solution of

X_jD_jX_j

−

A_jX_j

−

X_jB_j(Φ⁾

+

C_j

=

0

.

^(3.7)

Theorem 3.1contains our main result in this section.

Theorem 3.1. Suppose thatmare(3.1)is super-regular, i.e., W is a nonsingular or irreducible singular M-matrix.

(a)B

−

DΦis an M-matrix, and each A_j

−

_ΦjD_jis a nonsingular M-matrix for1

≤

j

≤

K . (b) eig(H_j(Φ⁾⁾is the multi-set union ofeig(B

−

DΦ^)andeig(

−

(A_j

−

_ΦjD_j)), and

eig(B

−

DΦ⁾

∩

eig(

−

(A_j

−

_ΦjD_j))

= ∅ .

^(3.8)

5

Thus H_j(Φ⁾has exactly n_jeigenvalues in the open left-half plane given byeig(

−

(A_j

−

_ΦjD_j))and the other m eigenvalues are in the closed right-half plane given byeig(B

−

DΦ). Moreover, if H_j(Φ⁾has an eigenvalue on the imaginary axis, then that eigenvalue is0and it is a simple eigenvalue.

(c) Each(3.7)is a super-regularmare, i.e., W_j(Φ⁾is a nonsingular or irreducible singular M-matrix, andΦjis the unique minimal nonnegative solution tomare(3.7).

Proof. The first claim in item (a) is due to Theorem 2.2(c,d). By Theorem 2.2(c), we know that A

−

_ΦD is either a nonsingular or irreducible singularM-matrix, and, therefore, eachA_j

−

_ΦjD_jis a nonsingularM-matrix byTheorem 2.1(e).

BecauseΦjis a solution to(3.7), it can be verified that H_j(Φ⁾

[I 0 Φj I ]

=

[I 0

Φj I

] [B

−

DΦ

−

D_j 0

−

(A_j

−

_ΦjD_j)

]

.

^(3.9)

In verifying(3.9), we used the fact B_j(Φ⁾

−

D_jΦj

=

B

−

DΦ. That eig(H_j(Φ)) is the multi-set union of eig(B

−

DΦ⁾ and eig(

−

(A_j

−

_ΦjD_j)) is a straightforward consequence of (3.9). Since A_j

−

_ΦjD_j is a nonsingular M-matrix and thus its eigenvalues are in the open right-half plane. So(3.8)holds. This completes the proof of item (b).

Since W is a nonsingular or irreducible singular M-matrix, we have(2.1a). Partition positive vector u₂

∈

_Rⁿ and nonnegative vectoru

ˆ

₂

∈

_Rⁿas

u₂

=

⎡

⎢

⎢

⎣

n₁ u_2,1

n2 u_2,2

... ...

nK u_2,K

⎤

⎥

⎥

⎦

,

u

ˆ

₂

=

⎡

⎢

⎢

⎣

n1 u

ˆ

_2,1

n2 u

ˆ

_2,2

... ...

nK u

ˆ

_2,K

⎤

⎥

⎥

⎦

.

^(3.10)

Expand(2.1a)to get Bu₁

−

K

∑

i=₁

D_iu_2,i

= ˆ

u₁

,

^(3.11a)

−

C_ju₁

+

A_ju_2,j

= ˆ

u_2,j forj

=

1

,

²

, . . . ,

^K

.

^(3.11b)

SinceΦ^u1

≤

u₂byTheorem 2.2(e), we have

u_2,j

−

_Φju₁

≥

0 forj

=

1

,

²

, . . . ,

^K

.

^(3.12)

Combining(3.11a)and(3.12), we get Bu₁

−

D_ju_2,j

−

∑

i̸=_j

D_iΦiu₁

= ˆ

u₁

+

∑

i̸=_j

D_iu_2,i

−

∑

i̸=_j

D_iΦiu₁

= ˆ

u₁

+

∑

i̸=_j

D_i(u_2,i

−

_Φiu₁)

≥

0

.

^(3.13)

Thus forj

=

1

,

²

, . . . ,

^K W_j(Φ⁾

[u₁ u_2,j

]

=

[B_j(Φ⁾

−

D_j

−

C_j A_j ] [u₁

u_2,j ]

=

[Bu₁

−

D_ju_2,j

−

∑

i̸=_jD_iΦiu₁

−

C_ju₁

+

A_ju_2,j ]

=

[_u

ˆ

₁

+

∑

i̸=_jD_i(u_2,i

−

_Φiu₁) u

ˆ

_2,j

]

≥

0

.

^(3.14)

BecauseW_j(Φ^{) is aZ-matrix,W}j(Φ^{) is an}M-matrix byTheorem 2.1(c).

IfW is nonsingular, thenu

ˆ

₁

>

^0,u

ˆ

₂

>

^{0 and}Φ^u1

<

^u2 byTheorem 2.2(e). Consequently, the inequalities in(3.12) and(3.13)are strict, and so is(3.14), which meansW_j(Φ) is a nonsingularM-matrix byTheorem 2.1(d).

Consider now thatWis an irreducible singularM-matrix. We know that allA_ifor 1

≤

i

≤

Kare nonsingularM-matrices byTheorem 2.1(e). The Schur complement of diag(A₂

, . . . ,

^AK) inW

[B

−

∑K

i=₂D_iA⁻_i ¹C_i

−

D₁

−

C₁ A₁ ]

is also an irreducible singularM-matrix byTheorem 2.1(e). ByTheorem 2.2(c), allΦi

>

0, and therefore

⎧

⎨

⎩

(s

,

^t)

:

s

̸=

t

,

[

B

−

K

∑

i=₂

D_iA⁻_i ¹C_i ]

(s,t)

<

⁰

⎫

⎬

⎭

⊆

⎧

⎨

⎩

(s

,

^t)

:

s

̸=

t

,

[

B

−

K

∑

i=₂

D_iΦi

]

(s,t)

<

⁰

⎫

⎬

⎭

,

6

implying that W₁(Φ⁾

=

[B₁(Φ⁾

−

D₁

−

C₁ A₁ ]

is irreducible

.

Moments ago, we showedW₁(Φ^{) is an}M-matrix. ThusW₁(Φ) is a nonsingular or irreducible singularM-matrix. ForW_j(Φ⁾ withj

>

1, we permute symmetricallyW to

⎡

⎣

B D_j

−ˆ

D_j

−

C_j A_j

−ˆ

C_j ˆA_j

⎤

⎦

,

whereˆC_jandˆD_jare obtained fromCandDwith theirjth block removed, andˆA_jfromAwith itsjth block row and column removed. Now use the same proof we had forW1(Φ) to conclude thatWj(Φ) is a nonsingular or irreducible singular M-matrix.

In summary,mare(3.7)is super-regular, and thus has a minimal nonnegative solution byTheorem 2.2. That solution, denoted byˆ_Φj, can be uniquely characterized by that

[I ˆΦj

]

is the basis matrix for the eigenspace ofH_j(Φ) associated with itsmright most eigenvalues ofH_j(Φ), given by eig(B

−

DΦ). That eigenspace is unique by item (b) we just proved. On the other hand, it follows from(3.9)that

[I Φj

]

is the basis matrix for the same eigenspace. Therefore_Φˆ_j

=

_Φj, as expected. □

Theorem 3.2. Assume(1.2). Given U

= [

U₁^T

,

^U₂^T

, . . . ,

^U_K^T

]

^T

∈

_R^n×mpartitioned in the same way as X in(3.2), if0

≤

U_i

≤

_Φi for1

≤

i

≤

K , then

X_jD_jX_j

−

A_jX_j

−

X_jB_j(U)

+

C_j

=

0 for1

≤

j

≤

K

are super-regularmares and thus each has a unique minimal nonnegative solution.

Proof. Recall(3.5). The condition of the theorem impliesB_j(U)

≥

B_j(Φ^{) and thusW}j(U)

≥

W_j(Φ^).Wj(Φ) is a nonsingular or irreducible singularM-matrix byTheorem 3.1(c), and, hence,W_j(U) is a nonsingular or irreducible singularM-matrix byTheorem 2.1(a,b). The proof is completed. _□

The next theorem is about the monotonicity in the minimal nonnegative solution of super-regularmare. Besidesmare (1.1), considermare

˜X˜D˜X

−˜

A˜X

−

˜X˜B

+

˜C

=

0

,

^(3.15)

where˜A

,

˜B

,

˜C, and˜Dhave the same sizes asA

,

^B

,

^{C, andDof}(1.1). Denote byW˜the corresponding defining coefficient matrix of(3.15).

Theorem 3.3. Suppose that bothare (1.1)and(3.15)are super-regular, and letΦ ^and˜_Φbe their minimal nonnegative solutions, respectively. IfW˜

≥

W , then˜_Φ

≤

_Φ.

Proof. SplitAandBasA

=

DA

−

NA andB

=

DB

−

NB, whereDA

=

diag(A) andDB

=

diag(B). The following iterative scheme

Z₀

=

0

,

D_AZ_k+₁

+

Z_k+₁D_B

=

N_AZ_k

+

Z_kN_B

+

Z_kDZ_k

+

C_kfork

≥

0

produces a sequence

{

Z_k

}

^∞_k=0that monotonically converges to the minimal nonnegative solutionΦ ^of(1.1)[3, Theorem 2.3]. The same idea applied to(3.15)yields a sequence

{˜

Z_k

}

^∞_k=0that monotonically converges to_Φ˜. Inductively, it is not hard to show˜Z_k

≤

Z_kfor allk, which leads to the desired conclusion. □

4. Doubling algorithm – accadda

In this section, we outline the doubling algorithm,adda[13] and its highly accurate implementation accadda[16] (see also [1]) for super-regularmare(1.1).addastarts by picking parameters

α

^and

β

that satisfy

0

≤ α ≤ α

opt

:=

( max

i

[

A

]

_(i,i))⁻1

,

⁰

≤ β ≤ β

opt

:=

( max

i

[

B

]

_(i,i))⁻1

,

^(4.1a)

max

{ α, β } ̸=

0

.

^(4.1b)

7

Often we take

α = α

opt and

β = β

opt for the fastest convergence [13, Theorem 3.3]. Then it computesE₀

∈

_R^m×m, F₀

∈

_R^n×n,Z₀

∈

_R^n×mandY₀

∈

_R^m×nby solving

[

α

^B

+

I

− β

^D

− α

^C

β

^A

+

I

] [E₀ Y₀ Z₀ F₀ ]

=

[I

− β

^B

α

^D

β

^{C I}

− α

^A

]

,

^(4.2)

which is followed by the doubling iteration: fork

=

0

,

¹

, . . .

^,

E_k+₁

=

E_k(I_m

−

Y_kZ_k)⁻¹E_k

,

^(4.3a)

Fk+1

=

Fk(In

−

ZkYk)⁻¹Fk

,

^(4.3b)

Z_k+₁

=

Z_k

+

F_k(I_n

−

Z_kY_k)⁻¹Z_kE_k (4.3c)

=

Z_k

+

F_kZ_k(I_m

−

Y_kZ_k)⁻¹E_k

,

^(4.3d)

Y_k+1

=

Y_k

+

E_k(I_m

−

Y_kZ_k)⁻¹Y_iF_k (4.3e)

=

Y_k

+

E_kY_k(I_n

−

Z_kY_k)⁻¹F_k

.

^(4.3f)

A detailed derivation of the formulas(4.2)and(4.3)can be found in [1, pp. 20–21]. The alternative expression(4.3d)vs.

(4.3c)and(4.3f)vs.(4.3e)can be useful at implementation, especially when eitherm

≪

norn

≪

m.

With

α

^and

β

^{satisfying(4.1),Z}kis monotonically increasing and converges toΦquadratically, except in the case when Hhas a double eigenvalue 0 coming from a 2

×

2 Jordan block, for which the convergence is only linear with the linear rate 1

/

2. For more detailed statements ofadda’s convergence, the reader is referred to [1, Theorem 6.3].

A highly accurate implementation ofaddawas discovered first by Nguyen and Poloni [15] for a singular but irreducible M-matrix W and then by Xue and Li [16] for a nonsingular M-matrix W. The key part in implementation is the computations of the inverses of provably nonsingularM-matrices

[

α

^B

+

I

− β

^D

− α

^C

β

^A

+

I ]

,

^I

−

ZkYk

,

^I

−

YkZk

,

^(4.4)

to almost full entrywise relative accuracy by the GTH-like algorithm [24,25]. It is made possible by a novel way, especially for the case whenWis a nonsingularM-matrix [16], to find triplet representations forI

−

Z_kY_kandI

−

Y_kZ_kto nearly full entrywise relative accuracy during the iterative process.

Xue and Li [16] started by assuming a triplet representation W

=

(

offdiag(W)

,

[u₁

u₂ ]

,

[u

ˆ

₁

u

ˆ

₂ ])

(4.5a) ofW is known to almost full entrywise relative accuracy, where

[u₁ u₂ ]

>

⁰

,

[u

ˆ

₁

u

ˆ

₂ ]

=

W [u₁

u₂ ]

≥

0

.

^(4.5b)

To accurately invert the firstM-matrix in(4.4), we have the following lemma.

Theorem 4.1([16]).

(a)If

α =

0but

β >

^{0, then} [I_m

− β

^D

0 I_n

+ β

^A ]−1

=

[I_m

β

^D(I

+ β

^A)−1 0 (I_n

+ β

^A)−1

]

,

and a triplet representation for I_n

+ β

A can be read off from (I_n

+ β

^A)u2

=

u₂

+ β

^(Cu1

+ ˆ

u₂)

.

(b) If

α >

^0and

β =

0, then [

α

^B

+

I_m 0

− α

^{C I}n

]⁻1

=

[ (

α

^B

+

I_m)⁻¹ 0

α

^C(

α

^B

+

I_m)⁻¹ I_n ]

and a triplet representation for

α

^B

+

I_mcan be read off from (

α

^B

+

I_m)u₁

= α

⁽u

ˆ

₁

+

Du₂)

+

u₁

.

(c) If

α >

^0and

β >

^{0, then} [

α

^B

+

I_m

− β

^D

− α

^C

β

^A

+

I_n ] [u₁

/α

u₂

/β

]

=

[u

ˆ

₁

+

u₁

/α

u

ˆ

₂

+

u₂

/β

]

,

which yields a triplet representation for

[

α

^B

+

I_m

− β

^D

− α

^C

β

^A

+

I_n ]

immediately.

8

Algorithm 4.1Highly Accurateaddaformare(1.1)

Input: Strongly regularmare(1.1), vectorsu₁

,

^u2andu

ˆ

₁

,

u

ˆ

₂that satisfy(4.5b);

Output:the minimal nonnegative solutionΦ^andz

=

u₂

−

_Φu₁.

1:

α =

(

max_i

[

A

]

_(i,i))⁻1

,

β =

(

max_j

[

B

]

_(j,j))⁻1

,k

= −

1;

2: computeE₀

,

^F0

,

^Z0andY₀ according to(4.2)by the GTH-like algorithm using the triplet representation provided by Theorem 4.1;

3: computew⁽⁰⁾1 andw⁽⁰⁾2 according to(4.8)by the GTH-like algorithm;

4: repeat

5: k

=

k

+

1;

6: computev^(k)1 andv^(k)2 according to(4.9)and generate the triplet representations forI

−

Y_kZ_kandI

−

Z_kY_kas in(4.7);

7: computeE_k+₁

,

^Fk+₁

,

^Zk+₁andY_k+₁according to(4.3)by the GTH-like algorithm using the triplet representations for I

−

Y_kZ_kandI

−

Z_kY_k;

8: computew^(k₁^{+1) and} w^(k₂⁺¹⁾ according to(4.9c) and(4.9d)(reuseE_k(I

−

Y_kZ_k)⁻¹ andF_k(I

−

Z_kY_k)⁻¹ that appear in implementing line 8 to reduce work);

9: untilconvergence;

10: return the lastZ_k

≈

_Φ, andz_k

=

w^(k)2

+

F_ku₂

≈

z.

To accurately invert the second and thirdM-matrices in(4.4), Xue and Li [16] introduced auxiliary vectors [w^(k)₁

w^(k)2

]

:=

[u₁ u₂ ]

−

[E_k Y_k

Z_k F_k ] [u₁

u₂ ]

,

^(4.6)

which are provably nonnegative and can be computed, not in the way as defined in(4.6), but recursively according to the following theorem.

Theorem 4.2([16]).The triplet representations for I_m

−

Y_kZ_kand I

−

Y_kZ_kare given by I_m

−

Y_kZ_k

=

(

offdiag(I_m

−

Y_kZ_k)

,

^u1

,

v^(k)₁ )

,

^(4.7a)

I_n

−

Z_kY_k

=

(

offdiag(I_n

−

Z_kY_k)

,

^u2

,

v^(k)₂ )

,

^(4.7b)

wherev^(k)₁ ^andv^(k)₂ are computed recursively as follows: solving [

α

^B

+

I_m

− β

^D

− α

^C

β

^A

+

I_n ] [w⁽⁰⁾₁

w⁽⁰⁾₂ ]

=

(

α + β

⁾ [u

ˆ

₁

u

ˆ

₂ ]

,

^(4.8)

forw⁽⁰⁾₁ ^andw⁽⁰⁾₂ , and for k

=

0

,

¹

,

²

, . . .

v^(k)₁

=

w^(k)₁

+

E_ku₁

+

Y_k(F_ku₂

+

w^(k)₂ ⁾

≥

0

,

^(4.9a)

v^(k)₂

=

w^(k)₂

+

F_ku₂

+

Z_k(E_ku₁

+

w^(k)₁ ⁾

≥

0

,

^(4.9b)

w^(k1⁺¹⁾

=

w^(k)1

+

E_k(I

−

Y_kZ_k)⁻¹

[

w^(k)1

+

Y_kw^(k)2

] ,

^(4.9c) w^(k2⁺¹⁾

=

w^(k)2

+

F_k(I

−

Z_kY_k)⁻¹

[

Z_kw^(k)1

+

w^(k)2

] .

^(4.9d)

With the help of these triplet representations inTheorems 4.1and4.2all three nonsingularM-matrices in(4.4)can now be inverted in a cancellation-free way, leading to accaddaof [16]. However, in using accaddalater in Section5to solve each smaller-sized super-regularmare(3.3)highly accurately, upon fixingB_j(X), we will need to be able to compute the vector

z

:=

u₂

−

_Φu₁ (4.10)

with high entrywise relative accuracy. This expression cannot be straightforwardly used to fulfill the task because, if computed asz

≈

u₂

−

Z_ku₁, potential cancellations will likely destroy entrywise relative accuracy in some of the computed entries ofz. We have to do something different. The next lemma is essentially [28, Lemma 5.1] which is stated for triplet representations.

Lemma 4.1. Letz_k

=

w^(k)2

+

F_ku₂. Thenz_k

=

u₂

−

Z_ku₁, and, as a result, z

=

u₂

−

_Φu₁

=

lim

k→∞

(u₂

−

Z_ku₁)

=

lim

k→∞

z_k

.

^(4.11)

Proof. It follows from(4.6)thatw^(k)2

=

u₂

−

Z_ku₁

−

F_ku₂, and thusz_k

=

u₂

−

Z_ku₁. Lettingkgo to

∞

yields(4.11). _□

9