A Structure-Preserving Doubling Algorithm for

Nonsymmetric Algebraic Riccati Equation

Xiao-Xia Guo∗ Wen-Wei Lin† Shu-Fang Xu‡

Abstract

In this paper we propose a structure-preserving doubling algorithm (SDA) for computingthe minimal nonnegative solutions to the nonsymmetric algebraic Riccati equation (NARE)based on the techniques developed in the symmetric cases. This method allows the simultane-ous approximation of the minimal nonnegative solutions of the NARE and its dual equation,only requires the solutions of two linear systems, and does not need to choose any initialmatrix, thus it overcomes all the defaults of the Newton iteration method and the fixed-pointiteration methods. Under suitable conditions, we establish the convergence theory by usingonly the knowledge from elementary matrix theory. The theory shows that the SDA iterationmatrix sequences are monotonically increasing and quadratically convergent to the minimalnonnegative solutions of the NARE and its dual equation, respectively. Numerical experi-ments show that the SDA algorithm is feasible and effective, and can outperform the Newtoniteration method and the fixed-point iteration methods.

Keywords: algebraic Riccati equation, minimal nonnegative solution, structure-preservingdoubling algorithm

AMS(MOS) Subject Classifications: 65F10, 65F15, 65N30; CR: G1.3

1 Introduction

In this paper we investigate a structure-preserving doubling algorithm (SDA) for computing the

minimal nonnegative solutions X ∈ Rm×n and Y ∈ Rn×m to the nonsymmetric algebraic Riccati

equation (NARE)

XCX − XD − AX + B = 0 (1.1)

and its dual equation

Y BY − Y A − DY + C = 0, (1.2)

where A, B, C, D are real matrices of sizes m × m, m × n, n × m, n × n, respectively.

Algebraic Riccati equations arise in many important applications, including the total least

squares problems with or without symmetric constraints [8], the spectral factorizations of rational

∗State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Sci-entific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O.Box2719, Beijing 100080, China (guoxx@lsec.cc.ac.cn).

†Department of Mathematics, National Tsing Hua University, Hsinchu, 300, Taiwan (wwlin@am.nthu.edu.tw).‡LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China (xsf@pku.edu.cn). This

research was supported in part by the National Center for Theoretical Sciences in Taiwan.

1

matrix functions [7, 11], the linear and nonlinear optimal controls [4, 31, 34, 35, 38], the contrac-

tive rational matrix functions [12, 21], the structured complex stability radius [18], the transport

theory [20], the Wiener-Hopf factorization of Markov chains [37], the computation of matrix sign

function [25, 32], and the optimal solutions of linear differential systems [22].

Symmetric algebraic Riccati equations have been the topic of extensive research. The theory,

applications, and numerical solution of these equations are the subject of the monographs [22]

and [27]. The NARE (1.1) has been studied recently by several authors, see [15, 16, 17, 20] and

references therein.

To compute the minimal positive solution of the NARE (1.1) under certain assumptions, Guo

and Laub [16] recently proposed a Newton iteration method and a fixed-point iteration method.

However, either the Newton iteration method or the fixed-point iteration method requires solving

a Sylvester equation at each step of the iterations. This is very costly and complicated in actual

applications, in particular, when the matrix sizes are very large, although several feasible and

efficient Sylvester-equation solvers, e.g., the Bartels-Stewart method [2] and the Hessenberg-Schur

method [13], are available. Also, good starting matrices are crucial for guaranteeing the fast

convergence, but are often not easily obtainable, for both iteration methods.

In order to avoid these defaults, in this paper we proposed a SDA algorithm for computing the

minimal nonnegative solution of the NARE (1.1). The doubling algorithms has attracted much

interests in 70s and 80s (see [1] and references therein). However, these algorithms were largely

forgotten in the past decade. Recently, they have been revived for Riccati-type matrix equations,

because their nice numerical behavior: quadratical convergence rate, low cost computational cost

per step, and good numerical stability (see [5, 6, 26, 29]). By employing the same technique as in

[5], we develop a SDA algorithm for solving the NARE (1.1). This method allows the simultaneous

approximation of the minimal nonnegative solutions of the NARE and its dual equation, only

requires the solutions of two linear systems, and does not need to choose any initial matrix, thus it

overcomes all the defaults of the Newton iteration method and the fixed-point iteration methods.

In addition, this algorithm only involves matrix operations and are, hence, more convenient for

being implemented in parallel computing environments. By employing the techniques developed

in [26], under suitable conditions, we establish the convergence theory, using only the knowledge

from elementary matrix theory. The theory shows that the SDA iteration matrix sequences are

monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the

NARE and its dual equation, respectively. Numerical experiments show that the SDA algorithm

is feasible and effective, and can outperform the Newton iteration method and the fixed-point

iteration methods.

The paper is organized as follows. In Section 2, we first introduce some necessary notations,

2

terminologies, and lemmas. Then, in Section 3 we generalize the structure-preserving transfor-

mation of a symplectic pencil to a more generalized matrix pencil, and develop its properties. In

Section 4, we present the SDA algorithm for computing the minimal nonnegative solution of the

NARE (1.1). The convergence theory is established in Section 5, and the practical implementation

is discussed in Section 6. Finally, in Section 7, some numerical examples are given to show the

feasibility and effectiveness of the SDA algorithm, and concluding remarks are given in Section 8.

2 Preliminaries

We first introduce some necessary notations and terminologies. For two matrices A = [aij ], B =

[bij ] ∈ Rm×n, we write A ≥ B (A > B) if aij ≥ bij (aij > bij) holds for all i, j and denote

|A| := [|aij |]. A matrix A ∈ Rm×n is called positive (nonnegative) if its entries satisfy aij > 0

(aij ≥ 0). A matrix A ∈ Rn×n is said to be a Z-matrix if all of its off-diagonal elements are

nonpositive. It follows that any Z-matrix A can be written as the form A = sI − B, with s a

positive real number and B a nonnegative matrix. A matrix A ∈ Rn×n is called a nonsingular

M-matrix if it is a Z-matrix (i.e., A = sI − B for some positive real s and nonnegative matrix B)

and satisfying s > ρ(B), where ρ(B) is the spectral radius of B. For a matrix A ∈ Rn×n, we use

σ(A) to denote its spectrum. For a matrix A ∈ Rm×n, we use ‖A‖1 to denote its matrix 1-norm,

i.e., ‖A‖1 = max1≤j≤n∑m

i=1 |aij |, and ‖A‖∞ := ‖A⊤‖1. We also use e to denote the vector all

components are 1, i.e., e = (1, 1, . . . , 1)T . The open right half-plane is denoted by C>.

The following results will play an important role in our subsequent discussions.

Lemma 2.1. [3, 10, 36] Let A ∈ Rn×n be a Z-matrix. Then the following statements are equivalent:

(a) A is a nonsingular M -matrix;

(b) A−1 ≥ 0;

(c) Av > 0 holds for some positive vector v ∈ Rn;

(d) σ(A) ⊂ C>.

The next result follows from Lemma 2.1.

Lemma 2.2. [3, 10, 28, 36] Let A ∈ Rn×n be a nonsingular M -matrix, and let B ∈ Rn×n be a

Z-matrix. If B ≥ A, then B is also a nonsingular M -matrix. In particular, for any positive real

γ, B = γI + A is a nonsingular M -matrix.

For the existence of the minimal nonnegative solutions to the NARE (1.1) and its dual equation

(1.2), the following result had been established in [15].

3

Lemma 2.3. [15] If

K =

[D −C

−B A

](2.1)

is a nonsingular M-matrix, then the NARE (1.1) and its dual equation (1.2) have the minimal

nonnegative solutions X and Y , respectively, such that D − CX and A − BY are nonsingular

M-matrices.

The condition that the matrix K is a nonsingular M-matrix is suggested by the Wiener-Hopf

factorization of Markov chains. In what follows we always assume that K satisfies this condition

without other statement.

3 Doubling Transformation

In [26], the authors introduce a structure-preserving transformation of a symplectic pencil, which

plays an important role in the establishment of the unified convergence theory for the SDA algo-

rithms for solving a class of symmetric Riccati-type matrix equations. This is also a corner stone for

us to develop the SDA algorithms for solving the NARE (1.1). Therefore, here we first generalize

this transformation to a more generalized matrix pencil, and develop its some basic properties.

Let M − λL ∈ Rℓ×ℓ be a matrix pencil, and define

N (M,L) ={

[M∗, L∗] : M∗, L∗ ∈ Rℓ×ℓ, rank[M∗, L∗] = ℓ, [M∗, L∗]

[L

−M

]= 0

}.

Since rank

[L

−M

]≤ ℓ, it follows that N (M,L) 6= ∅. For any given [M∗, L∗] ∈ N (M,L), define

M̂ = M∗M, L̂ = L∗L.

The transformation

M − λL −→ M̂ − λL̂

is called a doubling transformation.

Notice that the proof of the results (b) and (c) of Theorem 2.1 in [26] does not require the

pencil to be symplectic, we immediately get the following results.

Theorem 3.1. [26] Assume that the pencil M̂ − λL̂ is a doubling transformation of a pencil

M − λL. Then we have:

(i) if

MU = LUR or MV S = LV,

where U, V ∈ Rℓ×k and R,S ∈ Rk×k, then

M̂U = L̂UR2 or M̂V S2 = L̂V ;

4

(ii) If the pencil M − λL has the Kronecker canonical form

WMZ =

[Jr 00 Iℓ−r

], WLZ =

[Ir 00 Nℓ−r

],

where W,Z are nonsingular, Jr is a Jordan matrix corresponding to the finite eigenvalue of M−λL,

and Nℓ−r a nilpotent Jordan matrix corresponding to the infinite eigenvalues of M −λL, then there

exists a nonsingular matrix Ŵ such that

ŴM̂Z =

[Jr

2 00 Iℓ−r

], Ŵ L̂Z =

[Ir 00 N2ℓ−r

].

Theorem 3.1 shows us that the doubling transformation is eigenspace preserving and eigenvalue

doubling. Next we will show that this kind transformation can also preserve some special structures.

A pencil M −λL is said to be a standard symplectic-like form (SSF) if it has the following form

M =

[E 0

−H Im

], L =

[In −G0 F

], with ± E,±F,G,H ≥ 0, (3.1)

where E, H, G, F are real matrices of sizes n × n,m × n, n × m,m × m, respectively.

The following theorem shows that the SSF form can be preserved by an appropriate choice of

the doubling transformations.

Theorem 3.2. Let M −λL be a SSF form. If the matrices In−GH and Im−HG are nonsingular

M-matrices, then a matrix [M∗, L∗] ∈ N (M,L) can be constructed such that its corresponding

doubling transformation M̂ − λL̂ is still a SSF form.

Proof. Using the same techniques as in [6, 26] we are able to get

M∗ =

[E(I − GH)−1 0

−F (I − HG)−1H Im

], L∗ =

[In −E(I − GH)

−1G0 F (I − HG)−1

], (3.2)

such that

M∗L = L∗M,

i.e., [M∗, L∗] ∈ N (M,L). We then compute M∗M and L∗L to produce

M̂ = M∗M =

[Ê 0

−Ĥ Im

], L̂ = L∗L =

[In −Ĝ

0 F̂

]

whereÊ = E(In − GH)

−1E, Ĥ = H + F (Im − HG)−1HE,

F̂ = F (Im − HG)−1F, Ĝ = G + E(In − GH)

−1GF.(3.3)

It is clear that Ê, F̂ , Ĥ, Ĝ ≥ 0, since ±E,±F ≥ 0 and the matrices In − GH and Im − HG are

nonsingular M-matrix, and so the resulting pencil M̂ − λL̂ is still a SSF form.

The proof of Theorem 3.2 provided us with the well defined computation formula for construct-

ing the special structure preserving doubling transformation, which is the base for us to derive the

SDA algorithms for solving the NARE (1.1).

5

4 SDA Algorithm

Assume that X ≥ 0 is the minimal nonnegative solution of the NARE (1.1), it is easy to verify

that the NARE (1.1) can be rewritten as

H

[InX

]=

[InX

]R, (4.1)

where

H =

[D −CB −A

], R = D − CX. (4.2)

Since K is a nonsingular M-matrix, it follows from Lemma 2.3 that the matrix R is a nonsingular

M -matrix, by Lemma 2.1, which implies σ(R) ⊂ C>. Using Cayley transformation with some

appropriate γ > 0, we can transform (4.1) into the following form

(H− γI)

[InX

]= (H + γI)

[InX

]Rγ , (4.3)

where

Rγ = (R + γIn)−1(R − γIn). (4.4)

Since σ(R) ⊂ C>, it follows that ρ(Rγ) < 1 for any γ > 0. By Lemma 2.2, for any γ > 0 the

matrix K + γI is a nonsingular M-matrix, and hence, it follows that B, C ≥ 0 and the matrices

Aγ = A + γIm and Dγ = D + γIn (4.5)

are nonsingular M-matrix for any γ > 0. Let

Wγ = Aγ − BD−1γ C, Vγ = Dγ − CA

−1γ B. (4.6)

By Lemma 2.1, we have

0 ≤ (K + γI)−1 =

[Dγ −C−B Aγ

]−1

=

([In 0

−BD−1γ Im

] [Dγ −C0 Wγ

])−1

=

[D−1γ D

−1γ CW

−1γ

0 W−1γ

] [In 0

BD−1γ Im

]

=

[D−1γ + D

−1γ CW

−1γ BD

−1γ D

−1γ CW

−1γ

W−1γ BD−1γ W

−1γ

]. (4.7)

Applying the Shermann-Morrison-Woodbury formula (SMWF; see,e.g.,[14, p.50]) we have

D−1γ + D−1γ CW

−1γ BD

−1γ

=D−1γ[In + C(Im − A

−1γ BD

−1γ C)

−1A−1γ BD−1γ

]

6

=D−1γ (In − CA−1γ BD

−1γ )

−1

=(Dγ − CA−1γ B)

−1 = V −1γ .

This, together with (4.7), shows that W−1γ , V−1γ ≥ 0. Thus, by Lemma 2.1, it follows that the

matrices Wγ and Vγ are nonsingular M -matrices for any γ > 0.

Now let

L1 =

[D−1γ 0

−BD−1γ Im

], L2 =

[In 00 −W−1γ

], L3 =

[In D

−1γ C

0 Im

].

Then, with the help of the SMWF, direct multiplication gives rise to

L = L3L2L1(H + γI) =

[In −Gγ0 Fγ

],

M = L3L2L1(H− γI) =

[Eγ 0

−Hγ Im

],

(4.8)

whereEγ = In − 2γV

−1γ , Gγ = 2γD

−1γ CW

−1γ ,

Fγ = Im − 2γW−1γ , Hγ = 2γW

−1γ BD

−1γ .

(4.9)

Clearly, with this transformation the equality (4.3) is transformed into the following form

M

[InX

]= L

[InX

]Rγ , (4.10)

and moreover, it is easy to derive that if Y ≥ 0 is the minimal nonnegative solution of the dual

equation (1.2), then

M

[YIm

]Sγ = L

[YIm

], (4.11)

where Sγ = (S + γIm)−1(S − γIm) with S = A − BY a nonsingular M-matrix and ρ(Sγ) < 1.

Theorem 4.1. Assume that K defined by (2.1) is a nonsingular M-matrix, let Eγ , Fγ ,Hγ , Gγ , Rγ , Sγ

be defined as above. If the parameter γ satisfies

γ > max{ max1≤i≤m

aii, max1≤i≤n

dii}, (4.12)

where aii and dii are the i-th diagonal elements of the matrices A and D, respectively, then

−Eγ ,−Fγ ,−Rγ ,−Sγ ≥ 0, with −Eγe,−Fγe,−Rγe,−Sγe > 0; and moreover, the matrices Im −

HγGγ and In − GγHγ are nonsingular M-matrices.

Proof. By the definition of matrix Eγ , we have

Eγ = In − 2γV−1γ = V

−1γ (Vγ − 2γIn) = V

−1γ (−γIn + D − CA

−1γ B). (4.13)

Since CA−1γ B ≥ 0 and D is a nonsingular M-matrix, we have γIn − D + CA−1γ B ≥ 0 with all

diagonal elements nonzero if γ > max1≤i≤n dii. Notice that Vγ is a nonsingular M-matrix, it follows

from (4.13) that −Eγ ≥ 0 with −Eγe > 0, provided γ > max1≤i≤n dii.

7

Since C, X ≥ 0, we have D ≥ D − CX = R. Thus, if γ > max1≤i≤n dii, then γIn − R =

γIn − D + CX ≥ 0 with all diagonal elements nonzero, since D is a nonsingular M-matrix.

Consequently, −Rγ = (γIn + R)−1(γIn − R) ≥ 0 with −Rγe > 0, provided γ > max1≤i≤n dii.

Here we utilize the fact that γI + R is a nonsingular M-matrix.

Similarly, we can prove that −Fγ ,−Sγ ≥ 0, with −Fγe,−Sγe > 0, provided γ > max1≤i≤m aii.

Clearly, it follows from (4.9) that Hγ , Gγ ≥ 0. Thus, the matrices I −GγX and I −GγHγ are

Z-matrices. Comparing the blocks of the both sides of (4.10) yields

X − Hγ = FγXRγ , Eγ = (In − GγX)Rγ .

This, together with the first part of this theorem, gives rise to

X − Hγ = FγXRγ ≥ 0, (In − GγX)v = −Eγe > 0,

with v = −Rγe > 0. Consequently, by Lemma 2.1, the matrix In−GγX is a nonsingular M-matrix,

and the inequality In − GγX ≤ In − GγHγ , by Lemma 2.2, implies that the matrix In − GγHγ is

also a nonsingular M-matrix.

Similarly, using the equality (4.11) we can prove that the matrix Im − HγGγ is a nonsingular

M-matrix.

Theorem 4.1 tells us that by a simple choice of the parameter γ we are able to make the pencil

M − λL defined by (4.8) be a SSF form with Im −HγGγ and In −GγHγ nonsingular M-matrices.

Therefore, applying the doubling transformation defined by (3.3) to this pencil repeatedly gives

rise to the following structure-preserving doubling algorithm for computing the minimal negative

solutions of the NARE (1.1) and its dual equation (1.2):

Algorithm SDA.

E0 = Eγ , F0 = Fγ , G0 = Gγ , H0 = Hγ ,

Ek+1 = Ek(In − GkHk)−1Ek,

Fk+1 = Fk(Im − HkGk)−1Fk,

Gk+1 = Gk + Ek(In − GkHk)−1GkFk,

Hk+1 = Hk + Fk(Im − HkGk)−1HkEk.

Of course, to ensure that this iteration is well defined, the matrices In − GkHk and Im −

HkGk must be nonsingular for all k. In fact, in the next section we will prove that under the

hypothesis of Theorem 4.1 the iteration is well defined and the matrix sequences {Hk} and {Gk}

8

are monotonically increasing and quadratically convergent to the minimal nonnegative solutions

X and Y of the NARE (1.1) and its dual equation (1.2), respectively.

Clearly, This algorithm avoids all the defaults of the Newton iteration method and the fixed

point iteration method proposed in [16]. Each step of this algorithm only requires the solutions of

two linear systems, thus it is much faster than other iterations. In addition, this algorithm only

involves matrix operations and are, hence, more convenient for being implemented in parallel com-

puting environments. Numerical experiments show that this algorithm are feasible and effective,

and can outperform the Newton iteration method and the fixed-point iteration methods.

5 Convergence Theory

Now we establish the convergence theory of Algorithm SDA based on Theorem 3.1. The main

results are listed in the following theorem.

Theorem 5.1. Assume that K defined by (2.1) is a nonsingular M-matrix, let X,Y ≥ 0 be the

minimal nonnegative solutions of the NARE (1.1) and its dual equation (1.2), respectively, and let

Rγ = (R + γIn)−1(R − γIn), Sγ = (S + γIm)

−1(S − γIm), (5.1)

where R = D − CX, S = A − BY . If the parameter γ satisfies

γ > max{

max1≤i≤m

aii, max1≤i≤n

dii

}, (5.2)

where aii and dii are the i-th diagonal elements of the matrices A and D, respectively, then the

matrix sequences {Ek}, {Hk}, {Gk}, and {Fk} generated by Algorithm SDA are well defined, and

they hold that

(a) Ek = (In − GkX)R2

k

γ ≥ 0, with Eke > 0;

(b) Fk = (Im − HkY )S2k

γ ≥ 0, with Fke > 0;

(c) Im − HkGk and In − GkHk are nonsingular M-matrices;

(d) 0 ≤ Hk ≤ Hk+1 ≤ X and

0 ≤ X − Hk = (Im − HkY )S2

k

γ XR2

k

γ ≤ S2

k

γ XR2

k

γ ; (5.3)

(e) 0 ≤ Gk ≤ Gk+1 ≤ Y and

0 ≤ Y − Gk = (In − GkX)R2

k

γ Y S2

k

γ ≤ R2

k

γ Y S2

k

γ . (5.4)

9

Proof. (Apply Mathematical Induction.) Denote

Mk =

[Ek 0

−Hk Im

], Lk =

[In −Gk0 Fk

].

Firstly, by Theorem 4.1, we have −Eγ ,−Fγ ,−Rγ ,−Sγ ≥ 0, with −Eγe,−Fγe,−Rγe, −Sγe >

0, and Im − HγGγ and In − GγHγ to be nonsingular M-matrices. Therefore, it follows that

E1,H1, G1, F1 are all well defined, and moreover, we have

E1 = E0(In − G0H0)−1E0 = (−Eγ)(In − GγHγ)

−1(−Eγ) ≥ 0,

F1 = F0(Im − H0G0)−1F0 = (−Fγ)(Im − HγGγ)

−1(−Fγ) ≥ 0,

H1 = H0 + F0(Im − H0G0)−1H0E0 = H0 + (−Fγ)(Im − HγGγ)

−1Hγ(−Eγ) ≥ H0,

G1 = G0 + E0(In − G0H0)−1G0F0 = G0 + (−Eγ)(In − GγHγ)

−1Gγ(−Fγ) ≥ G0,

with

E1e = E0(In − G0H0)−1E0e = (−Eγ)(In − GγHγ)

−1(−Eγ)e > 0,

F1e = F0(Im − H0G0)−1F0e = (−Fγ)(Im − HγGγ)

−1(−Fγ)e > 0.

Since the pencil M1 − λL1 is a doubling transformation of M0 − λL0, where M0 = M and

L0 = L are defined by (4.8), applying (i) of Theorem 3.1 to (4.10), we get

[E1 0

−H1 Im

] [InX

]=

[In −G10 F1

] [InX

]R2γ . (5.5)

Equating the blocks of (5.5) gives rise to

X − H1 = F1XR2γ , E1 = (In − G1X)R

2γ . (5.6)

Noting that −Rγ , F1 ≥ 0, with −Rγe, F1e > 0, it follows from (5.6) that X − H1 ≥ 0, i.e.,

X ≥ H1, and (In − G1X)v = E1e > 0 with v = R2γe > 0, which, by Lemma 2.1, implies that the

matrix In − G1X is a nonsingular. Using X ≥ H1, we have In − G1X ≤ In − G1H1, therefore, by

Lemma 2.2, the matrix In − G1H1 is also a nonsingular M-matrix.

Similarly, utilizing (4.11) we can derive

Y − G1 = E1Y S2γ , F1 = (Im − H1Y )S

2γ , (5.7)

from which we can prove that Y ≥ G1 and Im − H1G1 is a nonsingular M-matrix.

Combining (5.6) with (5.7) yields

X − H1 = (Im − H1Y )S2γXR

2γ ≤ S

2γXR

2γ ,

Y − G1 = (In − G1X)R2γY S

2γ ≤ R

2γY S

2γ .

10

Thus, we have proved that this theorem is true for k = 1.

Next, assume that the theorem is true for all positive integers less than or equal to positive

integer k. Consider the case of k + 1. Since I −HkGk and I −GkHk are nonsingular M -matrices,

it follows that Ek+1,Hk+1, Gk+1, Fk+1 are all well defined, and, following the same lines as the

proof as in the case of k = 1, from the induction assumption we can easily derive that

Ek+1 ≥ 0, Ek+1e > 0, 0 ≤ Hk ≤ Hk+1,

Fk+1 ≥ 0, Fk+1e > 0, 0 ≤ Gk ≤ Gk+1.

On the other hand, since Mj+1 − λLj+1 is a doubling transformation of Mj − λLj for j =

0, 1, · · · , k, by using (i) of Theorem 3.1 to the equality (4.10) k + 1 times, we get

Mk+1

[InX

]= Lk+1

[InX

]R2

k+1

γ . (5.8)

Equating the blocks of (5.8) yields that

X − Hk+1 = Fk+1XR2

k+1

γ , Ek+1 = (In − Gk+1X)R2

k+1

γ . (5.9)

Notice that −Rγ ≥ 0 with −Rγe > 0 implies that R2

k+1

γ ≥ 0 with R2

k+1

γ e > 0, similar the proof as

in the case of k = 1, from the equalities (5.9) it can be proved that X −Hk+1 ≥ 0 and the matrix

In − Gk+1Hk+1 is a nonsingular M-matrix.

Similarly, by using (i) of Theorem 3.1 to the equality (4.11) k + 1 times, we can derive that

Y − Gk+1 = Ek+1Y S2

k+1

γ , Fk+1 = (Im − Hk+1Y )S2

k+1

γ , (5.10)

from which we can prove that Y − Gk+1 ≥ 0 and the matrix Im − Hk+1Gk+1 is a nonsingular

M-matrix, by using the fact that −Sγ ≥ 0 with −Sγe > 0 implies that S2

k+1

γ ≥ 0 with S2

k+1

γ e > 0.

Combining (5.9) with (5.10) gives rise to

0 ≤ X − Hk+1 = (Im − Hk+1Y )S2

k+1

γ XR2

k+1

γ ≤ S2

k+1

γ XR2

k+1

γ ,

0 ≤ Y − Gk+1 = (In − Gk+1X)R2

k+1

γ Y S2

k+1

γ ≤ R2

k+1

γ Y S2

k+1

γ .

This shows that the theorem is also true for positive integer k + 1. By induction principle the

theorem is true for all positive integers.

Noting that ρ(Rγ), ρ(Sγ) < 1 for all γ > 0 and that 0 ≤ A ≤ B implies that ‖A‖1 ≤ ‖B‖1,

from Theorem 5.1 we immediately get the following convergence result of Algorithm SDA.

Corollary 5.2. Under the hypothesis of Theorem 5.1, we have

(1) ‖Ek‖1 ≤ ‖R2k

γ ‖1 −→ 0, as k → ∞;

11

(2) ‖Fk‖1 ≤ ‖S2

k

γ ‖1 −→ 0, as k → ∞;

(3) ‖X − Hk‖1 ≤ ‖X‖1‖S2

k

γ ‖1‖R2

k

γ ‖1 −→ 0, as k → ∞;

(4) ‖Y − Gk‖1 ≤ ‖Y ‖1‖S2

k

γ ‖1‖R2

k

γ ‖1 −→ 0, as k → ∞.

Theorem 5.1 tells us that the matrix sequences {Hk} and {Gk} are monotonically increasing,

provided that the matrix K is a nonsingular M-matrix and the parameter γ satisfies the condi-

tion (5.2), while the above corollary shows that under the assumptions of Theorem 5.1, they are

quadratically convergent to the minimal nonnegative solutions X and Y of the NARE (1.1) and

its dual equation, respectively.

6 Practical Implementation

Selection of the parameter γ. We assume that the LU factorization of Aγ , Dγ , Wγ = Aγ −

BD−1γ C and Vγ = Dγ − CA−1γ B are computed by Gaussian elimination with partial pivoting

(GEPP). Let LA, UA, LD, UD, LW , UW , LV and UV be the corresponding computed LU factors.

Then we have the following relations (see e.g., [31, Theorem 9.3])

Aγ + △Aγ = LAUA, |△Aγ | ≤ γm|LA||UA|, (6.1)

Dγ + △Dγ = LDUD, |△Dγ | ≤ γn|LD||UD|, (6.2)

Wγ + △Wγ = LW UW , |△Wγ | ≤ γm|LW ||UW |, (6.3)

Vγ + △Vγ = LV UV , |△Vγ | ≤ γn|LV ||UV |, (6.4)

where γm = mu/(1−mu) and γn = nu/(1− nu). The quantity u is the unit roundoff or machine

precision. Consequently, it holds

f l(W−1γ ) = W−1γ + E1, |E1| ≤ cmu|W

−1γ ||LW ||UW || f l(W

−1γ )|, (6.5)

f l(V −1γ ) = V−1γ + E2, |E2| ≤ cnu|V

−1γ ||LV ||UV || f l(V

−1γ )|, (6.6)

where cm and cn are modest constants. From (6.5) and (6.6), the forward error bound in evaluating

Eγ and Fγ in (4.9) are, respectively.

f l(Eγ) = Eγ + E3, |E3| ≤ 4γcnu|V−1γ ||LV ||UV || f l(V

−1γ )| + u|Eγ | + O(u

2), (6.7)

f l(Fγ) = Fγ + E4, |E4| ≤ 4γcmu|W−1γ ||LW ||UW || f l(W

−1γ )| + u|Fγ | + O(u

2). (6.8)

Furthermore, from (6.2) we have

C̄ ≡ f l(2γD−1γ C) = 2γD−1γ C + E5, |E5| ≤ 2γcnu|D

−1γ ||LD||UD||C̄|, (6.9)

12

hence the forward error bound in evaluating Gγ in (4.9) is

f l(Gγ) = Gγ + E6, |E6| ≤ 2γcnu|D−1γ ||LD||UD||C̄||W

−1γ | + cnu| f l(Gγ)||LW ||UW ||W

−1γ |. (6.10)

Similarly, from (6.2) and (6.3), the forward error bound in evaluating Hγ in (4.9) is

f l(Hγ) = Hγ + E7, |E7| ≤2γcnu|W−1γ ||B̄||LD||UD||D

−1γ |

+ cnu|W−1γ ||LW ||UW || f l(Hγ)|, (6.11)

where B̄ ≡ f l(2rBD−1γ ).

For GEPP, we have in practice

‖|LD||UD|‖∞ ≈ ‖Dγ‖∞, ‖|LW ||UW |‖∞ ≈ ‖Wγ‖∞, ‖|LV ||UV |‖∞ ≈ ‖Vγ‖∞.

It follows from (6.1)-(6.11) that

‖ f l(Eγ) − Eγ‖∞ . 4cnuγκ∞(Vγ)‖ f l(V−1γ )‖∞ + u‖Eγ‖∞ + O(u

2), (6.12)

‖ f l(Fγ) − Fγ‖∞ . 4cmuγκ∞(Wγ)‖ f l(W−1γ )‖∞ + u‖Fγ‖∞ + O(u

2), (6.13)

‖ f l(Gγ) − Gγ‖∞ . 2cnuγκ∞(Dγ)‖C̄‖∞‖W−1γ ‖∞ + cnuκ∞(Wγ)‖ f l(Gγ)‖∞, (6.14)

‖ f l(Hγ) − Hγ‖∞ . 2cnuγκ∞(Dγ)‖B̄‖∞‖W−1γ ‖∞ + cnuκ∞(Wγ)‖ f l(Hγ)‖∞. (6.15)

Suppose that all quantities of ‖ · ‖∞ in (6.12)-(6.15) are bounded by O(1). In order to control

the forward error bounds of Eγ , Fγ , Gγ and Hγ , we consider the following min-max optimization

problem to determine an optimal value γ̂ > 0 :

min

{max

1≤j≤4(fj(γ)) : γ ≥ max{ max

1≤i≤maii, max

1≤i≤ndii}≡ γ0

}, (6.16)

wheref1(γ) = γκ∞(Vγ), f2(γ) = γκ∞(Wγ),

f3(γ) = γκ∞(Dγ), f4(γ) = κ∞(Wγ).(6.17)

Extensive numerical experiments on randomly generated matrices indicates the function F (γ) ≡

max1≤j≤4 fj(γ) is a strictly convex function in the neighborhood of the optimal value γ̂ where the

global minimum of F (γ) occurs. For illustration, we report a sample of F (γ) in Figure 6.1. From

Theorem 5.1 and Corollary 5.2, we know that if γ approaches ∞, the symplectic-like matrix pair

(M,L) as in (4.8) has eigenvalues close to one, leading to very slow convergence of SDA algorithm.

This can be avoided via the min-max optimization problem (6.16).

The optimization (6.16) can be solved by applying the Fibonacci search method, see e.g., [9,

p.272]. Our experience indicates that three to five iterations of Fibonacci search are adequate to

obtain a suboptimal yet acceptable approximation to γ̂. In practice, more convenient choice is to

13

0 5 10 15 20 25 30 35 40 45 5010

1

102

103

104

γ

F(γ)

Figure 6.1: The graph of F (γ) with γ0 ≈ 5 and γ̂ ≈ 9.81

take γ = γ0 + δ for some small positive number δ, where γ0 = max{max1≤i≤m aii,max1≤i≤n dii}.

Our numerical experiments show that such a choice is often a better one.

Error analysis of SDA. We consider the forward error bounds of the computed matrices Ek+1,

Fk+1, Gk+1 and Hk+1 in the SDA algorithm for one iterative step k. For simplicity, we suppose

that there is no forward errors in the matrix multiplication and addition. Let

Φk := I − GkHk, Ψk := I − HkGk. (6.18)

Assume Φk and Ψk have the LU-factorizations by Gaussian elimination with partial pivoting

(GEPP) :

Φk = LΦkUΦk + △Φk, |△Φk| ≤ γn|LΦk ||UΦk |, (6.19)

Ψk = LΨkUΨk + △Ψk, |△Ψk| ≤ γm|LΨk ||UΨk |, (6.20)

where γn = nu/(1 − nu) and γm = mu/(1 − mu). Then we have the following forward error

bounds in evaluating Gk+1 and Hk+1, respectively,

f l(Ek+1) = Ek+1 + EEk+1 , |EEk+1 | ≤ cnu|EkΦ−1k ||LΦk ||UΦk ||Φ

−1k ||Ek|

. cnu‖EkΦ−1k ‖∞κs(Φk)‖Ek‖∞, (6.21)

f l(Fk+1) = Fk+1 + EFk+1 , |EFk+1 | ≤ cnu|FkΨ−1k ||LΨk ||UΨk ||Ψ

−1k ||Fk|

. cnu‖FkΨ−1k ‖∞κs(Ψk)‖Fk‖∞, (6.22)

f l(Gk+1) = Gk+1 + EGk+1 , |EGk+1 | ≤ cnu|EkΦ−1k ||LΦk ||UΦk ||Φ

−1k ||Gk||Fk|

14

. cnu‖EkΦ−1k ‖∞κs(Φk)‖Fk‖∞, (6.23)

f l(Hk+1) = Hk+1 + EHk+1 , |EHk+1 | ≤ cnu|FkΨ−1k ||LΨk ||UΨk ||Ψ

−1k ||Hk||Ek|

. cnu‖FkΨ−1k ‖∞κs(Ψk)‖Ek‖∞, (6.24)

where cn is a modest constant and κs(A) :=∥∥|A||A−1|

∥∥∞

is the Skeel condition number.

When the Skeel condition numbers κs(Φk) and κs(Ψk) in (6.21)-(6.24) are bounded from above

by acceptable numbers, the accumulation of error will be dampened by the fast rate of convergence

at the final stage of iterative process. Danger, if any, lies in the early stage of the process before

the ‖R2k

r ‖1 and ‖S2

k

r ‖1 converge factors dominates. It is unlikely to have any ill-effect, as the

accumulated error in the matrix additions and multiplications should be of magnitude around a

small multiple of the machine accuracy. As the SSF properties are preserved in the SDA, any

error will be a structured one, only pushing the iteration towards a solution of a neighboring SSF

system. Thus, the SDA algorithm is stable in this sense, when the errors are not too large and

when the nonnegativity of matrices Ek, Fk, Gk and Hk are maintained.

7 Numerical Examples

In this section, we use several examples to show the monotone and quadratical convergence property

and the numerical effectiveness of the SDA algorithm. In particular, we compare the numerical

behavior of SDA with the Newton iteration method and the fixed-point iteration methods (i.e.,

FP1, FP2, and FP3) with respect to the number of iteration steps (IT), the computing times

(CPU), and the relative errors (RES), where

RES =‖X̃CX̃ − X̃D − AX̃ + B‖∞

‖X̃CX̃‖∞ + ‖X̃D‖∞ + ‖AX̃‖∞ + ‖B‖∞,

with X̃ the approximation solution to the minimal nonnegative solution of (1.1). In all the exam-

ples, we will take γ = [γ0] + 1 (see Example 7.1) without special statement, where γ0 is defined

as (6.16), and [γ0] denotes the integer part of γ0.

In our implementations, all iterations are run in MATLAB (version 6.5) on a personal Pentium

IV, with machine precision 10−16, and are terminated when the current iterate satisfies RES <

10−12.

Recall that the Newton’s iteration and the fixed-point iterations proposed in [16] can be de-

scribed as follows:

Newton’s iteration(NW)[16]. Given an initial X0 = 0. For k = 0, 1, 2 . . . until Xk converges,

compute Xk+1 form Xk by solving the following Sylvester equation

(A − XkC)Xk+1 + Xk+1(D − CXk) = B − XkCXk.

15

Fixed-point iterations[16]. Given an initial X0 = 0. For k = 0, 1, 2 . . . until Xk converges,

compute Xk+1 form Xk by solving the following Sylvester equation

A1Xk+1 + Xk+1D1 = XkCXk + XkD2 + A2Xk + B,

where A = A1 − A2 and D = D1 − D2, with A2,D2 ≥ 0 and A1,D1 being Z-matrices. By the

different splitting of A,D, the following three fixed-pointed iterations are proposed:

FP1. Let A1 = diag(A) and D1 = diag(D).

FP2. Let A1 = tril (A) be the low triangular of A and D1 = triu (D) be the upper triangular

of D.

FP3. Let A1 = A and D1 = D.

Example 7.1. Consider NARE (1.1), for which

A = D = tridiag (−I, T, −I) ∈ Rn×n

are block tridiagonal matrices,

C =1

50· tridiag (1, 2, 1) ∈ Rn×n,

is a tridiagonal matrix, and

B = AS + SD − SCS,

such that

S =1

50eeT ∈ Rn×n, with e = (1, 1, · · · , 1)T ∈ Rn,

is the minimal nonnegative solution of NARE (1.1). Here,

T = tridiag (−1, 4 +150

(m + 1)2, −1) ∈ Rm×m

and n = m2, m = 8.

In Figure 7.1, we plot the curves of the relative errors

ER =‖Hk − S‖∞

‖S‖∞and DRES =

‖GkBGk − GkA − DGk + C‖∞‖GkBGk‖∞ + ‖GkA‖∞ + ‖DGk‖∞ + ‖C‖∞

with respect to the iteration index k, respectively, where Hk and Gk are computed by SDA algo-

rithm. Clearly, we see that both of {Hk} and {Gk} are quadratically convergent, which further

confirms the quadratic convergence property of the SDA algorithm from numerical viewpoint.

In Figures 7.2 and 7.3, we plot the curves of the (i, j)-th entries Hk(i, j) and Gk(i, j) of the

iterates Hk and Gk with respect the iteration index k for (i, j) = (1, 1), (1, n), (n, 1) and (n, n).

16

1 1.5 2 2.5 3 3.5 410

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

k

ER

1 1.5 2 2.5 3 3.5 410

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

k

DR

ES

Figure 7.1: The quadratic convergence property of SDA algorithm for Example 7.1

Clearly, we see that both of {Hk}, {Gk} are monotonically increasing, and {Hk} converges to the

exact minimal nonnegative solution S of NARE (1.1). These results further confirm the monotone

convergence property of the SDA algorithm from numerical viewpoint.

In addition, in Table 1 we list the number of iteration for the stopping criterion to be satisfied

with respect to various parameter γ. From the table we see that the number of iteration is

monotonically increasing as the parameter γ is increasing. For this example, it is clear that the

best choice for the parameter γ is γ0 or slightly greater than γ0.

Table 1: The number of iteration with respect to various parameter γ for Example 7.1

γ γ0 γ0 + 1 γ0 + 2 γ0 + 3 γ0 + 4 γ0 + 13IT 4 4 4 4 5 6γ γ0 + 32 γ0 + 70 γ0 + 146 γ0 + 296 γ0 + 600 γ0 + 1200IT 7 8 9 10 11 12γ γ0 + 2500 γ0 + 6000 γ0 + 7000 γ0 + 8000 γ0 + 9000 γ0 + 10000IT 13 14 ≥ 50 14 ≥ 50 15

Example 7.2. Consider NARE (1.1), for which A and D are the same as in Example 7.1, B =

1

50· tridiag(1, 2, 1) ∈ Rn×n, and C = ξB, where ξ is a positive constant and n = m2.

We take m = 16, the iteration steps, CPU times and relative errors of SDA, Newton and

fixed-point iteration methods with respect to various ξ are listed in Table 2.

From Table 2 we see that all iterations can converge to the exact minimal nonnegative solution

of NARE (1.1) with high accuracy. According to the iteration step, the Newton method is the least;

SDA and FP1 is comparable with Newton; and SDA is much less than FP2 and FP3. According to

the computing times, SDA is the least; Newton is more than SDA; FP1 is more than Newton; and

17

S(1, 1), Hk(1, 1) versus k

1 1.5 2 2.5 3 3.5 40.0196

0.0197

0.0197

0.0198

0.0198

0.0199

0.0199

0.0199

0.02

0.02

S(1, n), Hk(1, n) versus k

1 1.5 2 2.5 3 3.5 40.0196

0.0197

0.0197

0.0198

0.0198

0.0199

0.0199

0.0199

0.02

0.02

S(n, 1), Hk(n, 1) versus k

1 1.5 2 2.5 3 3.5 40.0196

0.0197

0.0197

0.0198

0.0198

0.0199

0.0199

0.0199

0.02

0.02

S(n, n), Hk(n, n) versus k

1 1.5 2 2.5 3 3.5 40.0196

0.0197

0.0197

0.0198

0.0198

0.0199

0.0199

0.0199

0.02

0.02

Figure 7.2: S(i, j),Hk(i.j), i, j = 1, n, with respect to k for Example 7.1(solid: S(i, j), dashed:Hk(i, j))

.FP3 is the most. Therefore, with respect to the computing efficiency, SDA algorithm outperforms

the Newton and all fixed-point iteration methods including FP1, FP2, and FP3.

Example 7.3. Consider NARE (1.1), for which A, B, C and D are generated according to the fol-

lowing rule: First, generate and save a random 100×100 nonzero matrix R by using rand(100, 100);

then set W = diag(Re)−R, with e = (1, 1, . . . , 1)T ∈ R100; and finally, for a given positive constant

κ, define{

D = W (1 : 50, 1 : 50) + κI, A = W (51 : 100, 51 : 100) + κI,B = −W (51 : 100, 1 : 50), C = −ξW (1 : 50, 51 : 100),

where ξ is a positive constant. Note that the so-generated matrix W is a nonsingular M -matrix,

and A, B, C and D are 50×50 matrices with A, D being nonsingular M -matrices and B, C being

nonnegative matrices, respectively.

The iteration steps, CPU times and relative errors of SDA, Newton and fixed-point iteration

methods with respect to various ξ are listed in Table 3.

From Table 3 we see that all iterations can converge to the exact minimal nonnegative solution

of NARE(1.1) with high accuracy. According to the iteration step, the Newton method is the

same as SDA, and they are the least; FP1 is more than SDA and Newton and is less than FP2

and FP3. According to the computing times, SDA is the least; Newton is about 5 times that of

SDA; FP1 is more than Newton, but is comparable with FP2 and FP3. Therefore, with respect to

18

Gk(1, 1) versus k

1 1.5 2 2.5 3 3.5 43.992

3.994

3.996

3.998

4

4.002

4.004x 10

−3

Gk(1, n) versus k

1 1.5 2 2.5 3 3.5 41.65

1.7

1.75

1.8

1.85

1.9

1.95

2

2.05

2.1x 10

−6

G(n, 1) versus k

1 1.5 2 2.5 3 3.5 41.65

1.7

1.75

1.8

1.85

1.9

1.95

2

2.05

2.1x 10

−6

G(n, n) versus k

1 1.5 2 2.5 3 3.5 41.65

1.7

1.75

1.8

1.85

1.9

1.95

2

2.05

2.1x 10

−6

Figure 7.3: G(i, j), i, j = 1, 2, with respect to k for Example 7.1

the computing efficiency, SDA algorithm considerably outperforms the Newton and all fixed-point

iteration methods including FP1, FP2, and FP3 .

Example 7.4. Consider NARE (1.1), for which

A = D =

3 −1

3. . .

. . . −1−1 3

∈ Rn×n, B = In and C = ξIn,

with ξ a positive constant.

We take n = 256, the iteration steps, CPU times and relative errors of SDA, Newton and

fixed-point iteration methods with respect to various ξ are listed in Table 4.

From Table 4 we see that all iterations can converge to the exact minimal nonnegative solution

of NARE(1.1)) with high accuracy. According to the iteration step, the Newton method is the

same as SDA algorithm, and they are the least; FP1 is more than them; and much less than FP2

and FP3. According to the computing times, SDA is the least; Newton is much more than SDA,

is about 10 times that of SDA; FP3 is much more than Newton, but is less than FP1 and FP2,

FP1 is the most. Therefore, with respect to the computing efficiency, SDA algorithm considerably

outperforms the Newton and all fixed-point iteration methods including FP1, FP2, and FP3.

Moreover, from Examples 7.3 and 7.4 we see that when ξ, or C, becomes large, the iteration

numbers and the computing times of all fixed-point iterations increase quickly, and those of the

19

Table 2: Numerical results for Example 7.2 when m = 16

method FP1 FP2 FP3 NW SDAIT 5 197 104 3 6

ξ = 0.2 CPU 8.954 314.031 361.891 7.062 4.079RES 4.5E-14 9.9E-13 8.3E-13 4.4E-16 7.7E-15IT 5 198 104 3 6

ξ = 0.5 CPU 9.109 317.125 363.125 7.14 4.078RES 7.7E-13 9.3E-14 9.1E-14 4.6E-16 8.1E-15IT 6 199 105 3 6

ξ = 1.0 CPU 10.984 320.688 365.625 7.094 4.078RES 2.4E-13 9.2E-13 8.0E-13 4.3E-16 8.5E-15IT 7 201 106 3 6

ξ = 2.0 CPU 12.781 324.125 368.781 7.125 4.078RES 3.0E-13 9.1E-13 8.2E-13 4.7E-15 1.1E-14

Table 3: Numerical results for Example 7.3 when κ = 10

method FP1 FP2 FP3 NW SDAIT 9 33 21 4 4

ξ = 0.2 CPU 0.375 0.422 0.375 0.157 0.031RES 6.4E-13 7.6E-13 5.3E-13 1.8E-16 5.0E-16IT 13 38 26 4 4

ξ = 0.5 CPU 0.547 0.531 0.469 0.156 0.031RES 1.0E-12 8.2E-13 3.5E-13 1.9E-16 9.7E-15IT 22 50 35 5 5

ξ = 1.0 CPU 0.891 0.625 0.609 0.203 0.047RES 4.6E-13 7.5E-13 1.0E-12 1.7E-16 6.0E-16

SDA algorithm however, are almost fixed. This shows that SDA could successfully solve NARE(1.1)

of strong nonlinearity.

8 Conclusions

In this paper, we investigate a structure-preserving doubling algorithm (SDA) for computing the

minimal nonnegative solutions to the nonsymmetric algebraic Riccati equation based on the tech-

niques developed in the symmetric cases. This method allows the simultaneous approximation of

the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equation and its dual

equation, only requires the solutions of two linear systems, and does not need to choose any initial

matrix, thus it overcomes all the defaults of the Newton iteration method and the fixed-point

iteration methods. Under suitable conditions, we establish the convergence theory by using only

the knowledge from elementary matrix theory. The theory shows that the SDA iteration matrix

sequences are monotonically increasing and quadratically convergent to the minimal nonnegative

solutions of the nonsymmetric algebraic Riccati equation and its dual equation, respectively. Nu-

20

Table 4: Numerical results for Example 7.4 when n = 256

method FP1 FP2 FP3 NW SDAIT 8 26 19 3 4

ξ = 0.2 CPU 46.688 43.953 31.812 16.813 2.875RES 9.7E-14 6.7E-13 5.5E-13 2.8E-14 9.4E-18IT 10 28 21 4 4

ξ = 0.5 CPU 58.062 47.250 35.156 22.140 2.875RES 3.4E-13 5.6E-13 2.9E-13 1.4E-16 1.5E-16IT 13 31 24 4 4

ξ = 1.0 CPU 74.968 51.453 40.125 22.157 2.875RES 9.9E-13 9.2E-13 3.5E-13 1.4E-16 1.4E-16IT 22 42 32 5 4

ξ = 2.0 CPU 127.609 68.875 53.531 22.062 2.875RES 3.5E-13 5.8E-13 8.7E-13 9.6E-13 1.9E-15

merical experiments show that the SDA algorithm is around 2 to 10 times more efficient than the

Newton iteration method, and around 2 to 80 times more efficient than the fixed-point iteration

methods.

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