Effective Condition Numbers and Small Sample Statistical Condition Estimation for the Generalized...

SCIENCE CHINAMathematics

. ARTICLES . May 2013 Vol. 56 No. 5: 967–982

doi: 10.1007/s11425-013-4583-3

c© Science China Press and Springer-Verlag Berlin Heidelberg 2013 math.scichina.com www.springerlink.com

Effective condition numbers and small samplestatistical condition estimation for the generalized

Sylvester equationDedicated to Prof. Li Zi-cai on the occasion of his 75th birthday

DIAO HuaiAn1, SHI XingHua2 & WEI YiMin3,∗

1School of Mathematics and Statistics & Key Laboratory for Applied Statistics of MOE,Northeast Normal University, Changchun 130024, China;

2School of Mathematical Sciences, Fudan University, Shanghai 200433, China;3School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics,

Fudan University, Shanghai 200433, China

Email: [email protected], [email protected], [email protected]

Received December 21, 2011; accepted January 8, 2013; published online January 30, 2013

Abstract In this paper, we investigate the effective condition numbers for the generalized Sylvester equation

(AX − Y B,DX − Y E) = (C, F ), where A,D ∈ Rm×m, B,E ∈ R

n×n and C,F ∈ Rm×n. We apply the small

sample statistical method for the fast condition estimation of the generalized Sylvester equation, which requires

O(m2n+ mn2) flops, comparing with O(m3 + n3) flops for the generalized Schur and generalized Hessenberg-

Schur methods for solving the generalized Sylvester equation. Numerical examples illustrate the sharpness of

our perturbation bounds.

Keywords generalized Sylvester equation, Sylvester equation, effective condition number, perturbation

bound, small sample statistical condition estimation (SCE)

MSC(2010) 15A09, 65F10

Citation: Diao H A, Shi X H, Wei Y M. Effective condition numbers and small sample statistical condition esti-

mation for the generalized Sylvester equation. Sci China Math, 2013, 56: 967–982, doi: 10.1007/s11425-

013-4583-3

1 Introduction

The Sylvester and generalized Sylvester equations play an important role in the linear control

systems [6, 23]. Condition number [14, 36] is a fundamental research topic in numerical analysis, which

describes the worst case sensitivity of a problem with respect to perturbations on the input data. If the

condition number is large, then we may be facing an ill-posed problem [8].

In this paper, we apply the concept of effective condition number [4, 5, 37] to the Sylvester and gener-

alized Sylvester equations. As shown in numerical examples of Section 4, the effective condition number

can be much smaller than the orthodox condition number [17, 35]. It can reveal the true conditioning of

the Sylvester and generalized Sylvester equations, if there are no perturbations on the right-hand sides. In

order to better assess the accuracy of the computational solution, we present sharp perturbation bounds

for the Sylvester and the generalized Sylvester equations based on the effective condition number.

∗Corresponding author

968 Diao H A et al. Sci China Math May 2013 Vol. 56 No. 5

In matrix perturbation theory, the classical normwise perturbation analysis measures the errors both

on input and output data using norms, producing the classical normwise condition number. However,

the sparsity and scaling of the data are ignored. Alternatively, the componentwise perturbation analy-

sis [18] produces sharper perturbation bounds. There are two types of condition numbers, the mixed and

componentwise condition numbers. In practice, the problem of how to estimate the condition number

efficiently is very critical (cf. [18, Chapter 15]). Kenny and Laub [21] developed the method of the small

sample statistics condition estimation (SCE) applicable for general matrix functions, linear equations [27],

eigenvalue problems [29], linear least squares problems [22] and roots of polynomials [28]. We devise an

SCE algorithm to estimate the normwise, mixed and componentwise condition numbers of the generalized

Sylvester equation, which is used to estimate the perturbation bounds in real applications.

The generalized Sylvester equation [6, 23] is given by{AX − Y B = C,

DX − Y E = F,(1.1)

where A,D ∈ Rm×m, B,E ∈ R

n×n and C,F ∈ Rm×n are given, and X,Y ∈ R

m×n are the unknown

matrices. If D and E are chosen to be identity matrices and F the zero matrix, then (1.1) reduces to the

standard Sylvester equation

AX −XB = C, (1.2)

where A ∈ Rm×m, B ∈ R

n×n and C ∈ Rm×n. (1.2) arises in various applications in linear control theory

and stability analysis [2, 3, 6, 11, 13, 26], matrix differential equations, matrix difference equations [23],

block diagonalization of a block triangular matrix [18], and eigenproblems [17], etc.

The generalized Sylvester equation can be used to compute the stable eigendecompositions of the

matrix pencil [9]

M − λN =

[A −C

0 B

]− λ

[D −F

0 E

],

i.e., we need to find [X,Y ] which satisfies

P−1(M − λN)Q =

[A 0

0 B

]− λ

[D 0

0 E

], P−1 =

[I −Y

0 I

], Q =

[I X

0 I

].

The first m columns of P−1 and Q, respectively, span a pair of eigenspaces (deflating subspaces) asso-

ciated with the regular pair (A,D) [39]. By solving (1.1) for X and Y , we obtain a pair of complemen-

tary eigenspaces (deflating subspaces) corresponding to λ(B,E) spanned by the last n columns of P−1

and Q, respectively.

It is well known that the solution to the generalized Sylvester equation exists and is unique under the

assumption that the regular pencils A − λD and B − λE have disjoint spectra or that the matrix pairs

(A,D) and (B,E) are separated [38], i.e.,

Dif ([A,D], [B,E]) := inf[X,Y ] �=0

‖[AX − Y B,DX − Y E]‖F‖[X,Y ]‖F > 0,

where ‖ · ‖F is the Frobenius norm of a matrix. We can formulate (1.1) in terms of the equivalent

matrix-vector problem {(I ⊗ A)vec(X)− (BT ⊗ I

)vec(Y ) = vec(C),

(I ⊗D)vec(X)− (ET ⊗ I)vec(Y ) = vec(F ),

i.e.,

Z

[vec(X)

vec(Y )

]=

[vec(C)

vec(F )

], (1.3)

Diao H A et al. Sci China Math May 2013 Vol. 56 No. 5 969

where

Z :=

[I ⊗A −BT ⊗ I

I ⊗D −ET ⊗ I

],

A⊗B is a Kroncecker product [15], and the ‘vec’ operator stacks the columns of a matrix one underneath

the other [15].

The generalized Sylvester equation (1.1) has a unique solution if and only if Z is nonsingular. Under

this condition, it is shown that [9]

Dif([A,D], [B,E])−1 = ‖Z−1‖2,

where ‖ · ‖2 denotes the spectral norm of a matrix.

Kagstrom and Westin [20] proposed algorithms for solving (1.1), which extend the Schur method [1]

for (1.2). The algorithms include the generalized Schur and generalized Hessenberg-Schur methods. The

computational costs of the algorithms are O(m3 + n3) flops.

The generalized Schur method can be described as follows. Consider the generalized Schur decompo-

sitions of (A,D) and (B,E):

A = U1A1VT1 , D = U1D1V

T1 , B = U2B1V

T2 , E = U2E1V

T2 , (1.4)

where U1, U2, V1 and V2 are orthogonal, A1 and B1 are upper quasi-triangular andD1, E1 upper triangular

matrices. After solving the transformed system{A1X1 − Y1B1 = C1,

D1X1 − Y1E1 = F1,(1.5)

where C1 = UT1 CV2 and F1 = UT

1 FV2, one obtains X = V1X1VT2 and Y = U1Y1U

T2 . Because of the

special structures of A1, B1, C1 and D1, one can solve (1.5) efficiently, by means of Gaussian elimination

with partial pivoting [18, Chapter 9].

Let us consider the perturbed generalized Sylvester equation (1.1):{(A+ΔA)(X +ΔX)− (Y +ΔY )(B +ΔB) = C +ΔC,

(D +ΔD)(X +ΔX)− (Y +ΔY )(E +ΔE) = F +ΔF.(1.6)

Let

ε := min{ε | ‖ΔA‖F � εδ1, ‖ΔB‖F � εδ2, ‖ΔC‖F � εδ3,

‖ΔD‖F � εδ4, ‖ΔE‖F � εδ5, ‖ΔF‖F � εδ6}. (1.7)

Lin and Wei [35] showed that‖[ΔX,ΔY ]‖F‖[X,Y ]‖F � εκGSYL,

where the condition number κGSYL := ‖Z−1S‖2

‖[X,Y ]‖F, and

S :=

[H 0

0 H

], H := [δ1(X

T ⊗ I),−δ2(I ⊗ Y ),−δ3I], (1.8)

H := [δ4(XT ⊗ I),−δ5(I ⊗ Y ),−δ6I].

One can find the perturbation bounds for the generalized Sylvester equation in [19,35] and the references

therein. In Section 2, we derive the effective condition number for the generalized Sylvester equation,

which can be much smaller than the condition number κGSYL.

Lin and Wei [35] presented the normwise, mixed and componentwise condition numbers for the gener-

alized Sylvester equation. However, they did not propose an efficient method to estimate the condition


numbers. Formulae involving Kronecker products are not efficient to evaluate, even for (1.1) with the

coefficient matrices of small or medium order. Based on the SCE method, we devise some fast algorithms

to estimate the normwise, mixed and componentwise condition numbers for the generalized Sylvester

equation. The computational cost of Algorithm 3.2 is only O(m2n+mn2) flops, which is lower than the

cost O(m3 + n3) of the generalized Schur method.

The effective condition number was first proposed by Rice [37], and studied in [4, 5]. Recently, Li et

al. [31–34] developed the effective condition number, and applied it to the symmetric positive definite

linear equations from the finite difference method. In this paper, we apply the effective condition number

to the Sylvester and the generalized Sylvester equations. The effective condition number can give sharper

perturbation bound. As shown in [8, p. 267], if we use the finite difference method to solve two-dimensional

Poisson equation∂2v(x, y)

∂x2+

∂2v(x, y)

∂y2= −f(x, y),

on the unite square {(x, y) : 0 < x, y < 1}, with v = 0 on the boundary, then the problem reduces to

solving the Sylvester equation

AX −XB = C,

where

A = −B =

⎡⎢⎢⎢⎢⎢⎢⎣2 −1 0

−1. . .

. . .

. . .. . . −1

0 −1 2

⎤⎥⎥⎥⎥⎥⎥⎦ ∈ Rn×n, C = (cij) = h2 [f(ih, jh)] , h =

1

n+ 1.

The matrix C contains the term h2 with h being the discretization step size. Because of the tridiagonal

structure of the matrices A and B, the discretization error is dominant in solving (1.2). The condition

number κSYL given by (2.7) is misleading, as it allows perturbations on A and B for nonzero δ1 and δ2. The

effective condition number κSYLEff in Corollary 2.2 reflects the errors from the discretization better, which

can be much smaller than κSYL as shown in Example 4.2. We utilize the SCE method [21] to estimate

the normwise, mixed and componentwise condition numbers for the generalized Sylvester equation. From

the random test examples in Section 4, the SCE algorithms are very effective.

This paper is organized as follows. In Section 2, the effective condition number κGSYLEff for the gener-

alized Sylvester equation is determined by Theorem 2.1, and the error bounds pertinent to the effective

condition number are derived. In Section 3, the small sample statistical condition estimation algorithms

for the normiwse, mixed and componentwise condition numbers of the generalized Sylvester equation are

proposed. In Section 4, numerical examples are presented. We present some concluding remarks in the

last section.

2 Effective condition numbers

In this section, we define the effective condition number κGSYLEff , and derive the perturbation bounds for

the solution to the generalized Sylvester equation. We obtain the corresponding results for the Sylvester

equation in Corollary 2.1.

Theorem 2.1. For the generalized Sylvester equation (1.1) and the perturbed generalized Sylvester

equation (1.6), assuming that δ = ‖Z−1‖2‖ΔZ‖2 < 1 with

ΔZ =

[I ⊗ (ΔA) −(ΔB)T ⊗ I

I ⊗ (ΔD) −(ΔE)T ⊗ I

]and Z is defined in (1.3), we have the perturbation bound

‖[ΔX,ΔY ]‖F‖[X,Y ]‖F � 1

1− δ

(‖ΔZ‖2‖Z‖2 cond(Z) +

‖[ΔC,ΔF ]‖F‖[C,F ]‖F κGSYL

Eff

), (2.1)


where the condition numbers are

cond(Z) = ‖Z‖2‖Z−1‖2 and κGSYLEff =

‖[C,F ]‖FDif([A,D], [B,E])‖[X,Y ]‖F =

‖Z−1‖2‖[C,F ]‖F‖[X,Y ]‖F .

If we choose δ1, δ2, δ4 and δ5 arbitrarily in (1.7) and δ3 = ‖C‖F , δ6 = ‖F‖F , then

κGSYLEff � ‖[C,F ]‖F

min{‖C‖F , ‖F‖F }κGSYL.

Furthermore,‖Z‖2‖[X,Y ]‖F

‖S‖2 κGSYL � cond(Z) � ‖Z‖2‖[X,Y ]‖Fmin{‖C‖F , ‖F‖F}κ

GSYL.

Proof. Let [X, Y ] = [X+ΔX,Y +ΔY ] be the exact solution to (1.6). We denote the residual matrix R =

[AX− Y B−C,DX− Y E−F ]. Since [X,Y ] is the exact solution to (1.1) and [X, Y ] = [X+ΔX,Y +ΔY ],

we have

R = [AΔX −ΔY B,DΔX −ΔY E].

We rewrite (1.6) as follows:{AΔX +ΔAX +ΔAΔX − YΔB −ΔY B −ΔYΔB = ΔC,

DΔX +ΔDX +ΔDΔX − YΔE −ΔY E −ΔYΔE = ΔF.

Recalling the expression of R, we obtain

R = [ΔC −ΔAX −ΔAΔX + YΔB +ΔYΔB,ΔF −ΔDX −ΔDΔX + YΔE +ΔYΔE]

= [ΔC,ΔF ]− [ΔAX − YΔB,ΔDX − YΔE]− [ΔAΔX −ΔYΔB,ΔDΔX −ΔYΔE].

Using the Kronecker product, we arrive at

Z

[vec(ΔX)

vec(ΔY )

]= vec(R) = vec

([ΔC

ΔF

])−ΔZ

[vec(X)

vec(Y )

]−ΔZ

[vec(ΔX)

vec(ΔY )

]. (2.2)

Taking norms of both sides, we conclude that∥∥∥∥∥[vec(ΔX)

vec(ΔY )

]∥∥∥∥∥2

� ‖Z−1‖2(∥∥∥∥∥vec

([ΔC

ΔF

])∥∥∥∥∥2

+ ‖ΔZ‖2∥∥∥∥∥[vec(X)

vec(Y )

]∥∥∥∥∥F

+ ‖ΔZ‖2∥∥∥∥∥[vec(ΔX)

vec(ΔY )

]∥∥∥∥∥2

).

From ‖ΔZ‖2

‖Z‖2cond(Z) = ‖Z−1‖2‖ΔZ‖2 and∥∥∥∥∥

[vec(ΔX)

vec(ΔY )

]∥∥∥∥∥2

= ‖[ΔX,ΔY ]‖F ,

we deduce that

‖[ΔX,ΔY ]‖F‖[X,Y ]‖F � 1

1− δ

(‖ΔZ‖2‖Z‖2 cond(Z) +


Eff

).

Let λmax(·) be the maximal eigenvalue of a symmetric semi-positive definite matrix. Since S is defined

in (1.8), for δ3 = ‖C‖F , δ6 = ‖F‖F and SST = S1 + S2 + S3, we have

‖Z−1S‖22 = λmax(Z−1SSTZ−T) = λmax(Z

−1S1Z−T + Z−1S2Z

−T + Z−1S3Z−T),

where

S1 =

[δ21(X

TX ⊗ I) 0

0 δ24(XTX ⊗ I)

],


S2 =

[δ22(I ⊗ Y Y T) 0

0 δ25(I ⊗ Y Y T)

],

S3 =

[δ23I 0

0 δ26I

].

Since S1 and S2 are symmetric semi-positive definite matrices, assume that ‖C‖F � ‖F‖F , we obtain

‖Z−1S‖22 � λmax(Z−1S3Z

−T)

= max‖v‖2=1

vTZ−1

(‖F‖2F I +

[(‖C‖2F − ‖F‖2F )I 0

0 0

])Z−Tv

� ‖F‖2F max‖v‖2=1

vTZ−1Z−Tv = ‖F‖2Fλmax(Z−1Z−T) = ‖F‖2F‖Z−1‖22.

Similarly, if ‖C‖F � ‖F‖F , then we get ‖Z−1S‖22 � ‖C‖2F ‖Z−1‖22, i.e., ‖Z−1‖2 � ‖Z−1S‖2

min{‖C‖F ,‖F‖F } .We prove that

κGSYLEff =

‖Z−1‖2‖[C,F ]‖F‖[X,Y ]‖F � ‖Z−1S‖2‖[C,F ]‖F

‖[X,Y ]‖F min{‖C‖F , ‖F‖F} =‖[C,F ]‖F

min{‖C‖F , ‖F‖F}κGSYL.

It is easy to verify that ‖Z−1S‖2

‖S‖2� ‖Z−1‖2 � ‖Z−1S‖2

min{‖C‖F ,‖F‖F } , then we obtain

‖Z‖2‖[X,Y ]‖F‖S‖2 κGSYL � cond(Z) � ‖Z‖2‖[X,Y ]‖F

min{‖C‖F , ‖F‖F}κGSYL. �

The Sylvester equation (1.2) is a special case of the generalized Sylvester equation. We can formulate

(1.2) in terms of the equivalent matrix-vector problem,(I ⊗A−BT ⊗ I

)vec(X) = vec(C). (2.3)

For brevity, let

P := I ⊗A−BT ⊗ I.

It is obvious that P is nonsingular, if and only if A and B have disjoint spectra, and the solution to (2.3) is

unique. The existence and uniqueness of the solution to (1.2) are discussed in [7,25]. Sufficient conditions

are given in [16] for the existence of nonsingular solutions in the case of m = n. We denote ‘sep(A,B)’,

the separation of the matrix A and B is defined by [38, 39],

sep(A,B) := minX �=0

‖AX −XB‖F‖X‖F .

The relation between sep(A,B) and the smallest singular value of P is revealed in [39],

sep(A,B) = σmin(P ). (2.4)

In the rest of the paper, we assume that A and B have disjoint spectra, so that (1.2) has a unique

solution X [39].

We can obtain the general perturbation bound of the linear equation (1.3). However, without taking

into account any special structures, this will be pessimistic. To obtain an effective perturbation bound,

Higham [17] considered the perturbed Sylvester equation

(A+ΔA)(X +ΔX)− (X +ΔX)(B +ΔB) = C +ΔC (2.5)

and obtained the upper bound [17]

‖ΔX‖F/‖X‖F � ε√3κSYL, (2.6)


where

ε := min{ε | ‖ΔA‖F � εδ1, ‖ΔB‖F � εδ2, ‖ΔC‖F � εδ3}, (2.7)

κSYL := ‖P−1[δ1(XT ⊗ I),−δ2(I ⊗X),−δ3I]‖2/‖X‖F .

Then κSYL can be defined as the normwise condition number of (1.2). For other perturbation bounds of

the Sylvester and the Lyapunov equations, please refer to [3,10–12,24,30] and the references therein. Next,

we derive the effective condition number κSYLEff for the Sylvester equation, for which the corresponding

perturbation bound is much sharper than Higham’s [17]. We summarize the result in the following

corollaries, where the proofs are omitted.

Corollary 2.1. For the perturbed Sylvester equation (2.5), assume that δ = ‖P−1‖2‖ΔP‖2 < 1 with

P = I ⊗A−BT ⊗ I, where ΔP = I ⊗ (ΔA)− (ΔB)T ⊗ I, one can obtain the perturbation bound

‖ΔX‖F‖X‖F � 1

1− δ

(‖ΔP‖2‖P‖2 cond(P ) +

‖ΔC‖F‖C‖F κSYL

Eff

), (2.8)

where cond(P ) = ‖P‖2‖P−1‖2 and κSYLEff := ‖C‖F

sep(A,B)‖X‖F= ‖P−1‖2‖C‖F

‖X‖F.

If we select δ1, δ2 arbitrarily, and δ3 = ‖C‖F , then

κSYLEff � κSYL. (2.9)

Furthermore,

‖P‖2‖X‖F‖[δ1(XT ⊗ I),−δ2(I ⊗X),−δ3I]‖2κ

SYL � cond(P ) � ‖P‖2‖X‖F‖C‖F κSYL.

Corollary 2.2. For the perturbed Sylvester equation (2.5), if there is no perturbation on A and B,

i.e., ΔP = 0, then one achieves the upper bound

‖ΔX‖F‖X‖F � ‖ΔC‖F

‖C‖F κSYLEff , (2.10)

where κSYLEff = ‖P−1‖2‖C‖F

‖X‖F.

3 Small sample statistics condition estimation (SCE) for the generalized

Sylvester equation

The SCE method was proposed by Kenny and Laub [21], which is efficient for the estimation of the

condition numbers for linear equations, linear least squares problems, eigenvalue problems and roots of

polynomials. Let us review the general theory of SCE.

We are interested in the sensitivity of a function f : Rp → R at some point x. In the following, let f

be at least twice continuously differentiable. We derive the Taylor series at a point x ∈ Rp,

f(x+ δd) = f(x) + δ(∇f(x))Td+O(δ2),

where ∇f(x) = (∂f(x)∂x1, ∂f(x)

∂x2, . . . , ∂f(x)∂xp

)T is the Frechet derivative of f at x, δ is a small positive number

and d ∈ Rp with ‖d‖2 = 1. The local sensitivity can be measured by ‖∇f(x)‖2. The condition numbers

of f at x are mainly determined by the norm of the Frechet derivative ∇f(x) (see [21]). Suppose that

d ∈ U(Sp−1), i.e., d is chosen uniformly and randomly from the unit p-sphere Sp−1, it follows from [21]

that we have

E(|(∇f(x))Td|) = ωp‖∇f(x)‖2,


where E(·) is the expectation and ωp is the Wallis factor (cf. [21]) which only depends on p. The Wal-

lis factor

ωp =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, for p = 1,2

π, for p = 2,

1 · 3 · 5 · · · (p− 2)

2 · 4 · 6 · · · (p− 1), for p odd and p > 2,

2

π

2 · 4 · 6 · · · (p− 2)

1 · 3 · 5 · · · (p− 1), for p even and p > 2,

can be accurately approximated by using a result in [21],

ωp ≈√

2

π(p− 12 )

. (3.1)

Therefore, we use ν = |(∇f(x))Td|/ωp as a condition estimator, which can estimate ‖∇f(x)‖2 with high

probability for the function f at x, i.e., for γ > 1 we have

Prob

(‖∇f(x)‖2γ

� ν � γ‖∇f(x)‖2)

� 1− 2

πγ+O

(1

γ2

).

To improve the accuracy of the condition estimator, we prefer more samples [21] in the following way.

Firstly, one can select d1, d2, . . . , dk ∈ U(Sp−1) and then use a QR decomposition [18] to produce an

orthonormal basis {d1, d2, . . . , dk}. Thus the orthogonal vectors d1, d2, . . . , dk ∈ Sp−1 span V which is

uniformly and randomly selected from the space of all k-dimensional subspaces of Rp. Then the expected

value of the norm of the projection of ∇f(x) onto V is given by [21]

E(√

|∇f(x)Td1|2 + |∇f(x)Td2|2 + · · ·+ |∇f(x)Tdk|2)=

ωp

ωk‖∇f(x)‖2,

where ωp and ωk are Wallis factors with orders p and k, respectively. A subspace condition estimator

ν(k) =ωk

ωp

√|∇f(x)Td1|2 + |∇f(x)Td2|2 + · · ·+ |∇f(x)Tdk|2

has the following probability for the estimator accuracy (cf. [21]):

Prob

(‖∇f(x)‖2γ

� ν(2) � γ‖∇f(x)‖2)

≈ 1− π

4γ2,

Prob

(‖∇f(x)‖2γ

� ν(3) � γ‖∇f(x)‖2)

≈ 1− 32

3π2γ3,

Prob

(‖∇f(x)‖2γ

� ν(4) � γ‖∇f(x)‖2)

≈ 1− 81π2

512γ4.

For example, if we take k = 3 and γ = 10, then the estimator ν(3) has probability 1 − 323π2103 ≈ 0.9989

with a relative factor of 10 of the true condition number ‖∇f(x)‖2.Now we apply SCE to estimate the normwise, mixed and componentwise condition numbers for (1.1)

in the following two sub-sections. We firstly derive the directional derivative of (1.1) at the input data

(A,B,C,D,E, F ) with respect to the direction (A,B, C,D, E ,F). Suppose that [X + δX , Y + δY] is theexact solution of {

(A+ δA)(X + δX )− (Y + δY)(B + δB) = C + δC,(D + δD)(X + δX )− (Y + δY)(E + δE) = F + δF ,

where δ ∈ R. Consider the difference of the above equations and (1.1), divided by δ. When δ → 0, we

conclude that {AX − YB = C − AX + Y B,DX − YE = F −DX + Y E . (3.2)


The solution to (3.2) is the directional derivative of (1.1) at the input data (A,B,C,D,E, F ) with respect

to the direction (A,B, C,D, E ,F).

Remark 1. When we solve the generalized Sylverster equation (1.1) by means of the generalized

Schur decomposition [20], if the decompositions (1.4) of A,B,D and E are obtained, then (3.2) can be

solved efficiently.

3.1 Normwise perturbation analysis

By means of SCE, we devise the algorithms for the normwise condition estimation of (1.1). We introduce

the ‘unvec’ operation: for v = (v1, v2, . . . , vn2) ∈ R1×n2

, then A = unvec(v) sets the entries of A to

aij = vi+(j−1)n.

Algorithm 3.1 (One sample condition estimation for the solution [X,Y ] to the generalized Sylvester

equation (1.1)). 1. Let A, D ∈ Rm×m, B, E ∈ R

n×n and C, F ∈ Rm×n be selected randomly and

independently from a normal distribution with mean 0 and variance 1, i.e., each entry is in N (0, 1).

Denote η = ‖(A,B, C,D, E ,F)‖F and set A = A/η, B = B/η, C = C/η, D = D/η, E = E/η and F = F/η.

2. Let p = 2(m2 + n2 +mn). Approximate ωp by (3.1).

3. Solve the generalized Sylvester equation (3.2).

4. Calculate the condition number based on the small sample estimation,

KGSYLabs =

1

ωp|[X ,Y]|, nGSYL

F,SCE =1

ωp‖[X ,Y]‖F = ‖KGSYL

abs ‖F ,

where |[X ,Y]| is a matrix whose entries are the absolute values of the corresponding entries of [X ,Y].Define the relative condition number κGSYL

F,SCE = nGSYLF,SCE/‖[X,Y ]‖F .

Algorithm 3.2 (Subspace condition estimation for the solution [X,Y ] to the generalized Sylvester equa-

tion (1.1)). 1. Generate matrices (A1,B1, C1,D1, E1,F1), (A2,B2, C2,D2, E2,F2), . . . , (Ak,Bk, Ck,Dk,

Ek,Fk) with entry being in N (0, 1). With the QR factorization of⎡⎢⎢⎢⎢⎢⎣vec(A1) vec(A2) · · · vec(Ak)

vec(B1) vec(B2) · · · vec(Bk)

· · · · · · . . . · · ·vec(F1) vec(F2) · · · vec(Fk)

⎤⎥⎥⎥⎥⎥⎦form an orthonormal matrix [q1, q2, . . . , qk]. Each qi can be converted into the desired matrices (Ai,Bi, Ci,Di, Ei,Fi) with the unvec operation.

2. Let p = 2(m2 + n2 +mn). Approximate ωp and ωk by (3.1).

3. Solve the generalized Sylvester equations (3.2) with the matrix (Ai,Bi, Ci,Di, Ei,Fi), i = 1, 2, . . . , k.

4. Compute the condition number

nGSYL,(k)F,SCE :=

ωk

ωp

√‖[X1,Y1]‖2F + ‖[X2,Y2]‖2F + · · ·+ ‖[Xk,Yk]‖2F = ‖KGSYL,(k)

abs ‖F ,

KGSYL,(k)abs :=

ωk

ωp

√|[X1,Y1]|2 + |[X2,Y2]|2 + · · ·+ |[Xk,Yk]|2,

where the square root and the power operation of the second equation are applied at each entry of

[Xi,Yi], i = 1, 2, . . . , k. Note that every operation run in the second equation is meant to be performed

componentwise, and not in the sense of matrices.

Define the normwise condition number: κGSYL,(k)F,SCE = n

GSYL,(k)F,SCE /‖[X,Y ]‖F .

In Step 1 of Algorithm 3.2, if we use the Householder transformation [18] to compute the QR decompo-

sition to form an orthonormal matrix [q1, q2, . . . , qk], then we need 8k2(m2+n2+mn)− 43k

2 flops. To form

the right-hand side of (3.2), the total cost is 4m2n+ 4mn2 − 2mn flops. If we use the generalized Schur

decomposition [20] to solve (1.1), then A and B are upper quasi-triangular, and D, E are upper triangular

matrices. The total flops of solving (3.2) with A,B,D and E in standard form is 5m2n+ 5mn2 + 974 mn


at most according to [20]. Then the third step of Algorithm 3.2 needs 9m2n + 9mn2 + 894 mn flops

at most. The last step of Algorithm 3.2 uses 4kmn flops. At end the total cost of Algorithm 3.2 is

9m2n+ 9mn2 + (894 + 8k2 + 4k)mn+ 8k2(m2 + n2)− 43k

2 flops, which is lower compared with the total

cost O(m3 + n3) flops of the generalized Schur decomposition [20] to solve (1.1). Generally with k = 3,

the algorithm yields accurate estimates of condition numbers efficiently.

Remark 2. SCE method can be used to estimate the effective condition number κGSYLEff . Recall

that κGSYLEff = ‖Z−1‖2‖[C,F ]‖F

‖[X,Y ]‖F, the main task is to estimate ‖Z−1‖2. From the theory of SCE [21], we

can generate the orthogonal vectors l1, l2, . . . , lk ∈ Sp−1 uniformly and randomly with p = 2mn by

QR factorization. Firstly we calculate Z−1li, while Z−1li is the solution to a generalized Sylvester

equation (1.3). As pointed in Remark 1, Z−1li can be computed efficiently when we have the generalized

Schur decomposition [20]. Finally ‖Z−1‖2 can be estimated by

ωk

ωp‖√|Z−1l1|2 + |Z−1l2|2 + · · ·+ |Z−1lk|2‖2

with high probability.

3.2 Componentwise perturbation analysis

In componentwise perturbation analysis, for a perturbation Δa on a ∈ R, it should satisfy |Δa| � εa.

For a perturbation ΔA = (Δaij) on a matrix A = (aij) ∈ Rm×n, it is a componentwise perturbation, if

|ΔA| � ε|A| or |Δaij | � ε|aij |.

We can write ΔA = δ(A�A) with |δ| � ε and each entries of A being in the interval [−1, 1], where � is

a Hadamard product (cf. [26]). Based on the above observation, we obtain a componentwise sensitivity

of (1.1)’s solution [X,Y ] in the following manner. We can modify Algorithm 3.2 directly, i.e., after

generating and normalizing (or orthonormalizing as in Algorithm 3.2) the random elements of Step 1, these

elements are multiplied by the corresponding entries of A,B,C,D,E or F , respectively. The remaining

steps of the algorithm are unchanged. In the following, let ‖A‖max = ‖vec(A)‖∞ and AB = (

aij

bij) with A =

(aij) and B = (bij). Assume [Xi,Yi] is the solution to (3.2) with the random matrix (Ai,Bi, Ci,Di, Ei,Fi)

as in Algorithm 3.2, whereas (Ai,Bi, Ci,Di, Ei,Fi) are obtained by multiplying (Ai,Bi, Ci,Di, Ei,Fi) with

the associated entries of A,B,C,D,E or F , respectively, in the first step of Algorithm 3.2. Let us denote

the mixed condition number mGSYL,(k) and componentwise condition number cGSYL,(k) as follows:

mGSYL,(k) := ‖MGSYL,(k)‖max/‖[X,Y ]‖max, cGSYL,(k) :=

∥∥∥∥MGSYL,(k)

[X,Y ]

∥∥∥∥max

, (3.3)

where MGSYL,(k) = ωk

ωp

√|[X1,Y1]|2 + |[X2,Y2]|2 + · · ·+ |[Xk,Yk]|2.Remark 3. Since the algorithm of the estimation for mGSYL,(k) and cGSYL,(k) is only different in the

first step of Algorithm 3.2, an extra 2k(m2 + n2 + mn) flops for matrix componentwise multiplication

will be needed. Then the total flop count to estimate mGSYL,(k) and cGSYL,(k) is 9m2n + 9mn2 + (894+ 8k2 + 6k)mn+ (8k2 + 2k)(m2 + n2)− 4

3k2 at most.

4 Numerical examples

In this section, we present some examples to show the sharpness of our perturbation bounds. All numerical

examples are carried out by Matlab 7.0, with machine epsilon ε ≈ 2.2 × 10−16. Here we choose δ1 =

‖A‖F , δ2 = ‖B‖F , δ3 = ‖C‖F , δ4 = ‖D‖F , δ5 = ‖E‖F and δ6 = ‖F‖F to compute κSYL and κGSYL

in (1.7) and (2.7), respectively.

To illustrate the differences between the effective condition number κGSYLEff and the normwise condition

number κGSYL for the generalized Sylvester equation in Theorem 2.1, we test several numerical examples.

From the given examples, we see that the effective condition number captures the conditioning of the


problem well, if there are only perturbations on the right-hand sides [C,F ], without perturbations on

A,B,D, and E in (1.1), i.e., ΔZ = 0 in (2.1) of Theorem 2.1. We add some random perturbations on

the right-hand sides [C,F ] and compare the actual perturbation error ‖[ΔX,ΔY ]‖F

‖[X,Y ]‖Fwith the first order

perturbation bound given by the effective condition number


Eff .

From the numerical examples, the effective condition number κGSYLEff gives sharper perturbation bounds.

Example 4.1 (See [40]). Consider the real regular matrix pair (A,B) with

A = V

[A −C

0 B

]UT, B = V

[D −F

0 E

]UT,

where U and V are orthogonal matrices with

A =

⎡⎢⎢⎣1 0 0

0 −5 0

0 0 10−k

⎤⎥⎥⎦ , B =

⎡⎢⎢⎣10k 0 0

0 2 0

0 0 −10−l

⎤⎥⎥⎦ , C = −

⎡⎢⎢⎣2× 10k 1 0

0 −1 3

0 0 0

⎤⎥⎥⎦ ,

and

D =

⎡⎢⎢⎣−4× 10−k 0 0

0 3 0

0 0 −2× 10−l

⎤⎥⎥⎦ , E =

⎡⎢⎢⎣1 0 0

0 1 0

0 0 −10−k

⎤⎥⎥⎦ , F = −

⎡⎢⎢⎣7 2 0

0 5 1

0 0 −3× 10k

⎤⎥⎥⎦ .

In order to compute the eigenpair of the matrix pair (A,B), we need to solve the generalized Sylvester

equation (1.1).

In the following, we test different choices k and l to show that κGSYLEff is much smaller than κGSYL. From

Table 1, we find when k = l, κGSYL ≈ 10k+l, but κGSYLEff is always O(1). Let randn(m,n) denote an m×n

random perturbation matrix with each entry in N (0, 1). We generate some random perturbations on the

right-hand side C and F , where ΔC = ε · randn(m,n),ΔF = ε · randn(m,n) for different perturbation

parameters ε, then we compare the first order perturbation bound ‖[ΔC,ΔF ]‖F

‖[C,F ]‖FκGSYLEff given by the effec-

tive condition number with the actual error bound ‖[ΔX,ΔY ]‖F

‖[X,Y ]‖F. From the first to third columns, we see

that ‖[ΔX,ΔY ]‖F

‖[X,Y ]‖Fhas the same order of ‖[ΔC,ΔF ]‖F

‖[C,F ]‖FκGSYLEff , which indicates that κGSYL

Eff gives tighter error

bounds than κGSYL. For the fourth column, the perturbation bound by the effective condition number

is much larger than the true relative error, however we can trust the computational solution since the

problem is well-conditioned in the setting of the effective condition number. Even for the ill-conditioned

case k = 10, l = 1, the bound ‖[ΔC,ΔF ]‖F

‖[C,F ]‖FκGSYLEff is only 10 times that of the true error. For the last

column, in the normwise perturbation sense, the problem is very ill-conditioned as the normwise condi-

tion number κGSYL has the order of 20. If there are only perturbations on the right-hand sides, then we

Table 1 The comparison of κGSYL, κGSYLEff ,


Eff and‖[ΔX,ΔY ]‖F

‖[X,Y ]‖F

k = l = 2 k = 2, l = 5 k = l = 5 k = 10, l = 1 k = l = 10

κGSYL 1.5838e+004 1.4435e+004 1.5811e+010 5.0360e+010 1.5811e+020

κGSYLEff 2.4782 100.1887 2.5039e+003 4.0172e+009 2.4777

‖[ΔC,ΔF ]‖F‖[C,F ]‖F 1.8894e−011 9.9243e−012 1.0601e−014 8.0632e−020 1.2198e−019

‖[ΔX,ΔY ]‖F‖[X,Y ]‖F 2.3971e−011 1.7072e−010 1.0314e−014 7.5077e−011 5.3479e−020


compute the solution accurately, and the effective condition number κGSYLEff reveals the true conditioning of

the problem.

Next, we will test four numerical examples of the Sylvester equations from [8,41–43] to demonstrate the

effective condition number κSYLEff in Corollary 2.1 are much smaller than Higham’s normwise condition

number κSYL in (2.7). Also we add some random perturbations on the right-hand matrix C in the

standard Sylvester equation AX −XB = C with ΔC = ε · randn(m,n). We compare the true relative

errors ‖ΔX‖F

‖X‖Fwith the first order perturbation bound given by the effective condition number ‖ΔC‖F

‖C‖FκSYLEff .

The numerical examples show that ‖ΔC‖F

‖C‖FκSYLEff gives sharper perturbation bounds.

Example 4.2 (See [8, p. 267]). This example is from the discretization of the two-dimensional Poisson

equation as in Section 1, the coefficient matrices A,B ∈ Rn×n in the Sylvester equation AX −XB = C

have the tridiagonal structure as follows,

A = −B =

⎡⎢⎢⎢⎢⎢⎢⎣2 −1 0

−1. . .

. . .

. . .. . . −1

0 −1 2

⎤⎥⎥⎥⎥⎥⎥⎦ .

We choose C = (cij) = h2 exp (ih+ jh), i, j = 1, 2, . . . , n, where h = 1n+1 is the discretization step size.

Next we compare κSYL, κSYLEff , ‖ΔC‖F

‖C‖FκSYLEff and ‖ΔX‖F

‖X‖Ffor different choices of n (see Table 2).

Example 4.3 (See [41, 42]). The matrix in this example presents a Hamiltonian-like structure,

H = QT

[A −C

0 −AT

]Q, A =

⎡⎢⎢⎢⎢⎣1− α α

α 1− α

α 1− α

α 1− α

⎤⎥⎥⎥⎥⎦ ∈ R4×4,

where C = randn(4, 4), and Q is a random orthogonal matrix. In order to compute the eigenpair of the

matrix H , we need to solve the Sylvester equation

AX −XB = C, B = −AT. (4.1)

Next we compare κSYL, κSYLEff , ‖ΔC‖F

‖C‖FκSYLEff and ‖ΔX‖F

‖X‖Ffor different choices of α. It is shown that when

α approximates to 1/2, κSYL increases more, while κSYLEff changes a little bit (see Table 3).

Example 4.4 (See [43]). Let m = n, C = randn(m,n), and A = −B = I − wUn − θeneT1 , where

Un =

⎡⎢⎢⎢⎢⎢⎢⎣0 1

0. . .

. . . 1

0

⎤⎥⎥⎥⎥⎥⎥⎦Table 2 The comparison of κSYL, κSYL

Eff , ‖ΔC‖F‖C‖F κSYL

Eff and ‖ΔX‖F‖X‖F

n = 5 n = 10 n = 20 n = 50

κSYL 13.9146 65.9868 341.2454 3.1933e+003

κSYLEff 1.1338 1.2209 1.2824 1.3272

‖ΔC‖F‖C‖F 3.6812e−010 3.1934e−010 3.0396e−010 3.2062e−010

‖ΔX‖F‖X‖F 8.3033e−011 3.6238e−011 2.1271e−011 1.1388e−011


and the vectors e1, en are the first and last column of the identity matrix In, respectively. The matrix

A is an M -matrix [26] if 1 − θwn−1 > 0. We fix n = 10, next we compare κSYL, κSYLEff , ‖ΔC‖F

‖C‖FκSYLEff and

‖ΔX‖F

‖X‖Ffor different choices of w and θ. From Table 4 we can see that when w and θ approximate to 1,

κSYL grows greatly while κSYLEff varies a little.

From Tables 2–4, we see that the effective condition number κSYLEff is much smaller than the normwise

condition number κSYL for the Sylvester equation. For the first order perturbation bound ‖ΔC‖F

‖C‖FκSYLEff

and the actual error bound ‖ΔX‖F

‖X‖F, we have similar results of the generalized Sylvester equation in

Example 4.1, which is the effective condition number κSYLEff reveals the true conditioning of the problem

better, when there are only perturbations on the right-hand sides.

We next compare the actual error bounds with the bounds given by the small sample statistical

condition estimation. We test one thousand 10 × 10 random matrices. We generate A = randn(m,m),

B = randn(n, n), D = randn(m,m), E = randn(n, n), X = randn(m,n), Y = randn(m,n), and C,

F are determined by C = AX − Y B, F = DX − Y E. The perturbations are ΔA = ε(M1 � A),

ΔB = ε(M2 � B), ΔC = ε(M3 � C), ΔD = ε(M4 � D), ΔE = ε(M5 � E), ΔF = ε(M6 � F ), where

ε = 1e − 8, Mi (i = 1, 2, . . . , 6) are random matrices whose entries are uniformly distributed in (−1, 1).

Let A = A+ΔA, B = B +ΔB, C = C +ΔC, D = D+ΔD, E = E +ΔE, F = F +ΔF , then we solve{AX − Y B = C,

DX − Y E = F .(4.2)

The actual relative errors are

γκ =‖[X,Y ]− [X, Y ]‖F

‖[X,Y ]‖F , γm =‖[X,Y ]− [X, Y ]|max

‖[X,Y ]‖max, γc =

∥∥∥∥ [X,Y ]− [X, Y ]

[X,Y ]

∥∥∥∥max

,

respectively.

We employ the subspace condition estimation for the solution [X,Y ] in Algorithm 3.2 with k = 3. The

perturbation bound of the mixed condition number is

‖[X,Y ]− [X, Y ]‖max/‖[X,Y ]‖max � mGSYL,(3)ε.

The upper bound of the componentwise condition number is∥∥∥∥ [X,Y ]− [X, Y ]

[X,Y ]

∥∥∥∥max

� cGSYL,(3)ε,

Table 3 The comparison of κSYL, κSYLEff , ‖ΔC‖F

‖C‖F κSYLEff and ‖ΔX‖F

‖X‖F

α = 12− ( 1

2)10 α = 1

2− ( 1

2)20 α = 1

2− ( 1

2)30 α = 1

2− ( 1

2)50

κSYL 496.8907 5.2429e+005 5.3687e+008 5.5967e+014

κSYLEff 104.9469 114.6726 114.6735 116.3981

‖ΔC‖F‖C‖F 1.6835e−009 1.6835e−009 1.6835e−009 1.6835e−009

‖ΔX‖F‖X‖F 2.8714e−008 3.1369e−008 3.1369e−008 3.1369e−008

Table 4 The comparison of κSYL, κSYLEff , ‖ΔC‖F

‖C‖F κSYLEff and ‖ΔX‖F

‖X‖F

w = 0.9, θ = 0.9 w = 0.999, θ = 0.9 w = 0.999, θ = 0.999 w = θ = 0.999999

κSYL 28.9755 268.9324 3.1600e+003 3.1622e+006

κSYLEff 3.0291 19.7776 20.4311 20.4371

‖ΔC‖F‖C‖F 1.8894e−011 9.9159e−010 9.9159e−010 9.9159e−010

‖ΔX‖F‖X‖F 2.3971e−011 8.8392e−010 7.8660e−010 7.8634e−010


where mGSYL,(3) and cGSYL,(3) are given by (3.3) with k = 3. Recall that κGSYL,(3)F,SCE is defined in

Algorithm 3.2 with k = 3. The perturbation bound of the normwise condition number is

‖[X,Y ]− [X, Y ]‖F /‖[X,Y ]‖F � κGSYL,(3)F,SCE ε.

We denote three overestimate ratios as follows:

rn :=κGSYL,(3)F,SCE

γκε, rm :=

mGSYL,(3)

γmε, rc :=

cGSYL,(3)

γcε.

These ratios demonstrate the efficiency of small sample statistical condition estimations. The ratios

are displayed in Figures 1–3. Among 1000 tests, the ratios in most cases are of order 1, except a few

exceptional cases. The average values of rn, rm and rc are 1.6927, 2.4087 and 2.8434, respectively.

Usually an estimate of the condition number that is correct to within a factor 10 is acceptable, see [18,

Chapter 15] for details. We see that the small sample statistical method is quite effective for condition

number estimation. It also yields tighter error bounds on average.

5 Concluding remarks

We apply the concept of effective condition numbers to the Sylvester and the generalized Sylvester equa-

tions. Sharper perturbation bounds based on the effective condition numbers are obtained. From the

numerical tests, the effective condition numbers are much smaller than the normwise condition numbers.

The small sample statistical condition estimation algorithms for the normwise, mixed and componentwise

condition numbers of the generalized Sylvester equation are also proposed, which are very effective for

the numerical examples.

30

25

20

15

10

5

00 200 400 600 800 1000

Figure 1 The value of rn

0 200 400 600 800 1000

70

60

50

40

30

20

10

0

Figure 2 The value of rm


0 200 400 600 800 1000

60

50

40

30

20

10

0

Figure 3 The value of rc

In addition, the investigation of the mixed and componentwise effective condition numbers will be our

future research topic.

Acknowledgements The work was supported by National Natural Science Foundation of China (Grant Nos.

11001045, 10926107 and 11271084), Specialized Research Fund for the Doctoral Program of Higher Education

of MOE (Grant No. 20090043120008), Training Fund of NENU’S Scientific Innovation Project of Northeast

Normal University (Grant No. NENU-STC08009), Program for Changjiang Scholars and Innovative Research

Team in University, the Programme for Cultivating Innovative Students in Key Disciplines of Fudan University

(973 Program Project) (Grant No. 2010CB327900), Doctoral Program of the Ministry of Education (Grant No.

20090071110003) and Shanghai Science & Technology Committee and Shanghai Education Committee (Dawn

Project). The authors would like to thank Prof. Ya-xiang Yuan and three anonymous referees for their detailed

comments and helpful suggestions. They are also grateful to Prof. Eric King-wah Chu for carefully reading

the manuscript and improving the presentations of this paper. The third author would like to thank Professors

Higham N. and Kagstrom B. for their reprints and useful discussions. Partial work was finished when the first

author visited Shanghai Key Laboratory of Contemporary Applied Mathematics in 2012.

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