CONVERGENCE OF THE DOMINANT POLE ALGORITHM ANDRAYLEIGH QUOTIENT ITERATION!

JOOST ROMMES† AND GERARD L.G. SLEIJPEN†

Key words. eigenvalues, eigenvectors, dominant poles, two-sided Rayleigh quotient iteration,dominant pole algorithm, subspace accelerated Newton method, rate of convergence, transfer func-tion, modal model reduction

AMS subject classification. 65F15

Abstract. The dominant poles of a transfer function are specific eigenvalues of the state spacematrix of the corresponding dynamical system. In this paper, two methods for the computation of thedominant poles of a large scale transfer function are studied: two-sided Rayleigh Quotient Iteration(RQI) and the Dominant Pole Algorithm (DPA). Firstly, a local convergence analysis of DPA willbe given, and the local convergence neighborhoods of the dominant poles will be characterized forboth methods. Secondly, theoretical and numerical results will be presented that indicate that forDPA the basins of attraction of the dominant pole are larger than for two-sided RQI. The price forthe better global convergence is only a few additional iterations, due to the asymptotically quadraticrate of convergence of DPA, against the cubic rate of two-sided RQI.

1. Introduction. The transfer function of a large scale dynamical system oftenonly has a small number of dominant poles compared to the number of state variables.The computation of the dominant poles, that are specific eigenvalues of the systemmatrix, and the corresponding modes, requires specialized eigenvalue methods. In[12] Newton’s method is used to compute a dominant pole of single input singleoutput (SISO) transfer function: the Dominant Pole Algorithm (DPA). In two recentpublications this algorithm is improved and extended to a robust and e!cient methodfor the computation of the dominant poles and modes of large scale SISO [20] andMIMO transfer functions [19].

This paper is concerned with the convergence behavior of DPA. Firstly, DPAwill be related to two-sided or generalized Rayleigh quotient iteration [15, 17]. Alocal convergence analysis will be given, showing the asymptotically quadratic rateof convergence. Furthermore, for systems with a symmetric state-space matrix, acharacterization of the local convergence neighborhood of the dominant pole will bepresented for both DPA and RQI. The results presented in this paper are sharp (insome sense), in contrast to the ones found in literature as by Ostrowski [14, 15] forDPA, and by Beattie and Fox [5] for RQI. Secondly, theoretical and numerical resultsindicate that for DPA the basins of attraction of the most dominant poles are largerthan for two-sided RQI. In practice, the asymptotically quadratic (DPA) instead ofcubic rate (two-sided RQI) of convergence costs about two or three iterations.

The outline of this paper is as follows. Definitions and properties of transferfunctions and dominant poles, and further motivation are given in section 2. TheDominant Pole Algorithm and its relation to two-sided Rayleigh quotient iterationare discussed in section 3. In section 4 the local convergence of DPA is analyzed. Thebasins of attraction of DPA and two-sided RQI are studied in section 5. Section 6concludes.

†Mathematical Institute, Utrecht University, POBox 80010, 3508 TA, Utrecht, The Netherlands,http://www.math.uu.nl/people/{rommes, sleijpen}, {rommes,sleijpen}@math.uu.nl

!This work was supported by the BRICKS-MSV1 project.

1

2 J. Rommes and G.L.G. Sleijpen

2. Transfer functions and poles. The motivation for this paper comes fromdynamical systems (A,E,b, c, d) of the form

!Ex(t) = Ax(t) + bu(t)y(t) = c!x(t) + du(t), (2.1)

where A,E ! Rn"n, E may be singular but the pencil (A,E) is regular, b, c,x(t) !Rn, and u(t), y(t), d ! R. The vectors b and c are called the input, and output vector,respectively. The transfer function H : C "# C of (2.1) is defined as

H(s) = c!(sE "A)#1b + d. (2.2)

The poles of transfer function (2.2) are a subset of the eigenvalues !i ! C of thematrix pencil (A,E). An eigentriplet (!i,vi,wi) is composed of an eigenvalue !i of(A,E) and corresponding right and left eigenvectors vi,wi ! Cn:

Avi = !iEvi, vi $= 0,

w!i A = !iw!

i E, wi $= 0.

In the case of nondefective multiple eigenvalues, decompose b and c as

b =n"

i=1

mi"

j=1

"ji (Evj

i ) +m!"

j=1

"j$vj

$,

and

c =n"

i=1

mi"

j=1

#ji (E!wj

i ) +m!"

j=1

#j$wj

$,

respectively, where n is the number of distinct finite eigenvalues, mi is the multiplicityof !i, and "j

i , #ji are coe!cients (index % denotes the respective quantities for the

eigenvalues at infinity). The right (left) eigenvector vi (wi) of a pole !i of multiplicitymi is then identified by vi =

#mi

j=1 "ji v

ji (wi =

#mi

j=1 #ji w

ji ). Assuming that the pencil

is non-defective, the right and left eigenvectors corresponding to finite eigenvaluescan be scaled so that w!

i Evi = 1. Furthermore, it is well known that left and righteigenvectors corresponding to distinct eigenvalues are E-orthogonal: w!

i Evj = 0 fori $= j.

The transfer function H(s) can be expressed as a sum of residues Ri ! C [11]:

H(s) =n"

i=1

Ri

s" !i+ R$ + d, (2.3)

where the residues Ri are

Ri = (c!vi)(w!i b),

and R$ (which is often zero) is the constant contribution of the poles at infinity, andn & n is the number of finite first order poles (to be assumed to be numbered first).

Although there are di"erent indices of modal dominance [2, 8, 20, 26], the following[9] will be used in this paper.

Converge of DPA and RQI 3

0 2 4 6 8 10 12 14 16 18 20−80

−75

−70

−65

−60

−55

−50

−45

−40

−35

Frequency (rad/sec)

Gai

n (d

B)

OriginalModal Equiv.

−3 −2.5 −2 −1.5 −1 −0.5 0−10

−8

−6

−4

−2

0

2

4

6

8

10

real

imag

dominant polesother polesdpa to targetrqi to target

Fig. 2.1. The left figure shows the Bode plot of the transfer function (n = 66 states) of theNew England test system [12], together with the Bode plot of the k = 11th order modal equivalent,constructed by projecting the system onto the modes of the 6 most dominant poles, which may belongto complex conjugated pairs. The right figure shows part of the pole spectrum together with the initialshifts for which DPA (marked by circles) and two-sided RQI (x-es) converge to the most dominantpole ! = !0.467 ± 8.96i. Real (imaginary) part of initial shifts are at horizontal (vertical) axis.

Definition 2.1. A pole !i of H(s) with corresponding right and left eigenvectorsvi and wi (w!

i Evi = 1) is called the dominant pole if |Ri| > |Rj |, for all j $= i.An approximation of H(s) that consists of k < n terms with |Rj | above some

value, determines the e"ective transfer function behavior [22] and is also known astransfer function modal equivalent (assuming R$ = 0):

Hk(s) =k"

j=1

Rj

s" !j+ d.

More generally, a pole !i is called dominant if |Ri| is not very small comparedto |Rj |, for all j $= i. A dominant pole can be well observable and controllable inthe transfer function. Its presence can be observed in the Bode-plot corresponding to(2.2) (see fig. 2.1), which is a plot of |H(i$)| against $ ! R: in this example, peaksoccur at frequencies $ close to the imaginary parts of the dominant poles of H(s).The height of the peaks, and the controllability/observability of the (dominant) polethat causes the peak, also depends on the size of the real part of that pole (cf. (2.3)).Therefore, in the light of model order reduction by modal truncation, Definition 2.1may not be suitable, and a characterization in terms of |Ri|/|Re(!i)| might be moreappropriate. The purpose of this paper, however, is to analyze the convergence of theDominant Pole Algorithm [12], described in the following section, and compare it tothe convergence of Rayleigh Quotient Iteration. For this purpose, Definition 2.1 willdo. For an overview of model order reduction techniques, see [3].

The dominant poles are specific (complex) eigenvalues of the pencil (A,E) andusually form a small subset of the spectrum of (A,E). They can be located anywherein the spectrum, see also figure 2.1. The two algorithms to compute poles (eigenvalues)that will be discussed in this paper, the Dominant Pole Algorithm (DPA) and two-sided Rayleigh Quotient Iteration (two-sided RQI), both start with an initial shifts0, but behave notably di"erently: as can be seen in figure 2.1, DPA converges tothe most dominant pole for many more initial shifts than two-sided RQI (marked bycircles and x-es, respectively). In section 5 more of such figures will be presented andfor all figures it holds: the more circles (compared to x-es), the better the performance

4 J. Rommes and G.L.G. Sleijpen

of DPA over two-sided RQI. The typical behavior of DPA will be discussed in moredetail in sections 4 and 5.

In both DPA and two-sided RQI, the generalized two-sided Rayleigh quotient playsa central role. This quotient is defined as follows (cf. [15, 17]):

Definition 2.2. The generalized two-sided Rayleigh quotient %(x,y) is given by%(x,y) ' %(x,y, A, E) ' y!Ax/y!Ex, provided y!Ex $= 0.

Note that y!Ex can be zero even if E is nonsingular.Since the dominance of a pole is independent of d, without loss of generality d = 0

in the following.

3. The Dominant Pole Algorithm (DPA). The poles of transfer function(2.2) are the ! ! C for which lims%! |H(s)| = %. Consider now the function G :C "# C

G(s) =1

H(s).

For a pole ! of H(s), lims%! G(s) = 0. In other words, the poles are the roots ofG(s) and a good candidate to find these roots is Newton’s method. This idea is thebasis of the Dominant Pole Algorithm (DPA) [12] (and can be generalized to MIMOsystems as well, see [13, 19]).

The derivative of G(s) with respect to s is given by

G&(s) = "H &(s)H2(s)

. (3.1)

The derivative of H(s) with respect to s is

H &(s) = "c!(sE "A)#1E(sE "A)#1b. (3.2)

Equations (3.1) and (3.2) lead to the following Newton scheme:

sk+1 = sk "G(sk)G&(sk)

= sk +1

H(sk)H2(sk)H &(sk)

= sk "c!(skE "A)#1b

c!(skE "A)#1E(skE "A)#1b. (3.3)

The formula (3.3) was originally derived in [6]. Using xk = (skE " A)#1b andyk = (skE " A)#!c, the Newton update (3.3) can also be written as the generalizedtwo-sided Rayleigh quotient %(xk,yk), provided y!kExk $= 0:

sk+1 = sk "c!(skE "A)#1b

c!(skE "A)#1E(skE "A)#1b

=c!(skE "A)#1A(skE "A)#1bc!(skE "A)#1E(skE "A)#1b

=y!kAxk

y!kExk.

An implementation of this Newton scheme is represented in Alg. 1. It is also knownas the Dominant Pole Algorithm [12].

Converge of DPA and RQI 5

Algorithm 1 The Dominant Pole Algorithm (DPA).INPUT: System (A,E,b, c), initial pole estimate s0, tolerance &( 1OUTPUT: Dominant pole ! and corresponding right and left eigenvectors v and w1: Set k = 02: while not converged do3: Solve xk ! Cn from (skE "A)xk = b4: Solve yk ! Cn from (skE "A)!yk = c5: Compute the new pole estimate

sk+1 = sk "c!xk

y!kExk=

y!kAxk

y!kExk

6: The pole ! = sk+1 with v = xk and w = yk has converged if

)Axk " sk+1Exk)$ · )E!yk)1

|y!kExk|< &)A)$

7: Set k = k + 18: end while

The two linear systems that need to be solved in step 3 and 4 of Alg. 1 can bee!ciently solved using one LU -factorization LU = skE " A, by noting that U!L! =(skE"A)!. In this paper it will be assumed that an exact LU -factorization is available,although this may not always be the case for real-life examples, depending on the sizeand condition of the system. If an exact LU -factorization is not available, one has touse inexact Newton schemes, such as inexact Rayleigh Quotient Iteration and Jacobi-Davidson style methods [21, 10, 23], a topic that is described in [18]. Note that thestopping criterion in step 6 of Alg. 1 (and in step 7 of Alg. 2) guarantees a backwarderror of size & (see, e.g., [?, p. ??]).

3.1. DPA and two-sided Rayleigh quotient iteration. The two-sided Rayleighquotient iteration [15, 17] is shown in Alg. 2. The only di"erence with DPA is thatthe right hand sides in step 3 and 4 of Alg. 1 are kept fixed, while the right hand sidesin step 4 and 5 of Alg. 2 are updated every iteration.

While the use of the fixed right hand sides drops the asymptotic convergence ratefrom cubic to quadratic, it is exactly this use of fixed right hand sides that causesthe typical better convergence to dominant poles, as will be shown later. In thatlight the quadratic instead of cubic local convergence, that in practice only makesa small di"erence in the number of iterations, is even more acceptable. Moreover,based on criteria in [5, 24] for switching from inverse iteration to Rayleigh QuotientIteration, one could define similar criteria to switch from DPA to two-sided RQI in thefinal phase of the process, to save some iterations. However, such techniques are notconsidered in this paper, since the primary goal is to study the convergence behavior.

4. Local convergence analysis. The generalized two-sided Rayleigh quotient(Def. 2.2) has some well known basic properties, see [15, 17]:

• Homogeneity: %('x,"y, #A, (E) = (#/()%(x,y, A, E) for ', ", #, ( $= 0.• Translation Invariance: %(x,y, A" 'E, E) = %(x,y, A, E)" '.• Stationarity (all directional derivatives are zero): % = %(x,y, A, E) is station-

ary if and only if x and y are right and left eigenvectors of (A,E), respectively,

6 J. Rommes and G.L.G. Sleijpen

Algorithm 2 Two-sided Rayleigh quotient iteration.INPUT: System (A,E,b, c), initial pole estimate s0, tolerance &( 1OUTPUT: Pole ! and corresponding right and left eigenvectors v and w1: x0 = (s0E "A)#1b, y0 = (s0E "A)#!c, and s1 = %(x0,y0)2: Set k = 13: while not converged do4: Solve xk ! Cn from (skE "A)xk = Exk#1/)xk#1)25: Solve yk ! Cn from (skE "A)!yk = E!yk#1/)yk#1)26: Compute the new pole estimate

sk+1 = %(xk,yk) =y!kAxk

y!kExk

7: The pole ! = sk+1 has converged if

)Axk " sk+1Exk)$ · )E!yk)1

|y!kExk|< &)A)$

8: Set k = k + 19: end while

with eigenvalue % and y!Ex $= 0.

4.1. Asymptotically quadratic rate of convergence. In [17, p. 689] it isproved that the asymptotic convergence rate of two-sided RQI is cubic for non-defective matrices. Along the same lines it can be shown that the asymptotic conver-gence rate of DPA is quadratic. For the eigenvalue, this also follows from the fact thatDPA is an exact Newton method, but for the corresponding left and right eigenvectorsthe following lemma is needed, which gives a useful expression for (%k+1 " !) (usingsk+1 ' %k ' %(xk,yk, A, E) from now on).

Lemma 4.1. Let v and w be right and left eigenvectors of (A,E) with eigenvalue!, i.e. (A" !E)v = 0 and w!(A" !E) = 0, and w!Ev = 1. Assume that w!b $= 0and c!x $= 0, and let %k be given and not an eigenvalue of the pencil (A,E). Let)k,$k ! C be scaling factors so that the solutions xk and yk of

(%kE "A)xk = )kb and (%kE "A)!yk = $kc (4.1)

are of the form

xk = v + dk and yk = w + ek, (4.2)

where w!Edk = e!kEv = 0. Then with u ' (I"Evw!) bw"b and z ' (I"E!wv!) c

v"c ,it follows that

u = (%k " !)#1(%kE "A)dk * w and z = (%k " !)#!(%kE "A)!ek * v,

and with %k+1 = y!kAxk/(y!kExk), one has that

%k+1 " ! = (%k " !)µk, where µk =e!kEdk " e!ku1 + e!kEdk

. (4.3)

Note that u and z do not change during the iteration.

Converge of DPA and RQI 7

Proof. Substitution of (4.2) into (4.1) and multiplication from the left by w! andv!, respectively, gives

)k =%k " !

w!band $k =

(%k " !)!

v!c.

It follows that

(%kE "A)dk = (%k " !)(I " Evw!)b

w!b' (%k " !)u * w

and

(%kE "A)!ek = (%k " !)!(I " E!wv!)c

v!c' (%k " !)!z * v,

where u and z are independent of the iteration. With %k+1 = y!kAxk/(y!kExk), itfollows that

%k+1 " ! =y!k(A" !E)xk

y!kExk=

e!k(A" !E)dk

1 + e!kEdk.

Note that e!k(A"!E)dk = e!k(A"%kE)dk +(%k"!)e!kEdk = (%k"!)(e!kEdk"e!ku),which shows (4.3).

This lemma will be used in the proof of the following theorem, that shows theasymptotically quadratic rate of convergence of DPA, and expression (4.3) in partic-ular will be used to derive the local convergence neighborhoods of DPA and RQI insection 4.2.

Theorem 4.2. Let v and w be right and left eigenvectors of (A,E) with eigen-value !, i.e. (A"!E)v = 0 and w!(A"!E) = 0, and w!Ev = 1. Then limk%$ xk =v and limk%$ yk = w if and only if sk+1 = %k = %(xk,yk) approaches !. The con-vergence rate is asymptotically quadratic in case of convergence.

Proof. The proof is an adaptation of the proofs in [17, p. 689] and [10, p. 150].The main di"erence here is that for DPA the right hand-sides of the linear systemsare kept fixed during the iterations. Let the iterates xk and yk, see lemma 4.1, be ofthe form

xk = v + dk and yk = w + ek,

where w!Edk = e!kEv = 0 and w!Ev = 1. Put dk = (%k"!)dk with (%kE"A)dk =u, and ek = (%k"!)!ek with (%kE"A)!ek = z. Let V and W be the associated rightand left eigenspaces for !. Then there are two orthogonal decompositions of Cn:

Cn = V + (E!W)' = (EV)' +W,

and it can be shown that for all z ! C, one has (zE " A) : (E!W)' "# W' and(zE "A)! : (EV)' "# V'. Since these mappings are onto for all z su!ciently closeto !, there is a neighborhood N of ! and a constant m > 0 such that

)(zE "A)s) , m)s) and )(zE "A)!t) , )t),

for all z ! N , s ! (E!W)', and t ! (EV)'. It follows that if %k # !, then forsu!ciently large k

)dk) &|%k " !|

m)u), (4.4)

8 J. Rommes and G.L.G. Sleijpen

and similarly

)ek) &|%k " !|

m)z), (4.5)

and dk and ek, and dk and ek, are bounded. Hence, %k # ! if and only if xk # vand yk # w.

To prove the asymptotically quadratic rate of convergence, first note that

%k+1 " ! = %(xk,yk) = (%k " !)2e!k(A" !E)dk

1 + (%k " !)2e!kEdk

,

and hence

|%k+1 " !| = (%k " !)2|e!k(A" !E)dk|+ O((%k " !)4). (4.6)

By (4.4) and (4.6), it follows that as k #%, then

)v " xk+1) = )dk+1)

& )(A" !E)))u)m

)v " xk))w " yk)+ O((%k " !)4),

and similarly, by (4.5) and (4.6),

)w " yk+1) = )ek+1)

& )(A" !E)))z)m

)v " xk))w " yk)+ O((%k " !)4),

which proves the asymptotically quadratic convergence.

4.2. Convergence neighborhood. In this section it will be assumed that A isa symmetric matrix and that E = I. In [14] Ostrowski characterizes the convergenceneighborhood of the iteration

(A" %kI)xk = )kb, k = 0, 1, . . . , (4.7)

for symmetric matrices A, where %0 arbitrary, %k+1 = %(xk, A) ' %(xk,xk, A, E)(k > 0) and )k is a scalar so that )xk)2 = 1. It can be seen that DPA for symmetricmatrices (with E = I, b = c),

(%kI "A)xk = )kb, k = 0, 1, . . . , (4.8)

is similar and hence Ostrowski’s approach can be used to characterize the local con-vergence neighborhood of DPA for symmetric matrices A with c = b = (b1, . . . , bn)T .In fact, a larger convergence neighborhood of DPA will be derived here. This resultgives insight in the typical convergence behavior of DPA.

Since the two-sided Rayleigh quotient and (4.7, 4.8) are invariant under uni-tary similarity transforms, without loss of generality A will be a diagonal matrixdiag(!1, . . . ,!n) with !1 < . . . < !n. Note that Rj = )b)2 cos2 !(vj ,b) and the!J with J = argmaxj(cos(!(vj ,b))) is the dominant pole. The main results of thispaper, sharp bounds for the convergence neighborhoods of DPA and RQI, respec-tively, are stated in theorem 4.3 and theorem 4.4, respectively. The proofs are givenin section 4.2.1.

Converge of DPA and RQI 9

Theorem 4.3. Let (!,v) be an eigenpair of A. In the DPA iteration for A andb with initial shift %0, let xk and )k be such that

)xk) = 1, (%kI "A)xk = )kb, with %k+1 ' x!kAxk, (k , 0),

and put # = min!i (=! |!i " !|. If

|%0 " !|#

< 'dpa '1

1 + *2with * ' tan!(v,b), (4.9)

then, with c ' cos !(v,b), it follows that

%k # ! and|%k+1 " !|

c2#&

$|%k " !|

c2#

%2

(k , 0).

Since c2 = 11+"2 , Condition (4.9) is equivalent to |#0#!|

$c2 < 1. In the setting of thispaper, Ostrowski’s convergence condition [14, p. 235 (eqn. (19))] is given by

|%0 " !|#c2

& 12

min(1

2(1" c2),

1c2

).

Because 12 min( 1

2(1#c2) ,1c2 ) & 3

4 < 1, it is clear that the convergence neighborhood{%0 | Condition (4.9) holds} that follows from Theorem 4.3 is larger than the onefrom Ostrowski’s condition.

In [14, p. 239], the convergence neighborhood of standard RQI,

(A" %kI)xk = )kxk#1, k = 0, 1, . . . (4.10)

where x#1 arbitrary, %k+1 = %(xk, A) (k > 0) and )k is a scalar so that )xk)2 = 1, isderived. Here a sharper bound is derived.

Theorem 4.4. Let (!,v) be an eigenpair of A. In the RQI iteration for A andb with initial shift %0 and x#1 = b, let xk and )k be such that

)xk) = 1, (%kI "A)xk = )kxk#1, with %k+1 ' x!kAxk, (k , 0),

and put # = min!i (=! |!i " !|. If

|%0 " !|#

< 'rqi '1

1 + *with * ' tan!(v,b), (4.11)

then |%1 " !| < #/2, and

%k # ! and|%k+1 " !|

# " |%k+1 " !| &$

|%k " !|# " |%k " !|

%2

(k > 0).

In particular, the results of theorem 4.3 and 4.4 are sharp in the following sense:Theorem 4.5. The convergence statement in both Theorem 4.3 and Theorem

4.4 is sharp: to be more precise, there is a vector b such that for each %0, the %k ofTheorem 4.3 (Theorem 4.4) converge towards ! if ond only if |%0"!| < 'dpa (< 'rqi,respectively).

Proof. Let b = cv+cjvi0 where i0 is such that # = |!i0"!|. For ease of notation,write (!1,v1) = (!,v) and (!2,v2) = (!i0 ,vi0).

10 J. Rommes and G.L.G. Sleijpen

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cj2

α

αrqiαdpa

Fig. 4.1. Bounds of the local convergence neighborhood for DPA (dashed) and best-case RQI(solid). If, with " = mini"=j |!i ! !j |, one has |!j ! #!| " $" for some %, then the sequence of #k

convergences to !j . Here, $ = $dpa or $ = $rqi. Along the horizontal axis c2j is varied between 0

(b is orthogonal to vj) and 1 (b is a multiple of vj): cj # cos !(vj ,b).

With 'i0 ' |%0 " !i|/# and *(i)

0 ' tan!(vi,b) (i = 1, 2), one has that '(2)0 = 1"

'(1)0 and *(1)

0 = 1/*(2)0 . Therefore, if '(1)

0 > #/(1+ (*(1)0 )p) then '(2)

0 < #/(1+ (*(2)0 )p)

(p = 1, 2), and Theorem 4.3 (take p = 2) and Theorem 4.4 (take p = 1) guaranteeconvergence towards !2.

If '(1)0 = #/(1 + (*(1)

0 )p) then '(2)0 = #/(1 + (*(2)

0 )p), and, as can be seen in theproof of the corresponding theorem, the contraction statement in the theorem holdsfor both ! = !1 and ! = !2. This implies stagnation of the sequence of |%k " !|.

Note that it is actually proved that Theorem 4.5 is correct for any non-trivial bin the two-dimensional subspace spanned by v and vj with j such that # = |!j " !|.

In [5, Thm. 1] it is shown that, with #b = ""' a known gap in the spectrum of A(for instance, # = min!i (=! |!"!i|), if %1 < ('+")/2 and )r1) = )Ax0"%1x0) & #b,then %k < (' + ")/2 for k , 1, and similarly for the case %1 > (' + ")/2. The firstcondition of this theorem implies that |%0 " !| < #/2, while )r1) & # is possible onlyif !(v,x0) < 45). In other words, * < 1 and, |%0 " !|/# < 1/2 < 1/(1 + *). Ascan be learned from its proof, in Theorem 4.4, this is the situation only after oneiteration step. Theorem 4.4 seems to allow a weaker start. To see this, consider thetwo-dimensional example A = diag("1, 1). With %0 = 0.01, x#1 = b = [

-2/2,

-2/2]

and x0 = (A"%0I)#1x#1, it follows that |!"%0| < 1 and condition (4.11) is satisfied,while )r1) = )Ax0 " %1x0) . 1.03 > 1. Hence, the result in theorem 4.4 is sharper.

In Figure 4.1, 'dpa and 'rqi, see equations (4.9) and (4.11), respectively, areplotted for c2

j , where cj = cos !(v,b). As c2j increases, i.e. as mode j becomes more

dominant, both local convergence neighborhoods increase and '# 1, while the boundfor the DPA neighborhood is larger for c2

j > 1/2, or !(v,b) < 45).The price one has to pay for the cubic convergence, is the smaller local convergence

neighborhood of the dominant pole, as it becomes more dominant, for RQI. WhileDPA emphasizes the dominant mode every iteration by keeping the right hand-side

Converge of DPA and RQI 11

fixed, RQI only takes advantage of this in the first iteration, and for initial shifts too farfrom the dominant pole, the dominant mode may be damped out from the iteratesxk. In that sense, RQI is closer to the inverse power method or inverse iteration,which converges to the eigenvalue closest to the shift, while DPA takes advantage ofthe information in the right hand-side b.

Because the results are in fact lower bounds for the local convergence neighbor-hood, theoretically speaking no conclusions can be drawn about the global basinsof attraction. But the results strengthen the intuition that for DPA the basin ofattraction of the dominant pole is larger than for RQI.

4.2.1. Proofs of theorem 4.3 and theorem 4.4. The following two lemmasprovide expressions and bounds that are needed for the proofs of theorem 4.3 andtheorem 4.4.

Lemma 4.6. Let v be an eigenvector of A = AT with eigenvalue ! with )v) = 1,and let )k ! R be a scaling factor so that the solution xk of

(%kI "A)xk = )kb

is of the form

xk = v + dk, (4.12)

where v!dk = 0, and let z = (%k " !)#1(%kI " A)dk. Then %k+1 = x!kAxk/(x!kxk)satisfies

%k+1 " ! = (%k " !)µk,

where

µk =d!kdk " d!kz1 + d!kdk

. (4.13)

Proof. The result follows from lemma 4.1, by noting that A = AT and E = I.

Lemma 4.7. Under the assumptions of Lemma 4.6, put # = min!i (=! |!i " !|,c = cos !(v,b), * = )z), 'k = |#k#!|

$ , and 'k = 'k/(1 " 'k). The followingstatements hold:

'k < 1 / )dk) & 'k*. (4.14)

If 'k & c = 1/&

1 + *2, then

'k+1 & '2k(1 + *2), (4.15)

and 'k < 1 / 'k+1 ''k+1

1" 'k+1& ('k*)2. (4.16)

Proof. Put *k = )dk). Then by (4.13)

|µk| & +(*k) where +()) ' *) + )2

1 + )2() ! R).

12 J. Rommes and G.L.G. Sleijpen

The function + is increasing on [0, )max], where )max = (1 +&

1 + *2)/*, or, usingc ' cos !(v,b) = 1/

&1 + *2, )max =

&(1 + c)/(1" c), and 0 & + & 1+c

2c on (0,%).Since )(A" %k)#1|v#) & |1/(# " |!" %k|)|, it follows that

*k ' )dk) & |%k " !|)(A" %k)#1|v#))z) &|%k " !|

|# " |%k " !|| ='k

1" 'k*,

which proves (4.14). The statement: if 'k & c = 1/&

1 + *2, then *k & 'k* & )max

and

|µk| & +(*k) & +('k*) & 'k + '2k

1 + '2k*2

*2 ='k*2

(1" 'k)2 + '2k*2

, (4.17)

now follows from the observation that 'k*/(1 " 'k) & )max if and only if 'k &1/

&1 + *2 = c. Furthermore, if 'k & c = 1/

&1 + *2, then

|µk| & 1 if 'k*2 & 1 0 'k &1

1 + *2= c2. (4.18)

This follows readily from +('k*) & 1, statement (4.14), and the definition 'k ='k/(1" 'k).

For statement (4.15), first note that (1"'k)2 +'2k*2 , *2/(1+*2) for all 'k , 0,

and therefore |µ| & 'k(1 + *2). Hence, with 'k+1 ' |%k+1 " !|/#, inequality (4.15)follows by (4.13).

Finally, statement (4.16) follows the fact that (4.17) and (4.16) imply 'k+1 &('k*)2/(1 + ('k*)2).

Note that it is essential that the function + is increasing, since this allows to useupper bound (4.14) also to handle the denominator in (4.13), leading to (4.17).

In the two-dimensional case, the estimate in (4.17) is sharp (equality), since bothz and dk are in the same direction (orthogonal to v). Furthermore, in statement 2,|µ| & 1 if and only if 'k*2 & 1.

Proof. [Proof of theorem 4.3] Note that * is the same in all iterations, and recallthat 'k ' |%k " !|/#. Since c2 = 1/(1 + *2), condition (4.9) implies '0(1 + *2) < 1,and by induction and (4.15) of lemma 4.7, 'k

&1 + *2 & 'k(1 + *2) < 1. Again by

(4.15) of lemma 4.7, it follows that

'k+1(1 + *2) & ('k(1 + *2))2,

which implies the quadratic convergence.Note that result (4.16) implies 'k+1*2 & ('k*2)2, which guarantees quadratic

convergence as soon as '0*2 < 1. This condition is equivalent to '0 < 1/(1+ *2), thecondition (4.9) of the theorem.

Proof. [Proof of theorem 4.4] Note that *k = tan !(v,xk) changes every iteration,and recall that 'k ' |%k " !|/#, and 'k = 'k/(1 " 'k). Condition (4.11) implies'0 < 1/(1 + *), or, equivalently, '0*0 < 1. By (4.16) it follows that '1 < 1/2, or,equivalently, |%1 " !| < #/2. Since 'k*k < 1 implies that 'k < 1/

&1 + *2

k , results(4.14) and (4.16) of lemma 4.7 can be applied to obtain

'k+1 & ('k*k)2 and *k+1 & 'k*k,

if 'k*k < 1. It follows that *k+2 & 'k+1*k+1 & ('k*k)3 < 1. Therefore, since '0*0 <1, the sequences ('k) and (*k) converge dominated cubically, and, since *1 & '0*0 < 1and *k+2 < 1 (k , 0), it follows that 'k+1 & '2

k for k > 0.

Converge of DPA and RQI 13

4.3. General systems. Theorems 4.3 and 4.4 can readily be generalized fornormal matrices, but it is di!cult to obtain such bounds for general matrices withoutmaking specific assumptions. To see this, note that it is di!cult to give sharp boundsfor (4.3) in lemma 4.1. However, the following theorem states that DPA is invariantunder certain transformations and helps in getting more insight in DPA for general,non-defective systems (A,E,b, c).

Theorem 4.8. Let (A,E) be a non-defective matrix pencil, and let X, Y !Cn"n be of full rank. If DPA(A,E,b, c, s0) produces the sequence (xk,yk, sk+1), thenDPA(Y !AX, Y !EX, Y !b, X!c, s0) produces the sequence (X#1xk, Y #1yk, sk+1), andvice versa.

Proof. If x = xk is the solution of

(sE "A)x = b,

then x = xk = X#1x is the solution of

(sY !EX " Y !AX)x = Y !b,

and vice versa. Similar relations hold for y = yk and y = yk = Y #1y. Noting that

sk+1 =y!Y !AXxy!Y !EXx

=y!Axy!Ex

= sk+1.

completes the proof.Let W and V have as their columns the left and right eigenvectors of (A,E),

respectively, i.e. AV = EV # and W !A = #W !E, with # = diag(!1, . . . ,!n). Fur-thermore, let W and V be scaled so that W !EV = $, where $ is a diagonal matrixwith (ii = 1 for finite !i and (ii = 0 for |!i| = %. According to theorem 4.8,DPA(A,E,b, c) and DPA(#,$,W !b, V !c) produce the same pole estimates sk. Inb = W !b and c = V !c, the new right hand-sides, one recognizes the contributions tothe residues Ri = cibi = (c!vi)(w!

i b). The more dominant pole !i is, the larger thecorresponding coe!cients bi and ci are, and, since (#,$) is a diagonal pencil, thelarger the chance that DPA converges to the unit vectors v = ei and w = ei, thatcorrespond to the right and left eigenvectors vi = V ei and wi = Wei, respectively.

As observed earlier, DPA emphasizes the dominant mode every iteration by keep-ing the right hand-sides fixed, and thereby can be expected to enlarge the convergenceneighborhood also for general systems, compared to two-sided RQI. In practice, thequadratic instead of cubic rate of local convergence costs at most 2 or 3 iterations. Nu-merical experiments confirm that the basins of attraction of the dominant eigenvaluesare larger for DPA, as will be discussed in the following section.

5. Basins of attraction and typical convergence behavior. It is not straight-forward to characterize the global convergence of DPA, even not for symmetric ma-trices (see [14, p. 236-237]). Basins of attraction of RQI in the three-dimensionalcase are studied in [1, 4, 16], while in [5, 24] local convergence neighborhoods aredescribed. Because the DPA residuals rk = (A " %kI)b are not monotonically de-creasing (in contrast to the inverse iteration residuals rk = (A" ,I)xk and the RQIresiduals rk = (A" %kI)xk, see [5, 16, 17]), it is not likely that similar results can beobtained for DPA. Numerical experiments, however, may help to get an idea of thetypical convergence behavior of DPA and may show why DPA is to be preferred overtwo-sided RQI for the computation of dominant poles.

14 J. Rommes and G.L.G. Sleijpen

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

b2

ρ 0

dpa borderrqi borderαdpa border

0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

b2

ρ 0

dpa borderrqi border

Fig. 5.1. Convergence regions for DPA (solid borders) and RQI (dashed), and the theoreticalDPA border (dash-dot, see Thm. 4.3), for the matrix A = diag(!1, s, 1), for s = 0 (left) and s = 0.8.The regions of convergence to !2 = s for DPA and RQI respectively are enclosed between the lowerand upper borders of DPA and RQI respectively. The regions of convergence to !1 = !1 (!3 = 1)are below (above) the lower (upper) border.

An unanswered question is how to choose the initial shift of DPA. An obviouschoice is the two-sided Rayleigh quotient s0 = (c!Ab)/(c!Eb). This choice will workin the symmetric case A = A!, E = I, c = b. In the general nonsymmetric case thischoice will not always be possible: the vectors b and c are often very sparse (onlyO(1) nonzero entries) and moreover, it may happen that c!Eb = 0. In that casethe initial shift should be based on heuristics. For two-sided RQI, an obvious choiceis to take as the initial vectors x0 = b and y0 = c, but similarly, if y!0Ex0 = 0,this fails. Therefore, in the following experiments an initial shift s0 will be chosenand the (normalized) initial vectors for two-sided RQI are x0 = (A " s0E)#1b andy0 = (A" s0E)#1c, see Alg. 2.

All experiments were executed in Matlab 7 [25]. The criterion for convergencewas )Axk " sk+1Exk)2 < 10#8.

5.1. Three-dimensional symmetric matrices. Because RQI and DPA areshift and scaling invariant, the region of all 313 symmetric matrices can be parametrizedby A = diag("1, s, 1), with 0 & s < 1 due to symmetry (see [16]). In order to computethe regions of convergence of RQI and DPA (as defined in (4.7, 4.8)), the algorithmsare applied to A for initial shifts in the range ("1, 1)\{s}, with c = b = (b1, b2, b3)T ,where 0 < b2 & 1 and b1 = b3 =

&(1" b2

2)/2. In figure 5.1 the results are shown fors = 0 and s = 0.8. The intersections % = %!1 and % = %!3 at b2 = b with the bordersdefine the convergence regions: for "1 & %0 < %!1 there is convergence to !1 = "1,for %!1 & %0 < %!3 there is convergence to !2 = s, while for %!3 & %0 & 1 there isconvergence to !3 = 1.

For the case s = 0 it can be observed that (see vertical lines) for 0 & b2 ! 0.5,the convergence region to the dominant extremal eigenvalues is larger for DPA. For0.5 ! b2 & 1/

-3 . 0.577, the point at which !2 becomes dominant, the convergence

region of RQI is larger. However, for b2 " 0.5, the convergence region of !2 isclearly larger for DPA. Note also that the theoretical (lower bound 'dpa# of the)local convergence neighborhood for DPA (Thm. 4.3) is even larger than the practicalconvergence neighborhood of two-sided RQI for b2 " 0.8.

A similar observation can be made for the case s = 0.8. There, due to thedecentralized location of !2, the figure is not symmetric and the region of convergence

Converge of DPA and RQI 15

−8 −7 −6 −5 −4 −3 −2 −1 0 1 20

1

2

3

4

5

6

7

8

9

10

real

imag

dpa to targetrqi to targetdpa high resrqi high respoles

Fig. 5.2. Convergence regions for DPA and two-sided RQI, for the example of section 5.2. Thecenter of the domain is the pole ! $ !2.9 ± 4.8i with residue norm |R| $ 3.0 · 10#3. Circles (x-es)mark initial shifts for which convergence to the target takes place for DPA (two-sided RQI). Squares(+-es) mark initial shifts for which convergence takes place for DPA (two-sided RQI) to a moredominant pole outside the depicted area. Real (imaginary) part of initial shifts are at horizontal(vertical) axis. Horizontal and vertical stride are both 0.25.

of !2 is clearly larger for DPA. For 0 & b2 ! 0.35, DPA and RQI appear to be verysensitive to the initial shift. While the convergence region for !1 was similar to thecase s = 0, convergence for "0.1 ! %0 ! 0.8 was irregular in the sense that forinitial shifts in this interval both !2 and !3 could be computed; hence the regions areonly shown for b2 " 0.35. Because the theoretical lower bounds are much smaller,since d = mini (=j |!i " !j | = 0.2, and make the figure less clear, they are not shown(the theoretical DPA border still crosses the practical two-sided RQI border aroundb2 . 0.9).

It is almost generic, that apart from a small interval of values of b2, the area ofconvergence of the dominant eigenvalue is larger for DPA than for RQI. The followingexample discusses a large scale general system.

5.2. A large scale example. This example is a test model of the BrazilianInterconnect Power System (BIPS) [20, 19]. The sparse matrices A and E are ofdimension n = 13, 251 and E is singular. The input and output vectors b and c onlyhave one nonzero entry and furthermore cT Eb = 0; the choice x0 = b and y0 = cis not practical, see the beginning of this section. The pencil (A,E) is non-normaland the most dominant poles appear in complex conjugated pairs. It is not feasibleto determine the converge regions for the entire complex plane, but the convergencebehavior in the neighborhood of a dominant pole can be studied by comparing thefound poles for a number of initial shifts in the neighborhood of the pole, for bothDPA and two-sided RQI (Alg. 1 and Alg. 2). The results, for two areas of the complexplane, are shown in figure 5.2 and figure 5.3.

Initial shifts for which DPA and two-sided RQI converge to the target (the mostdominant pole, in the center of the domain) or its complex conjugate are marked bya circle and an x, respectively. In figure 5.2, occasionally there is converge to a more

16 J. Rommes and G.L.G. Sleijpen

−30 −28 −26 −24 −22 −20 −18 −16 −14 −12 −10−4

−3

−2

−1

0

1

2

3

4

5

6

real

imag

dpa to targetrqi to targetpoles

Fig. 5.3. Convergence regions for DPA and two-sided RQI, for the example of section 5.2. Thecenter of the domain is the pole ! $ !20.5± 1.1i, with residue norm |R| $ 6.2 · 10#3. Circles (x-es)mark initial shifts for which convergence to the target takes place for DPA (two-sided RQI). Real(imaginary) part of initial shifts are at horizontal (vertical) axis. Horizontal and vertical stride are0.5 and 0.25.

dominant pole outside the depicted area (marked by a square and a +, respectively).Grid points with no marker denote convergence to a less dominant pole.

Figure 5.2 shows rather irregular convergence regions for both DPA and two-sided RQI. This is caused by the presence of many other (less) dominant poles in thispart of the complex plane. Nevertheless, the basins of attraction of the dominantpoles are notably larger for DPA. Moreover, it can be observed that even for initialshifts very close to another less dominant pole, DPA converges to a more dominantpole, while two-sided RQI converges to the nearest pole. For example, for initial shifts0 = "2+4.5i, DPA converges to ! . "2.9+4.8i with |R| . 3.0·10#3, while two-sidedRQI converges to ! . "2.1 + 4.6i with |R| . 1.0 · 10#5.

In figure 5.3 the target is the most dominant pole of the system. It can be clearlyobserved that for DPA the number of initial shifts that converge to the dominantpole is larger than for two-sided RQI. The basin of attraction of the dominant pole islarger for DPA: except for regions in the neighborhood of other relatively dominantpoles (see, for instance, the poles in the interval ("28,"24) on the real axis), thereis convergence to the most dominant pole. For DPA typically the size of the basin ofattraction increases with the relative dominance of the pole, while for two-sided RQIthe e"ect is less strong, cf. theorem 4.3, theorem 4.4 and the discussion in section 4.2.The symmetry with respect to the real axis can be explained by the fact that if forinitial shift s0, DPA (two-sided RQI) produces the sequence (xk,yk, sk+1) convergingto (v,w,!), then for s0 it produces the sequence (xk, yk, sk+1) converging to (v, w, !).

In both figures it can be seen that for many initial shifts DPA converges to themost dominant pole, but two-sided RQI does not. On the other hand, for a verysmall number of initial shifts, two-sided RQI converges to the most dominant polewhile DPA does not. This is a counterexample for the obvious thought that if two-sided RQI converges to the dominant pole, then also DPA converges to it.

Converge of DPA and RQI 17

The average number of iterations needed by DPA to converge to the most dom-inant pole was 7.2, while two-sided RQI needed an average number of 6.0 iterations.The average numbers over the cases where both DPA and two-sided RQI convergedto the most dominant pole were 6.1 and 5.9 iterations, respectively.

Similar behavior is observed for other systems and transfer functions. Althoughthe theoretical and experimental results do not provide hard evidence in the sensethat they prove that the basin of attraction of the dominant pole is larger for DPAthan for two-sided RQI, they indicate at least an advantage of DPA over two-sidedRQI.

5.3. PEEC example. The PEEC system [7] is a well known benchmark systemfor model order reduction applications. One of the di!culties with this system of ordern = 480 is that it has many equally dominant poles that lie close to each other in arelatively small part, ["1, 0]1 ["10i, 10i], of the complex plane. This explains why infigure 5.4 for only a relatively small part of the plane there is convergence (markedby circles and x-es for DPA and two-sided RQI, respectively) to the most dominantpole ! . "0.14± 5.4i (marked by a *).

Although the di"erence is less pronounced than in the previous examples, DPAstill converges to the most dominant pole in more cases than two-sided RQI, and theaverage residue norm of the found poles was also larger: Rdpa

avg . 5.2 · 10#3 vs. Rrqiavg .

4.5 · 10#3. Again a remarkable observation is that even for some initial shifts veryclose to another pole, DPA converges to the most dominant pole, while two-sidedRQI converges to the nearest pole: e.g., for initial shift s0 = 5i DPA converges to themost dominant pole ! . "0.143 + 5.38i with |R| . 7.56 · 10#3, while two-sided RQIconverges to less dominant pole ! . "6.3 · 10#3 + 4.99i with |R| . 3.90 · 10#5.

The average number of iterations needed by DPA to converge to the most dom-inant pole was 9.8, while two-sided RQI needed an average number of 7.9 iterations.The average numbers over the cases where both DPA and two-sided RQI convergedto the most dominant pole were 9.4 and 7.7 iterations, respectively.

6. Conclusions. The theoretical and numerical results confirm the intuition,and justify the conclusion, that the Dominant Pole Algorithm has better global con-vergence than two-sided Rayleigh quotient iteration to the dominant poles of a largescale dynamical system. The derived local convergence neighborhoods of dominantpoles are larger for DPA, as the poles become more dominant, and numerical experi-ments indicate that the local basins of attraction of the dominant poles are larger forDPA than for two-sided RQI.

Both DPA and two-sided RQI need to solve two linear systems at every iteration.The di"erence between DPA and two-sided RQI is that DPA keeps the right hand-sides fixed to the input and output vector of the system, while two-sided RQI updatesthe right hand-sides every iteration. The more dominant a pole is, the bigger thedi"erence in convergence behavior between DPA and two-sided RQI. The other wayaround, for considerably less dominant poles, the basins of attraction are much smallerfor DPA than for two-sided RQI. This could be observed in cases where the initialshift was very close to a less dominant pole and DPA converged to a more dominantpole, while two-sided RQI converged to the nearest, less dominant pole.

The fact that DPA has a asymptotically quadratic rate of convergence, against acubic rate for two-sided RQI, is of minor importance, since this has only a very locale"ect and hence leads to a small di"erence in the number of iterations (typically adi"erence of 1 or 2 iterations).

18 J. Rommes and G.L.G. Sleijpen

−3 −2.5 −2 −1.5 −1 −0.5 04

4.5

5

5.5

6

6.5

7

real

imag

dpa to targetrqi to targettarget poledominant poles

Fig. 5.4. Convergence regions for DPA and two-sided RQI. The center of the domain is thepole ! $ !0.14±5.4i, with residue norm |R| $ 7.6 ·10#3. Circles (x-es) mark initial shifts for whichconvergence to the target takes place for DPA (two-sided RQI). Real (imaginary) part of initial shiftsare at horizontal (vertical) axis. Horizontal and vertical stride are both 0.2.

Acknowledgments. We are grateful to Nelson Martins for fruitful discussionsabout DPA [12] and its follow-ups SADPA [20] and SAMDP [19]. He also providedus with the New England and BIPS test systems. We thank Henk van der Vorstfor useful comments on earlier versions of this paper, and suggestions that helpedus to improve the presentation of the paper. Finally, we are much indebted to theanonymous referees, whose comments and suggestions proved their careful reading ofthe manuscript and led to improved readability and presentation.

REFERENCES

[1] P. A. Absil, R. Sepulchre, P. Van Dooren, and R. Mahony, Cubically convergent iterationsfor invariant subspace computation, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 70–96.

[2] L. A. Aguirre, Quantitative Measure of Modal Dominance for Continuous Systems, in Proc.of the 32nd Conference on Decision and Control, December 1993, pp. 2405–2410.

[3] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, 2005.[4] S. Batterson and J. Smillie, The dynamics of Rayleigh quotient iteration, SIAM

J. Num. Anal., 26 (1989), pp. 624–636.[5] C. Beattie and D. W. Fox, Localization criteria and containment for Rayleigh quotient

iteration, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 80–93.[6] L. H. Bezerra, Written discussion to [12], IEEE Trans. Power Syst., 11 (1996), p. 168.[7] Y. Chahlaoui and P. van Dooren, A collection of Benchmark examples for model reduction

of linear time invariant dynamical systems, SLICOT Working Note 2002-2, 2002.[8] M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, 1995.[9] A. M. A. Hamdan and A. H. Nayfeh, Measures of modal controllability and observability for

first- and second-order linear systems, J. Guid. Contr. Dyn., 12 (1989), pp. 421–428.[10] M. E. Hochstenbach and G. L. G. Sleijpen, Two-sided and alternating Jacobi-Davidson,

Lin. Alg. Appl., 358 (2003), pp. 145–172.[11] T. Kailath, Linear Systems, Prentice-Hall, 1980.[12] N. Martins, L. T. G. Lima, and H. J. C. P. Pinto, Computing dominant poles of power

system transfer functions, IEEE Trans. Power Syst., 11 (1996), pp. 162–170.[13] N. Martins and P. E. M. Quintao, Computing dominant poles of power system multivariable

Converge of DPA and RQI 19

transfer functions, IEEE Trans. Power Syst., 18 (2003), pp. 152–159.[14] A. M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of

the characteristic roots and vectors. I, Arch. Rational Mech. Anal., 1 (1958), pp. 233–241.[15] , On the convergence of the Rayleigh quotient iteration for the computation of the char-

acteristic roots and vectors. III, Arch. Rational Mech. Anal., 3 (1959), pp. 325–340.[16] R. D. Pantazis and D. B. Szyld, Regions of convergence of the rayleigh quotient iteration

method, Num. Lin. Alg. Appl., 2 (1995), pp. 251–269.[17] B. N. Parlett, The Rayleigh quotient iteration and some generalizations for nonnormal ma-

trices, Math. Comp., 28 (1974), pp. 679–693.[18] Joost Rommes, Methods for eigenvalue problems with applications in model order reduction,

PhD thesis, Utrecht University, 2007.[19] J. Rommes and N. Martins, E!cient computation of multivariable transfer function dominant

poles using subspace acceleration, IEEE Trans. Power Syst., 21 (2006), pp. 1471–1483.[20] , E!cient computation of transfer function dominant poles using subspace acceleration,

IEEE Trans. Power Syst., 21 (2006), pp. 1218–1226.[21] G. L. G. Sleijpen and H. A. van der Vorst, A Jacobi-Davidson iteration method for linear

eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 401–425.[22] J. R. Smith, J. F. Hauer, D. J. Trudnowski, F. Fatehi, and C. S. Woods, Transfer function

identification in power system application, IEEE Trans. Power Syst., 8 (1993), pp. 1282–1290.

[23] A. Stathopoulos, A case for a biorthogonal Jacobi-Davidson method: Restarting and correc-tion equation, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 238–259.

[24] D. Szyld, Criteria for combining inverse and Rayleigh quotient iteration, SIAM J. Num. Anal.,25 (1988), pp. 1369–1375.

[25] The Mathworks, Inc., Matlab.[26] A. Varga, Enhanced modal approach for model reduction, Math. Mod. Syst., (1995), pp. 91–

105.