Computer Methods in Applied Mechanics and Engineering 105 (1993) 122 NorthHolland CMA 347
Lanczos algorithm for the quadratic eigenvalue problem in engineering applications C. Rajakumar Swanson Analysis Systems, Inc., Houston, PA 15342, USA Received 28 March 1991
A new approach of employing the Lanczos twosided recursion to solve the quadratic eigenvalue problem is presented. The methodology employed retains the n order quadratic p~',~blem as posed, without the need to use the method of matrix augmentation traditionally used to cast the problem as a linear eigenvalue problem of order 2n. Pursuing the concept of finding a 'characteristic matrix' to yield an "cigenvalue square matrix', the Lanczos recursion has been devised to solve the quadratic e'genvalue problem. Appropriate proofs showing the Lanczos biorthogonal transformation are included. The example problems presented validate the algorithm and show the effectiveness of the method for large quadratic eigenproblems.
1. Introduction 1.1. The quadratic eigenvalue problem
The quadratic eigenvalue problem is encountered in structural dynamic applications whenever there is damping in a problem. The usual approach employed to extract the eigenvalues A~ of the damped system matrix dynamic equation 2
Kx i + A~Cx i + A i Mx i = 0
(1)
is to transform it into a linear eigenproblem format through matrix augmentation [111]. In (1) the system stiffness, damoing and mass matrices, K, C and M, respectively, are of order n. The interest is to devise an algorithm that deals with the quadratic eigenvalue problem without resorting to the linear transformation, thereby avoiding the need to work with matrices of order 2n. More specifically, in this paper, we seek to devise the Lanczos recursion that can be directly applied to the quadratic eigenvalue problem given by (1), where K, C and M are real and unsymmetric.
1.2. MotiwTtionfor the present approach Recently a nonaugmentation method of solving the quadratic eigenvalue problem was presented by Zheng et al. [12] using the inverse iteration technique. Using the concept of an Correspondence to: Dr. Charles Rajakumar, Swanson Analysis Systems, Inc., Johnson Road, P.O. Box 65, Houston, PA 153420065, USA. 00457825/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved
2
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
"eigenvalue square matrix transformation' yielded by a 'characteristic matrix of the quadratic eigenproblem', the authors have devised an algorithm called the Generalized Ritz Vector Iteration (GRI) method. Motivated by their successful implementation of the algorithm which directly applies to the quadratic eigenproblem, the Lanczos algorithm that applies to the quadratic eigenvalue problem is derived in the present paper. Also included are appropriate proofs of the Lanczos biorthogonal transformation. In order to keep the derivations general, we treat the unsymmetric quadratic eigenvalue problem where the matrices K, C and M in (1) are unsymmetric and real. Application of the Lanczos twosided recursion to the unsymmetric generalized eigenvalue problem was presented by Rajakumar and Rogers [13]. The motivation then was to employ the Lanczos algorithm to certain fluidstructure interaction problems [14] where the finite element matrices of the coupled problem are unsymmetric. Also, in boundary elementbased eigenvalue problems the Lanczos algorithm is found to be powerful in accurately finding a few of the eigenvalues of large unsymmetric fully populated matrices [15]. In [13], the twosided Lanczos recursion was also shown to be efficient in extracting the complex eigenvalues of unsymmetric generalized eigenproblems. In the present paper, as a logical extension of our previous work, the damping matrix that arises in nonconservative dynamical systems is included and we employ the Lanczos twosided recursion to solve the unsymmetric quadratic eigenproblem. Typical applications of the present work are rotor dynamic stability investigations, and damped system eigenfrequency computation of fluidstructure problems. Furthermore, when dealing with nonconservative structural or acoustical systems, the boundary element formulation will, conceptually, lead to unsymmetric quadratic eigenproblems, although this application is not readily found in the current literature. 1.3. Review of the literature Gupta [1] formulated an inverse iteration technique using complex arithmetic to iterate and find the roots of the linearized quadratic eigenvalue problem. Utku and Clemente [2] dealt with the spinning structures problem using inverse iterations in complex space. They show the shift and iterate process as applied to the generalized eigenproblem obtained from the linearization of the quadratic problem. The subspace iteration technique was applied to the quadratic eigenvalue problem arising in finite element fluidstructure analysis by Olson and Vandini [4]. Taking advantage of the block partitioned nature of matrices in the linearized problem, they have presented the subspace algorithm where the 2n order augmented matrices are not explicitly formed. Based on the same idea, their paper also presents the inverse iteration algorithm. Rodrigues and Gmuer [5] have presented the subspace iteration algorithm for undamped gyroscopic systems, where only n order submatrices of the augmented linear eigenvalue problem are employed in the iteration process. The use of the Lanczos algorithm to solve quadratic eigenvalue problems encountered in structural dynamics is dealt with in [611]. Borri and Mantegazza [6] have demonstrated that the partitioned nature of the augmented matrices in linearized quadratic eigenvalue problems can be taken advantage of in setting up the Lanczos recursions, without having to explicitly form the 2n order augmented matrices. Bauchau [7] has presented the Lanczos algorithm that applies to the special case of undamped gyroscopic structural systems, where the augmented
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
3
stiffness and mass matrices of the linearized eigenproblem are symmetric and skewsymmetric, respectively. Kim and Craig [8] and Gupta and Lawson [9] present block Lanczos algorithms for the linearized quadratic eigenvalue problem, the former treating the case of unsymmetric system matrices, and the latter dealing with the symmetric and skewsymmetric matrices that arise in spinning structures. Chen and Taylor [10] and NourOmid and Regelbrugge [11] have given the Lanczos recursions for damped dynamic systems, also using the linearized form of the quadratic eigenproblem.
1.4. Present work In the sections that follow, we present the twosided Lanczos recursions that apply to unsymmetric quadratic eigenvalue problems. The linearization process is not used explicitly. However, looking at the linearized form of the quadratic problem, alternative recursion schemes can be conceived, and this is illustrated in the paper. In addition, a proof of the Lanczos biorthogonal transformation that leads to the tridiagonal subspace matrix is shown. A complete derivation of the coefficients that are computed in the Lanczos twosided recursion is included in Appendix A for the sake of completeness. Implementation details of the algorithm such as the use of shift and the rebiorthogonalization of Lanczos vectors are also included. Example problems illustrating the accuracy of the algorithm in extracting the complex eigenvalues of the unsymmetric quadratic eigenvalue problem are presented. Also, a practical application of the method to extract a few of the eigenvalues of a damped system eigenvaiue problem is shown. 2. Preliminaries of the Lanczos algorithm
2.1. Transposed eigenproblem In the references by Cullum and Willoughby [16] and Wilkinson [17], the Lanczos algorithm is given for the unsymmetric standard eigenvalue problem, Ax~ = A~x~. The Lanczos recursion for the unsymmetric problem is twosided, since the original matrix A and its transpose AT are used in generating two sets of vectors, namely the right and lefthand Lanczos vectors. The unsymmetric generalized eigenproblem was treated in [13], where the Lanczos twosided recursion is presented considering the transposed eigenproblem along with the original problem. In order to derive the Lanczos recursion for the unsymmetric quadratic eigenvalue problem, we consider the transposed problem
K'rz~ + Aac'rz~ + A~M'rz~ = 0 ,
(2)
along with the original problem given by (1). The zeros of the determinant equation I r + A,C + = 0 = ICr + A,c + A M) I are the eigenvalues A, of (2). Since the determinant of a matrix is the same as that of its transpose, it is evident that the eigenvalues of the original and the transposed problems, given by (1) and (2), respectively, are the same. The associated eigenvectors x; and z; are called the right and lefthand eigenvectors, respectively, of the unsymmetric quadratic eigenproblem. Therefore, the Lanczos twosided recursion is devised considering the original eigenproblem, (1), and the transposed eigenproblem, (2), in
4
C. Rajakumar, Lanczos algorithmfor the quadratic eigenvalueproblem
order to biorthogonally transform the n order quadratic problem into a standard eigenvalue problem of order m, m ~
2.2. Biorthogonality of the eigenvectors We next have to determine the orthogonality relationship of the right and lefthand eigenvectors. If Ai and Aj are'two distinct eigenvalues for i ~ j, let us consider (1) and (2) for Ai and Ai, respectively. Corresponding to the eigenvalue Aj, (2) is rewritten as
KTzj + ,~jcTzj + A~MTzj
= O.
(3)
Premultiplying (1) by zjT and (3) by x~r and transposing it, we write down the equation obtained by taking the difference between the two resultant equations: 2 T ( ~, A~)zTCx, + ( A~ ~j)zjMx, =0.
(4)
Factoring our (A~ Aj) in (4), and noting that At ~ Aj, the biorthogonality relationship is found to be + (
+
=
o .
(5)
Using the following definition of two additional eigenve~tors,
xi = A,x, ,
(6)
Zj ~zj,
(7)
=
(5) is rewritten as
z~Cx i + ~.~Mx~ + z~M.~ ffi O.
(8)
Equation (8) represents the generalized biorthogonality condition of the right and lefthand eigenvectors of the unsymmetric quadratic eigenproblem. When i = j, if the eigenvectors are normalized such that
z~Cx~ + 7.'fMx~ + z~M.~ = 1,
(9}
the generalized biorthonormal condition is written as
z T c x + ZTMX + ZTMf( = I .
(lO)
Normalization as in (9) may not always be possible due to the possibility that the sum of the three scalar products yields a null value. X and Z are the right and lefthand eigenvector matrices. For a given eigenvalue At, it is seen from (6) and (7) that the vectors x~ and z~ are basically obtainable from the independent eigenvectors x i and z~. So, we call the vectors x~ and
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
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z~ the dependent right and lefthand eigenvectors. Furthermore, the biorthogonality condition given by (8) suggests that, in order to biorthogonally transform the quadratic eigenproblem, two independent and two dependent right and lefthand Lanczos vectors need to be formed through a twosided recursion. Pursuing the motivation to derive an algorithm to generate four sets of Lanczos vectors, the quadratic eigenvalue problem and the transposed problem, given by (1) and (2), respectively, are rewritten as follows: Kx, = A,(Cx, + M,~,),
(11)
KTzi" ,~,(cTzi dr"MT~i).
(12)
The dependent eigenvectors are given by "ri = Aix~ ,
(13)
~'i = Aizi "
(14)
The generalized biorthogonalization presented in this section can be extended to higher order eigenvalue problems. For example, proceeding on the same lines for a cubic eigenprobiem given by Kx~ + A~Cx~ + A~Mx~ + A~Dx~ = 0, the generalized biorthogonality condition can be shown to be zTCx~ + (zTM.~ + ~.TMx~)+(zTDx~ +.. zTDx~ + fTD£,) = 0," The dependent ! ! ! ! I ~/ eigenvectors in this expression are given by x~ = Aix~, z i = Aiz j, x i = A~£~, z'~ = Aj~.
3. Lanczos twosided recursion
In this section, the Lanczos twosided recursion for the unsymmetric quadratic eigenproblem posed in (11)(14) are derived. A proof of the resulting biorthogonal transformation of the quadratic problem, in the subspace formed by the right and lefthand Lanczos vectors generated by the twosided recursion, is included. Detailed derivation of the Lanczos recursion coefficients that form the subspace tridiagonal matrix is given in Appendix A. 3.1. Krylov sequence
We start by seeking the Krylov sequence of vectors [16] that will be used to generate the Lanczos vectors. Two sets of arbitrarily chosen vectors (t~! , ~1) and (,61, q l) are used to start the two pairs of Krylov sequence of vectors that apply to (11)(14). From (11) and (12), the right and lefthand sequence of vectors that will map the independent eigenvectors xi and zi, respectively, are
{o,, o2 = K'(CO, + Mp,), {1~1, I~2 =
KT(cTI~I
"4
(15)
K'(COz + MP2) ,... },
MTt~,), I~'3 " KT(cTI~'2 + M T t ] 2 ) , . . .
}.
(16)
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C. Ra/akumar, Lanczos algorithm ['or the quadratic eigenvalue problem
From (13) and (14), the right and lefthand sequences that will map the dependent eigenvectoi's x~ and z~, iespectively, can be written as {p,, p, =
=
(17)
,... },
{ q l ' q2 = I~l' q3 = ]~2 ' ' ' " } "
(18)
Notice that the primary Krylov sequences in (15) and (16), respectively, are coupled to the secondary sequences in (17) and (18). The Krylov sequences can also be written by looking at the following linearized augmented form of the quadratic eigenproblem:
[ K~ 0
xi
C IM]
Xi

(19)
•
Furthermore, there are other choices for the Krylov sequences based on the format of the quadratic eigenproblem in (1). For example, if (1) is expressed as
x, + A,KtCx, + A~KtMx, = O,
(20)
then the transposed problem will be
z~ + A~CTKTz~ + A~MTKrz~ =0.
(21)
Proceeding to set up the problem as in (11)(14), or writing in the following augmented form: I
_.0_
~
x
__A,
[~]
o L ,J
(22)
it can be seen that the Krylov sequences will be different from those given in (15)(18). The lefthand eigenvectors z~ in (21) and z~ in (2) are related by z~  KTz~. The particular choice of sequences given by (15)(18) is used in our formulation, because the problem reduces to generating only the two sequences given by (15) and (17) when the quadratic eigenproblem is symmetric.
3.2. Definition of the eigenvalue square matrix and characteristic matrix Zheng et al. [12] used the following definition in their development of a Generalized Inverse Iteration procedure to reduce the quadratic eigenvalue problem into a subspace matrix T whose eigenvalues will map the eigenvalues of the original problem. Suppose K, C, M E R "×" where R "×n represents the original space of the matrices K, C and M in (1). If there exists a matrix P E S "~" satisfying (a) rank(T)= m and (b)
MP + CPT + KPT~= O,
(23)
then T is called an m order eigenvalue square matrix of the quadratic eigenproblem. P is called the characteristic matrix and S "x'' is the subspace where the standard eigenvalue
C. Rajakumar, Lanczos algorithm ]'or the quadratic eigenvalue problem
7
problem
1
,2,,
is obtained. Using the above definition as a basis, we seek to set up the Lanczos recursion that will generate the characteristic matrix P, and yield the biorthogonally transformed eigenproblem, given by (24), with a tridiagonal eigenvalue square matrix T. Two primary sets of Lanczos vectors V = [v~ v 2 . . . Vm], W = [w 1 w 2 . • • Win] and two secondary sets of Lanczos vectors P = [p~ P2  Pm], Q = [q~ q2 . . . qm] are generated from the primary and secondary Kryiov sequences, respectively, given in (15)(18). The primary right and lefthand vectors v~ and w~ will map the right and lefthand eigenvectors x~ and z i, respectively. The secondary right and lefthand vectors p~ and q~ will map the dependent right and lefthand eigenvectors ~i and z~, respectively. Along the lines of the generalized biorthonormal condition of the eigenvectors given in (10), the Lanczos vectors are generated such that they are biorthonormal as follows: wTcv
+ QTMV + WTMP = I.
(25)
3.3. L a n c z o s recursion
The generalized Lanczos twosided recursion that applies to the quadratic eigenvalue problem as posed in (11)(14) are first presented here, and then the appropriate proofs of the transformation as defined in (23) and (24) are shown. Two sets of vectors 0~, ff~ and/~, ~ are arbitrarily chosen to start the recursions. Since they are arbitrary, we set 0~ = ff~ and/~ = ~,. The first two sets of Lanczos vectors v t, w~ and p~, q, are obtained from the starting vectors through normalization such that (wr~Cvl + qTMv~ + w T M p ~) = 1. For j  1, 2 , . . , m (m < n), 6j, I =  K  ' ( C v / +
Mp/)
ajvj  13iv~_ I ,
l~/+ I = KT(CTw/
+ MTq/)  Of~W~ " 8/W/_ I ,
/~j+, = v / _ a i p j  ~jpj_, , qi+l = wj  aiq i  8/qi_ I
(26a)
(26b) (27a)
(when j = 1,/31v o =/3,p o = 81Wo = 81qo = 0 ) ,
aj =   ( w T C + q T M ) K  ' ( C v i
+ MPi) + wTMvj ,
(27b)
(28)
4 + t = wT+,c6J + , + qT+'Mvj +' + vvT+'MPi + ' '
(29)
6j+, = 14+1l,/2,
(30)
/3i÷ , = 8i+ , sign(A/÷,),
(31)
m
01+I " 
vj÷, ~/+I
(32) '
8
C. Ra]akumar, Lanczos algorithm for the quadratic eigenvalue problem i
W j+ !
, w,+, = ~t3j+,
(33)
=/~J+'
(34)
qJ+' " q'+' = Oj+,
(35)
The recursions presented above are also derivable starting from the augmented matrices of the generalized eigenvalue problem that results from the linearization of the quadratic problem. This approach was presented by Borri and Mantegazza [6]. As can be observed from the recursion equations, the quadratic eigenproblem requires an extra set of right and lefthand vectors when compared with the recursion for the linear eigenvalue problem presented in [13]. The coefficients of the recursion %, ~j and 8j are used to form the Lanczos tridiagonal matrix as follows:
0[2 03 83 0[3 ""
T= 0
•
(36)
". 8"£ 0m 0[m
For m < 2 n , the inverse of the eigenvalue of the standard eigenproblem in (24) will approximate the eigenvalues of the original quadratic eigenvalue problem. As the subspace size m is progressively increased, the eigenvalues of (24) will converge to yield the eigenvalues A~of (1). Theoretically, when m  2n, all of the eigenvalues of the quadratic eigenproblem will be extracted. However, for large problems, often only the first few lowest magnitude eigenvalues are of interest, and usually they will converge for m << 2n. Spectral transformation of the eigenproblem to improve convergence of the eigenvalues in a specified part of the eigenvalue magnitude spectrum is presented in Section 5 describing the details of implementation. The rebiorthogonalization scheme that is an essential part of the recursion is also shown. 4. Generalized biorthogonal transformation 4.1. Proof of biorthogonal transformation
Proof of the Lanczos biorthogonal transformation of the quadratic problem to the standard eigenvalue problem in the subspace spanned by the Lanczos vectors is shown first. In the quadratic eigenvalue problem as posed in (11), the righthand eigenvectors x~ and x~ are projected onto the subspace spanned by the Lanczos vectors as follows: X i ~ gy i ,
X i ~"
PYi •
(37a) (37b)
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
9
Substituting (37) into (11) and (13), we multiply (11) by (W+C + Q + M ) K ~ and (13) by W+M, and add the two equations to obtain [W+CV + Q+MV + W+MP + X,(W+C + a + M ) g  l ( c v
+ M P )  ;~iW+MVIy, = O.
(38) Since the Lanczos vectors generated are biorthonormal, satisfyin~ (25), we obtain (39)
Ty, = P,iYi,
where T is the subspace matrix given by T= (W+C + Q+M)K'(CV + MP)+ WTMV
(40)
and the eigenvalue/z~ is defined as 1
/xi= A~"
(41)
Equation (39), which is the same as (24), is the biorthogonaily transformed eigenproblem, whose eigenvalues and eigenvectors relate to that of the original eigenproblem through (37) and (41). 4.2. Proof o f the characteristic and eigenvalue square matrices
Next, we show that P and T are the approximate righthand characteristic matrix and eigenvalue square matrix, respectively, of the quadratic eigenproblem by proving the condition given in (23). It will also be seen that T is tridiagonal. After substituting from (32) and (34) for 6~+t and ,5j+t, (26a) can be written in matrix form by grouping together m recursion steps as follows: [0
0
...
8,,+~v,.+~l=rtC[vt v5 [V, V~ . . .
...
v,.IK'M[Pl
P2
vmlr.
...
Pro] (42)
The T matrix here is the same as defined in (36) and so is tridiagonal. Using the matrix notation for the Lanczos vectors, (42) is rewritten as  K  I C V  K  t M P  V T = ~m+lV,,+le + ,
(43)
where e a"= (0 0 . . . 1) of size m. Considering (27a) and proceeding along the same lines, the following matrix equation results: V  P T = 6,,,+IP,,,+ le + .
Substituting for V from (44) into (43) and after some rearranging, we obtain
(44)
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C. Rajakumar, Lanczos algorithm ]'or the quadratic eigenvalue problem
KiMP + K'CPT + PT 2= 6m+,[(V,,,+, + KtCPm+,)e T + pm+,eTT].
(45)
Premultiplying (45) by K leads to MP + CPT + KPT 2= F.
(46)
The matrix F consists of m  2 columns of zeros and is given by 0
...
L,
L],
(47)
where fm~ = 6mKPm+t and fm= KV,,+~ + CPm+~+ otmKPm+t. The matrix F will be zero when 8m+~= 0, and (46) will satisfy the condition stated in (23). Then the matrix P will be an exact characteristic matrix, and the matrix T will be an exact eigenvalue square matrix of the quadratic eigenproblem. In general F # 0, and the matrices P and T obtained will be the approximate characteristic matrix and eigenvalue square matrix, respectively. Therefore, the eigenpairs/z~, y~ of the standard eigenproblem in (30) will yield the approximate eigenvalues and eigenvectors of the quadratic problem through the transformation equations (41) and (37). With the increasing subspace size m, the A~ and x~ found from (41) and (37) will converge to the exact eigenpairs of the quadratic problem. Starting from the recursion equation (26b) and proceeding along the lines of the derivations in this section, it can be shown that Q and T "r are the lefthand characteristic matrix and eigenvalue square matrix, respectively. 4.3. Eigenvalues of the subspace matrix
Since the subspace matrix T in the standard eigenvalue problem given in (39) is tridiagonal, its eigenvalues can be extracted very efficiently by any one of the standard eigenvalue extraction procedures such as the QR algorithm and Jacobi iterations. These methods iteratively zero out the lower diagonal elements of the tridiagonal Tmatrix. Once diagonalized, the diagonal elements of T are recovered as the eigenvalues, /z~, of the standard eigenvalue problem (39). In the present work, the QR algorithm [17] is employed. For the general case of the unsymmetric quadratic eigenproblem, the T matrix will also be unsymmetric and complex arithmetic is employed in the OR iterations to extract the complex conjugate eigenvalue pairs. The algorithm is very efficient in extracting both the real and ,complex conjugate pairs of eigenvalues of an unsymmetric tridiagonal matrix. A shift logic with successive deflation of the Tmatrix, as found in [18], has been implemented to accelerate convergence of the OR iterations. The eigenvectors y~ of the Tmatrix are found by inverse iterations.
5. Implementation details 5.1. Inverse of K and K T
In (26a) and (26b) of the Lanczos generalized twosided recursion, the inverses of K and g T are not explicitly computed. Instead, equations of the type K0~+~ Cv~ + Mp~ and KTI~'j+I  
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
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C Twj + MTq~ are solved by a triangularization and back substitution process on K and K T, respectively. The matrix K r, however, is not triangularized separately. The triangularized K is used in the computation of wi+ t = KT(CTw~ + M'rqj)" This will result in appreciable saving in computing time when an automated spectral transformation procedure is employed in the implementation. Details of the steps involved in recovering K T from the triangularized K are given in [13]. 5.2. Rebiorthogonalization o f Lanczos vectors
Equations (26) and (27) are the biorthogonalization steps of the recursions, where the set of (j + 1)st vectors are orthogonalized against the jth and ( j  1)st vectors. In the proofs presented in Appendix A, this is shown to be all that is needed to generate the series of Lanczos vectors that satisfy the biorthogonality condition given by (25). However, in finite precision computations, the loss of biorthogonality of the Lanczos vectors is well known. In order to retain generalized biorthogonality of the right and lefthand pairs of Lanczos vectors, Oi+l, Pj+l and Wj+I, qj+t, respectively, rebiorthogonalization is an essential part of the recursions. The ( j + 1)st vectors computed in the Lanczos recursion step given by (32)(35) are checked for their level of biorthogonality with respect to all of the previous vectors computed. For i = 1, 2 , . . . , j, the biorthogonality coefficients are calculated as follows: 0i = w~Coj+ ! + q~Moj+ I + w~Mp/+l,
(48)
T I Co i+ W/+ x I Mp,+q'~+ I Mv i • ~i = Wi+
(49)
If the magnitude of any of the coefficients, say the kth coefficient, exceeds a predetermined small value ecj, then the vectors are biorthogonalized with respect to the kth set of vectors: vj+, "* vi+ ,  Okvk ,
(50a)
wj+,, wj+,  6 , w , ,
(50b)
Pj+ I * Pj+ I  OkPk ,
(51a)
qi+l "> q~+1
(51b)

t~kql
•
The value of eo is dependent on the machine precision of the computer used. In the present implementation on a VAX 8550 computer, an ~0 value of 10 s has been found to work well. A much smaller value for c0 will yield Lanczos vectors with a higher level of biorthogonality. However, since the algorithm also depends on the finite precision of the machine to capture repeated eigenvalues [20], the value of eo should be chosen at least slightly higher than the machine precision. The present choice of the value for e0 is based on numerical experimentation, so that a repeated eigenvalue converged within five to ten Lanczos recursions after the first has converged. The complex eigenvalues always converged in conjugate pairs. 5.3. Spectral transformation
To accelerate convergence of the eigenvalues in a specified part of the eigenvalue magnitude spectrum, a spectral transformation is incorporated. A shift of the eigenvalue is
12
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
introduced as follows: Ai = vi + o'.
(52)
Substituting (52) into (1) leads to the spectrally transformed eigenproblem
K,,x i + viC~,x~ + ~.2Mx~ = O,
(53)
where Ko = (K + orC + cr2M) and Co = (C + 2crM). The Lanczos algorithm has the property of converging first to the largest magnitude eigenvalues in the spectrum. Therefore, in view of (J,1), the lowest magnitude eigenvalues will be extracted first. The use of a real valued shift or accelerates convergence of the eigenvalues that are close to its magnitude. Also, the shift is needed to extract the rigid body modes in structural dynamic applications.
5.4. Eigenvalue search The eigenvalues in a specified magnitude range are searched by introducing a shift ¢r equal to the lower limit of the eigenvalue magnitudes sought. A subspace size m is chosen such that m = r + s, where r is the number of eigenvalues desired. The value of s is chosen such that the r eigenvalues with magnitude higher than that of the shift or will have converged without missing any in the magnitude range [crl < IA, I IA, I. Based on our experience with damped eigenvalue problems in structural dynamics, a value of s = 16 is found to work well. In order to ensure that no eigenvalue is missed, a Sturm sequence check [19] tailored to detect missed complex eigenvalues will be necessary.
5.5. Breakdown of Lanczos recurslons It was pointed out earlier that the normalization of the eigenvectors, as shown in (9), may not always be possible due to the possibility of the sum of the scalar products resulting in a null value. For the same reason, the normalizations in (32)(35) will not be possible if the computed coefficient Aj+ I in (29) is zero. In this event the recursion breaks down and the remedy is to restart with a new set of starting vectors. In our experience this occurrence is rare. However, if [Aj+ l[ drops close to zero we detect two possibilities: (a) the aforementioned breakdown has occurred, and (b) the eigenvalues have all converged in the range of [or[ < [Ai[ ~ [A~[ (k < j) and the magnitude of the next eigenvalue Ak+! is much higher than that of Ak, often by more than two orders of magnitude. In either case, the recursions are restarted employing a new shift of magnitude slightly lower than that of the largest converged eigenvalue from the terminated recursion.
6. Example problems Three examples are presented to demonstrate the application of the Lanczos generalized twosided recursion. In the first two problems, randomly generated matrices from known
c. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
13
eigenvalues form the K, C and M matrices of the quadratic eigenproblem. The third problem is a damped system eigenvalue problem of a simply supported beam. In the first two examples, the matrices K, C and M are generated from a set of known complex eigenvalues Ai = cri  J o J i, Ai+ ~ =o'i + J o~i (J = ~ :  i ) . First, three diagonal matrices are formed as follows: Kd, ' = or~ + ¢a~, Cdii = 20ri and Mdi ~ = 1. It is easy to show that the eigenvalues of the diagonal 2 quadratic eigenvalue problem KdX ~ + A~Cdx ~ + A~ Mdxi = 0 are the ones that were preselected. Then, using integer arithmetic, the diagonal matrices are premultiplied by an arbitrary matrix A whose elements are randomly filled. The quadratic eigenproblem formed by the product matrices K = A k d, C = A C d and M = A M d have the same eigenpairs as that of the diagonal matrix problem. 6.1. R a n d o m matrix problem 1
This is a small size problem to validate the algorithm. Input matrices generated by the random process described above are of order six and are shown in Table 1. The computed eigenvalues along with the known eigenvalues are given in Table 2. The 12 eigenvalues are the complex conjugate eigenvalue pairs of the quadratic eigenproblem. As is evident, the subspace size m in this problem is double the order of the eigenproblem, and hence all the six conjugate pairs of eigenvalues have converged. In spite of the wide eigenvalue magnitude spectrum in this problem, the computed eigenvalues are in good agreement with tbe actual eigenvalues of the problem.
Table 1 Input matrices for random matrix problem 1 KMatrix
324 72 1152 612 900 1728
2190 2482 657 1314 2409 2263
23160 1930 13510 13510 52110 88780
10864415 4205580 8411160 2803720 18224180 2102790
4182340 13383488 14219956 12547020 11710552 3764106
109027740 7268516 32708322 85405063 61782386 47245354
CMatrix 108 24 384 204 300 576
180 204 54 108 198 186
216 18 126 126 486 828
62 24 48 16 104 12
940 3008 3196 2820 2632 846
161760 10784 48528 126712 91664 70096
MMatrix 9 2 32 17 25 48
30 34 9 18 33 31
12 1 7 7 27 46
31 12 24. 8 52 6
10 32 34 30 28 9
60 4 18 47 34 26
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
14
Table 2 Eigenvalues of random matrix problem
1
Eigenvalues No. 1 2 3 4 5 6 7 8 9 10
11 12
Known 6+ 63+ 39+ 91+ 147 + 47  1348 +  1348 
0J 0J 8J 8J 43 J 43 J 592 J 592 J 645 J 645 J 5J 5J
Computed 6.0004588 5.9995408 2.9999997 2.9999997 9.0000022 9.0000022 1.0000667 1.0000655 46.999961 46.999959  1348.0007  1348.0007
0.99258436E  06 J  0 . 9 8 9 7 9 8 1 2 E  06 J 8.0000007 J 8.0000007 J 42.999999 J 42.999998 J 591.99991 J 591.99991 J 644.99963 J 644.99963 J 5.0045941 J 5.0045945 J
6.2. Random matrix problem 2
This is a larger size problem with the matrices generated by the random process described at the beginniag of this section. The quadratic eigenproblem of order 18 has 36 eigenvalues and these are listed under known eigenvalues in Table 3. The eigenvalues in this problem have been computed using shifts and are presented in this table. The values of the shift or were chosen to improve convergence in a specified range of the eigenvalue magnitude spectrum as indicated. Eigenvalues shown in bold face are recovered as the converged eigenvalues within each eigenvalue magnitude range searched. 6.3. Beam with proportional damping
A simply supported beam of square crosssection, shown in Fig. 1, is discretized by finite elements to compute its damped eigenfrequencies. Its damping matrix is derived from the proportional damping expression given by C = aM + ~ K, where a and/3 here are the scalar coefficients that bring a fraction of the mass and stiffness matrix, respectively, of the beam
y
l
Material Properties: E, p Crosssection Properties: A, I
_.t1.0 X
["
L'30 [ Fig. 1. Simply supported beam of square crosssection (Young's modulus E = 1 2 E l 0 , density p = 1E4, area A = 1, moment of inertia ! = 1 / 12, proportional damping coefficients: a = 0 . 1 2 , / 3 = 0.003).
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem into the damping matrix. The analytical expression g i v e n b y t h e f o l l o w i n g e x p r e s s i o n [6]
+ a,x,,
i¥ where
for the eigenfrequencies
15 of the beam
is
4 E1
=
/
pA '
n = 1,2,...,~,
An a r e t h e c i r c u l a r e i g e n f r e q u e n c i e s
(54)
of the beam.
E and p are the Young's modulus
and
Table 3a Eigenvalues of random matrix problem 2 Computed eigenvalues with shifts ~ No.
Known eigenvalues
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
1+ 2J 1 2J 8 8 4 + 9 J 4  9 J 4 + 9 J 4  9 J 23 + 7 J 23  7 J 23 + 7 J 23  7 J 7 + 23 J 7  23 J 32 + 8 J 32  8 J + 49 J  49 J + 49 J  49 J 73 73 73 73 52 + 66 J 52  66 J 10 + 120 J 10  120 J 243 + 9 J 243  9 J 243 + 9 J 243  9 J 800 + 50 J 800  50 J + 1700 J  1700 J
o'=O(O<[aa
o" =  4 0 (10< [AA< 100)
2.0000000 J 2.0000000 J 0.11426959E 0.11427266E 9.0000000 J 9.0000000 J 9.0000002 J 9.0000003 J 6.9999794 J 6.9999795 J 7.0000211 J 7.0000211 J 23.000018 J 23.000018 J 8.0000048 J 8.00{)0046 J 49.000000 J 49.000000 J 49.000015 J 49.000015 J 0.54786469E 0.54780921E 0.78792279E 0.78792278E 65.999947 J 65.999947 J 120.00013 J  120.00013 J 9.0050936 J 9.0050936 J 9.0044968 J 9.0044966 J 50.263474 J 50.263470 J 1700.0140J  1700.0140 J 
02 J 02 J
0.99999903 0.99999907 7.9999938 7.9999936 3.9999993 3.9999993
4.0000000
03 J 03 J 01 J 01 J
4.0000000 23.000000 23.000000 23.000013 23.000013 6.9999994 6.9999993 32.000001 32.000001 0.00000000E + 00 O.O0000000E + 00 0.00000000E + 00 0.00000000E + 00 72.999999 73.000000 73.000046 72.999951 52.000000 52.000000 10.000010 10.000010 243.00000 243.00000 243.00016 243.00015 800.00074 800.00073 0.29220623E  02 0.29212371E  02
1.9999983 J  1.9999982 J 0.45616818E 0.45616807E 8.9999995 J  8.9999995 J 9.0000000 J 9.0000000 J 7.0000000 J 7.0000000 J 7.0000065J 7.0000065 J 22.999999 J 22.999999 J 7.99999$3 J 7.99999S3 J 48.999999 J 48.999999 J 49.000000 J 49.000000 J 0.42M4$$3E 0.42M4$SSE 0.31263770E 0.31263716E66.000000 J 66.000000 J 119.99999 J  119.99999 J 9.0000027 J 9.0000029 J 8.9973039 J 8.9973029 J 49.998918 J 49.998917 J 1699.9889 J  1699.9890 J
a Bold faced values are recovered as 'converged' eigenvalues in the magnitude range shown.
02 J 02 J
02 J 02 J 01 J 01 J
C. Ra]akumar, Lanczos algorithm for the quadratic eigenvalue problem
16
Table 3b Eigenvalues of random matrix problem 2 Computed eigenvalues with shifts a No. 1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23 "~ ~ 26 27 28 29 30
Known eigenvalues 1+ 2J 1 2J 8 8 4 + 9 J 4  9 J 4 + 9 J 4  9 J 23 + 7 J 23  7 J 23 + 7 J 23  7 J 7 + 23 J 7  23 J 32 + 8 J 32  8 J + 49 J  49 J + 49 J  49 J 73 73 73 73 52 + 66 J 52  66J 10 + 120 J 10  120 J  243 + 9 J 243  9 J
31
243 + 9 J
32 33 34 35 36
243  9 J 800 + 50J 800  50J + 1700 J  1700 J
cr = 400 (100 < I~i[ <~ 1000) 0.99998852 0.99998868 8.0635284 7.9364372 3.9999857 3.9999947 4.0000089 4.0000002 22.999984 22.999984 22.999968 22.999969 7.0000136 7.0000136 31.999997 31,999997 0.16295413E 0.16232779E 0,24658263E 0,24615132E 72,999963 72,999966 73,000040 73,000039 52,000001 52,000000 10,000007
04 04 04 04
10,000007
o" = 4000 (1000 < n,~i[~< 10000)
2.0006092 J 2.0006091 J 0.00000000E + 00 J 0.92016557E  07 J 9.0000115 J 9.0000142 J 9.0000028 J 9.0000001 J 7.0001461 J 7.0001461 J 7.0005490 J 7.0005492 J 22.999948 J  22.999948 J 8.0000084 J 8.0000083 J 48.999943 J 48,999943 J 49.000021 J 49,000021 J 0,10912903E  01 J 0,10912947E  01 J 0,48654808E  01 J 0,48654725E = 01 J 65,999988 J 65,999988 J 120,00001 J
 120.00001 J
243,00000
9,0000018 J
243,00000 243,00000 243,00001
9,0000018 J 8.9997624 J 8.9997624 J
800,00001
49,999999 J
800,00001
49,999999 J
0,21695793E  03 0,20521643E  03
1700,0000 J  1700.0000 J
2.2590900 0.66025755 0.66026314 4.5141538 4.5141552 17.264034 17.264047 23.084297 23.084298 6.8564092 6.8564065 27,930836 31.575778 31.575779 0.10592243 0.10592388 0.34246225E  02 0,34262372E  02 65,549001 73,011581 73,011534 79,515642 51,494009 51,494020 10.114031 10. i 14032  243.00504 243.00504
242.94268 242.94269 799.99941 799.99940 0,21499226E  03 0,210905$6E  03
0.87024078E 2.5599129 J  2.5599235 J 10.857838 J  10.857837 J  3.1142596 J 3.1142628 J 8.1505006 J 8.1505061 J  23.706376 J 23.706382 J 0.37418943E 7.1219173 J 7.1219172 J 48.813535 J 48.813532 J 48.992257 J 48.992260 J 0,00(O00(R)E + 0.59628271 J 0.59628269 J O.l~OE + 65,825288 J 65,825287 Y 120.51599 J  120,51599 J 9.1787784 J 9.1787767 J 9.0165364J 9.0165382 J 50.000347 J 50.000340 J
05 J
05 J
00 J
00 J
1700.0002 J 1700.0002 J
Bold faced values are recovered as 'converged' eigenvalues in the magnitude range shown.
density of the beam material, respectively, L, I and A are the beam length, crosssection area and moment of inertia, respectively. The beam shown in Fig. 1 is divided into 50 elements, resulting in a total of 151 degrees of freedom to solve. Analytical frequencies and computed frequencies are presented in Table 4. The first ten frequencies computed are in good agreement with the analytical frequencies, and the slight differences are partly attributable to the discretization error inherent in the finite discretization of the continuum.
C. Ra]akumar, Lanczos algorithm for the quadratic eigenvalue problem
17
Table 4 Eigenfrequencies of the beam with proportional damping Frequency (Hz) Mode 1 2 3 4 5 6 7 8 9 10
Analytical 0.38258801E01 0.38258801E01 0.46890141 0.46890141 2.3350194 2.3350194 7.3591831 7.3591831  17.952991  17.952991
Computed 1.7449094 J  1.7449094 J 6.9655503 J 6.9655503 J 15.533437 J  15.533437 J 26.938125 J 26.938125 J 39.768680 J 39.768680 J
0.38231453E01 0.38231453E01 0.46721040 0.46721040 2.3159626 2.3159626 7.2530238 7.2530238  17.551764  17.551764
1.7440785 J  1.7440785 J 6.9527743 J 6.9527743 J 15.471081 J  15.471081 J 26.757246 J 26.757246 J 39.410940 J 39.410940 J
7. Conclusions
A Lanczos generalized twosided recursion scheme has been devised to solve quadratic eigenproblems and has been validated. The algorithm presented is one of the different ways a generalized recursion can be set up for the quadratic eigenvalue problem, provided that the conditions of the eigenvalue square matrix and characteristic matrix stated in Section 3.2 are satisfied. Since the matrices of the quadratic eigenproblem are retained in their original order n, unlike the linearized problem with 2n order matrices, large size problems can be handled efficiently. The twosided recursion presented here can easily be specialized for symmetric matrices where only a singlesided recursion will be needed to generate just the two sets of righthand Lanczos vectors. The Lanczos algorithm is found to be very effective in extracting a few of the eigenvalues, say less than 50, of large quadratic eigenproblems, both in accuracy and speed of computation. When large numbers of eigenvalues are requested, however, the full rebiorthogonalization adopted in the present implementation will slow down the computation. This can be improved by devising a selective rebiorthogonalization scheme on the lines of [20,211.
Appendix A. Derivation of Lanczos coefficients
The coefficients a t,/3j and 8j in the Lanczos generalized twosided recursion presented in Section 3.3 are derived here. In the recursion equations (26) and (27), the first terms on the righthand side, vj+l =  K  ~ ( C v j
+ Mpj) ,
ff ~+1 =  K  ' r ( c ' r w j
+ Mrqj) ,
(A.la) (A.lb)
C. Rajakumar, Lanczos algorithm .for the quadratic eigenvalue problem
18
(A.2a)
Pj + 1 = Oj ,
(A.2b)
A
q j+ 1 = wj ,
are the (j + 1)st vectors of the Krylov sequence from which the new set of Lanczos vectors oj+l, wj+l, Pj+t, qj+l are computed. The j Lanczos vectors computed already are in generalized biorthogonality as follows. For i, k = 1, 2 , . . . , j, T wTCv, + qkMV, + w~Mp, =0
(for i ~ k)
= 1 (for i = k).
(A.3)
In (26) and (27), the (j + 1)st set of vectors are obtained by biorthogonalizing 6j+~, ~]+~, ,~j+~ and ~j.t against the preceding two Lanczos vectors. For the purpose of this derivation, we will assume that the 'hat' vectors contain the components of all the j Lanczos vectors in addition to the pure biorthogonal components v~+~, ~j. !, ,Sj+~ and ~j +t as shown below. t~j. t = 6j+ t + %vj + fljvj_ 1 + yjvj_ 2 + . . . ,
(A.4a)
wj÷l = wi+l + a~wj + 8jwj_ 1 + rljwj_ 2 + . . . ,
(A.4b)
pj., •,Sj+, +
(A.5a)
ajpj + jSjpj_, + "YiPj2 + ' " . ?
#j,t = qs*= + •Jqj + 8/qj_= + rljq/2 + ' " .
(A.5b)
The coefficients %, ~j, yj,,..., and a~, 8j, vii,.., represent the amplitudes of the previous Lanczos vectors containea m the right and lefthand Krylov sequence vectors oj, t,/~j,t and ff],=, ~j,~, respectively. To evaluate these coefficients, a systematic process of premultiplieation by appropriate vectors and addition of (A.4) and (A.5) are carried out. Coefficients a i and a~
To find the coefficient aj, (A.4a) and (A.Sa) are premultiplied by ( w T c + q ~ M ) and w iTM , respectively, and added together. Then using the biorthonormal relationship shown in (A.3), the resulting equation reduces to T ^ + wjMPi+, = wjCv]+= r  + q~Maj + , + wTM.Sj+I + oti . wTco/+, + q~Mvj+,
(A.6)
Since 5j, = and ,Sj, t have been defined as biorthogonal vectors, the first three terms on the righthand side sum up to zero. Substituting for the hat vectors from (A.1) and (A.2), the expression for % is found, % = (wTC + q ~ M ) K  t ( C o i + Mpj) + w ~ M v , .
(A.7)
To find the coefficient ~ , (A.4b) and (A.5b) are premultiplied by (vTC T + p T M T ) and v T M r respectively, and proceeding on the same lines as for %, the expression for a; is found,
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
19
a; = (vTc T + pTMT)KT(CTwj + M t q j ) + u t M t w j .
(A.8)
Taking the transpose of (A.8), it can be seen that a~ = a i.
Coefficients [3/and 6/ First, we derive the expression for/3j and then write down the expression for 6j observing symmetry. Premultiplying (A.4a) and (A.5a) by (w~_~C + q/_~M) T T . and wj_~M, respectwely, the two equations are added together. Then using the biorthonormal relationship of (A.3) we obtain
T COt+! + q Wj!
T
T
,,
tMOj+l + WjIMpj+ I
.._
W T
T

T

j_,Cffj+, + qj_lMvj+, + Wj_lMffj+, + ~j. (A.9)
Since v~+~ and/~/+ ~ are biorthogonal vectors, the first three terms on the righthand side sum to zero yielding p , = w •,r_ ,
COj+, + qT_,M6j+, + Wt, _ ,
(A.10)
Mp,+ , .
Now substitution for the hat vectors from (A.la) and (A.2a) is made. Also, from (A.lb) and t and qj_~ T can be found and substituted in (A.10) to give (A.2b), expressions for wj_~
aj =  ¢ ~ c r  ' ( c ~ j + Mpj) + (~,~r + ¢ ~ c ) m  ' m r  ' ( c o j
+ Mpj) + OyMoj.
(A.11)
After expanding the product terms, certain cancellations occur and the expression for /3j becomes
,8~ = ,~co, + o,'rM,,, + *~Mr,.
(A.12)
From (A.4b) and (A.fb), expressions for };'i and ,~j can be substituted in (A.12). Furthermore, using the biorthonormal relation given by (A.3), the following result is obtained:
z, =
~7co, + ¢)Mo,
+ *TMP, •
(A.13)
Starting from (A.4b) and (A.5b), a similar expression for 8j can be found. Here, we write it down by inspecting (A.13) and observing a certain symmetry that can be seen in the derivations so far.
8j = 6~cTw, + : ~ M t w , + 6tMtq/ .
(A.14)
Now, (A.13) and (A.14) can be rewritten for the (j + 1)st vectors as follows:
/3j+, = }~+,Cvj+, + ~+,Mvj+, + ~t+,Mpj+, ,
8,+, eT+,c%+, + p~+,m%+,
+ oT+l M~q j + l
(A.15) •
(A.16)
C. Rajakumar, Lanczos algorithm for the quadratic eigenvalue problem
20
The vectors v~+t, ffj+~,/ij+~ and tij+~ have been defined as biorthogonal to the j Lanczos vectors generated so far. Therefore, from (A.15) and (A.16), the normalized vectors can be written as m
Or+ t UJ+I  (~j+l '
(A.17a)
WI+ 1 Wj+I ~" ~ j + ,
(A.17b)
/~j+, Pi+l = Sj+ 'l '
(A.18a)
qj+! qj+l = ,8.j+l ,
(A.18b)
the (j + 1)st right and lefthand Lanczos vectors are given by (A.17a), (A.18a) and (A.17b), (A.18b), respectively. To find the coefficients fl~+~ and ~$j+t, (A.17a) and (A.18a) are substituted back into (A.15) to give 4+, =
+
+
(A.19)
where A t + t = ~/+ 18j + t. By choosing
aj+, = 14+, I1'2
(A.20) ,
/3~+, = 8j+, sign(At+t),
(A.21)
the coefficients will be real valued. Coefficients % a n d ~i
Starting with the premultiplication of (A.4a) and (A.Sa) by (wX~_2 C + q ~ _ 2 M ) and wXi_2M, respectively, and proceeding along the same lines as for finding the expression in (A.13) for /3~, the expression for % can be derived. Here we write down the expressions for )'i and vii by inspection of (A.13) and (A.14), (A.22) ~j = 6T_,CTw~ + 6T_tMTwj + 6~_,MTqj .
(A.23)
From (A.17) and (A.18), expressions for vi_ t, %_t and Pjt, qjl can be found and substituted into (A.22) and (A.23). Then using the biorthogonal property of the Lanczos vectors, we obtain r "/i = [3~_,( w Tj_, C v ~ + qj_zMoj + w jr_ z M p j ) = O ,
(A.24)
C. Ra]akumar, Lanczos algorithm for the quadratic eigenvalue problem rlj ~i_l(viTlCTwj + p jT_ l M T Wj'b
j_lMqj)=O.
VT
21
(A.25)
Therefore, the fourth term in (A.4) and (A.5) will be zero. By induction it can be observed that all the rest of the terms beyond the fourth will also vanish. Then using the result that a~ = %, (A.4) and (A.5) yield the Lanczos recursion equations, Ol+ ! ~" U]+ if, j + ,
=
1
,j÷, 
Ol/Oj 
(A.26a)
p]Uj_I ,
%wj

6jwj_,
,
(A.26b)
Pj+l = Pj+i 
otjpj fljPjI,
(A.27a)
qi÷,
%qj
(A.27b)
= cij÷l


aiqj1.
In view of (A.1) and (A.2), the above equations are the same as the Lanczos recursion step presented in (26) and (27). To summarize, the Lanczos coefficients are given by (A.7) and (A.19)(A.21). The set of biorthonormal vectors generated at the (j + 1)st step is given by (A.17) and (A.18). Finally, the generalized recursion equations for the quadratic eigenproblem are given by (A.26) and (A.27).
Acknowledgment The author would like to thank Swanson Analysis Systems, Inc. for the support provided for this work. The excellent work of Jackie Williamson and Charmaine Grunick in the preparation of the manuscript are appreciated.
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C. Ra/akumar, Lanczos algorithm for the quadratic eigenvalue problem
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