Power iterations and the dominant eigenvalue problem

26 Pages·1993·1.2 MB·en_US

by  Leader, Jeffery J.

NPS-MA-93-007
NAVAL POSTGRADUATE SCHOOL
Monterey, California
POWER ITERATIONS AND THE DOMINANT
EIGENVALUE PROBLEM
by
Jeffery J. Leader
ft
Technical Report For Period
March 1992 - June 1992
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NAVAL POSTGRADUATE SCHOOL
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Rear Admiral T.A. Mercer .H-arri.son S_.hul.,l,
Superi. nt. endent
Provost
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1. TITLE (Include Security Classification)
ower Iterations and the Dominant Eigenvalue Problem
PERSONAL AUTHOR(S)
2.
effery J. Leader
3a TYPE OF REPORT 3b TIME COVERED 14 DATE OF REPORT {Year, Month. Day) 5 PAGE COU.V
echnical Report from 3-92 to 6-92 12-15-92 11
SUPPLEMENTARY NOTATION
5
COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block numoer)
FIELD GROUP SUB-GROUP Power iteration, Power method
J ABSTRACT (Continue on reverse if necessary and identify by block number)
The orbits of an iterative numerical method for the dominant eigenvalue problem are
lalyzed from a discrete dynamical systems perspective. It is shown that the method
m extract more information thant the standard power method but at greater computational
)St.
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POWER ITERATIONS AND THE DOMINANT EIGENVALUE PROBLEM
JEFFERY J. LEADEF:
—
r^,i - 4- «--<r- + t-+- m-.* v- -„-_ 4 ; .—r- w2Va1 r-.-.»-4. ^-»^ - .-i,T- + —. •—-i .,—
Abstract. Tht
>i i^-x «—-^
dominant eigenvalue problem are analyzed from a ciscrete
dynamical systems perspective. It is shown that the method
can extract more information than the standard power method
but at greater computational cost.
Key words. Power iteration, Power method
.
NTn».a'UuiiL;i
The paxoe-r iteration is the matrix iterauor,
Vn+l = B«Vn/liV i! CI . 1
ri
Wii*=Tte- j '.r o O-L Vfe-'j i r>^-n^Le. o TTi \tr'L *w-Oi"~ •> i_> -r «. l «^ca. it.kjti rr.at,r i
ii
O-,.II-.1^--Ai lMi : I»Il J,'. r.=-> +(^VH..C-. tI^TU-k.--lJ.J. l-.!'*rC».rl-l -V.^C=.V—4VV—l-»- Tl-lll!>—•l»-TJi^l rL "_lL OV_' > "_'L is similar
to the pouter mj&th.od. for finding the dominant eigenvalue of a
real maJ rix,
A* - v s\j CI. 2D
;r.+l ' r.-*l r.
-here A is a re;.] matrix: with a dominant ei aenval ue v is an
, o
m
i "<-•! aj eswim«i-e oi an :i Qfnvei-tor soci ate J wi t h trie
domi nan' eigenvalue of A id /,• is an element or with
the property that
H Ii'; .vr>+l IIa.
Csep il p. 144] D. We will show that although the power
,
ciene; iy slower than the power method
j
can provide extra information about the dominant
eiaenvaj ubCs'J of a matrix in certain cases Ve take
cie. scre\-1 u
syst,trm dnd l ncjui
CattractorsD in various cases Cin the spirit of [93D.
Thie iteration CI. ID is considered in a different- context a:
a special case of the [R -»[R map
V = A*V + B*V /IIV CI 3D
n+l r> n nI .
in [101- based on work in [2] Calso reported in C31D. Further
details on the iteration C 1 3D may be found in [10,12,133. and
.
t he for thcomi ng [41. Al though C 1 3D is onl y a 1 i near
.
perturbation of the well -behaved iteration CI. ID, it exhibits
—
SITaiiyfe aHJ 3oX.*Ji i cLIisj ajjpci cutj. _/ Ciic^iwJ^x^ u^iiCLjjLu.^^. •"-•i
course, CI. ID can also be viewed as CI. S3 with a change of
normal i r.ati on , and much is known about the numerical method
given by CI 2D Csee also IS, p. 3623 D.
.
2. The Pover Method
The power method CI. 2D has the property that if A is a
nondefecti ve matrix with a dominant ei aer.value, say ?, , and vo
i
has a nonzero projection on an eigenvector associated with
this dominant, eigenvalue, then
—
>a dS n >oo
fj
r. 1
and v converges to an eigenvector associated with X and with
r> l
unit / norm. If u does not have a nonzero compfonent alongv
a o
an gic^'uVcCucr associ a^ e*~ wi un A- ano lni lnit.@ Precision
<
crithine* ic is used then we must consider the eigenvalue of
laraest modulus along which does have a nonzero component.
v>
In actual computations, however, a component along an
eigenvector associated with X would almost certainly be
introduced eventually and magnified in successive iterations
CI, p. 1453. The same results are found if A is defective
Cconsidering now principal vectors [7, p. 3] rather than just
eigenvectors} but the convergence is much slower.
When the dominant eigenvalue is real, the method converges
to a fixed point. However, when the dominant eigenvalue is a
complex conjugate pair, the method generally fails to
converge. Methods exist to recover information in such cases
[6, p. £57] but they tend to be somewhat involved.
5
—
Vr'- - -- / - <- — i<O i»-c; »- —- 1 »«-• + •— - r»-.-.t-,w-..- .-." »--I —•• ,...- i ,,-
n+i — A* vn- i vnli 00 , c>n+1 = A**"yn+1,-iiA**'ynII00
and in this formulation v need not be calculated until it is
r>
actually needed Cto estimate X. J Then the iteration for v
1 . "n+i
is the siine as CI. II) except for the particular / norm used in
r
the normalization. For this reason we sometimes refer to the
quantity /j in C 1 . 2J as the sif,ed / norm.
LONIC L*BITS
12
If t : s nonsinqular then the points V , V , V3 , . . . of the
T
G = CB*B D *
VT*G*V = 1
for all i ^I . For ,
T T
Vri+l*G*Vr>+l = CVn*bV|IVri l!I)*G*CB*Vn/II Vr.11}
= VT*C BT*G*B}*V /IIV 11°
n ri n
= VT*7*V /IIV II"
n n n
for every n>l and for any V which is nonzero. Clearly G is
o
positive definite symmetric, and so the points V V V
,.
, , . .
must all lie on the hyperell ipse defined by
T
V *G*V = 1 C 3 1
.
in K
.
If B is singular, a similar result holds. In order to
handle simultaneously both the case where B is simple and the
case where B is defective we state the result in terms of
.
pi i n>^i p< J. >c^^-^i ^>. Vic: iid » c i^iic i.v>a.x*^wj-»i^ ^uCoi cr,ii.
Theorem 1: Suppose Bis a real square mxm matrix with 0<q<rr,
null eicenvalues. If q> and V has a nonzero component alona
a principal vector associated with a nonzero eigenvalue, then
all orb ts of the power iteration CI. 13 are constrained to a
hyperel lips© in Cm—q3 —dimensions Cfor all but finitely many
n3 Otf r-rwise, the orbit reaches the origin in finitely many
.
terati ens
i
.
Proof: First note that
,
V = B*V /WW
o o II
1
11
y = b*V y V
2 || II
= B*CB*Vo •••li Vo 113/11 B*Vo/liVo li li
= &Z*V /!IB*V
o oI!
and. in qener
V = Bn*V /'IIBr' **V C3.£3
o OII
r>
for n>i Nov let
.
J = R *B**R
be the Jordan normal form of B for some nonsingular R
Substituting this into C3. £3 gives
V = CR*J*P"13 r'^V /IIBr'-1*V
n o oII
= R*Jr'*R-1*V /IIBr'~1*V
o oII
n
= at R* |J *Z C3. 33
where Z =R *V and
o o
n
a = l/IIB **V
n OII
is a scalar Cfor each n3 CI earl v, if Z has no nonzero
o
component along a principal vector of J C equi val entl y if V
,
has no nonzero component along a principal vector of B3 that
is associated with a non-null eigenvalue, the term Jr'*Z in
o
undtf iiH d &.r»d the iteration stops. Othwr wise, for n
sufficiently large Cit suffices that n>m) all Jordan blocks
,
in J associated with a null eigenvalue will have become blocks
of entirely zeros in J Csince these Jordan blocks are
nilpotent}. Now, the principal vectors belonging to a given
Jordan "lock do not interact with the remaining principal
vectors. in the sense that if is a principal vector
x>
j
associated with J CW, a Jordan block of J, then J '*u
v J
involves only a linear combination of principal vectors of J
that, an also associated with J C \~> Therefore for n large
.
enough that all nil potent Jordan blocks have become entirely
zei o submatnces, the vector
n
j *z
o
can b^ written in terms of a basis consisting of only the
remaining m-qD principal vectors. Thus the iteration lies in
C
a C m-q) -dimensional subspace of CP , and a suitable change of
variables c*r then be used to transform the iteration into one
of the or m
:
w = C*V x-HW
n-'i n nII
where V is a C m-qj -vector for every n and C is a. real
n
nonsingular Cm-q) xC m-qD matrix. Hence, in this subspace, the
iteration is constrained to the hyperellipse determined by the
matri x
_T -1
CC*C
3
Cas was shown previously for the nonsingular case) for all but
finitely many n. a
We emphasize that this is not an asymptotic result; after a