nps-ma-93-009 naval postgraduate school · 2011. 5. 14. · lis2 now r-1 i cu/p] and a second...

AD-A262 297

NPS-MA-93-009

NAVAL POSTGRADUATE SCHOOLMonterey, California

0., STATv$

S e 1992 1993

a A PARALLEL DIVIDE AND CONQUER

Ln ALGORITHM FOR THE GENERALIZED REAL

0 SYMMIETRIC DEFINITE TRIDIAGONALcv~~ EIGENPROBLEM

by- Carlos F. Borges

* William B. Gragq

Technical Report For Period* September 1992 - December 1992

Approved for public release; distribution unlimited

Prepared for: Naval Postgraduate School

93 3 6 091 Monterey, CA 93943

NAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943

Rear Admiral R. W. West, Jr. Harrison ShullSuperintendent Provost

This report was prepared in conjunction with research conductedfor the Naval Postgraduate School and funded by the NavalPostgraduate School.

Reproduction of all or part of this report is authorized.

This report w repared by:

Assistant Pr Aessor

WILLIAM B. GRAGVProfessor

Reviewed by: Released by:

DepCHARD FRt PAoa JcMARTOchairman Dean of! ResearchDepartment of Mathematics

UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE

Form Approved

REPORT DOCUMENTATION PAGE OffNO OAPP -0oV

la REPORT SECURITY CLASSIFICATION lb RESTRICTIVE MARKINGSUnclassified

2a. SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION/AVAILABILITY OF REPORT_Approved for public release;

2b. DECLASSIFICATION/DOWNGRADING SCHEDULE Distribution unlimited

4 PERFORMING ORGANIZATION REPORT NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBER(S)

NPS-MA-93-009 NPS-MA-93-009

6a. NAME OF PERFORMING ORGANIZATION 6b OFFICE SYMBOL 7a- NAME OF MONITORING ORGANIZATIONI (if applicable)Naval Postgraduate School MA Naval Postgraduate School

6c. ADDRESS (City, State, and ZIP Code) 7b ADDRESS(City. State, and ZIPCode)

Monterey, CA 93943 Monterey, CA 93943

Ba. NAME OF FUNDING /SPONSORING Wb OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION (If applicable)

Naval Postgraduate School MA O&MN Direct Funding

Sc. ADDRESS (City, State, and ZIP Code) 10 SOURCE OF FUNDING NUMBERS

PROGRAM PROJECT "ASK WORK UNITMonterey, CA 93943 ELEMENT NO NO NO ACCESSION NO

11 TITLE (include Security Classification) A Parallel Divide Conquer Algorithm for the Generalized Real

Symmetric Definite Tridiagonal Eigenproblem

12 PEISONAL AUTHOR(S)CarlosF. Borges, William B. Gragg

13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year, Month, ODay) 15 PAGE COUNTTechnical Report FROM 9-92 TO 12-92 921216 19

16 SUPPLEMENTARY NOTATION

17 COSATI CODES 18 SUBJECT TERMS (Continue on reverse of necessary and d•entify by block number)

FIELD GROUP SUB-GROUP Parallel divide and conquer algorithm, Tridiagonaleigenproblem, eigenvectors

19 ABSTRACT (Continue on reverse if necessary and identify by block number)

We develop a parallel divide and conquer algorithm, by extension, for the generalizedreal symmetric definite tridiagonal eigenproblem. The algorithm employs techniquesfirst proposed by Cu and Eisenstat to prevent loss of orthogonality in the computedeigenvectors for the modification algorithm. We examine numerical stability and adaptthe insightful error analysis of Gu and Eisenstat to the arrow case. The algorithmincorporates an elegant zero finder with global monotone cubic convergence that hasperformed well in numerical experiments. A complete set of tested matlab routinesimplementing the algorithm is available on request from the authors.

20 DISTRIBUTION/AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION

EJUNCLASSIFtEDIUNLIMITED 0 SAME AS RPT ODTIC USERS Unclassified22a NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE (Include Area Code) 22c OFFICE SYMBOL

Carlos Borges (408)656-21 A/Bc

0D Form 1473. JUN 86 Previous editions are bsole te SECURITY CLASSIFICATION Of THIS PAGE

S/N 0102-LF-014-6603

A Parallel Divide and ConquerAlgorithm for the Generalized Real

Symmetric Definite TridiagonalEigenproblem

Carlos F. Borges*and William B. Graggt

Abstract.We develop a parallel divide and conquer algorithm, by extension, for the generalized

real symmetric definite tridiagonal eigenproblem. The algorithm employs techniquesfirst proposed by Gu and Eisenstat to prevent loss of orthogonality in the computedeigenvectors for the modification algorithm. We examine numerical stability and adaptthe insightful error analysis of Gu and Eisenstat to the arrow case. The algorithmincorporates an elegant zero finder with global monotone cubic convergence that hasperformed well in numerical experiments. A complete set of tested matlab routinesimplementing the algorithm is available on request from the authors.

1 Introduct'on

We consider the problem of finding a matrix U E K'" such that

U T (T - SA) U -= A - IA,

is diagonal, or equivalently

UTSU = I and UTTU = A, (1)

where

*Authors address: Code Ma/Bc, Naval Postgraduate Sciool, Monterey, CA 93943. Email:borgesOwaylon.math.nps.uavyjntii

a Authors address: Code Ma/Gr, Naval Postgradate School, Monterey, CA 93943. Email:gragg~guinness.inath.nps.navy.nul

| • . . 1

-I - . l IIII . , I I. * I

2 Carlos F. Borges and William B. Gragg

01 02 12 71 62 72

T=, 2= S2 If 2

•.-1'i -. 7 1 6,.

and S is assumed to be positive definite. This generalized eigenvalue problem hastwo special cases that are of interest in themselves. They are:

1. S = 1, the ordinary tridiagonal eigenproblem.

2. S = I and oa (j, the bidiagonal singular value problem (Bsvp), by perfectshuffle of the Jordan matrix

10 BT

with B upper bidiagonal [16].

There are two phases to the divide and conquer algorithm, the divide (or split)phase, and the conquer (or consolidate) phase. We shall address these in order.

2 The algorithm

2.1 The divide phase

Denote by ej the ith axis vector where the dimension will be clear from the context.Let s, I < s _< n, be an integer, the split index, and consider the following blockforms:

r T, e._. 1T = [0,,- 0.eT I

ejjO, T:

S = 7,- leTj 6. 7.eT .e17,0 S2

Note that e = n is possible; then T2 , S2 , and el are empty [9, 10). Suppose wesolve the subproblems

UT(Tk - SkA) Uk Ak - IA (k-= 1,2). (2)

Generalized Divide and Conquer 3

The form of the subproblems is preserved. In particular, the matrices S1 arepositive definite and, if T has a zero diagonal, so do the matrices Tk. Let

U/2

Then

OT (T - SA) 0 =S UT(Ti - Si A) U UlTe.-.(1h.. -7 .'-A) 1

I (/,- - 7,-iA)eT_.IU1 a. - 6,A (13,- ',A)e*U 2 •

LU~e I(1A - t,,\) U14(T2 - S2A)U2 J

2.2 The conquer phase

The conquer phase consists of solving the subproblems (2) from the divide phase,consolidating the solutions, and finally, solving the consolidated problem. Let

u --- IU~e,-1, U2 =Ul~el,

where the Uk are solutions to (2). Then

OT (T -SA) - (/,I-7-Au ,6,A (/,- 7A)uI .

u2(fl. - Y, ) A2 - AThe right side is the sum of a diagonal and a Swiss cross:

(IT(T- SA)C= z + x +

This can be permuted to an arrow matrix by a permutation similarity transfor-mation with P. = [ej,e2, .- ,e,-t,e,+t ... ,e,,e,]. Thus

OeOssio For

A(A) := OT (T - SA) 0Ps $ RA&i

Al 1u -I1 'TAB ciA2 u 2/0, - I u27# A trMI onoed 3

2 [ [ U IT -Y. Ur btD Bu I Cu ,

utT B at UT C 7with .ributo _ j,,

f lAbliity Cedes

Dst DOlal

'•.., -," Auam i ajj


U 2U

12B ~ ~ ~ 1 Old C V -1YI

Since S and I UT Cu ] are congruent the latter inherits positive definiteness

from the former. Its Cholesky decomposition is

[I CU] [ ][ C]=~ CC =RTR,

with p2 = - uTC 2 u > 0 the Schur complement in S of

LIS2Now

R-1 I Cu/P]

and a second congruence transformation with R-1 gives

A(A) R-r.A(A)R-'R -_TT BD Bui R-' - ,X

I R XT[ B Q]RI A

" D wT A A -Al

with

(B - DC) ip

, -T (2B - DC) Cup 2

We have reduced the conquer step to the problem of solving an ordinary eigen-problem for a symmetric arrow matrix. If V is an orthogonal matrix with

AV = VA

and A diagonal, then (1) holds with

U = UPR-IV= [VU, -Uiury._.,/p~lp V

I/ -V


It is useful that vk = Ukuk can be computed in O(n) time by solving Svy1 =e.-.1 and S2 v 2 = el using the Cholesky factorization S, = LILT and the reverseCholesky factorization S2 = L L2 . In the case that only the eigenvalues arewanted it is only necessary to compute the first and last rows of the U-matriceswhich constitutes a further savings.

In summary, the conquer phase proceeds by consolidating the subproblems andbuilding a full eigenproblem for an arrow matrix.

3 Solving the eigenproblem for the arrow

In this section we consider the solution of the eigenproblem for a real symmetricarrow matrix

A= (bT ")

where A E Rn Xfl is symmetric, D = diag(a), a = [a1 , I a. >_ cr2 > ... >an-1, and b = [/, ., 1 1 T > 0. When A arises from the BSVD then a is oddand b is even, that is a + Ja = 0 and b = Jb, with J the counter-identity, theidentity matrix with its columns reversed, and - = 0.

If any f# = 0 then it is possible to set Aj = aj and deflate the matrix sinceej is clearly an eigenvector [28). We shall call this O-deflation and note that if,j < tolhjblJ where told is a small tolerance then a numerical deflation occurs.We derive a precise value for tolij in section 4.4.

A second type of deflation occurs if applying a 2 x 2 rotation similarity trans-formation in the (j, j+ 1)-plane that takes Oj to zero introduces a sufficiently smallelement in the (j, j+ 1) position of the matrix. This will be called a combo-deflation(see [151). At each consolidation step we perform a sweep to check for #-deflationsfollowed by a sweep to check for coynbo-deflations. The combo-deflation can bearranged so that the ordering of the aj is preserved whenever one occurs. After

deflation the new I:= j + /+i> Žfl+i and hence no further #-deflationcan occur. The combo-deflations can be disposed of with a single pass by backingup a single element whenever one occurs. Note that deflation is backward stablesince it results in small backward errors in A. Deflation for the BSVD is moredelicate involving a simultaneous sweep from both ends of the matrix. Care mustbe exercised at the center of the matrix.

After deflation the resulting matrix can be taken to have all Oj > 0 and theelements of the arrow shaft distinct and ordered, that is ac > a 2 > ... > an-1.An arrow matrix of this form will be called ordered and reduced. Henceforth, weshall assume A is of this form.

The block Gauss factorization of A - Al is


1 0 D-Al bA-Al bT(D-AI)l ) jI OT -f(A)

where f, the spectral function of A, is given by

f(A) = A- -Y + Ej=1

This is a rational Pick function with a pole at infinity [1]. The most general formof a rational Pick function is

A(•) = 6A-'v+Z - 6>o. (3)

In relation to the various divide and conquer schemes, the case 6 > 0 correspondswith eztension, 6 = 0 with modification, and 6 = 7 = 0 with restriction [7].

Inspection of the graph of the spectral function reveals that the elements ofthe shaft interlace the eigenvalues

A] > (VI > A2 >...- > Qn1 > An- (4)

Moreover, in the present case, the derivative of the spectral function is boundedbelow by one so that its zeros are, in a certain sense, well determined.

3.1 The zero finder

The fundamental problem in finding the eigenvalues of an arrow is that of providinga stable and efficient method for finding the zeros of the spectral function. Wenow examine this problem in some detail.

The zero finding algorithm we present is globally convergent in the sense thatthe iteration will converge to the unique zero of f in (aLa, atk-.) from any startingvalue in the closed interval [k., ak-.], where we put a•0 = +o and a, = -oo.

The zero finder converges monotonically at a cubic rate and applies to a generalPick function as given in formula (3).

3.2 Interior intervals

The iterative procedure for finding the unique zero of f in one of the interiorintervals ((k,ak-1) proceeds as follows. Let zo, "k < zo < ak_1, be an initialapproximation to Ak. If rj is known choose

_(_) __+ +

so that


€• (x) -=f(')(xi), i = O,1,2. (5)

Thus a, wo, and w, must satisfy

0 (ak-I )-2 (ak - Xc)'2 1o f=(Xz)0 (tk- r-j )-3 (ak _T)- w3 W f'(z" j)

(I~ I [~Y (ai 1 =~) i J f(I )j

and we find

a' = 3x - (-y+ok-i + Ok) + X1i -? :k-l Cki

ujk-I,k ai- Xi j - Xj Vi- -j

WO= 21 +(fLIk. XA " ai - ak'

i(Ak-lXti ± ? a Vi

Since w0 > 0 and w1 > 0 it follows that oj is a Pick function. Thus Oj has aunique zero xj+I E (a L, akL-). Also

1,o > /•j_ ý > 0 > j32>/ > 0.

The error function

(X)-O(x)=x-( h+')+ •) ,- a;-W- +ik-,k-I

k-

has n zeros, counting niultiplicities. There are n-3 zeros exterior to (ak, or_.-1) andthree more at xj. Thus the error function crosses zero exactly once in the interval

(akak_1). Hence xj+i lies between xj and Ak, and the iteration is monotonicallyconvergent from any starting guess 2-o0 E [Ok, ark-1) as claimed. The cubic rate of

convergence follows from (5).Successive iterates can be found by solving quadratic equations. Rather than

solve O,(z) = 0 for zj+l it is better to solve

€j(a.' - A) = 0

for the increment A = zj - xj+,. Some rearrangement using (5) reduces this to

A2 + OA• -- f -- 0, (6)

with


-- or (7)

S= f'(:j) -(l x-- x+ k-xj j (8)

When shifts of the origin to the nearest pole [15] are used then one of Qk-I or akis zero. The computation of = 3(xj) should account for the fact that it has onlysimple poles at ak-1 and at.

If we start at the midpoint of the interval, z 0 = (ak -I + ak)/2, then we alwayshave 0 = A(zj) > f'(xi) > 1. This can be seen by noting that P(x0 ) = f'(Zo)and that when x0 > ,k then for all of the succeeding iterates f(xj) > 0, bymonotonicity, and ±, + ý ' Ls negative. If xo < Ak a similar argumentapplies. It follows that the increment can always be computed stably as

= 2f (9)

3.3 Exterior inte:vals

The treatment of the two exterior intervals is geoinetrically the same as above.Again, the approximating function has poles at the endpoints and the residues atthese poles, and the constant term, are chosen to satisfy (5). We present the casefor the interval (a 1, oc), the case for the other exterior interval being similar. Now

CtI - X

with

It-I+ E 13' a+ - -Vi > 1

,='. aj - t) X'• - (,V -

= n - U I> o-+= X - Uti )

The inequalities are strict, unless n= 2. Again we find (6) where now

X; - Ct1

f (3 .j )Pl=f(Jxi) + Xj - ,,--'--6

These are limiting cases of (7) and (8) ( introduce another pole ao > a, and let00 - +00). If xu > \, then f(A) > 0 so (3 > f' > 1 and A is again computedstably using (9). WVe obtain global monotone cubic convergence as before.


Contrary to the algorithms of 111, 12, 15) our algorithm is well-defined when

starting at the endpoints of the intervals. The algorithm of [23] can start at the

endpoints but has only quadratic convergence.To guarantee that z0 > AI we take xO to be the iterate in (a,, +oo) from +oo.

As xO - +oo the approximate Pick function tends to

(X) = X -+ jb (10)

Our actual starting guess is the zero of (10) in (cr , +oo):

+/ + + >ai,

{-+ / +

When shifts are used we have o I = 0.

3.4 Orthogonality of the eigenvectors

It is essential that the conmputed eigenvectors of the arrow matrix be numerically

orthogonal. As a point of entry into the furither analysis of the algorithm we nowexamine the orthogonality of the eigenvectors followilg [15].

Consider the divided difference

f(A ~ f (A) - f- 00________ (ii)fA,1-+1: i (11)

j=( 1 -A)(nj )

= + t)T(D - A)-'(D -t)-lb.

Note that p = A gives f'(A) = I + Ii(D - ,I)- hlj•. If f(A) = 0 then

v(A) [ ~ - (Al - D)-'b]v(,A) A 1 0 1

is an eigenvector of the arrow zr Atrix A = 1) associated with tTe eigen-

value A, and

t,(A(A)

is the normalized eigenvector whose last cloiiient is positive. The ordering of A

implies that its matrix of eigenvectors cati be taken positive below and on the

diagonal, and negative above.


Let f(Ao) = f(pi0 ) = 0 with A• ý jiu, Thus Au and po are distinct eigenvaluesof A. The eigenvectors t(Au) and u(I•o) are orthonorinal:

u(Ao)7"uC(,tp) = f(Ao, po) = 0.

Let A and p be approximate eigenvalues in time sense that

-6 A - A- , 6

-A- , +6'

(12)

(It -J, 1 +6Here 6 > 0 is hopefully, bmt iit n cvcýsarify, cluse to the miachine uiiit c. Note that(12) is equivalent. with

A - A (I - it,

These conditions imply that Ith appruxincate eigcnvectors u(A) and u(p) arenearly orthogonal. For we have,

Vf77(A)f'(P)uA)7"u(/,) = f(A, p,) f(Au, p(u)'- ' z j-. (n"- A)(c•- .o

k -c,,o - p)- -

3=[T

- (,,j -A)(,-m -- )

Since

26 62+ 6, + 26

then

\/f'(7)/'(A,),,C ,,,) = 261h" () - AI)-'O(D- it)-1b

with 10 _< 1. Thus

v/'TT~f'JiU)1u A)Tiip= "2II(D - Al)- ' I)11t(D - pJ)-'btl2.and so

I u(A)Tu(Io) < 26.

Condition (12) is stringent. If we let ih. - 0 then it is easy to show that Acan have an eigenvalue A0 = A 0(,Y4ý) = oL + O(/3j); (12) then requires that theapproximate eigenvalue A salisfies a bound

(eneralizetl Divide and Conquer 11

JA - A0, _< 0(6;12),

which is difficult if Ok/l flil is o0ly somewi:a largv'r than machine precision, sayC3/4. Two techniques are used to att-?,'pt to satisfy (12) - shifts of the origin [15],and simulated extended precision (sEP) arithimetic [26, 141. Condition (12) meansthat

JA- A0j < 6 min{Ao - "•., i.-I - Aol.

When shifts are used it means that A is niarly fl(Ao).

4 Numerical stability of the algorithm

We now give a partial analysis of the stabilhiy of this approach to the eigenproblemfor the symmetric arrow matrix. Observe t[h:a

f(A) - -p(A) , - n (A A j)

The following inv•rse eigenvaltmu problcmn [6] is important: given {a. } and {Aj)satisfying (4), find {13j) and( I, so that A(A) = {Aj }. This problem is simply solvedby computing the residues of the partial fraclion decomposition of f. In particular

".- "(AJ)

7= A - _____._

-T-=

j=1 )=I

For fixed {fj }, the elements of the arrow buad, {Jh ammid -, are explicitly knownfunctions of the eigenvalues.

Now let {A,} be a set. of appriomiunnm e•ignvalues of A satisfying (4). Then

i3•,l-= ,(" L - A,)oi - 0( >0), (13)

= - ,(14)j=l *j=i

define a modified matrix A with A(A) = {A3}. To obtain a backward error analysis

for the complete eigenvalue problem we bumid the differences ýk - flk and ; - 7.

12 Carlos F. Borges and William It. (ragg

r

4.1 Error analysis for the Don garra-Sorensen condition

We give an error analysis usIitig th, l)oizgarra-Sorensen condition

As - ,

, A = b,.. - 16j, LI S 6, (15)

where 6 = 0(() is of the order of the ciadiinc mnit, siuplifying that in [6].Rearrangement of (15) gives

0t - .= (A - W,)(l + 6 jk).

It follows that

=.di. * 1+flu .)=;i + j.k )

and

with the b6,k and 6J7 at most only slightly larger than the 6i,k. Thus

14i, - 3k. < 16 11,

where 6" = 0(t) i.s oinly slightly ltor, r than c.Now (14) bevoitv-,

= ~+ >3(A1j=I

with Ok(ji OWe of hle poles of f. Thlis

2=1

To minimize this bound we chot.is, rq~j) to be a pole off closest to Aj. Clearly,ak(i) =(iI and Ca .(,) = 0 SOi

- ((A1, i + [Aj t~t + (~~ 1 -A)

For I < j < na a closest pole to Aj is either flj or j 1 The distance

IA1 - ni.,J = win {Aj - fl1,ctj_ - A})

(;eIirMiazed Divide and Conquer 13

is maximized when Ai is the tidtpuit. of the it erval (o, o J1), and tile value ofthe maximum is (a, + aj-t)/ 2 . Thus

< b(At(k +2 ij- - fi ) + (o,- -Ai

- 6 (A A, (J-2

_ (Al -4 Ar) 26Kj--",,-

In summary, the Dongarra-Sorense -, couidit ti imiil.es small relative errors ineach fk and a small absolute error iii ý. I Vr ite Hs\'I) this implies small element-wise relative errors since the condiion ou = - = I is enforced by Aj + A,+ 1 -. = 0(only half of the eigenvalue-s are act utally cotiputed, t he re't follow from this con-dition).

4.2 Rounding error analysis of the computation of f(A)

The choice of a termination criterion del0-ciu H,1 ' careful rotunding error anal-ysis of the particular mratnner it which wem. t'ttlut,' f(A). Let {Jai}, {fl), and 'Ybe floating point numbers. We r-eprc.-cifl A as the ordered pair of floating pointnumbers (a, p) where the shift a is a pole d-ost to A, aild A := a + p. For theexterior intervals we have a = (I or a = .,, -. For t li, interior intervals a' can bedetermined by evaluating f at flt. m id puin•i ad a lntil~ i og the sign. We computef(A) as

n--

f., 00, + (, -'

with the standard operatioit pre'ethviwe ride,., whur,

0I = cj - a amd c=r--".

We use Wilkinson's notation: fi(.r * yj) (j' * y)(I + b) with 161 < (/(1 + c)and f = 2'' the machine unit. More getierally, ( (letiotcs tutmbers not essentiallylarger than 2-' [27] and the rounding errors 6 satisfy 161 <

We define

flCaj - A) := fI(,' - ) = fl((j - a) -

If a' = ka then

fl(ok - A) = Ok - ,

with no rounding error. For j k$

14 Carlos F. Borges and William B ,

fl(j- A) = (,- A) I + +- A

and since ok is a pole chmtst to A Ihell [ < 2. Thus all terms aj - A are

computed with stiall relhivte err,,-

f((- A) = (,,- A)(I + <) VAl< . (16)

When computing f(A) = f x(vc) a,.tld IhOw temr A - = (A c)- ( - a) last.

A routine error analysis using (1(i) .am

lA - -,I S Ijj'A)I + • 1•-h

to eliminate the term JA - -1 1 fr,m hli. error huumd gives

ift(f(A)) - f(A)j I •' (A)I + I a - AfI + (?I + -5) E . Al

which implies

lft(f(A))l< :(S + +S )lf(.) i ± - Al + (,, + E t- ) (17)

4.3 Termihiation

Our goal is to chouos, a itrtiuh•;s ii le-Ii ,rist, (.i Ihat we stop when A is as closeto the true eigenvalt AL. a s' ... I,"t' p AL- h1i (01) with f(Ak) = 0. Nowak < Ak < (t4-1. A sk II ',i A (. 'liht lt Irh termis Ok - A and oj -- Ak

have the saite sign muid

JA - Ajl < lf(A)l (18)I + .. l I,, • ,, - k

To obtain an upper bound for IA - Ak, I we it-ed an upper bound for If(A)I and alower bound for the denominatour. I;ur the hlater we have

"I- l(,J - All,,i - ,.I + maxi ,"j - ,X, (

Let us determine how small lf(A)l is whit A is the roundcd representation ofAk. This is

hmri:dit•,d )ivide and Conquer 15

a +P+f1j T+.)= 1+40(l +6)= A4- + lui = A,• + (A•. - a)b

and we have

1( - = nz - Ad.

Thus

E"' (11 - \")(,t•j :- )!IA[= lA-A,1 (I+

= I~o~ Ak ) ~A)(',, - Ak))

From (17),

[lf ( I( A)) < f la r- A k][ + 1 A " + (it + 3j!•'-

Since Ak - = tA - a)/( + a) 1hi

ifI '(f (J )im ,•1-(T + (i ,t + t;) - ". ,) = i, - Al)

We terminate and set )• := A 4. eu

jfI(f(A))f < 2 21A- a!+(,,+(6) Al

Inequality (17) also hold.s if f(A) arid flj.(A})) arv, iw,'rchaiiged. Thus

I/(Aoi S, ,Ik + o:o, ÷ -T)• (20)

- = 1, • - )From (18) and (19)

-" - I a

"'- A1 + (:3, + 17) -_Ak - Ak.I .< , , ax IA4. - ,.,I

$ II;1N, i,\L - n ij I + = ,-

16 Carlos F. Borge.- and WVilbt.m H. (; agA

Since ka-A.j :l_ ja-A.I+ A;1- -•[ aid t -As.f _ naxa lk -AkI the computedeigenvalues sati.Jfy

JAk - Ak I_ (3n + I7)t max Ij - ALIj

(;(It + 6)(11411.'q ' (21)

4.4 Error analysis for the Gu-Eisenstat condition

From - = "=>(Aj - A,) ;II, (21) w' find

I - < _ (,,(, + 6), 11.112,

We have noted thai tle I), ,iyi i-r;r-.Sortvn,'i ,il(ition (15) is stringent. It is

natural to ask for siztall I,.(,tuil, , ij lit, ili.. If we replace 6J,k by 6bj,k/fk inthe analysis in sect ion 4.1 we lind ih.,t

I t~13L h IJ•L. ; . -= = 3k + k '1i,.,

and

- iI +0(

are implie(d by tin G(;.-L't.,, ,ta 1, 4 ,,hin,,,,

A., -- <j

We must bound 6.From (20)

-Ak + ( -A - Ak AI) < ( 6 -" +Zj= j,=,I" - I

with rn = 3(n + 6). Usi,,g

and the Giu-Eisenstat invqiualiI.\

we, - AetI (n - A)(tj - Ak)we get

(1v~rai17d Divide and Conquer 17

j~~~k ~ -~ ((ii :5I 4 7

1(A - _< ,,,,A,. A- r.+, - .A

where c has been increased to t/(I - w,, }By Cauchy's inequality.

/ , I,- , J)

_ 4 - ., +-

for every j. The arithmlhenc-g,,,ii,.i ii•c f .i , i ,,.•wjwuli :t•it the lrianigle inequalityyield

++,,i" ,,: _ Akl. + .•IAA. Ak-I ))JAL. - 4.1 <5 ,, 1A,. - 0•1 + Ad--7 1A- A

Thus

IA- -Ad :S I, 7 + - ).kI li4. )

1 2 - A] - <•- + JAL- c-lI ]

3 1,l,-, I - ib.<-- , A4.1

.I

If mclIbl, < j4 for all j, then,

and consequently

Thus Lold is in. If i3L. < :ien + 6), I[/i %N, rcp1hl', 1,14. Iby zro and accept ak asan eigenvalue with normalized eigeIiVe.tr Vt..

The computed eigenveciors of m, ar, ;&, i i,, h, tho., (if the nearby matrix A.Because of (13) anl (163) the(y are ruii,-ti.id i,, high relat ive precision elementwiseand hence are numerically orthogowdl 1211]

18 Carlos F. Borges and Williamn I. (;ratgg

5 Ackiiowledgements

We wish to thank Ming (Gu and Suam Elsenstat for providing a preprint of theirmanuscript [20]. It is our ulmhrisrt;rinlhg that they have independently extendedtheir results to include the arrow case in [21]. Both authors were supported bydirect grant, from the Naval lke•t dat. School. The second author also acknowl-edges support fromt thl• hiterdic•'ijiiiary I'oju'ct Ceitter for Supercomputing at theETIH, Ziirich.

References

[1] N. I. AKHIEZmt, The t . iw..Io tit pridde pi m.I .xotue related questions in anal-ysis, flafner Pulikhiiig ( -mlp..itir -m, I.

[2] G. S. AMslAR, L. I4"icim'i. ox 1 C. Soit .NsEN. A• implementation of a divideand co quci- ahplorith li•i fm I/, iti q q op riapolt. ,A. A(:M Trans. Math. Software,

[3] P. ARIIENz. Conluiiain t :v, ,itrd., , ,f ihrioi,d sytis -tric Toeplizt matrices, SIAMJ. Sci. Statist. C(ol put.. 12 1 I lI). ip. 7-13--7T51.

[4] - , Dividt an•u imirtm r ,i/l,,ihO.h, lii, th, batitsutiestric eigenmilue problem, inParallel Co(mptiig *"9. lJ.1I. C Xu... ( . It. Juloirt, and 11. Liddell, eds., ElsevierScience Publisher,. II. V,..Ai.\tvtri.ai, I miJ, pp. pp 1-15. .

[5] P. ARBENZ AND I; 11. GOI IK. 01H p, (Indl idcemtipuition of hermitian matricesrtiodified by low rank toii rt,.-i,, iti• . ppihicatinr, SIAM J. Matrix Anal. Appl.,9 (1988), pp. 40-58.

[6] J . L. BARLOW'. Error iap.I/ ). i,] ii.w thods for thei syuauretric cigenvolue prob-lern, SIAM .1. Natrix :.\A ,d \p14I. 'it) :;pj•'. -

[7] C. BA•,TIrL AN 1) W). \.I ()\. •himrlni, ,04( •nd the Weisltein.Aronszajnthlfor• for modifti mtrii i, ,, 1.udw pjlohh gu linear Algebra Appl., 108 (1988),pp. 37-61.

[8] C. BEATTIF A,, (" J. It Iil'iutmfitl h lui,,dim (of a qcneralizcd symmetr•c matrixeigenvaluc problei, \\Workig Ingllvir. ID'p.irltinint ld Mathluematics, Virginia Polytech-nic Institute and State U vir[it.%. Ilatcksbilig, Virginia, 1992.

[9] A. A. DEEX AND NI. I'. F.AW LSw I/.llthly 1,urllcl rec ursive/itertavie Toeplitzeigensixice deconrpostiiou. I F F. F Trans. Acoustics, Speech, Signal Proc., 37 (1989),pp. 1765-11768.

[10] A. A. BEEX, D. M. Wi\ ES.Is AN .•N•. I'. FAHU;CES, Tht C,.RISE algorithm and thegeneralizrd rigv'nmoluf irobl, it. Signal Proccssinig. Suibmiitted.

[11] J. IL. B3UNCI AND ( ' NI.I sI., I plottltq I/u xiuudlar value decomposition, Numer.Math., 31 (1979). pi,. I I I - •I

lit ria-rdd Dit %ide anid Conqluer 19

(12] J. R1. BUNCH, C. NIELSEN. AND I) RutI~Nhnk ouri modification of thesymmetric eigenipt-oblum, N mici, MI;,t It- 31 I !S7-, ppo. 31 -48,

h [13] J,. J. MI. CUPPEN, A dit'sd lild ("Isqm 1 lit, 1-d f,,,t/ .4 qiwiiictric ti-idiagorial eigen-

problent, Nuiner. Math., 36 (19,] 1), 1-1. 177,- I ~5

[14] T. 3. DEKKER, .4 Jloating-ptitinl It-Iaisique fwn rslo nlId,! Ilii javuiljbio pr-rcisiors Nut-met. Math., 18 (1971), pp. 224422

[15] J. J. DONc.ARRA AND 1). C. SuU~l--\,i.*%. .1 fulolq jontilb a~flotlainifor the symrmetriceigenvalur probi-vit. SI AM -) .sti. Soil Ill CMILiiipnt N 19S 1'. 139-154.

[16] G. GOLUB AND W. ll.*A lA.N ('ucillilulqimi ~Ih uvpiI.u ,ual, ti.4 find fiscudo-inverse ofa mtatrix, J. Soc. Iiiduisi. Appi. NI;,ih,. .,, JI It Nwl.. 2 ( 1965).

(171 G. W1 G;OLUB, So:;ar ,nndifJdi naislhI, qi ii oi-iiat iu .t-d,i Ins~. SI AM Rev ., 15 (1973),pp. 318-334.

[18] W. B. (4RAGG AND I.-RK1,I -i~i A. .1 id uInn) omlu caii uic HIM for unitary andorthogonal cgrilprobl~ii s NIn llnej. Nkh u7,7 (I H'ij.) pp. 9- 8

[19] W. B. GRAGG, J. It. 'flwaR'I ON. \Ni 1). 0. :Ri.w I'atmliel dzivide and conqueralgorithms -for tJc sylnnii Irw Iidi.i-,it I iw tipi ;qa.it, mill hidiauyonaa singular valueproblem, ill Nodcliig anti Sk jnjifalWi. \,)I 23. parti- 1. W, C . Vogt atid M. 11. Mlickie,eds., Univ. Pittsburgh Sdliuul ul lKVi iigmt,, pI1.p.*lti -19

[20] N1. GU AND S. C( LIEitN S A] - .1 m111 ud (fiit i t/ uilqp'il/nnt for the rank-onemodification of I/au symnit fill *it pi-14, Iin. "ýIA A- NII Iatii Awna. A pp1. To appear.

[21] Mi. GU AND S. C'. LISEN: 1 4-1t, .1 rf iv-ml nqu r- uutp'ruthin for Ili(- symMetrictridiagotaul etc zgcpvntd ,u \\nrkig u. Ik I 'Al lilui t 1.1 11 Cumpaulp cr Science, YaleUniversity (1992).

[22] E. R. JESSUP ANt) D. C- So itvKNsiKN\. A1 pa dlbl ulqiizI/nut f coinpuiting the singularvalue decomnpositiont of it nunthux. SIOA NI I. NI at ix A nal. A ppI. To appear.

[23] R.-C. Ll, Sotlving rn/tcu cquaiunto h1u id,j iwn! (flit i Ilyu. \Vorkiiig paper, Depart-

ment of M~athemuaatics,I'ivri uI(.linni (1'2)

124] K. L6W'NER, (,b ,t rnioni,,tuu./iIIu j 1A~im1,is~, it. Nhili Z..I TS 931) pp. 177-216.

[25] D. P. O'LEARY AN!) G. %Vi StlAA .I uptiliiq Hitu uty toptrt ic arid eigenvectorsof arrowhead miati ct ,;, J. ( unm,. PI'.\ H..lII II il, jpi. 1975(5

[26] D. C. SORENSEN AND) P. T. T.ANG. ()i I/i orl. h utunpuiitij (if u ugractu'ors comuputedby divide arid con qruts 1rhI/niqna .%, 5IA \1N .1. NaIMIm:. ,:u. 2X (1991), pp. 1752-1775.

(27] 3. 11. WILKINSON, lloiundiiw; ct rors tin ftup limuit litoir. as, ['relltice- Hall, 1963.

( 28] J. 11. WILKINSON, Tia. nlgcbrruic tp tyuualuur pridA lin. Oxford University Press, 1965.

DISTRIBUTION LIST

Director (2)Defense Tech Information CenterCameron StationAlexandria, VA 22314

Research Office (1)Code 81Naval Postgraduate SchoolMonterey, CA 93943

Library (2)Code 52Naval Postgraduate SchoolMonterey, CA 93943

Professor Richard Franke (1)Department of MathematicsNaval Postgraduate SchoolMonterey, CA 93943

Dr. Neil L. Gerr (1)Mathematical Sciences DivisionOffice of Naval Research800 North Quincy StreetArlington, VA 22217-5000

Dr. Richard Lau (1)Mathematical Sciences DivisionOffice of Naval Research800 North Quincy StreetArlington, VA 22217-5000

Harper Whitehouse (Code 743) (1)NCCOSC RDT&E Division271 Catalina Blvd.San Diego, CA 92152-5000

Keith Bromley (Code 7601) (1)NCCOSC RDT&E Division271 Catalina Blvd.San Diego, CA 92152-5000

John Rockway (Code 804) (1)NCCOSC RDT&E Division271 Catalina Blvd.San Diego, CA 92152-5000

Professor Carlos Borges (15)Department of MathematicsNaval Postgraduate SchoolMonterey, CA 93943

nps-ma-93-009 naval postgraduate school · 2011. 5. 14. · lis2 now r-1 i cu/p] and a second...

Documents