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REPORT DOCUMENTATION PAGE OMB No.048_________________O____No. 0704-0188

Public reporting burden for this collection of Information is estimated to average 1 hour per response, including the time for reviewing Instructions, searching existing datasources, gathenng and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or anyother aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operationsand Reports, 1215 Jefferson Davis Highway, Suite 1204, Adington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC. 20503.1 AGENCY USE ONLY (Leave Blank) 2 REPORT DATE 3. REPORT TYPE AND DAFES COVERED

August 2005 Research, June-July 20054. TITLE AND SUBTITLE 5- FUNDING NUMBERS

On Improving the Convergence of the Solution of a System of LinearEquations (U)6 AUTHOR(S)

Surendra Singh (U. Tulsa, OK), Klaus tialterman, and J. Merle Elson7 PERFORMING ORGANIZATION NAME(S) AND ADDRESSIES) B. PERFORMING ORGANIZATION

REPORT NUMBER

Naval Air Warfare Center Weapons Division NAWCWD TP 8608China Lake, CA 93555-6100

9. SPONSORINGIMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

Office of Naval Research800 N QuincyArlington, VA 2220111. SUPPLEMENTARY NOTES

12a DISTRIBUTIONIAVAILABILITY STATEMENT 12b DISTRIBUTION CODE

Approved for public release; distribution is unlimited.13. ABSTRACT (Maxmum 200W or&)

(U) The computation of electromagnetic scattering from a nanowire requires the solution of a system oflinear equations of the form, Ax = b, where the dimension of the matrix A increases with the nunber ofunknowns. Previously, we had implemented a bi-conjugate gradient algorithm to iteratively solve thissystem of equations. However, this method converges very slowly when the convergence criterion is madestringent. An improved version of the algorithm, namely stabilized bi-conjugate gradient algorithm, isimplemented to overcome this drawback. The new version provides very good convergence for convergencefactors up to 10-4 but the convergence is still slow for convergence factor in the 10-s - 10-' range. In orderto accelerate the convergence of the solution, we make use of the Levin transform to accelerate theconvergence of the solution. This transform is tested in the numerical solution of three different types ofintegral equations using the method of moments.

14 SUBJECT TERMS 15 NUMBER OF PAGES

Iterative Methods, Electromagnetic Scattering, Stablized Bi-Conjugate 18Gradient Algorithm 18 PRICE CODE

17 SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF ABSTRACT OF THIS PAGE

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED SARNSN 7540-01-280-5500 Standard Form 298 (Rev, 2-89)Preribed by ANSI Std 239-18

29-102

UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

Standard Form 298 Back (Rev. 2-89) SECURITY CLASSIFICATION OF THIS PAGE

UNCLASSIFIED

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CONTENTS

Introdu ction ................................................................................... . . 3

Bi-Conjugate Gradient Stabilized Method (beg-stab) ................................. 3Subroutine bcgstab (b,nunkns,nitm,conv,x,nit,rerr) ................................... 4

L evin T ransform ................................................................................ 9

N um erical E xam ples ............................................................................ 11

R eferen ces ................................................................................... .... 15

Appendix: FORTRAN Code for Bi-Conjugate Gradient Stabilized Method ............ 17

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INTRODUCTION

The work described in this report was performed in collaboration with ProfessorSurendra Singh, who came to the Naval Air Warfare Center Weapons Division(NAWCWD), China Lake, California, during June and July 2005. Professor Singh, afaculty member in the Electrical Engineering Department, University of Tulsa, Tulsa,Oklahoma, was visiting as an Office of Naval Research - American Society of ElectricalEngineers (ONR-ASEE) Summer Faculty Fellow. He worked in the Optics, RF, andMaterial Physics Section of the Physics and Computations Sciences Branch. This workinvolves implementing the stabilized version of the bi-conjugate gradient algorithm inorder to iteratively solve a linear system of equations resulting from integral equationsarising in electromagnetic scattering problems. An implementation of a convergenceacceleration transform to speed up the convergence of an iterative scheme is includedhere as the appendix to the report.

The solution of a system of linear equations can be obtained from direct methods,such as matrix inversion, and indirect methods that include a variety of iterative schemes.The conjugate gradient method is one such iterative method that provides an additionaladvantage: the coefficient matrix need not be stored in its entirety in memory. Thisprovides a much needed flexibility in very fine discretizations of the geometry underinvestigation without taxing the memory requirements. We implement the stabilizedversion of the conjugate gradient method and provide some convergence studies. Notedthat the stabilized version converges very rapidly to achieve a specific precision, and thentends to oscillate if higher precision is needed. To overcome this situation, we implementa transform to accelerate the convergence of the iterative scheme. The results of applyingthis transform to three different types of integral equations are provided.

BI-CONJUGATE GRADIENT STABILIZED METHOD (bcg-stab)

The bcg-stab method is an iterative technique to solve a system of linear equations ofthe form, Ax = b. The matrix, A, is a known matrix of order NxN, where N is thenumber of unknowns, b is the known forcing function of length N, and x is thesolution vector (length N). The method can be implemented without storing the matrixA. The algorithm can be coded such that only a row or a column of the matrix is neededat a time. This can be very helpful in cases where storing the entire matrix would pose a

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significant storage problem. The drawback of not storing the matrix is that it getscomputed twice in each iteration of the method. So we gain advantage in storage but wepay a price in increased computation time.

Here is the pseudo-code for the beg-stab method (Reference 1):

Initialization: alpha=l; rerr=100; conv=10-4 r; = b; Vo = Po = 0 ; P0 =o co)o 0O

nitm=nunkn;val I=dot(b,b)for i = 1: nitm

if (rerr > conv)p, = dot(b, r,-)

8 = (p, / p, )(a,/ , )

p, =,_ + /8(p,- - ao,_lvi1 )

v, zAp1 Note: This operation can be split up so that we use one row at a time.

a= p, /dot(b,v,)

s = ri_, - aV,

t = As Note: This operation can be split up so that we use one row at a time.aoi =dot (t, s) / dot (t, t)

Xi = X1.1 +--iP + O'+S

r, =s--ot

rerr = abs(dot(r, r ) vail)

endend

SUBROUTINE begstab(b,nunkns,nitm,conv,x,nit,rerr)

Input Variables

b (complex vector): Right-hand side or excitation vector of length nun kns.nunkns (integer): Number of unknowns (N),nitm (integer): Maximum number of iterations for bcg-stab (typically set equal tonunkns).

conv (real): Convergence factor to stop computation for bcg-stab (typically 10-4 to 10-1).

Output Variables

x (complex). Solution vector of length nunkns.

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nit (integer): Number of iterations taken by beg-stab method to converge.rerr (real): Residual error. The bcg-stab method will stop when the residual error (rerr)becomes less than the convergence factor (conv).

Lippman-Schwinger Integral Equation

We implement the beg-stab method in the numerical solution of the Lippman-Schwinger integral equation (Reference 2):

E(r) = E" (r) + k0 JG " (r, r') -Ae(r') E(r') dr' (1)S

In Equation 1, the symbols have their usual meaning: r = (x, y), E is the total electric

field, E"' is the incident electric field, GB is the two-dimensional (2-D) free-spaceGreen tensor and its explicit form is given in Reference 2, k0 =2.-/2, and

Ag(r) £ '-1, when r is entirely within the nanowire with permittivity c, and 0

otherwise. The nanowire is assumed to be infinite in the z direction and all spatialvariations occur in the x - y plane. The integration is carried out over the cross section

of the nanowire that is embedded in vacuum. We solve this equation numerically for ananowire 30 rn in size (side dimension) illuminated by an incident field of wavelength,2 = 400 nm, with the electric field vector perpendicular to the axis of the nanowire. Inthis example we solved for zero absorption, that is, the imaginary part of the permittivityis zero.

Table 1 convergence comparison of beg and the beg-stab for two values of theconvergence factor. nrx is a geometry discretization parameter such that the number ofunknowns is N = 2*nrx*nrx. # is the number of iterations taken by the method toconverge.

Table 1. Convergence Comparison, nvx and N.

Parameter No. of conv = 10-4 conv = 10-5unknowns

nrx N beg no. beg-stab no. beg no bcg-stab no.

20 800 90 57 127 17430 1800 320 37 424 57140 3200 589 19 777 688

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Figures 1 and 2 show the convergence behavior of the bcg-stab method as a functionof the iterations for cony 10-' and cony = 1i-5, respectively. We see that the methodconverges extremely rapidly for cony = 1 0-' in comparison to bcg, which takes 589iterations to converge, as indicated in Table 1. The numerical results shown in Table Iwere obtained on a Pentium IV 32-bit PC. The performance and precision of the bcg-stabmethod may improve if the code is run on a 64-bit machine.

-1.

o -2.50)0

-3.5

-4'

0 2 4 6 8 10 12 14 16 18

iteration number

FIGURE 1. Convergence of bcg-stab Method Showing Log of ResidualVersus Iterations for cony 10-4 and nrx = 40.

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-1 IF

-21

:3

V) -3 I

-5

-6 L I .... .. ..0 100 200 300 400 500 600 700

iteration number

FIGURE 2. Convergence of beg-stab Method Showing Log of ResidualVersus Iterations for conv = 1 0-' and nrx = 40.

When the imaginary part of permittivity is non-zero, the convergence of thealgorithms is quite rapid. Table 2 is convergence comparison beg and beg (bcg stab) as afunction of wavelength, /1, and convergence factor (conv). The results are obtained fromFORTRAN code: eigen2. The silver nanowire is solid and the the imaginary part ofepsilon (ei) # 0, nrx = 40, (and N = 3200). The convergence of beg stab as a functionof the iterations is shown in Figures 3 and 4.

Table 1. Convergence Compairson, k and conv.

/,I= 300 nm 2 = 400 nm 2 = 500 nmNumber of iterations Number of iterations Number of iterations

conv bcg beg stab beg beg stab beg beg stab10- 8 5 66 41 152 83

10t- 11 7 101 75 249 205

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0

-2

-3

._-4-

"0-5

-6

-7

-8

-9 ------- - --- --------0 10 20 30 40 50 60 70

iteration number

FIGURE 3. Convergence of bcostab Method for nrx = 40 (Number ofUnknowns = 3200), A = 400 nrm, Convergence Factor = 10 8 (ResultObtained From MATLAB Version of the Code).

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-2

-3

-4

PDU -5C',V

-8-

- 9 . ......... .. .... --- I]

0 20 40 60 80 100 120 140 160 180iteration number

FIGURE 4. Convergence of bcgstab Method of nrx= 40 (Number of

Unknowns = 3200), A = 500 nm, Convergence Factor (conv) = 10-(Result Obtained From MATLAB Version of begstab).

LEVIN TRANSFORM

As we have seen in the results shown in Figure 2, the bcg-stab method oscillates for along time before converging. Our goal is to apply a transform such that the sequence ofvectors given by the iterative scheme (bcg-stab) will converge faster to the solution, x, ofthe system Ax = b. There are a number of convergence acceleration methods, includingLevin's transform, Wynn's epsilon algorithm, Chebyschev-Toeplitz algorithm, andBrezinski's 0 algorithm. Here we focus on the Levin transform that provides significantenhancement in convergence of a vector sequence. To begin with, we start with thetransform as applied to a scalar sequence and then extend it to be applicable to a vectorsequence. Let S, be the partial sum of n terms of a series such that S" --> S as n --> cc,

where S is the sum of the series. If the partial sums are coming from an infinite seriesthat converges extremely slowly, then the scalar sequence S., S1, S2, ... , will take a

long time to converge. This is when we employ techniques to enhance the rate of

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convergence to arrive at S with only a small value of n. To this end, the k"' order Levintransform, tV(S), which provides an estimate of the sum of the series, S, may becomputed as (Reference 3):

k ( k)()n +

(n.) 1=0,• • i n\ ) S,,+i+l - ,, / k =0,, .. 2k k ,(k)( n + i)(k-1) k ý ,,2'2

A significant advantage of the Levin transform is that the higher order iterates arecomputed from the partial sums, S0 , S1, S2 , ... , rather than the lower iterates as in thecase of Wynn's algorithm, thereby providing some immunity from the accumulation ofround-off errors.

Consider the solution of the following:

Ax = b (3)

where A is a known matrix (moment) of order N x N; A = [a. I, and x is the unknown

column vector of order N xI; x = [x X2 ,... X N]T, and b is the known forcing function

column vector of order N x I ; b = [b1 b2 .. .bN],. With A and b known, we calculate theinitial estimate of x, designated as Y by the use of Gauss-Seidel relaxation, thus arrivingat vectors {SO }, {S1 }, {S 2 }, and so on. Each of these vectors is of order N x 1. Once we

have three initial vectors, we can apply the Levin transform using Equation I to obtain anestimate of the solution vector, Y. A numerical example of this process to a scalarsequence is given in Reference 4. The application of Equation 1 to a vector sequencerequires a bit of bookkeeping, as the scalar Levin transform, t'"', now becomes a vector,

{ýt) }, of length N. As with any iterative scheme, the computation process should stopwhen a convergence criterion is met. We employ the following measure:

(AC - b)11 < Cf (4)

~bj12

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The computations are stopped when the above criterion is met for a pre-defined convergencefactor, C1 , and we have the estimate of the solution, Y. We should point out that the initial

vectors can be generated from a variety of schemes. It may be possible to use the iterationsfrom a bcg algorithm and then subsequently use the Levin transform or any other vectoracceleration technique to enhance the convergence. One significant advantage of using thebcg estimates is that the moment matrix, A, does not need to be stored entirely in memory,as the algorithm can be implemented in such a way that only a row or column at a time isneeded in the computations. This alleviates, to a great extent, any storage problemsroutinely encountered when solving a very large system of equations.

NUMERICAL EXAMPLES

For our first example, we consider the solution of the integral equation for the chargedistribution, q(y), on a conducting strip of width = 2w with excitation, V(y):

"Jre f q(y') ln~y y' dy' = V(y), y c (-w, w) (5)

For illustration purposes, we take the permittivity of the medium, I = F/rn. Theintegral equation is cast into a system of linear equations of the form given in Equation 2by using method of moments (MOM) with pulse basis functions and point matching.

Figure 5 shows the charge distribution for N = 150, and a forcing function, V(y) = y 2 .

Note that the Levin transform result matches very well with direct matrix inversion.

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40~

30I

25

S20 MoM with Matrix Inversiono MoM with Levin Transform

151 /

- 10

5,-

. .. .. .. ...

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1y (m)

FIGURE 5. Charge Distribution on a Conducting Strip of Half-Width w = Im, Forcing Function V(y) = y2 , Convergence Factor Cf = 10-', Number of

Unknowns N = 150.

The second example to illustrate the Levin transform involves a thin wire antenna.The current distribution on a thin wire can be computed by the numerical solution ofHallen's equation or Pocklington's equation. Pocklington's integral equation for thecurrent distribution, I(y), on a thin conducting wire of radius = a, length =2h, orientedalong the y - axis is given by:

j 4• +ko I(y')K(y - y') dy' = E' (y), y e (-h, h) (6)d Kd 2 Y)

where

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1r '-j'OR

K(y - y') -2)_T J-R-o (7)

-7t

Here the distance between the observation and source point, R, can be written as:

R = [(y y,)2 + 4a 2 sin 2 0' ],/2 (8)2

The incident electric field is given as Ey (y) = V3(y - yg) for the antenna case, where

V is the feed voltage and yg is the feed point. In Equation 6, r/ is the intrinsic

impedance and k0 is the wave number of the medium. Once again, with the applicationof the method of moments, with suitable basis and testing functions, we can cast Equation6 in the form of a system of linear equations: Ax = b. The current distribution on a center-fed, half-wavelength (h = 2/4) thin wire antenna with V =1 volt, yg =0 m,

a = 0.0070222, CQ = 10-3, and 2 = 1 m is shown in Figure 6. A detailed account ofthese examples can be found in Reference 4.

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x 10-310 - T -T I

- 5-< ,Levin Transform (Real)

Levin Transform (Imag.)r" Matrix Inversion (Real)

--- "Matrix Inversion (Imag.)D

7O7

"-5I ----------- - _ _...... - ----

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25Length in meters

FIGURE 6. Current Distribution on a Center-Fed, Half-Wavelength, ThinWire Antenna of Radius = 0.007022 2, Voltage V = 1 Volt, Number ofUnknowns N = 21, and Convergence FactorC- = 10.

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REFERENCES

1. H. A. Van Der Vorst. "Bi-CGSTAB: A Fast and Smoothly Convergent Variant ofBi-CG for the Solution of Nonsyrmmetric Linear Systems," SIAM J Sci. Stat. Comput.,Vol. 13, No. 2 (March 1992), pp. 631-44.

2. K. Halterman, J. M. Elson, and S. Singh. "Plasmonic Resonances andElectromagnetic Forces Between Coupled Silver Nanowires," Phys. Rev. B, Vol. 72(2005).

3. D. Levin. "Non-linear Transformations of Divergent and Slowly ConvergentSequences," Math Comput., Vol. 16 (1973), pp. 301-22.

4. S. Singh, K. Halterman, and J. M. Elson. "An Iterative Method for the NumericalSolution of Integral Equations Using the Method of Moments," submitted to Microwaveand Opt. Tech. Lett. (2005).

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Appendix

FORTRA]N CODE FOR BI-CONJUGATE GRADIENT STABILIZED METHOD

FORTRAN listing of SUBROUTINE bog-stab:

subroutine bog-stab(b,nurikns,nitm,oonv,oi,n~i,rerr)

imnplicit noneoomplex, intent(in), dimension(l:nunkns) :bcomrplex, intent (out), dimension(l:nunkns) 01

integer, intent(in) ::nunkns,nitrninteger, intent(out) ::nitreal, intent(in) : onvreal, intent(out) ::rerroomplex,ciimension(l:nunkns) :: p,r,v,s,tcomplex : : beta,alpha,vall,rho, rhoi,omegainteger :: 1complex :: CDOTU

ci=O ;p=O ;v=rO; alpha=l; omega=l; rho=l; r=b; s=0; t=OrerrzdDOvall=dot produot (b, b)

do nit=1,nitmif (rerr>oonv) thenrhoi=dot-produot (b, r)

beta= (rhoi/rho) *(alpha/omega)

p=~r+beta* (p-ornega*v)

do i=l,nunknsv(i)=~CDOTU(nunkns,arow(i),l,p,1)

enddo

alpha=~rhoi/dot-produot (b, v)s~=r-alpha*v

do i=l,nunkns

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t (i) CDOTU (nunkns, arow (i),l,s, 1)enddoorrega=dot product (t, s) /dot-product (t, t)ci=~ci+alpha*p+omega*sr~s-omega*trho=rhoirerr=abs (dot product (r,r) /vall)!print *, 'iteration#=',nit, 'residual=',rerr

elseexitend ifend do

end subroutine bcg stab

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