Generation and analysis of high order strong coupling series for SU(2) lattice gauge theory

Computer Physics Communications 58 (1990) 199—209 North-Holland

199

GENERATION AND ANALYSIS OF HIGH ORDER STRONG COUPLING SERIES FOR SU(2) LATHCE GAUGE THEORY * Caries AYALA and Maria BAIG Grup de Fisica Tebrica, Universitat A utbnoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Received 24 April 1989

We present a package of VAX FORTRAN programs that analyze the information coming from two sets of FORTRAN routines. The first set of routines perform strong coupling series expansions in pure SU(2) lattice gauge theory mixed actions. The most general mixed action we are able to analyze is a mixing of the spin ~, 1, ~ and 2 representations. This set is a symbiosis between the use of algebraic outputs from REDUCE and its execution using the VAX FORTRAN language. The second set of FORTRAN routines perform the Padé approximants analysis of the series. Making a systematic study of the Padé table will give the mean value of the significant real (and complex) poles put together with its standard deviation. The programs perform the series analysis and graphical representation of the results for all the cases.

PROGRAM SUMMARY Title of the program: STRONG

SU2

CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue)

between the use of outputs coming from the algebraic manipulator REDUCE and the use of the FORTRAN language. The second program performs the Padé approximants analysis of the series using some NAg routines. Remaining programs perform the series analysis and graphical representation of the results for several cases. They use GKS-3D standard graphical routines [3] and, optionally, TOPDRAWER output is made [4].

Computer for which the program is designed: VAX 8800

Restrictions on the complexity of the program

Operating system: VAX/VMS

Series are generated along “radial” lines in the four-parametric phase space of couplings. However, those “radial” lines are the most natural way to study such expansions.

Catalogue number: ABLQ Program obtainable from:

Programming language used: VAX EXTENDED FORTRAN Typical running time No. of bits in a word: 32

mixed actions, phase transitions, D-dimensional theories

The test run output that performs the study of a single radial line in the four couplings phase space takes around I CPU minute on a VAX 8800. The previous order in the development of the action (including only spin ~, 1, ~) has a special treatment, it can be obtained in 10 CPU seconds on a VAX 8800.

Nature of physical problem

References

We present a package of FORTRAN routines and programs able to study strong coupling series expansions in pure SU(2) lattice gauge theory mixed actions [1]. The most general mixed action that can be analyzed is a mixing of the spin ~, 1, ~ and 2 representations [2].

[1] J.M. Drouffe and J.B. Zuber, Phys. Rep. C 102 (1983) 1. [2] C. Ayala and M. Baig, Preprint UABFT-203 (February 1989), Ann. Phys., in press. [3] GKS/GKS-3D Primer, CERN/DD/US/110, Geneva (May 1988). Guide to Computer Graphics at CERN, CERN/DD/US/ 111, Geneva (August 1987). [4] R.B. Chaffee, TOPDRAWER, SLAC Computation Group, CGTM No. 178.

No. of lines in combined program and test deck: 4.440 Keywords: lattice gauge theories, strong coupling expansions,

Method of solution

The set of routines generating the series is a symbiosis *

Work partially supported by Research Project CICYT.

0010-4655/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

200

C. Ayala. M. Baig / High order strong coupling series for SU(2) lattice gauge theory

LONG

WRITE-UP

1. Introduction

Lattice gauge theories play a fundamental role in our understanding of the fundamental interactions of particle physics [1]. Lattice formulation converts an original quantum field theoretical problem into a classical statistical mechanics systern that allows a wide range of new treatments. In particular, Monte Carlo simulation methods have offered an unique way to study long-range nonperturbative properties of quantum chromodynamics. However, in addition to “brute force” Monte Carlo simulation experiments, different analytical studies can also be performed in the lattice regularized theory. Indeed, strong coupling (high temperature) expansions are well known in statistical mechanics for a long time. Its application to lattice gauge theories, proposed by Wilson, allowed an analytical treatment of the confined region [1]. In addition, high order series extrapolate up to the neighbourhood of the continuum limit, allowing the study of the intermediate region where deconfining phase transitions may be present. Strong coupling expansions in pure lattice gauge theories are a major achievement [2]. Those expansions are analytical, an advantage due to the strong coupling regime. In spite of this analytical status, when one studies mixed actions one gets (working till the available order: the 16th) such huge numbers (in numerator and denominator more than 80 digits) as coefficients in the series that a standard algebraic manipulator as REDUCE cannot handle

MIXEDONES

NOIU1A

.~Ic0EF_c

them. Thus it is unreasonable and unuseful to get a result in this way. This is the first motivation that moved us to write the set of FORTRAN routines that perform the series [2]. The exact analytical result can be qualified as “esthetic”, but as we desire to make a numerical Padé analysis of the series we do not essentially need it. Explanations for the more physical or mathematical details can be found in ref. [21.

2. Package structure The package of programs contains two basic routines. The first, named MIXEDONES, performs the generation of the series for a given radial line in the four couplings phase space. It is called as a subroutine from the main program. The second basic routine is PADEMAKER that performs the Padé analysis of the series generated by MIXEDONES. It must also be called as a subroutine. Both routines have a complex structure that is summarized in figs. 1 and 2. The main program, named STRONG.SU2, will request first the direction to be studied and then calls MIXEDONES and PADEMAKER giving, as output, the mean Padé pole on that direction. The test run output is performed using this main routine. To be more useful, a set of complementary routines have also been incorporated. They perform the series generation and Padé analysis on several sets of lines in the phase space.

PADEMAKEFL

I I

TOLERANCES_DEF

___________________________ NAE.LIBRARY

—

________I ~IC0EF_5~i1ED_~IDPR1

ISTATIS

DPR5_1_r~H~D1

PRQD5

_________

ISTRU —.IPMIX2

,...,

TURE

[~~~III

PMIX6

Fig. I. Structure of the subroutines called from MIXEDONES.

Fig. 2. Structure of the subroutines called from PADEMAKER.

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201

High order strong coupling series for SU(2) lattice gauge theory

The first set of complementary routines analyze all the two-dimensional planes on the phase space. The main program BIDIM performs the series generation, asking for a given plane and dimension and calling MIXEDONES routines. The output is a file that stores radial series in the plane. Main program PADEBD performs the Padé analysis of the above series giving as output a file containing the stable poles in a suitable form to be plotted by TOPDRAWER [4] routines. This output, however, can easily be modified to be used by any graphics program. The last set of routines analyze surfaces in a three-dimensional phase space. Main program WONDER performs the task of generating the series and Padé analysis calling the basic routines MIXEDONES and PADEMAKER. Main program PLOT using standard three-dimensional GKS-3D [4] routines performs the graphical representation of the results generated by WONDER. Figures 1 and 2 summarize the structure of routines of the package. In the following chapters the main facts of the routines of the package are described in detail. In addition, the program deck contains a wide set of comments and details of the routines’ usage.

Then, we will have expansions in the “radial” variable

R

=

+

~/[$5/2}2

If we define

S(U) R

~1r

+

2

[si]

+

[53/212

ED

~i.rXr(U),

=

r—

1/2.1,3/2,...

remembering r+I

Xr(U)Xi(U)

~

=

k

=

X~~(U),

(3.1)

r— I I

we can also define

( ) s( U) R

nL

n

~nrXr(U).

=

r=0,1/2,1 .3/2,...

Using eq. (3.1) one can derive the following recursive relation which is very easy to implement in a FORTRAN program L ~n+i.r

r+m ~1.,n

=

m= 1/2,1,3/2,...

~

~nk

k=Ir—mI

This ~ is a very useful object, because 3. The

eS~= ~5Xr(U)~ {r}

series

We explain here the algorithms and the functions of all the FORTRAN subroutines called from the top routine MIXEDONES. The action we are dealing with is

s(v~)=~$r

nL

~nrXrW)~

=

fl0

\ r=O,1/2.1,3/2,...

~

)

n!

So

/~=~

(2r+1)’

(r)

/

(S(U))~ n!

where Xr(U) tr~(U)is the trace of the matrix u in the { r } representation of the group (U E su(2)). This XrW) is the so-called character of the { r } representation of the group (Xr(i) dr 2r + 1) The first subroutine to be commented is COEF_B. We choose “radial” directions such as =

=

The rest of the subroutine presents no additional obstacles. It performs in a very direct way the inversion of a series to be able to perform br /3r/((2r + 1)/~~)and also the log J~. The subroutine COEF DAB is the realization of the formula =

r± t ($I/2’

~ $

3/2,...)=R~

~

~=

—~b1+

2r±1 21±1 m=Ir-/l ~

(2m±1)bm,

202

C. Ayala, M. Baig

/

High order strong coupling series for SU(2) lattice gauge theory

Table 1 The relation between the mathematical tools and physical quantities with the arrays and variables in the MIXEDONES set of routines The MIXEDONES routines

This article

R=2r=1,2 12 L=2/=1,2 8 M=2m=1,2,3,4

r=-~,1,~ ~ ~

a)

6 4

The functions play a remarkable role in the program. We want to compute the product of series in each term of the the free energy (all of them are in general the product of 2 3 4 series). The procedure is straightforward: if A —BC, ~

and we can understand these quantities as series

fir

CB(n, R)

~

CBB(n, L, m)

[(b

DAB(n, L, M) CLNA(n)

[ab/a$,,,]~ [log$ 0]~

E(n,0) E(n,M)

[F],, [E ]

D

D

1)”],,

in R (we define, for all 1 = 1B1 rCl IA L

in

k=0

IaAl

Jn—kL

~ =

Q, Q ~~[Q]~R~), then =

Jk

aC

IaBl

k=O {axiflk

+

[BI~k{~J}.

~ In general A=l,,[A],,R~

where the derivatives of b~are obtained doing an operation including birt1,..., br+i. The subroutine COEF C performs the coeficients of the topologically different contributions. Those are polynomials in the space—time dimensionality. In fact, that one is the sole subroutine in which the space—time dimensionality plays a role. For all the above explained (both mathematical and physical) quantities we have an equivalent in our MIXEDONES routine. The relations between them are exposed in table 1. Now we are facing the calculation of the proper series. The final subroutine COEF ENERGIES performs the contributions of all the topologically different terms. Most of them are REDUCE output. This REDUCE output is also a manipulation of a previous REDUCE result. We can say only good things of the ON FORT command in REDUCE and the several devices that enable us to manipulate the FORTRAN output. The set of REDUCE programs that we wrote will not be extensively commented in here. We must say that the computation of the expression for 4~ (PHI111444(I) in the program) needed one order of magnitude more, both of work and CPU time, than the rest. The reason is the extremely cornplicated structure of the integrals over the group. The way to solve these integrals is explained in appendix A of ref. [2]. -

-

4. The Padé analysis The second main routine, named PADEMAKER, performs all Padé extrapolation analysis of a given radial series. The input is the series constructed by MIXEDONES and the output is the mean value of the stable significant poles. For an explanation of the Padé extrapolation techniques applied to lattice gauge theory see ref. [5]. The subroutine structure of PADEMAKER is shown in fig. 2. The first step is to construct Padé approximant’s table. Secondly, it performs the numerical analysis of the zeroes and poles of the approximant, filtering the non-significant poles. Then the procedure is repeated for all diagonal and next-to-diagonal terms in the Padé table [6]. The final step is to detect a congruence in the position of all significant poles. Subroutine PADEMAKER controls all the Padé procedures. For each selected line in the phase space of couplings, a DO loop over thermodynamical cases and a DO loop over the Padé table are performed. Subroutine PADE constructs a given approximant of the table. It calls the NAg routine that performs the approximant construction and the NAg routines that evaluate the zeroes and poles of the approximant. After this evaluation, it calls the subroutines FILTER and STATIS, returning the

C. Ayala, M. Baig

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203

High order strong coupling series for SU(2) lattice gauge theory

Table 2 Parameters of PADE.DEF

deviation of the poles. This procedure is done to eliminate spurious accumulations of poles that can

Variable

modify the mean value if one performs a simple statistical analysis of the locations of poles. Concerning complex poles with a small imaginary part compared with the real one, since we have few of them, a straightforward procedure is done to define the average position.

Value

Effect

RTOL RZER

lx iO~ tolerance to zero-pole pair 0.4 minimum distance to zero of the real part of a pole PRMAX 6.0 maximum distance to zero of the real part of a pole DITOL 1.5 maximum distance to real axis of complex poles NPP 5 minimum number of stable poles to be significant TMVAR 0.6 maximum variance to be a narrow pole

control to PADEMAKER to analyze another term of the Padé table. Subroutine FILTER performs the filtering of spurious poles for a given approximant. It firstly eliminates the coupled zero—pole pairs, the poles near the origin and the anomalously large poles. This routine also eliminates all complex poles with the same characteristics explained for the real ones, but, in addition, it eliminates also complex poles with an imaginary part much bigger that the real one. This procedure is controlled by a set of tolerances that can be changed by the user. They are stored in an external file named PADE.DEF and are read by the program through the subroutine TOLERANCES DEF, called once by PADEMAKER. Those parameters are summarized in table 2. Subroutine STATIS accumulates in a vector all significant poles for all approximants that have survived the filtering procedure. Actually, the program stores separately the real poles and the cornplex poles, of these last only the poles with positive real part. When the DO loop over the Padé table is finished (for a given thermodynamical quantity) the program has a vector of significant poles. Subroutine STRUCTURE detects a congruence in the location of poles giving as a result the mean position of the accumulation of poles (if any). The auxiliary subroutine ORDER first orders the poles in increasing values. Then a histogram is filled looking for the highest channel. Then the program eleminates all channels far from that maximum and computes the mean value and the standard

Some further tests are also included. First there is a cut-off in the number of poles to be a significant accumulation. Second, a maximum value for the standard deviation is also included to assure that the observed accumulation is significant. If these tests are not accomplished or there are no poles output from the filtering, a pole at zero value is given as output. This will mean that there are no significant poles from the Padé analysis. As a result of the PADEMAKER routine analysis, one obtains a real vector containing the average position of poles for each thermodynamical quantity and a real vector containing the standard deviations. A similar set of complex vectors is also obtained. Note that all the output from PADEMAKER including the file STRONG SU2. RES is in terms of /3-variables. The vector of poles is normalized to x-variables by PADEBD and WONDER routines that produces a graphycal output in terms of these variables. —

—

5. Using the program The two basic routines MIXEDONES and PADEMAKER can be called from three different main programs, according to the case to be studied. The possibilities are as follows. 5.1. STRONGSU2

Main program STRONG. SU2 performs the complete study of a given line in the four-dimensional phase space. Initial data for the run are the values of the four couplings determining a direction in the phase space and the value of the selected space—time dimension. Subroutine QUESTIONS asks interactively those values. Then, the program calls subroutines MIXEDONES and PADEMAKER to generate the series —

—

204

C. Ayala, M. Baig

/

High order strong coupling series for SU(2) lattice gauge theory

Table 3 Summary of the INPUT and OUTPUT corresponding to the STRONG - SU2 program

Table 4 All the possible bi-parametric cases studied of the four parameters action, chosen by IOPT

INPUT

IOPT

Option

Array containing the phase space

1

(P1/2, fit)

direction (interactive) Space— time dimension (interactive) Output mode [0/1] [short/full] (interactive) Parameter definition for Padé procedure (file)

2 3 4 5 6

(P1/2’ P3/2) (Pu /2’ P2) (PiP P3/2)

X(I) D 1W PADE.DEF

(Pu

$2)

(P3/2’

$2)

OUTPUT STRONG SU2.RES SUMMARY.RES MAIN.DXE

Full information on the Padé analysis (file) Summary of pole location (file and interactive) The coefficients of the series generated (file)

analysis of the above generated series, giving as output a file with the location of poles in the plane in a form suitable to be processed by the TOPDRAWER routine. However, since the out.

_______________________________________________

put file includes basically the list of (x, y) coordinates for the poles, such a file can be easily adapted to any graphics package. Of course, a

and compute the poles. Results are stored in a file named STRONG_ SU2.RES containing all the information of the filtering procedure. Two auxiliary output files contain more details. Table 3 summarizes the INPUT required and the OUTPUT produced by the STRONGSU2 program. Before starting the program, the values of the tolerance parameters needed by PADEMAKER must be stored in the file PADE.DEF. At the end of this paper an output obtained from STRONG. SU2 for a selected line in the phase space is included as a test run of the program.

definition of the tolerance parameters in PADE.DEF file must be done before running PADEBD. We have chosen to perform this two-step run to make handy the study of the Padé tolerance parameters, generating once the series for each two-parametric combination of couplings.

5.2.

BIDIM and PADEBD

The second main program included in the package is BIDIM. This intends to perform a study of a given plane. The input required is the particular combination of two couplings chosen (table 4), the space—time dimensionality and the angular step of the radial lines in the plane (in degrees). Main program BIDIM performs the generation of the series according radial lines in the chosen plane, being the series stored in the intermediate file S.CASE~DIM.DAT. Main program PADEBD performs the Padé

5.3. WONDER and PLOT To study the cases involving three-dimensional phase spaces (the selected triad), we developed two mighty programs named WONDER and PLOT. WONDER is a data generator program. Using a triad of ~8’s, previously chosen out of the four possibilites (to take three out of four), will construct an angular lattice created by subroutine NET. It consists in M + 1 divisions in the spherical 0 angle (0 <9 < ~-‘rr) and N + 1 divisions in the spherical ~ (0 <~ < ~‘rr). For all those points WONDER will perform the computation of the Padé poles and give, at the end, a file for each thermodynamical quantity containing 3D points that give the pole structure of the case. Those files will be named according to their content by subroutine NAMING. That output scheme is fulfilled by the subroutine OUT DATA.

C. Ayala. M. Baig

5,,

/

High order strong coupling series for SU(2) lattice gauge theory

~,,,

Es

~

205

GKS-3D. Those will be read by subroutine DATASAMPLES. The output from WONDER is actually read by DATA_WONDER. In the

.1~ ~

~~ S

~

caseallthat for thewedata available make average between in thea results given triad, the willaninternal be read energies and averaged by DATA AVERAGE. Apart from this average PLOT has other switches to manipulate the out-

5,,,

Es,

put. Subroutine DEPILATOR will make disapare extremely big. All those devices can be switched

a Fig. 3. A sample of the result of a run for a certain triad and a thermodinamical quantity represented by PLOT (in the laser printer ouput mode).

WONDER will also control the status of the actual case. So, if we did a previous run for the actual NET (for the same entries), it will control where it finished and will continue from there. If the actual run was performed previously, it will stop the run with a clarifying message. All those are marvelous functions of the subroutine CONTROL DATA. PLOT does the data presentation. PLOT will read the output of WONDER in the same way it was written and will plot them in a 3D plot using the GKS-3D package. It has a default file, named PLOT.DEF, that is charged any time it will run. With a unique executable image it will help us to change the view in which we see this 3D object. This will help a lot in order to get the best representation of it. PLOT can also plot several examples produced by the program SAMPLES. This will help to see how adequate the projections are generated by

off. pear certain sudden has all also and a extravagant main cosmetic radii routine: R that AXISPLOTI will draw the axis and write all the labelling. VIEW will draw all the lines between neighbouring points that have a pole assigned. If one of the two neighbour points in the NET does not have an assigned pole nothing will be drawn. For isolated points with an assigned pole it will draw a point, just to remind about it. PLOT is able to send output to a Tektronix terminal and to a PostScript laser printer. A sample of the latter is shown in fig. 3.

-

References [1] K. Wilson, Phys. Rev. D 10 (1974) 2445. [2] 1989); C. Ayala and M. in Baig, Preprint UABFT-203 (February Ann. Phys., press. [3] R.B. Chaffee, TOPDRAWER, SLAC Computation Group, CGTM No. 178. [4] GKS/GKS-3D Primer, CERN/DD/US/110, Geneva (May 1988). Guide to Computer Graphics at CERN, CERN/DD/US/ 111, Geneva [5] J.M. Drouffe (August and J.B. 1987). Zuber, Phys. Rep. C 102 (1983) 1. [6] G.B. Baker, Essentials of Padé Approximants (Academic, New York, 1975).

-

C. Ayala, M. Baig / High order strong coupling series for SU(2) lattice gauge theory

206

TEST RUN OUTPUT -Output of program

STRONG..SU2

DATA OF THE RUN

PHASE SPACE DIRECTION B1/2 31 B3/2 32

0.5000000000000000 0.5000000000000000 0.5000000000000000 0.5000000000000000

SPACE TIME DIMENSION D

=

4.000000000000000

VALUES OF PARAMETERS

(TOLERANCES DEF)

l.0000000000000000E—04 TOLERANCE ZERO—POLE 0.4000000000000000 MINIMUM DISTANCE TO 6.000000000000000 MAXIMUM DISTANCE TO 1.500000000000000 MAXIMUM DISTANCE TO 5 MINIMUM NUMBER OF STABLE POLES 0.6000000000000000 MAXIMUM VARIANCE TO

FREE

ENERGY DETERMINATION

LIST OF SELECTED REAL POLES 1.931090261714341 1.977884269190465 1.978352724152991 2.001599998388237 2.276772402677772 2. 37 911940 6788 936 2.400170718688920 2.460726883140318 2.537515477071757 2.593864268619464 2.752150454823299 2.902177256466036 3.954362173883564 3.987865402617098 4.205391506148626 4.211620645749397 4.514544756146303

PAIRS ZERO ZERO THE REAL AXIS BE A NARROW POLE-

/ High order strong coupling series for SU(2) lattice gauge

C. Ayala, M. Baig RELATIVE COEF

=

0.2352941176470588

SURVIVAL POLES 1.931090261714341 1.977884269190465 1.978352724152991 2. 00 159999 838 82 37 2.276772402677772 2.379119406788936 2. 4 0 017 07186 88 920 2. 4 6072688 3140 318 OF STABLE REAL POLES MEAN REAL POLES 2.175714583092748 NO COMPLEX POLES NUMBER

8 SD

0.2095961166347174

SD

0.1169699472135681

E1/2 ENERGY DETERMINATION

LIST OF SELECTED REAL POLES 1.835814412379422 1.858791943591890 1.860166423803628 1.865828409880489 2. 15 847 6080140 582 2.170282658228173 2. 172 32144 58 984 64 2.186536008083929 2.199668564695931 2.203612619363550 2.227273018774417 2. 24 88 3 32962 087 32 2.558021260894763 3.322715545670494 3.629432758358803 3. 850979 89 647 6727 3.933966411629220 4.128925802738641 4.884185200182115 RELATIVE COEF =

0.4210526315789474

SURVIVAL POLES 2.158476080140582 2.170282658228173 2.172321445898464 2.186536008083929 2. 199668 564 6 95931 2.203612619363550 2. 22727301877.4417 2.248833296208732 2. 558 02 12608 947 63 NUMBER OF STABLE REAL POLES MEAN REAL POLES 2.236113883587616 NO COMPLEX POLES

9

theory

207

208

El

C. Ayala, M. Baig

/

High order strong coupling seriesfor SU(2) lattice gauge theory

ENERGY DETERMINATION

LIST OF SELECTED REAL POLES 1.672147545247284 1.836657603982504 1.859612726281827 1.869083168959083 1.871128080768789 1.985506525496811 1. 9 92112 0 91153741 2.000719974376022 2. 034322230657277 2.215292548812034

2.436492931643633 2.538403181684803 2.625395416051311 2.954943919550458 3.539762842141775 4. 151102 161903 979 4.279790068150534 RELATIVE COEF =

0.2352941176470588

SURVIVAL POLES 1.672147545247284 1.836657603982504 1.859612726281827 1.869083168959083 1.871128080768789 1.985506525496811 1.992112091153741 2.000719974376022 2.034322230657277 2. 2152 92 54 8 8 12 034 NUMBER OF STABLE REAL POLES MEAN REAL POLES 1.933658249573537 NO COMPLEX POLES

E3/2 ENERGY DETERMINATION

LIST OF SELECTED REAL POLES 1. 7 38660 04 58 96 520 1. 74 0034 88 9600818 1. 75 6109 3338 8164 9 1.756130104974030 1.877094292550425 1. 87768 8 3517 81119 1.931209572585783 2.084184787532522 2.094066710236182 2.116135075329824 2.128325745190495

2.129218371799264 2.136118220826465 2.164070582382942

4.324022944593554 4.381838098539748

10 SD

0.1383521456751993

C. Ayala, M. Baig RELATIVE COEF =

/ High order strong coupling series for

SU(2) lattice gauge theory

0.3750000000000000

SURVIVAL POLES 1.738660045896520 1.740034889600818 1.756109333881649 1.756130104974030 1.877094292550425 1.877688351781119 1.931209572585783 2.084184787532522 2.094066710236182 2.116135075329824 2.128325745190495 2.129218371799264 2.136118220826465 2.164070582382942 NUMBER OF STABLE REAL POLES MEAN REAL POLES 1.966360434612003 NO COMPLEX POLES

E2

14 SD

0.1651462974200945

SD

9.9081843611816500E—02

ENERGY DETERMINATION

LIST OF SELECTED REAL POLES 1.443659463863726 1.502734548354827 1.588453422061654 1.593726286607190 1.670696382525655 1.894839727084408 1.915177958508314 1.954741017576610 2.009385641611723 2. 051372739965809 2.076254581036532 2.083033105021123 2.103412541571046 2.230702138624650 2.519531322118681 RELATIVE COEF =

0.4000000000000000

SURVIVAL POLES 1.894839727084408 1 915177958508314 1.954741017576610 2.009385641611723 2.051372739965809 2. 076254581036532 2.083033105021123 2.103412541571046 2. 2307 02138624 650 NUMBER OF STABLE REAL POLES MEAN REAL POLES 2.035435494555579 NO COMPLEX POLES

9

209