Subroutine for calculation of matrix padé approximants

Computer Physics Communications 11(1976)325—330 ~ North-Holland Publishing Company

SUBROUTINE FOR CALCULATION OF MATRIX PADE APPROXIMANTS Yair STARKAND Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel Received 20 May 1976

PROGRAM SUMMARY Title ofprogram: PADE APPROXIMANTS Catalogue number: ACWV

ber of coefficients in the perturbation expansion of a matrixvalued function of any given dimension, the program computes the approximant at a given point x.

Program obtainable from: CPC Program Library, Queen’s

Method of calculation

University of Belfast, N. Ireland (see application form in this issue)

The calculation is carried out using a closed formula obtained in ref. [1] and completely proved in ref. [2].

Computer: IBM 370/165; Installation: Weizmann Institute

Restrictions on the complexity of the problem

of Science, Israel

Calculations are done in double-precision mode (each number occupies two 32-bit words in memory). All the quantities calculated are kept in the active area. If matrices of large dimensions are to be used or approximants of high order calculated, the program could be easily modified to keep most of the information on tape or disk to reduce memory use.

Operating system: 0S360 Program language used: FORTRAN IV High speed storage required: 63000 words No. of bits in a word: 32 Overlay structure: none No. of magnetic tapes required: none Other peripherals used: card reader, line printer

Typical running time Running time depends roughly on the square of the order of the matrices involved and on the square of the order of the approximant requested. To give an order of magnitude for the running time we mention that calculating the [6,5] Pad~approximant for a three by three-dimensional problem needed 0.90 sec on an IBM 370/165. The test program enclosed needed 4.63 sec for calculating 66 Padé approximants ranging from [1,1] to [11,7] Pad~approximant.

No. of cards in combined program and test deck: 686 Card punching code: EBCDIC Keywords: General purpose, Pad~approximant, perturbation series, expansion coefficients.

Nature of physical problem This computer program contains subroutines which calculate any desired Pads approximant. Given the appropriate num-

References [1] H.M. Hofmann, Y. Starkand and M.W. Kirson, Perturbation theory and Pad~approximants in realistic largematrix models of the nuclear effective interaction, Nuclear Phys. A 266 (1976)138. [2] Y. Starkand, Weizmann Institute preprint.

1’. Starkand/Calculation of matrix Fade approximants

326

LONG WRITE-UP 1. Introduction

N TN(x)=1_~Bixi.

(5)

Many physical problems [1] are concerned with quantities given by their formal perturbation expansion

Then, using the quantities above, one finds:

W(x) =

[N+m,N] W(x)

N+m— 1

(1)

Fnxn.

+ [FN+m+

pansion coefficients Fn are then matrices of the appropriate order. The series defined in eq. (1) is said to be formal because it need not converge for every value of x. Nevertheless in many cases the first few terms of the series are the only information available on W(x). The Padé approximant technique is a popular procedure for dealing with such series in order to extract from them some approximation to the value of W(x), especially outside the radius of convergence of the series. The [N+m,NJ Padé approximant [2] is given by

in which:

=

SN+m(X) [TN(x)] -1

(2)

,

where SN+m and TN are polynomials in x of order N+m and N, respectively. Their coefficients are determined by equating to zero the coefficients of the first 2N+m+1 powers of the expression W(x) TN(x) SN+m(X). The program calculates the coefficients of TN and the Padé approximant itself, using the first 2N+in+l expansion coefficients Fn as input,

~I

i0 N—i N+m—i

The function W can be a scalar function or an operator-valued function of any desired order and the ex-

N] W(x)

=

~ ~

FiX’

(6)

X ‘F~B~.H k+m] xN+m[TN(x)1 —1,

i~’1k—i+m

BN

=

N—2 i—i N—2 [F2N_1+m,N+m,m+1J [1’2N_1+m,N+m N+m+1I

B,,

=

n—2 —1 [Fn+N+m_i,N+m,N_n+m+1I

for N> 1

[

N

X F~ n+N+m—1,N+m,N+m+1 2

I. mn+N+m_1,N+m,N_k+m~1Bk] ‘V~ ,-~n—2 kn+1

N

B

1

=

(7)

,

[FN+m11 [FN+n~1 —

k=2

for 1 1

B1

=

[FN+m]’ [FN+m+1J

(8)

N= 1

,

,

.

(9a) (9b)

—

2. The formula and method of calculation [2,3] Let F,, (n 0) be the expansion coefficients of the series and define Fm 0 (m <0). We define the following quantities: 2,K,M = FL_K+M FL Fk1FM, (3) ~‘

—

F

~

=

F~~j~j — F

K_n(F+,i,K,K_n

lRn—l - K+n,K,M’

for n > 0. Let B~(1

(4)

i ‘~ N) be the coefficients of the denominator polynomial TN(x), i.e. ~

The set of eqs. (6)—(9) defines [N+m,N] W(x) cornpie tely in terms of the input coefficients F,,. We will assume in the following that the inverse of all the needed matrices exist. The program is not in general expected to operate otherwise (we will show, however, in the test run that the program can sometimes overcome such difficulties). This formula is programmed to yield the Padé approxirnant. Using the Fn values it computes all the necessary quantities F2K,M for 0 ~ n ~N—2, beginning = 0. With every quantity F~KJ~,J we associatewith the ninteger NUM = M + K X 102

+L X

l0~+ n X

106.

(10)

FrKM is then stored in a three-dimensional matrix

FD(IA, IA, NMAX). Here IA is the dimension of the matrix FRKM as well as the dimension of the resultant approxirnant. NMAX is the number of quantities F~KMcalculated throughout the computation. The various integers NUM are stored in the vector LIST of

Y. Starkand/Calculation of matrix Fade approximants

length NMAX in ascending order. The element FrKM (i,j) is stored in FD(i,j, k) where LIST(k) = NUM and NUM is calculated according to eq. (10). The quantities F2KM are then used to calculate the coefficients B 1 ~ i ~Nwhich are used to compute the denominator and numerator of the approximant. -

3. Common variables The only common variable is the blank common variable NMAX (see text).

327

Subroutine NLKM: calculates NMAX and copies the the numbers NUM and the quantities FrKM to LIST and FD respectively. Subroutine SERCHF:extracts the quantitiesF2KM from FD on request. Subroutine PADE: calculates the Padé approxirnant and returns it and its denorninator for further use in the main program. 5. Machine-dependent features

4. Program subroutines The following subroutines perform various operations with square matrices, for which any other available (and maybe better) subroutines can be used instead. Subroutine COPY: Subroutine Subroutine Subroutine Subroutine

Subroutine Subroutine

Subroutine

copies an input matrix A into a returned matrix B. SUB: returns the result of subtraction of two matrices. ADD: returns the result of adding two matrices. MULT: returns the result of multiplying two matrices. XMULT: returns the result of multiplying every element of an input matrix by a constant x. ZERO: returns a matrix whose elements are all zeroes. MATIN1:returns the inverse of a matrix and was kindly given to me by H.M. Hofmann. INV: transfers control to subroutine MATIN1.

The other subroutines are:

Subroutine EXMA: extracts a two dimensional matrix out of a three dimensional one. SubroutineFOLKM: calculates the quantities defined in eq. (3). Subroutine FNLKM: calculates the quantities defined in eq. (4).

This program was developed and checked on the IBM 370/165 computer using small matrices, mostly of dimension 3, and calculating Padé approximants up to 2N+m 40. Memory difficulties may arise when many matrices of large dimensions are used or approximants of higher order calculated. In either case modifications can easily be made to keep most of the matrices on tape or disk, thus reducing the data kept in the active area to a minimum, and calling them whenever necessary. Nevertheless, for most anticipated uses no such difficulties are expected.

6. Use of subroutine

—

input and output

The program is invoked by a call statement from the main program (not supplied should be freely constructed by the user) to subroutine PADE, i.e. “call PADE (N,M,IA,PA,DPA,F,B,FD,LIST,NDIM, NSERI,A,C,D,E,X). The subroutine PADE has three kinds of arguments: —

6.1. Input arguments These arguments should of course be defined and transferred from the main program and are the following:

N,M

—

IA

—

NSERI

—

Integers which define the [N+M,N] Padé approximant to be calculated. The dimension of the expansion coefficients F,, and the resultant Padé approximant, all being square matrices. Number of expansion coefficients F~to be used. NSERI should be at least 2N+M+1.

Y. Starkand/Calculation of matrix Fade approximants

328

NDIM

—

F

—

x

—

Number of quantities F2KM which are expected to be calculated in the program. NDIM should be not less than N3—N. A three-dimensional matrix of dimensions (IA,IA,NSERI). The expansion coefficients Fn should be read into F in the main program so that the element F,,(i, /) n ~ 0 will be placed in F(i, j, n+l). The value of x at which the approximant is to be calculated,

6.2. Output arguments

PA

—

DPA

—

a matrix of dimensions (IA,IA) in which the Padé approximant will be returned. a matrix of dimensions (IA,IA) in which the denominator of the Padé approximant will be returned,

6.3. Working arguments B

a three-dimensional matrix of dimensions (IA,IA,NSERI) in which the coefficients B~will be stored, FD a three-dimensional matrix of dimensions (IA,IA,NDIM) in which the quantities F2KM will be stored (see text). LIST a vector of length NDIM. A,C,D,E working square matrices of dimension IA. No special preparations need be made for using the other subroutines accompanying subroutine PADE. —

The advantage of such a series i.e. 1+F+F2+F3+ is that we can calculate the exact expected sum using ...

.

~

F’~= 1/(1—F) ~0 Every Padé approximant should give the exact answer for this special case which is indeed the result as can be seen comparing the exact sum with the various approximants calculated in the test run. This further confirms the accuracy and correctness achieved in the computation. However, we draw attention to the fact that the quantities F&m are equal to zero for this series. These matrices are thus singular and cannot be inverted. The program will not be able to calculate F~mfor n > 0. As a result the program is in general not expected in such cases to be able to calculate [N+m,N] approximants for N> 2. This has become possible here because of rounding errors of the order of 10—14 in the calculation of the F~mquantities which made the inversion possible. This last example thus shows that the program can sometimes deal with pathological cases and give correct results even though we recommend not to use it when uninvertable matrices are expected.

—

Acknowledgements

— —

7. The test run In the test runs we show results for various Fade approximants. The input for the first one consists of 20 three-dimensional expansion coefficients of a perturbation series [denoted in ref. [2] by LRW (11w)]. The output are approximants of common use such as [2,1] [3,2] and other [N+l ,N] approximants, as well as many other [N+m,N] approximants for a great variety of possibilities for m and N. The results thus demonstrate the great flexibility and generality of the current program. Some of the approximants which appear in the test run were calculated separately with other existing codes [4] which gave the same results. The second example is that of a geometric series,

I am deeply indebted to Professor M.W. Kirson for very helpful and fruitful discussions and for reading the manuscript. I should also like to thank Dr. H.M. Hofmann for introducing me to the subject and for supplying me his codes which were very useful in checking the current program.

References [1] G.A. Boker, Jr., Advances in Theory. Physik, Vol. 1 (Academic Press, New York, 1965); ~ (Institute of Physics, London, 1973). [2] H.M. Hofmann, Y. Starkand and MW. Kirson, Perturbation theory and Pad~approximants in realistic largematrix models of the nuclear effective interaction, Nuclear Physics A 266 (1976) 138. [3] Y. Starkand, Weizmann Institute preprint. [4] H.M. Hofmann, private communication; M.W. Kirson, private communication.

Y. Starkand/Calculation of matrix Fade approximants

TEST RUN OUTPUT IA.NS0Pl.NPl,NPA0E2.I1r~.j0Uo 3 FXPANSIUN COFIFICIENT N4OE* 0.56400000000+0I 0.0 0.0 0.9*0fl0O0~’J0L,*0I 0.0 0.0 F4DANSIUN COEFFICIENT IJUMOFA I —0.12363000000+OI —X.b2~qoOo0Ooo4on 0.625901100000400 —0. 20493000000*01 —0.3000 b000 000 + 01 —0. SPbP 00) flU 00400 EXPANSION CUEFFICIFNT FJUJIBEJ) P —0.4335703I06000I —Q.00370310/70000 —Q.34013190820000 —0. /7b3139fl14040I —0.5832102?350400 —X.16472000300400 EXPANSION CUFIFICIENT NU~4~JJ 3 —0.50958115260+ufl —0.2II0FS~C5Cj1L,+00 —0.10635577100+00 —0.83900356151,000 O.1058100030040) 0.02000507000_UI E0PANS1U~I COE#F IC JEIJT N000TA 0 001b4000) o.7750)a404404Ufl 0. lPJP1E444l0*00 0.162+0346+40+00 0.+4595499060+Ofl 0. 14050 Q.9747341155u—0) FAPANSIoN CUFFFICIFNT 500J*EJ) S 0.15503500090—02 0. l513703007U+00 0.53002551140—01 0.52991345d70000 —0.45b55020350400 —0.1,6045)30)30—OI

EXPANSION COFFFICIFNT N008FA IA —0.2I909728800401 —0.45979614290+00 —0.1142029188040I —0.01110330380*00 —0.816651 78480+01 .0.90580014900400 EXPANSION COEFFICIENT NUMOER 19 0.2181 1600880+01 U.46930512800400 •0.53584592130400 —0.60350049710—OI 0.46460898680401 Q.99b71135410+00 I 1 P806 AP070!MANT 0.73749555670+01 0.76835441390+0fl 0.74960540430400 Q.99094017000+0I 0.1305551220040l 0.74535353240+00 2 I DADE APRPXIMANT —0.26977368350+01 —0.1306605?800+OI —0.10902436910+01 —0.17563943640+01 —0.22391207940+0I —Q.5616954591L)400 3 1 DADE APOnOIMANI —0.23491184330401 —O.1149978A990+0I —0.10266995A00+0I —0.1127.’735670+0l —0.20198651860+0I —Q.58682972060+00 6 I DADE APWO*IMANT —0.22592092060+01 —0.1183403635L1+0l Q.34887634850+00 0.L1962473650401 0.1783551607D+0I 0.1574694283000I 5 I DADE ADWI)XTMANT —0.16377013850401 —O.1050418942000I —0.10163166310401 O.170249b3410—0I —0.3129487856D+0I —Q.8I96329651L)+0fl I 2 DADE APWOXIMANT 0.18952460620+01 —0.72226059680+00 —0.6763073196D+00 0.07b93553030+01 —0.23635936810401 —0.50323514820+00 2 2 DADE AP000!MANT —0.16975090600+0I —O.104A8597750+0I —0.90214569080+00 —0.13138074350+00 —0.25243126750+01 —0.05330138570+00 3 2 DADE APR04 IMAN1 —0.15720648500+01 —Q.97161038760+00 —0.97807604600+00 —0.95651245030—02 —0.30083096190+OI —0.05352135130000 4 2 DADE AP404I4ANT —0.16688776470401 —0.10407704010401 —0.I0121725760+0I —Q.2327b829900+00 —0.28123103880+01 —0.61671630300+00 5 2 DADE ADADAIMANY —0.16736684930+01 —O.10605065850+OI —0.10247671700+01 —0.27903064920+00 —0.27265777370+01 —0.592167b4050+90 6 2 DADE AP400TMANT —0.17103975090+0I —0.10521918370401 —0.10392253720+01 —0.0579169331D+00 —0.2746860$16D+OI —0.50074754501)400 1 3 DADE AP+OXIMANT —0.87b118b113D+00 —0.1604I953950+0I —0.24450667I30+0I Q.5A070904640+0I —0.3697406I840+0I —0.13bA6217790,0I 2 3 DADE AD004IMANT —0.17780496130+01 —0.10006531 110+01 —0.76122534850400 0.13461306420—01 —0.28747036050+01 —0.61308088730+00 3 3 DADE APROXIMANT —0.16325311810+0I —0.102?4930b00+0I —0.9671 1560960400 —0.971 14382130—01 —0.26301151000401 —0.1,3440161940+X0 4 3 DADE AP+OXIMANT —0. 167649439AI)+01 —X.1064b829170+Ul .0.1027I052450401 —0.088673058904*0 —0.26864026930+0I —Q.57233250800400 5 3 DADE ADWITOIMANT —0.16666A70040+0l —0.105718b0850401 —0.10263#16590+0I —0.27119334000+00 —0.2712b342820+0l —0.S0335708540000 5 3 DADE APWflXIMANT —0.1666455539000I —U.10704707991)+01 —0.I0007267610401 —O.25843174400+00 —0.27331836b6040I —Q.4175315245L)400 7 3 DADE 40400TMANT —0.17146885920+01 —0.10041A2A530+0I —0.105059b9960401 —0.08907685410+00 —0.27030)SPSTL)X0I —0.61)821425070+00

.~O 0.0 0.18380000000+02 •U.300Ab0000l’000I —0 .52b2C)000000000 —0.07000000000—0 I —0.10175078140+XI —U.053481b7980000 *.10438336070000 0.I1I+5675800401 —0.018507020311—0I 0. LI539Qb77~0*0l 0.04579770070+X0 0.17043011530.XI O.449)bP4S7R0401 .JJ.NS3I5200)40+OO 0. 1303U00~o7L)+0fl 0. IP4ISI8057040I

—0.10091011370+0? 0.13940245180+0I —Q.21048806700+02 —Q.12524787390+02 —0.12160902190+01 0.24760225330407 0.14o38794o20401 0.77358395110+00 0.204671I5770+02 —0.0416153412L)+UI —0.86894007931)400 Q.598A8*1877040I •0.34b0258645)3+0l —0.80306673670+00 O.6806497*500+0l —0,3000776586L)—01 —0.20123585120+01 O.1601074531D+0l —0.63962048800+00 U.40343820060+00 0.I*703772780+02 —0.24080829900+01 —0.58254017190+00 0.11079A94550+02 —O.1A445320780+0I —0.45183974070+00 0.10288014780+02 —0.22664280530+01 —0.1f461536360—01 0.10653828210+02 —0.27202135440+01 —Q.I7295527940+Q0 0.95203622660+0I —0.26939385080+0I —0.21335351610+00 0.93b64#6I470+0I —0.23083654620+01 —0.12865737170—01 0.96322715110+01 —0.33903081200+01 —O.16061331200+OI 0.10085625960+02 —0.21616793030+01 —Q.1631233184L)+00 0.10415933760+02 —0.17041390330+01 —0.31b53063760+00 0.11115004030+02 —Q.26936389090+0l —0.21956612950+00 0.92708416150401 —0.27701786840+0I —0.20933040170+09 0.92I399577AL)+0l 0.07074709930+0I —Q.290A649A1I30+00 O.94501450210+01 —O.24330186480*OI —0.86353405960—01 0.93070635160+0I

329

330

Y. Starkand/~’alcu1ationof matr& Padé approximants

I 7 DADE APWOXI’4ANT 0.132l7676200—0I —0.25886793540+01 —0.13845390470+01 0.40092144440+OI —0.21374180540+01 —0.I1001?96I30+OI 2 7 PADE AP+OXIMANT —0.1958b646440+QI —0.1070+462710+Ql —0.11977038910+0I —0.26046195540+00 —0.28457167340+01 —0.b0427731140+00 3 7 DADE APMDXTMANT —0.185?378934D+OI —0.I0784572860+Q1 0.11006075440+0I —0.04128906840+00 —0.29172055230+01 —0.54394019140+oO 4 7 DADE AD400IN0NT 0.22IIb9Q29A0+OI —0.1119904873LJ+OI —0.131314070.20+01 —0.3212468I140000 0.26650657400+0I —0.57042286270+00 5 7 DADE API+OXIMAFoT —0.273I0180780+Ql —0.13257496610+01 —0.19446504870+01 —0.46680316800+00 —0.00167137150+01 —0.41526?2IASD+Qfi 1), 7 DADE AP+OOIMANI 0.158I7004I00+QI —0.11)31 7689430+0! —0.8800060060)3+00 —0.02260766690+00 0.26038061350+0l —0.b0b44344420*00 7 7 DADE APAOXII4ANT —0.17119732520+01 —0.I0051?39300+ol 0.10460196020401 —0.25962796300+00 —0.270A2359950+0I —0.5901 7008b00A00 8 7 DADE ADROXIM0NT —0.1584570704D*QI —0.I0521200210+0l —0.9776960507L)+0fl —Q.2A965+3553L)AQ0 —0.2758000690)3+01 —0.59711#82930+00 0 7 DADE AD000IMANT —0.15533024020401 —O.I03054b4390401 —Q.82968557560+00 —0.2257+362360+00 —0.08252075I40001 —0.o0d5305~040+00 tO 7 P406 APWC)OIM0NT —0.I4IEE64QIAU+Ul 0.10089?03330+0I —0.b3419680700+00 —0.1952I604300+UT —0.2886Q973751)+QI —0.61 763A23560000 11 7 DADE AP+TXIMXOT —0.I4OIYSJ?570+Ul —0.LOONOMbXU6O+ol —0.63388995040*00 —U.1954IOb703O400 —0.26781553900+0I —0.b1070415020+00 MPLI OF A GEOI*FIRIC SERIES EXA EXACI SUM —0.00000000000+01 X.J.0000000000+01 0.10000000000+0I U.10090000000+0l O.20000000000401 -0.3000000000000l EXD4NSION COEFFICIENT NUMAFA 0 0.10000000000+0I 0.0 0.0 0.I0000000000+0l 0.0 0.0 EXPANSION COFFFICIENT NUMBER I —0.10000000000+0l 0.10000000000+01 0.10000000000A01 0.10000000000+01 —0.10000000000+01 —0.20000000000+QI EADANSION COEFFICIENT NUMBER 2 0.50000000000+0I 0.6000000000040I 0.10000000000+0l 0.00000000000+QI 0.40000000000+OI 0.I0000000000UOI EXPANSION COEFFICIENT NUMBER 3 —0.IA000000000+02 —0. 13000000000+02 0.80000000000+OI 0.30000000000+OI —0.11000000000+0? —0.15000000000+0? EXPANSION COEFFICIENT NUMBER 4 0.59000000000+0? 0.6I000000000++2 0.21000000000+02 0.25000000000+02 0.45000000000+02 Q.34000000000+02 EXPANSION COEFFICIFNT NUMbER S —0.21500000000+03 —0. lA800000005+03 —0.83000000000+02 —0. 10000000000+0? —0.I4600000000003 ~0.14500000000+03 EXPANSIUN COEFFICIENT NUMBER 6 0.75600000000+03 0.6030000000)3*03 0.27800000000+03 0.2630000000)3+03 0.52700000000+03 0.4730000000)3+03 EXPANSION COEFFICIENT NUMBER 7 —0.25950000000+04 —0.0315000000)3+04 —0.96500000000+03 —0.90300000000+03 —0.18190000000+04 —0.16920000000+04 EXPANSION COEFFICIENT NUMBER A 0.90370000000+04 0.83540000000+00 0.34290000000+04 0.3I640000000+04 0.63840000000+04 0.58730000000+04 ERPAN5IUN COEFFICIENT WUM8FR S —0.316I8000000+05 —0.00I37000000+05 0.11902000000+04 —0.1I063500000+Q5 —0.22279000000+05 —0.00555000000+05 I I DADE APRI)XTMXN0 0.20000000000+OI 0.4000000000000I 0. I000000000000I 0. I00000tJ000L)*0l 0.2000000000040I —0.3000000000L)+0I 2 I DADE AP000IMANT —0.20000000000+01 0.40000000000+0l •0.10000000000+OI 0.1000090000U+0I 0.20000000000+0I —0.30000500000001 3 I DADE 42400740100 —0.20000000000+0I Q.o0000000000+0I —0.I000000000000I 0. l0000000000AUl 0.20000000000+0l 0.30000fl0”000401 4 I DADE 4M40010057 ..0.20000000001)+OI U.00000000000+0) —0.10000000000A01 U.100000000710*U) 0.2000000000))+UI U.3000000200000l 5 I DADE AP*0UI*ANT 0.20000000000+UI U.0000000000040) —0.I0000000000+OI 0. I0000000000A0) 0.2000000000D~07 —U.3000000000U+0)

0.18669+60230+0I —o.boAMlaASo4o—oI 0.+9033759400+QI •0.17194261770+QI 0.25300096b40+00 0.10071659340+02 —0.20612312260+01 —0.I6158631550—0l 0.10520304550+02 —0.$0942177II0400 0.IOI9I6QIO2O+OI 0.92I76694I90+OI •

0.107479279+0+01 O.3804b965410+01 0.66369094b60+Ql —0.3001 7230330+01 —0.9644I9184I0+00 0.97830946230+01 —0.01598775450+Ol —0.72537613220—01 0.93632383630+01 —0.2S4579277S0401 —0.37382430100+00 0.90842488620+QI —0.31 121383420401 —0.10I39544860+OI O.98741441740+0l 0.37069303I40A01 —U.I86A52I2570+OI 0.10130066960+02 —0.--377235883A0+OI —O.1867447157040I 0.I0104972250*02

0.I000000000U+UI 0.I0000000000+QI —0.I0000000000*QI 0.0 0.0 0.10000000000+0l —0.3000000000U+0I —0.10000000000+0I —0.I0000000000+0l 0. 10000000000+0l 0.30000000900A01 0.60000fl0000040I —0.08000000005+02 —0.10000000000+02 —0.19000000000402 0.9500000000)3+02 0.37000000000+02 0.67000000000+02 —0.33300000000+03 —0. 12500000005+03 —0.23600000000+03 0. IISF’OOOOOOO+OM 0.44400000000+03 0.81 900000000+03 —0.40670000000+04 —0.15410000000+04 —0.26730000000+04 0.I4027000000+05 0.53990000000+04 0.10022000000+05 —0.49690000000+05 —0.18850000000+05 —0 • 35047000000+05 0. I000000000040I 0.I0000000000+OI —0.10000000000+01 0.I0000000000+OI 0.I0000000000+0l —U.I0000000000+Ql 0.I0000000000,0l 0.I0000000000+01 —0.L0000000000+01 0.10000000000+OI 0.I000000000L3+0I —U.I0000000000+0I 0.I0000000000+0l 0.10000000000+0l —0. I0000000000+0I