SIPSOL — A suite of subprograms for the solution of the linear equations arising from elliptic partial differential equations

Computer Physics Communications 17 (1979) 383—391 © North-Holland Publishing Company

SIPSOL A SUITE OF SUBPROGRAMS FOR THE SOLUTION OF THE LINEAR EQUATIONS ARISING FROM ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS -

C.R. JESSHOPE Dept. of Computer Science, University of Reading, Whiteknights, Reading, UK Received 13 October 1978; in revised form 2 February 1979

PROGRAM SUMMARY Title of program: SIPSOL

Nature of physical problem SIPSOL is a suite of subprograms for iteratively inverting a set of linear equations obtained from the discretisation of an

Catalogue number: ACZL Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue)

arbitrary elliptic partial differential equation over a rectangular grid of mesh points. Method of solution A strongly implicit procedure using Stone’s algorithm [11.

Computer: CDC 7600; Installation: University of London Computer Centre Operating system: SCOPE 2.1.4

Restrictions on the complexity of the problem In this version the mesh must be rectangular, although arbitrary boundary conditions may be specified. A five point difference operator must be used, giving a sparse five diagonal matrix system. The mesh may be non-uniform. A subprogram is supplied for calculating iteration parameters for the method, these are calculated for the general anisotropic diffusion equation; as a function of the mesh, or as a scalar.

Programming language used: FORTRAN High speed storage: iON + C words, where N = number of mesh points and C moo Number of bits in a word: 60 Overlay structure: none

Typical running time For a mesh of 1024 points, a reduction in the L2 residual norm of 6 orders of magnitude can be obtained in 0.2—1.0s on a CDC 7600.

No. of magnetic tapesrequfred: none Other peripherals used: card reader, line printer

Unusual features of the program The surface output routine LIST in the test procedure uses AlO format (CDC 7600).

No. of cards in combined program and test deck: 1021 Card punching code: IBM 029 Keywords: general purpose, Stone’s algorithm, strongly implicit procedure, iterative, elliptic, partial, differential equations

Reference [1] H.L. Stone, Siam J.Numer. Anal. 5 (1968) 530.

383

CR. Jesshope / SIPSOL

384

—

Subprograms for solution of linear equations

LONG WRITE-UP 1. The Strongly Implicit Procedure (SIP) The SIP method solves a set of linear equations obtained from the discretisation of an arbitrary elliptic problem using a five point difference operator. The

~

°1ri r 0

N

ATCI)

=

(A +E)x’~’ (Ax’— b) —

where A + E has a sparse LU factorisation as shown in fig. 2 and eq. (3): A +E=LU. 0 and defining Given a trial solution x and an initial residual by (4) and (5),

an

I

(3) increment

XCI) X)I.1)

I

=

[

B(I)

~x)I~n)J

[IA,]

(2)

,

I

~

(1)

The method of solution uses the implicit iteration procedure: (A +E)Xi

ABlI)

0

matrix system thus obtained is shown in fig. 1 and eq. (1): Axb.

\

AL)I)1OARW

Fig. 1. Matrix

system to be solved.

instability. It has been shown [1—3]that a suitable maximum parameter for the diffusion operator (6) can be calculated from (7),

ax

ax

al/f ay fy—, ay

(6)

+~

(4) R° Ax° b,

(5)

the iterative procedure becomes: For lIR’~lI 2/IIxi~112
2.0 (1.0— amax)= minf 2 1(x) ~f~(~2’ ~ \byJ 1x~2 y 2 ÷ fx fy \bxl (bY)

—

J

—

1 j

(7)

— — —

The method for calculatingL and U is described in ref. [11. The coefficient matrix A is augmented by an error matrix E, which has two diagonals lying inside and adjacent to the outer diagonals of A. The algorithm is designed so that terms in Ex are small. An iteration parameter a is used to calculate L and U such that 0 ~ a < 1, this may be a function of the mesh. For a = 1, the cancellation in the terms of Ex is second order, but the iterative scheme is unstable. It has been shown [1—3]that a cycle of iteration parameters gives an improved rate of convergence. This cycle should contain a relatively dense distribution of values near am ax, where amax is the maximum parameter which can be used before the onset of

where x andy are the total region sizes and ox and by are the If frshown and fy[2,31, or Ox that and the by vary over themesh mesh,spacings. it has been rate of convergence can be improved by using a vector of parameters calculated from (7), for all points on the mesh. Alternatively, if the store requirement dictates it, an average of the values over the mesh given by (7) can be used as a scalar parameter.

I ~

TL2

TURD

~

0

1

~

1.OTU1

TU2

~~ =

0 [LI

AT

AL~OAR ~AB

~

[u] Fig. 2. LU factorisation of A

[~4+E] + E.

1

CR. Jesshope /SIPSOL

—

Subprogramsfor solution of linear equations

The recurrence relations programmed for steps 1, 2 and 3 of the iterative procedure are given below, for mesh point i in a mesh with n points in the first direction scanned in the mesh. Step 1

385

2. The SIPSOL suite of subprograms The SIPSOL suite of subprograms consists of the following FORTRAN subroutines.

(8a)

SIPNIT initialisation and checking procedure SIPARM parameter generation procedure SIPERR error handling procedure

(8b)

SIPSOL

(8c)

These routines are all well documented, but a fuller description of each routine, its input/output and action is given below. 2.1. SIPSOL(ND, NSIP, AT, AL, AR, AB, B, X,

—

—

TL2~= AT~/(1.0+ atTUl~_~), TL1, = AL1/(1.0 + a1TU2~i), TLD1

=

1.0

+

TL11(a~TU2~1 TU111) —

+ TL2~(a~TU l~_~ — TU2~~),

TUl~= (ARe

—

a~TL2tTUlt_~)/TLDj,

(8d)

TU21

—

atTLltTUl~_~)/TLD~,

(8e)

=

(AB,

—

=

This is the routine which performs the solution of

(R~ TL1~V~i TL2iVi_n)/TLD, —

(9)

the set of linear equations. It may be used in conjunction with SIPERR only, if the required iteration parameters are supplied on

(10)

input. However, it is recommended that SIPNIT also be used faulty as this input provides against data.some measure of protection

Step 3 Ax,

=

V,

—

TUl~Axt÷i TU2,~Ax~+~ —

.

The procedure is optimised by combining steps 1 and 2 within “one” loop, thus performing the LU decomposition and forward solution element by element. This saves 3N storage locations as the lower triangular matrix elements are stored as scalars within the loop. The backward solution is performed in a separate loop. Note that the coefficient matrix elements are normalised so the major diagonal is assumed to be unity, The loops are coded in parts so that at the edges of the mesh the array indexing does not go out of bounds. Depending on a program switch, either a vector or scalar parameter is calculated. If a scalar parameter is selected, an average or scalar value is calculated from (11) amax

=

SIP solution procedure

ALPHA)

Step 2 V,

—

1.0— mini 2 ~l/n

2.0 2) fy/(fx m

+

1

2.0 fx/(fyn2) +l/m2j

(11)

where n is the number of mesh intervals in the x-direction and m is the number of mesh intervals in the y-direction (i.e. N, the number of mesh points = (n + 1)(m + 1)). If a vector of parameters is chosen the elements are calculated from (7).

Input to routine ND this is the dummy dimension parameter for arrays AT-, AL, AR, AB, B and X. These arrays should be dimensioned in the main or calling program and ND should be set to the dimension of the array, NSIP ditto for array ALPHA. This array is treated separately as it may in fact be a scalar (dimension = 1) see description of SW1 in common block /SIPCON/, AT, AL, these four arrays contain the coeffiAR, AB cients of the linear equations to be solved, the ith element of these arrays gives ith fig. row1).ofThe the major coefficient matrixthe (see diagonal is assumed to be unity and the off diagonal coefficients and right hand side must be normalised accordingly, —

—

—

B

—

X

—

this array contains the right hand side of the coefficient equations, this array contains the trial solution to the iterative procedure. A good trial solution is not necessary as fundamental errors decay very rapidly,

C.R. Jesshope / SIPSOL

386

ALPHA

—

Subprograms for solution oflinear equations

this parameter may be a scalar or an array, depending on the value of SW1, see common block /SIPCON/. It contains iteration parameters, a scalar for global use over the entire mesh, or a vector for local use at each mesh point,

NCALL

=

I

—

post-initialisation call. In this mode, various checks are made on the control parameters. In addition the iteration parameters are set by a call to SIPARM. This call must therefore be made after the mesh for the problem has been set.

Output from routine

ND, NSIP AT, AL,

—

unchanged,

—

unchanged,

AR, AT B

—

X

—

ALPHA

—

this array is overwritten with the current residual vector, this array is overwritten with the current approximation to the solution of the coefficient equations, unchanged.

Line printer output The level of output is controlled by SW2, see common block /SIPCON/. On full output residual norms and accuracy are printed at the termination of each iteration cycle. On minimal output the same information is output only at the termination of the

Input to the routine NCALL = 0 the only input required are the —

NCALL = 1

—

dummy dimension parameters ND and NSIP, in addition to ND and NSIP, the only input required is the anisotropy coefficient, which is passed to SIPARM via the array/scalar ALPHA. See section on SIPARM.

Output from the routine NCALL = 0 on exit, all arrays contain default default data, NCALL = 1 on exit, ALPHA will contain the required iteration parameter(s). —

—

solution.

External references SIPERR SIPSOL suite error handler, SQRT system library routine, IFIX system library routine. — —

Line printer output Display messages are output in both modes. If full output has been selected a checklist of the control parameters will also be output.

—

Common blocks accessed

1/

/NGEOM/ /NCHAN/ /SIPCON/

— — — —

blank common workspace, mesh geometry, FORTRAN channel numbers, control information.

External references SIPARM SIPSOL suite, SIPERR SIPSOL suite, IFIX system library routine. — — —

Common blocks accessed

These are described in more detail in the section headed Common blocks to avoid repetition.

/NGEOM/ /NCHAN/ /SIPCON/

2.2. SIPNIT(ND, NSIP, AT, AL, AR, AB, B, X, ALPHA, NCALL)

2.3. SIPARM (NSIP, ALPHA)

This routine is used for initialisation. It has two actions, depending on the value of the parameter NCALL NCALL = 0

—

pre.initialisation call. In this mode the control parameters and arrays are preset to default values,

— — —

mesh geometry, FORTRAN channel numbers control information.

This routine calculates the iteration parameter(s) for the operator given in (6) by eq. (7). The iteration parameter can be either a scalar value or a vector of values, depending on SW1 in common block /SIPCON/, see section on common blocks. If the mesh spacing is non-uniform or the anisotropy coefficient (=fx/fy) varies over the mesh, it is recommended that a vector

C.R. Jesshope /SIPSOL

—

Subprograms for solution of linear equations

of parameters be used. In this case two common blocks must be supplied (/XMESH/ and /YMESH/) which give the mesh spacings in both coordinate directions, for a vector of parameters this holds even if the mesh is uniform. If a scalar parameter only is reqdired then these common blocks are not accessed. In this case NSIP should be set to 1 and ALPHA dimensioned accordingly.

Error

387

Severity

Description

FATAL

ERROR IN GEOMETRY— (NOx+I)(NoY+1) N or N exceeds the array bounds.

no.

1

See common block /NGEOM/. 2

FATAL

ERROR IN PARAMETER CONTROL SWITCH SW1 is not 0.0 or 1.0 or SWI = 1.0 and N exceeds the bounds of array ALPHA. ERROR IN INITIALISATION CALL PRE-INITIALISATION ASSUMED NCALL is not 0 or 1. ERROR IN SEQUENCE OF INITIALISATION CALLS —

Input to the routine NSIP dummy dimension parameter for array alpha, ALPHA must contain the anisotropy coefficient on entry. Either a scalar value representing the constant or averaged value over the mesh, for a scalar parameter; or an array of values representing the value at each mesh point, for a vector parameter. The default value of 1.0 is set in the pre-initialisation call to SIPNIT (NCALL = 0). N.B. This input must be passed to SIPARM via the post-initialisation call to SIPNIT (NCALL = 1). —

—

Output from the routine NSIP unchanged, ALPHA contains the iteration parameter(s). —

3

WARNING

—

WARNING

—

post-initialisation call occurred before a pre-initialisation call. 5

WARNING

—

6

WARNING

—

Line printer output None. External references FLOAT system library routine, IFIX system library routine. — —

INITIAL IMPLEMENTATION ARRAY BOUNDS EXCEEDED BY N blank common must be redimensioned in subroutine SIPSOL. PARAMETER SWITCH RESET SCALAR PARAMETER USED FROM LOCATION ALPHA(1) parameter switch = 0.0 but dimensions —

7

WARNING

of ALPHA exceeds 1, a scalar parameter is used from location ALPHA(1). FAILSAFE EXIT REQUIRED ACCURACY NOT OBTAINED iteration loop exit was made on iteration count, required accuracy was not obtained. —

Common blocks accessed /NGEOM/ mesh geometry /SIPCON/ control information /XMESH/ mesh spacing x-direction /YMESH/ mesh spacing y-direction.

2.5. Common block description

2.4. SIPERR (NERR)

Blank common and labelled common are accessed by the SIPSOL suite of subroutines.

This is the SIPSOL suite error handler. A list of error messages and their meaning is given below.

Blank common This is used only as scratch workspace. In this

— — — —

388

CR. Jesshope /SIPSOL

—

Subprograms for solution of linear equations

release of routines blank common is dimensioned to 3072 elements in subroutine SIPSOL. The amount of space required is 3N elements where N is the number of mesh points, Labelled common Each labelled common block with the exception of /XMESH/ and /YMESH/ contains a ten word record. Not all of these elements are used, which allows flexibility for future expansion and/or interfacing requirements. FORTRAN real/integer naming conventions are used in the common block labels to indicate the word type of all ten elements in the block. The common blocks are listed below, giving the elements in sequential order, /NGEOM/ contains mesh geometty N number of mesh points in the mesh (=(NOX31)(NOY+1)). NOX number of intervals in the X co-ordinate direction, NOY number of intervals in the Y co-ordinate direction. + 7 unused elements, —

—

/XMESH/ DX(100)

/YMESH/ DY(l00)

—

— —

/SIPCON/ CHECK ACC FAILCN

SW1

—

contains mesh spacings as /XMESH/ for the Y co-ordinate direction. NOY values are required.

—

2. SIPNIT (NCALL = 1) is the second call in the sequence. This call must be made after the mesh and problem specified. Checking is performed, and ahave call been is made to SIPARM to calculate the iteration parameters. This requires that mesh and problem dependent information be available at the time of call. Any change in default parameters, mesh spacing or anisotropy coefficient must be followed by this call before entering SIPSOL itself. 3. SIPSOL is the last call in the sequence. This may be called repeatedly, without re-initialisation provided that none of the above information has been changed.

contains SIPSOL control information

— — —

—

initialisation check variable, fractional exit accuracy (default = l0~), failsafe exit for convergence failure, limits the number of iteration cycles attempted (default = 20.0), controls local/global iteration parameter(default 1.0), 1.0= local parameter, array ALPHA must be dimensioned to at least N elements, 0.0 = global parameter, array ALPHA be dimensioned as a scalar, ELSEmay = fatal error,

SW2

—

1. SIPNIT (NCALL = 0) is the first call in the sequence. This may be performed before any other action is taken in the program. It sets the default control parameters and initialises all arrays used in the SIPSOL suite.

—

—

contains mesh spacings contains the mesh spacings in the X coordinate direction. These must be supplied if local iteration parameters are used. NOX values are required.

—

2.6. Calling sequence

—

/NCHAN/ contains FORTRAN channel numbers NIN card input (default = I). NPM printer monitor channel (default = 2). NPO main printer output channel (default = 6). + 7 unused elements.

—

—

controls level of output (default 1.0 = full output, 0.0 ELSE

= =

minimal output, action undefined.

=

.0),

2. 7. The test program A test program has been written to set up and solve a series of diffusion problems (12), with no interior sources or sinks.

a

a~ a ax ay

a~ ay

~—fx—~ +—fy—--0,

ax

where fx and fy are given by:

(

(12)

(

j~ocos( 2xx \ iT)’ \\ (13) fx = fy = exp —) cos —--—I) 2Ymax/ and Xmax and Ymax arexmaxj normalised to unity. The range of problems is achieved by the variation of the parameter ,f showingstiffness. how the method can cope with problems of0,different f0

=

~0, 10, 20, 30)

C.R. Jesshope / SIPSOL

—

Subprograms for solution of linear equations

These problems are solved twice, once with a scalar parameter and a second time with a vector parameter. Although all program paths have been tested, this test case checks the main body of the code concerned with the solution procedure. The results from the two sets of problems should match as fx = fy and the mesh is uniform, A unit square mesh is used with 1024 uniformly spaced mesh points, i.e.

NOX = NOY = 31, N = 1024, DX = 1/31, DY = 1/31.

These values are assigned to the program variables in the subroutine INPUT, so no card input to the test program is required. The boundary conditions specifled in the test problem are that the normal derivatives vanish on all of the boundary except for two outer quarters of the top side, which are constrained to take on the solutions i/f =f0 and i/f = exp(f0), respectively, i.e.

forx

=

0 andx

fory

=

1:

fory

=

0,

=

1.0:

al/i/ay

=

0,

aip/ax = 0,

0.25
aip/ax = 0,

fory0,x~0.75:

/ff0.

i/i =

this contains initialisation and problem subroutine calls only, and outputs headings to each channel. S/R INPUT — input routine, this sets program variables by assignment. S/R ANAL — analysis routine, this contains the main program loop setting up and solving the different problems. S/R EQNS — equation set up, this sets the four off diagonal coefficients of the matrix of equations. S/R BNDRY — boundary correction, this routine overwrites the equations with the boundary corrections. S/R DNORM — diagonal normalisation, this routine normalises the equations so that the diagonal is unity. N.B. For this problem the diagonal element is equal to minus the sum of the other four elements. Functions FX and FY, these calculate the diffusion coefficients specified as a function of X and Y.

exp(f0), References

A brief description of the test routines follows: PROGRAM TEST

—

main program,

389

[1] H.L. Stone, Siam J. Numer. Anal. 5 (1968) 530. [2] CR. Jesshope et a!., Electron. Lett. 11(1975) 285. [3] C.R. Jesshope and J.W. Eastwood, Reading University Department of Computer Science Report, RCS 89.

C.R. Jesshope / SIPSOL

390

—

Subprograms for solution of linear equations

TEST RUN OUTPUT

SIPSOL TEST PROGRAM

SIPSOL SUITE I INITIALISED :5:5355:525.

******PSCI..

ERROR ~tj,

SIPSOL SUITE

6 WARNING •

PARAMETER

SWITCH RESET

I CHECKED

CHECKLIST I EXIT ACCURACY • MAXIMUM CYCLES S 2B.i ONE CYCLE a 5 ITERATIONS GLOBAL ITERATION PARAMETER PROBLEM CYCLE CYC.LE cyc~F CYCI.E CYCLE CYCLE

~ 1 2 I 4 S

1 FO : L2 NORM • L2 NORM = L2 NORM L. NOR’ Li NORM Li NOB’ ~

PROBLEM CYCLE CYCLE CYCLE CYCLE

0 I 2 3

2 Fi 10.0 Li ‘4ORM : 2,RR5E—O1 ‘lAX Li NORM 5.~’)8F~24‘lAX Li NORM 8.39YF—05 lAX Li NOR-I X.821E_~I6 ‘lAX

2,652E—01 4,BOBE—04 I .2681—04 3,279E—~5 1 .273E05 2.461 E_nlN

~isx MAX MAX MAX MAX MAX

ALPHA •

NOR~1 • NORM NORM NOB’l NORM

• NI)RM S

•99895R4

0, 3,344E—03 6~188E04 2,8571—04 6•083135 3~b56E—05

N~R’4 =

1,1041.03

‘4’~’1 NORI

3,7741—02 =

8,045F03 1.R5RF—13

PROBLEM CYCLE ‘l CYCLE I CYCLE 2 CYCLE 3

3 CLI 5 20,0 Li LOB” = 3..1711..iI NORM 5 9F .34 lAX lAX NOPI Li NORM ‘JOB 6,45 Li 3.951E—i6 MAX NORM S Li NOH = 2.55iE~R6 “~x ‘:~p~

1.8031+137 2. Bt6F—0t 3.5741—132 7,666E—~3

PROII)F’l CYCLE CYCLE CYCLE CYCLE

4 Ccl 5 Li NORM L2 NOR” Li NOR Li ‘JOllA

3,2441+11 4.582F—OI 7,504E—02 1.409E~’l2

13 1 2 3

SIPSOL. 5’’I CE

114,2 S 3,628E—.~I = 5,877E—04 = O•pI3C~Mb S I,422F..36

NORM

lAX NJf)R,l -‘AX ‘0RU ~4AA TriM” ‘lAX

‘J’lPN

S

S r

CHECKED

CHECKLIST EXIT ACCIJRACY S I .~3~E—’i5 ‘IAXIft)M CYCLES 20,0 ONE CYCLE = 5 ITERATIONS LOC4L ITERATION PARAMETERS PROBLEM CYCLE 0 CYCLE I CYCLE 2 CYCLE I CYCLE 4 CYCLE 5

I Li Li Li Li L2 L2

CO S NORM ‘10PM NORM N~1Il1 ‘JORI ‘10PM

PROBLEM CYCLE 0 CYCLE I CYCLE 2 CYCLE 3

‘0 FO Li NORM S Li NORM 5 L2 NØRM a L2 ‘JOR’l 5

PROBLTM CYCLE 0 CYCLE 1 CYCLE 2 CYCLE 3

3 FlU Li NORM L2 NORM Li NORM Li NORM

2~652E~iIMXX NORM

0, S

3,344E03

a 5 S

6,189E04 2,857E—04 8,083Eaø5 3.5561.05

MAX NORM a MAX NORM a MAX NORM a MAX NORM 5

1,184E1’03 3.774E—R2 6.045E03 1.969E03

•

3,371E—01 MAX NORM a 6.4591.04 MAX NORM O,95i135 MAX NORM a

1.893E+07 2.616E01

a

2.5511.06

7,666E03

= 5

= a

4.808E—24 ‘AX i.268E~i4 MAX 3.279E05 -lAX 1.27~E~5 MAX 2.462E.06 MAX

2,0951.01 6,5981—44 8.399E—05 8.8211—136

NORM NORM NORM NORM NORM

20.0 S

MAX NORM

PROBLEM 4 10 a 30.0 CYCLE 0 Li NORM a 3,628E.01 MAX NORM I CYCLE I L2 NORM a 8,e77E—~~MAX NORM a CYCLE 2 L2 NORM a 3,233E—05 MAX NORM a CYCLE 3 L2 NORM a 1.422t.06 MAX NORM •

3,5?4r02

3,244E411 4.5S2t.01 704~02 1.4S9Ea02

SCALAR PARAMETER USED pROM LOCATION ALPHA(1)

C.R. Jesshope /SIPSOL OUTPUT BLOCK

YB YB Ys V. YB YB YB Vi Va YB Y YB Y~ ys Ya Va Vi VS

1 2 3 4 5 6 7 8 9 10 11

12 13

OF X,F0S0,0.GLOBAL 1

OF X,F0:0.0.GLOBAL 1 1,0001+00 9.6091—01 9.2281—01 8.8601—01 8.5151.01 8,1941.01

—

Subprograms for solution of linear equations

PARM PARM

7.6291.01 7.383E01 7.161E—01 6,9591—01 6,778E—01

2 1 .001’E+40 9.6041—01 ‘).218E.LII 8.8491—131 8,5031—141 8,1821—41 7,8861—01 7.6171.141 7.372E—OI 7.1501—01 6,9541.131 6.7891—01

3 1 .OIILiE+04 9.5911—01 9.1931—01 8.8161—141 8.4661—01 8.1411—01 7.8481—01 7,5801—01 7.’l371.01 7.1181—01 6,0201.01 6.7421—01

1 1,0001+013 9.5651—01 Q,147E—0l 8.7571~01 8.4041—01 P.13761—01 7,7831.01 7,5181—01 7.2791.01 7.0651—01 6.8711—01 6.6971—01

7.8991—01

6,6141—01

6.6051—01

6.5811—01

6.5401—01

14 15 16 17 18

6.4651—01 8.334E—0I 6.2141—01 6.1081.01 b.012E—01

6,45~E—~1 5,3271—01 8,2081—01 5,1021—01 6.0(471—01

Y1 19

5,9271—01

5,9231—01

6.4361—01 6.3081—01 5.1901—01 6,0851—01 5.9921—01 13,9001—01 5.8341—01 5,7601—01 5.7111—01

6.3991—01 6.2731.141 6.1591—01 6.3581.01 5.0651.131 5.8851.01 5.8131.01 5,1491—01 5.6931—01

V.23 VU 24 YB 25

5.5731—01 5,6201.01 5,5891—01 5,5571.01 5,5291.01 5.5071*01

5,6701.01 5,6251—01 5.586E.0I 5,5541—01 5,5271—01 5.5051.141

5,6601—141 5.616E—~1 5.5781—01

9,643E—~1 5,6001.03 5.5631—01

5.5451,01

5.4911.01 5.4791—01 5.4721—01 5,4691.01

5,4881.01 5,4751—01 5.4691—01 5,4671—01

YB 20 Y~ 21 YB 22

V.26 YU 27

Va 28 YB 29 YB 30 YB 31 YB 32

5.851E—01 5.784E—01 5.7251—01

391

5.8471—01 5,7801*01 5,7221—01

5 1, ‘IOclE+0fl ~,521E—41 9.c172E~1AI 8,6661.131 14.3021.01 /.‘378F”L” 7,6891.01 7.4311—141 7.I9B1—~1 6.0901.0! 6,8031~141 b.635E—01 5,4841.01 6.3481—01 5.2251—01 5,1171—141 b.1llYE~0l 5.9311—01 5.8531—01 5,7841—01 5,722E—01 5.66131.141 5.5201—01

5,5781—01 5.543E—01

5.5321.01

5.11131.01

5,5101—01

5,5061.141

5,4881.01

5.4971.01 5.4811—141 5.4601—01 5.4621—01 5.480E—01

5.4651.141 5.4601.01 5.457E.01 5.4511—01 5,4481—01

5.4881.01 5.4571.01 5.4411—01 5.4351.01 5.4321—01

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