An algol program of the simplex method

Report No. 507

aprii 1971

E

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

An ALGOL Progrea of the Simplex Method

bi

G

Usually, the Simplex Method. je known as a method to obtain the optimum

sOlution of the Linear Programming, which

is to obtain th.

optimum point

(vector X°

the aet of values 11° *20, , x°) to minimize

(or maximize) the value of the linear function,

J

_{«C' .%}

under the constraints of

AX-b

where A : nra matrix

x,C,. z n - dim. vector

transpose of vector C

b

_'m-dle

_vector

On the other hand, the Simplex Method in this program, which is developed by Neider and Miad (i), is used to solve the Non-Linear Programming which is

defined in

gen.ra].

the method to obtain th. minimum point of the non-linear

function

J - f(X)

and the f'unotion value of this point under the nonlinear constraints of

9(X)-o

Ther. are many kinds of methods to solve thee. problems, such as the steepest descent method, which is the most popular on.. Btt this method is said to b. not ff.otiv. when the function has the steep valley.

mi

Simplex Method ii known as one of the most effective method when

the problem has no oonstraiuts. It is defined that n + 1 vectors in

n-dimensional

vector apac. make a simplex, and these points are chosen for estimated v*luse of the minimum point. This m.thod is carried out to move

thes.

points according to th. proper algorithm so as to mov, the simplex near the exact minimum point.

This program is oonatruot.d with sight kinds of procedures, four of

them

ax'. used L'or the preparation of the movement of the simplex.

1, Procedure P I N

In this procedure the function valuei of the + i of a simplex are

evaluated, W. d.fine these values

2

Procedure MflIMAX

The maximum end the minimum value of th. ut of fi (i i, n + 1)

are obtained in this procedure. These points and values are defined,

f

..f(X).aNax{f(XÚ/i_].,

fi. úf(X)-Min{f((i)/i_1,

r.sp.otiv.ly.

Procedure SECMAX

The second mazlmum value and point are obtained

f

uuf(X)

'sMZ

fÇi) /iil,

, fl+l, %i(j }

4 Procedure

CDTROTh

The oentroid of the n

pointa of the simplex, .goluding the point X14, is

obtaiRM,

and we name it si

x4xt

The other four kinds of procedur.. are used to move the simplex1

5 Procedure R!1LEX

Aa the worst point of the aimplex with which w. are oonoerned is the point %4, we should concentrate our a.tt.ntion to this point. It would be reasonable to consider that w. can get a bettsr point by moving this point in the direction of the other points. For this puxpos., the

point is refl.ot.d against )(Ìj to

Xt.

e.g. (ese Fig. i)

Xi'i+ o(XM Xi)

(1+d)-

t,&

i

and the function value f of this point is tr&Luatsd. fk

-6. Procedure EXP}SION

If

f ii

mall.r than f, we may consider that the direction of »1 - )C

i. v.r ±fotiv., Therefore, it may be better to oontiu th. movement

in this direstion than to atop at this point, This movement is oafled oxpane ion and defined

X

(XR-.)

and the function value

f

of this point is obtained (Fig. 2).

7 Procedure COW]flTCP

On the other hand, when

f,

la larger than

f

it is considered that

the movement by the reflection is too large, and we should choose a áall.r displacement.

Before contraction, fj and X are replao.d by f and X respectively

when f is ealler than fit. The contraction le defined. (Fig.3)

Xc; 'Xj+

(%M-XH)

0'T<1

and the function value f i. evaluated. -

f(Xc)

0. Procedure NEWSThI'

When all of these movement have failed (f)f). we should consider that ti.: simplex with uhioh ve are d.aling ii too large and the optimum point i. situated Inside of the simplex, so that we should make our simplex smaller by moving the points in the direction of XL.

Thi procedure NEWSIMP te carried out by

Xi-Xi +S4-Xi)

-i - 1,

(Pig.

₄₎

Thee. movement, are illustrated in Figuree i - 4 in the case of n M 2.

The values of (, and Sar. chosen 1.5, 2, 0.5 and 05 respectively

in this program.

Utilizing these procedure., the flow chart of of the Simplex Method is shown in Fig. 5.

The oouvergenoe condition in this program is

fki-f1<

(s.. Appendix,

Sri

00088)

The condition itself or the value of hould be changed it it i. not

suitable Cor each program.

Per an example, an application of this program for the Rosenbrocl function (2) ii shown in Pig. 6.

4,

The Roasnbroolc

function baa a form of

f(X, X2) -

loo (x

X1)2 + (ir1)2

The minimum point and the function value at this point are of ocume

- x

-i

f(l, i) - o

respectively. This function vta uM. specially for the pump... of d.mon.tration, and as w. can se. it .asi]y, it bas a very sharp and

curved valley,

.0

it i. almost impossible to get 4h. optimum point by

th. not so adaptive method such as the .te.p.st descent method.

Starting from the

initial

simplex, (6, 1), (4, 2) and (5, 3), we get the

point first by the reflection of

_EH.

A. th. valu, was smaller than1,

the poo.d.

CPARSI0N van carried out and ve got th. point

F.

Next, we got FR b the reflect ion of lFg, _Fc. by the contraction of PL

and P by the reflection of F. and ?W2. were obtained by the

pro6*ure NTh1SD'.

At thi. point, ve got the vinioity of the bottais of valley of the

Rocen-brook function,

end

our simplex had to b. aide smaller by the several

times of .xeoution of the proo.dur. CONTRACT.

After our simplex had been made email enough, it begin to mov. again,

following the valley of the function to th. optimum point. Sev.rsl simplex..

are shown in this process Sometimes th.y became very small and some

times they became rather large.

Th. final

simplex we got was

(

1.0112, 1.0230)

( 0.9790, 0.9579)

( 0.9994, 0.9969)

They are cloue enough to the exact point (i, i). If it is not sufficiently

clos. to the exact point, we can get mol. exact result by

m1ring

the oonv.r..

genoa condition more strict.

The computation tim. to get these points was 6

seconde by IBM

360/65.

5.

(i) JA. Neider aM R. Mead, A Simplex Method for Function Minimization,

Oomput.r 3., 7, 1965.

(2) H.E. Roi.nbroolc, An Automatio Method for Finding the Greateat or

Leaat Value of a. Function, Coraputer 3, 3, 1960

XL

FiG i

REFLECTION

FIG 2 EXPANSION

Xli

(Xr-Xh)

FIG 3 CONTRACTiON

FIG 4 NEW SIMPLEX

---;Xs

Xc)

--Xh

XL

íiTIAL 8DLEZ

PIN

3

Plow Ohs't of' th Simplex Method

V..

p1M out

s

Fr2

-2

0

2

4

6

FIG 6 AN EXAMPLE OF THE ROSENBROCK FUNCTION

/OPTIMAL POINT

I

LEVEL 1JUL67 -...00048 00050 00050 00051 - ...0O51 00052 00053-00054 -. 0055... 00056 -...--. 0005a.-00058 00059---00060 . 00062.... 00062 SC ...SOURCE-STATEMENT... OS ALGOL F SOURCE PROGRAM DATE FEB 18 1971 PAGE 001 . . 00000 ... 'BEGIN' ...

...SIMO1AOC

00000 'COMMENT' SIMPLEX METHOD; _SIMÚ1BOO

00000 _{tINTEGERII,J,N,IL,1I1,IND,LS,K;}

SIMO1CPO

00001 N:=2; _SIMC)1DC0

00002 ...'BEGIN' ...'REAL'S,FKT,FL,FH,FS,FR,FE,FC; ... _SIMO1EOO

00003 _{'ARRAY'x(/l:N,l:N+l/),xM,xR,xE,xc,xu(/l:N/),f-x(,j:N+l,); SIMOIFCJO} 00004 _{'PROCEDURE'BLANK(D,N);IVALUEID,N;IINTEGERtD,N;ICOOE.;} .. 00008 _{'PROCEDURE'FIX(0,M,N,X);IVALUEID,M,N,X;tINTEGFRID,M,f,;IRFA1IX;ICODE,;} 00013 ...'PROCEDURE'PIN(XU,N,FKT ... . . SIMCIJOO 00014 'INTEGER'N;'REAL'FKT;'ARRAy'xu; SIMO1KOO .00017 ...FKT:=1OO*(XU/2/)_xU(/1/)**2)**2+(1_u(,1,)p**2 _...SIMOILCO 00018 'PROCEIJURE'MINMi\x(FX,N,FL,FH,IL,JH); SIMOIMOO .00019 ...'INTEGER'N,LL,IH ;'REAL'FL,FH;',ARRAY"FX ...SLMC1N00 00022 'IIEGIN'FL:=+'7;FH:=-'7; _SIMO1000 00024...-...'FOR'L:=I'STEP'I'UNT!L'N-i-l'[)ç]' - _STMO1POO 00024 'BEGIN' _SIMO1000 00024 - _. S=FXii/; _{...- ....}

00025 'IF'S >FH'THEN''HEGtN'FH:=S; IH:=I;'END';SIMOjROO

00028 ... _{'IF'S...<FL'THEN''BEGJN'FL:=S; IL:=1;'END';SIMOlSOo}

00031 'END'; _SIMO1TOO

00032 ...- 'END; ...- _siioiuoo

00033 _{'PRUCEBURE'SECMAX(FX,FS,N,[H,IS); SIMÛIVOO}

00034 'INTEGER'N, IH,IS;'REAL'FS;'ARRAY'Fx; . S1M010O

00037 'BEGIN' FS:=-'7;

00038...'FOR'I:1'STEP'l't.JNTIL'N4-I'DO' S'lfrlGlxOO

00038 'BEGIN''IF'I=IRTt-IEN' 'GOTO'SKIP;S:=FX(/I/l; SIMOIXUI

00040... --.-... 'IF'S )FS'THEN''BEGIN'FS:=S; IS:=1;'END'; SIMO1YOO

00043- SKIP: 'END';'END'; _SIMÛILOO

00045...'PROCEDURE'CENTRUID(X,XM,N,IH ...' SIMO2AOO 00046 'INTEGER'N,IH;'ARRAY'x,xM; _SIMc-2BOO 'BEGIN''REAL'S;'INTEGER'l,j; _SIMO2COO 'FOR'l:=Ì'STEP'I'UNTIL'N'DO' SMO2D00 ...'BEGIN'S:=O; ....--..

... --

-SIMC2EOO 'FOR'J:=I'STEP'l'UNTIL'N+l'Do' SIMO2FOO S:=SX(/!,J/ ...,

--

''SIMO2000 XM(/I/I:=(S-X(/1,JH/))/N; SIMO2HOO tENDU; . SIMO2IOO 'END'; _SIMO2JOO 'PROCEDURE'REFLECT(XM,XH,N,XR); - SIMC2K00

'INTEGER'N; 'ARRAY'XM,XH,XR; S!MO2LOO

-'FOR'-I:=l'STEP'L'UNTIL'N'DO' SIMO2MOO

XR(/I/):=2.5*XM(/I/)-1.5*xI1(/!/); SIMO2NOO

-.

'INTEGER'N; 'ARRAY'XM,XR,xE; SIMO2POO

'FOR'1:=1'STEP'1'UNTILN'OD' _SIMÛ2000 XE(/I/):=2*XR(/I/)-xM(/I/); SIMO2ROO

:4--

--:i

1_ Ç.-.'rt' .

-1) (. C, C - C-C

.c.

C C C SC.. SOURCESIATEMENT 00122 ...L5: _{'FOR'I:=l'STEP'I'uNTIL'N'Do'xo(/l,):=x(,I,IL,);} V 00123 NEWSIMP(X,XU,N); 00124 _{SY-5ACT(1,14,3).;QUTSTRING(1,'(INEw...sJMpLFx)); ...} 00126 SYSACT( 1,14,1); 'GOTO'Sl; 00128 ...PRINT.:...SYSACT(1,14,3); ... 00129 _{'IF'IND=1'TIiENb0UTSTRING(1,'(IExpANslJN));} 00130...'IF'IND=2'THE'OUTSTRING(1,'('REFLECTI)I ... 00131 _{IIFIIND3hTHENtOUTSTRING(1,I(tCQNTRACTI)I);SYSACT(1,14,1);} 00133_{...'F0R'I:=I'sTEP'L'urTIL'NeDouxu(,I/):.x4,I,jH,)...} 00134 FIX( 1,10,0, IN); 00135. _{'F[JR'K:=1'STEP'I'UNT!L'N'DO'FIX(1,5,4,XU(,K,)). ...} 00136 BLANK(1,1C);CiUTREAL(1,Fx(/IH/)); 00138 'G0TCJ'52 00139 STOP: _{SYSACT(1,14,5);QUTSTRJNG(1,'('CQNvERGEI)I);} 00141 . . 'ENO'; . ... 00142 'END';

SOURCE PROGRAM _{PAGE 003}

SI MO4POO S 1M04000 SI MO4ROO SIMG4SOO S 1MO4000-SIMO4VOO SI M 04 WOO SIMO4XOO S IMP4YOO S IMO4LOO SI M C 5 A 10 SI M05A20 SI MO5BOO SI MO5COO S IMG5000 -, - V. t. _t-LA .. . - '

Sc SOURCE STATEMENT PAGE 002

00063 _{-'PROCEDURE'CONTRACT(XM,XH,N,xc);} ... _SLMO2SOO

00064 'INTEGER'N; 'ARRAY'XM,XH,XC; _SIMO2TOO

-...00066 ...- ... 'FOR'1:=1'STtP'1'UNTjL'iN'DO' . . . SIMC'2U00

0O06 _{XC(/I/):=0.5*(XH(/I/)4-xM(/t/));}

SIMC2VOO

-

00067....-..--'PROCEDURE'NEWSIMp(x,xL,N; - _S1MO2w00

I 00068 'ENTEGER'N; 'ARRAY'x,xL; SIMO2XOO

.. . 00070 ...'FOR'J:=I'STEp'I'UNTIL'NfI'DO' _sIMe2YOO 00070 _{'FUR'I:=I'STEP'I'UNTIL'N'DÙ'} SIMO2ZOfl (...X(/!,J/):=0.5*(XL(/l/)+x(/r,J/)); SIMC3AOO 00071 _{SYSACT(1,B,6Q);sysACT(1,12,1); S1M03F00} ..- ...00073 _{'cQMMENT'MAIN PROGRAM; ... ...} SIMO3OflO ( 00073 'FOR'Jl'STEP'l'UNTIL'N+I'D(]' _SIMO3EOO ., 00073 ...'FOR'I:=1'SrEP'1uNTILINIDouINREAL(ox(,L,J,)); _SIMO3FOO

00074 OUTSTRING(1,'([N!TJAL SIMPLEX')');SYSACT(1,14,1); _SIMÜ3GOO

I....: Si:...'FOR'J:=I'STEp'lIUNTJL'Ni-l'DQ'. .

5!MO3HOO 000m 'BEGIN''FUR' I:=1'STEP'1'UNTIL'N'Do'xu(/j/:x(/,J/); SIMO3IOO

...-00017 PIN(XU,N,FKT);Fx(/J/):=FKr; . SIMC3JÜ0 . 1 'END'; _SJMO3KOC) ..T . 00080...'FQR'J:=j'STEp'1!UNTjLN+1II)Ql ... _...SLM03LO . 00080 'BEGEN''FURJ:=1STEP1uNrjL'NDl)'xu(/I/):(/J,J/); SHIO3MOO ; ... 00081...FIX(1,1O,C,J); .. . . STMC3NC0 00082 _{'F(]R'K:=I'STEP'1'UNTII'N'flo'FIx(1,5,4,xu(,K,),; S11Ü3N10} 00083 6LAiNK1,iC); _SIM('3N20 00084 _{OUTREAL(1,FX(/J/));SYSACT(l,14,1J; SIMr3UOO} -

.-00086 ...'FND' ...

.-.-. SIMC3000 00087 S2: _{MINMAX(FX,N,FL,FFj,JL,IH);} SIMÜ3ROO

00088 ...'IF'FH-FL<'-3 'THEN''GOT[J'SJOP; _.SIMO3SOO

00089 SECMAX(FX,FS,N,IH,IS); _SJMU3TOO ...00090 CENTRUiD(X,X,N,IH); _SJMC3UOO C £0091 'FOR'j:=l'STEP'l'UNTIL'N'Qo'XU(/I/):x(,I,IH,,; SJMO3WQQ -...00092 ...REFLECT(XM,XU,NX); ... . - SIMO3XOO 00093 _{PJN(XR,N,FKT);FR:=FKT;IIFIFR>FLtTHENIIGOTOIL1;} . SIMO3YOO ...:00096 ...EXPANS!ON(XM,XR,N,XE); .. . SIMO3ZOO

00091 PIN(XE,N,FKT) ;FE:=KT; ' IF'FE>FR'THEN' 'GOTO'12; _SIMC4A00

...-00100 ...'FOR'I:=Ì'STEP'I'UNTIL'N'Do'x/I,IHI:xE/In; _SIMÛ4BC.0

00101 FX(/IH/):=FE;1ND:=1;'GoroupRjT; SIMO4COO 00104

LI: 'IF'FR>FS'THEN''GOTO'L3; ...SLMÜ4000

00105 12: _{'F0R'I:=1'STEP'l'uNTIL'j4'DO' SIMO4EOO}

:..c ... -- 00E105₀₀₁₀₉ ...--- X(/I,!H/):=XR(/r/);Fx(/IH/):=FR;JND:=2;GoToupRINr; ...SIMO4FOO

L3: 'IF'FR>FH'THEN''GUTO'L4; _SIMÜ4GOO - 00110 _{IFORIIlJSTEI1IUNTILINIOot} SIMC4HCO 00110 X(/!,IH/J:=XR(/!fJ;Fx(/IH/):=FR;FH:FR; SIMO4IOO - ...00113-_...L4 ...'FOR'I-:=i'STEP'1'UNTILIN'Do'xU(/I/):=x(,1j1H,...-- 51M04J30 00114 CONTRACT(XM,XU,N,XC); _SIMO4KOO ...---- 00115....-...-PIN(Xc,N,FKT);Fe:=FKT SIMO4LOO 00117 'IF'FC>FH'THEN''GUTO'L5; _SIMO4MOO ---00E18...----_{...--'F0RJ:=1'STEp'F'UNTiL'N'QO'x(/IIH/).:xC..(,,);} ... 00119 FX(/IH/):rFC;rND:=3;GoTcJ1pRNr; SIMO4000 -..---..-...._-_.-...-- .... SOURCE PROGRAM

H

3.

o'

.-:.- .-.

SOURCE PROGRAM

_{PAGE 001}

Sc SOURC STAT.E.MENT

0.0.00.0 ' BEGIN '

_SiM0iA0O

00000

'COMMENT' SIMPLEX METHOD;

_SIMO1BOO

00.00.0 e INI...GER' I ,.J.,.N,.4-L.,..I.H....IND,.I..S,K;

_SiM0iC0O

00001 N:=2;

_S!MO1DOO

0000.2

I ß E. GIN I 'RE.AL..'..S..,. . F.S.,.FR.,.F.E..,.FC ;

..

_SIMO1E00

00003

_{'ARRAV'X(tl:N,l:N+l/),xM,XR,XE,xc,xu(/l:N/),Fx(,l:N+l/);}

_SIMO1FOO

00004

PRO CE- DURE...BLAN.K1.D...N);.'.VALlJE ' D...N;'INT..N;'GODE';

00008

_{'PROCEDURE'FIX(D,M,N,X);'VALUESD,M,N,X;IINTEGÈRID,M,N;$REALIX;ICODEI;}

O.00.1..3 e PROCE.DUR.E..'.PI.N.(.X.0...t'41F.KT.) ; _SiM0...JOG

00014

..

'INTEGER'N;'REAL'FKT;'ARRAy'xu;

SIMO1KOO

00017

F ..

S.i:M0...LOO

00018

'PROCEDURE'MINMAX(FX,N,FL,FH,IL,IH);

SIMOLMOO

0.0.0..9 q INT...GER'N,lL....¡.H. ;..R..AL..l..F.L.,..F.H.;.S.A.R.R4.v..L.F.x..; _S1MO1NOO

00022

. 'BEGIN'FL:='7;Fi-i:=-'7;

S!MO1000

00024

_SIM0I.POO

00024

'BEGIN'

_SIMO1QOO

0002.4

S :

=FX(íI--/);

00025

. 'IF'S >FH'THEN''BEGIN'.FH:=S; .

IHIEND'SJM01ROO

0002.8

t

_;

_{IL:=i;'ENO....;.S.IMO1..s,o.o}

00031

. 'END';

SIMO1TOO

00032

END..L.; ... . _SIMO1U0O

00033

_{'PROCEDURE'SECMAX(FX,FS,N,IH,IS);}

_SIMOIVOO

000.34

'-INTEGER'N,

I _SIMO1WOO

00037

'BEGIN' FS:=-'i;

00038

.

I FO...'i-:....1-'STEP'4'UNTIL-'N+1'DO'

SIMO..XO0

00038

'BEGIN'' IF'I=IH'TH[N' 'GOTO'SKIP;S:=FX(/I/) ; SIMO1XO1

00040

I IfS

_{IS:=I;'END....;}

_51MO..OO

00043

SKIP: 'END' ; 'END' ;

_SIMOILOO

00045

_{e PROCEDU'&EN.T.Ro.I.D....X-,XM,N,IH) ;}

_SIMO2A0O

00046

'INTEGER'N,IH;'ARRAY'X,XM;

_SIMO2BOO

00048

'-BEGI*'-'REAL-'S....'INTEGER'I,J;

S1MG2COO

00050

_{'FOR'I:=l'STEP'l'UNTIL'N'Do'}

_SIMO2000

00050

e BEG:fN'SO"

_SItIO...ECO 00051

_{'FOR'J:=l'STEP'l'UNTIL'Nl'Do'}

_SIMO2FOO

00051

.

S:=S-X±/-F,-J/-) ;

SIMO2G00

00052

_{XM(/I/):=(S-X(/J,JH/))/N;}

_SIMO2HOO

00053

'END' ;

_SIMO...F00

00054

'END';

_SIMO2JOO

00055

'PROGEDURE'REFLEC-T(-XM,XH,*,.XR-)-;

SIlO2KOO

00056

'INTEGER'N;

'ARRAY'XM,XH,XR;

_SIMO2LOO

00058

_&FMO2M0O

00058

XR(/I/):=2.5*XM(/J/)-105*XH(/I/p;

_SIMO2NOO

OOO59 p PRQ.. _{; SFMO200O}

00060

. 'INTEGER'N;

'ARRAY'XM,XR,XE;

SIMO2POO

00062

_{'FUR'-I:=F'S.T..EP.'..1..'.UN.T..IL'N'Dt3}

_SIMO..00

00062

_{XE(tI/):=2*XR(/I/)-XM(/I/);}

_SIMO2ROO

SOURCE PROGRAM

PAGE 002

Sc

SOURCE SfATEME*T

0006-3 'PROcEDURE'eONfRÂ&T*XM , XH,N , xeì ; S i MO'2 S 00

00064

'JNTEGERN;

'ARRAY'XM,xH,xC;

SIMO2TOO

00066

LFOR.F.:=.FSfEPIi..I.UN.T..I*.I.ND ' .

SIMO2U00

00066

xc(/1/):=o.5*(xH(/I/)+xM(/I/n;

SLMO2VOO

00067

PR0&EDURE'NEWSIMP(-X-fXL N);--- SFMO2 WOO

00068

'TNTEGER'N;

ARRAY'X,XL;

S!MO2XOO

OOCH0

'FOR'J:=F'SfEP'F1UNTI±'N+i'00'

. SIMO 2 YO0

00070

'FOR'I:=I'STEP'l'UNTIL'N'DO'

SIMO2ZOO

OOO?O

Xí/I,Jí)1Ow5*EXLf/j/)+X(/ 1,J / )) ;

_SFMO3AO0

00071

SVSACT(1,8,60);SYSACT(1,12,1);

_SIÑO3BOO

000 +3 'eoMMENT' MAfNPROG R AMI

_{SIMO 000}

00073

'FOR'J:=l'STEP'l'UNTIL'N+I'DO'

.

SIMO3EOO

O0O3

F0RII:=i.'.S±EPI..HUNTH.'..NI.DO fNREAtO,Xi1+,J-/)); ... SIMO 3 F00

00074

OUTSTRING(1,'('INITJAL SIMPIEX')');SYSACT(I,14,1);

.

SIMO3GOO

000m

S1 :

_S1M&3H00

00076

'BEGIN''FOR' I:=1'STEP'l'UNTIL'N'Do'xu(/I/):=x(/!,J/);

SIMO3IOO

000 T 7 P+N(»XU, NFK-T;FX(/J / ):=FKT;

StM03JO0

00079

'END'; .

SIMO3KOO

00080

I

FOR'J=i"STEP'1'uNTIL"N+-1'Do'

SIM03t00

00080

. 'BEGIN' 'FOR' I :=i' STEP' i 'UNTIL'N'DO'xUL/I/) :=X(/I,J/) ;

SIMO3MOO

00 08 i FF(i»i , 10 , O, J); SIM0 3 NO0

00082

_{'FOR'K:=1'STEP'l'uNTIL'N'Do'FIx(1,5,4,xu(/K/));}

_SIMO3NIO

00083

BtANK(»1 p 1 ) ; S-1MO 3N20

00084

_{OUTREAL(1,FX(/J/));SYSACT(1,14,1);}

_SIMO3000

00086

'fND ' ; .

SiM03Q00

00087

S2:

MINMAX(FX,N,FL,FH,IL,IH);

.

SIMO3ROO

00088

'IF'FH-FL('-3 'THEP4G&T0TOP;

S I M03S00

00089

SECMAX(FX,FS,N,IH,IS);

SIMO3TOO

00090

ENfR&iDtX , XM*N , 1H11 S I M03U00

00091

.

'FOR' I:=l'STEP'1'uNTL'N'oO'xu(/I/):=x(/I,IH/);

SIMO3WOO

00092

REFtECT(»XftXU , N,XR)1

_SIMO3X00

00093

PIN(XR,N,FKT);FR:=FKT;11F'FR>FL'THEN''GOTO'Ll;

SIMO3YOO

00096

EXPÂNS»IONtXMxR , N , X E)'; S IM03i00

00097

_{PIN(XE,N,FKT);FE:=FKT;'IF'FE)FR'THEN''GoTo'L2;}

.

SIMO4AOO

O0i.O.o _{'FOW't:=t'SPEP'1-'uNTiL'N-'Do'xfij , H 1 )1=XEt11»/1»; S I MO 4 BOO}

00101

FX(/IH/):=FE;IND:=1;'GOTO'pRINT;

SIMO4COO

0O104

L 1: fF'-FR)FS'1HEN''GU 'T O't3;

S1MO4D0O

00105

L2:

_{'FOR'I:=l'STEp'l'UNrIL'N'Do'}

.

SIMO4EOO

00105

Xt/ I , FHí)XR(1i/11FX(/ «1'Ht):=FR;FND : =2iGO T U'PRTNt;

SiMc4FOO

00109

L3:

'!F'FR>FH'THEN''GÚTO'L4;

_SIMO4GOO

0o1.1O 'FOR'I:=t'STEP '. t'UNTIt'N'DO' . . &iMcY4HOO

00110

X(/I,IH/):=XR(/I/);FX(/JH./):=FR;FH:=FR;

SIMO4IOO

00113

L4 : _{'POR'I»:=i"STEP't'tjNT1t'N'D...Xti'(/ i /1:X(tt , I Hi); S I MO .4 JOO}

00114

CONTRACT(XM,XU,N,XC);

_SIMO4KQO

00-115 _{PIN(XC,NF-K-T1iFC:=fKf;} .

SiMO4L0O

00117

'IF'FC>FH'THEN''GOTQ'L5;

SIMO4MOO

00 i LB _{s i MO4N 00}

o

SOURCE PROGRAM

PAGE 003

Sc

SOURCE STATEMENT

00.122 L.5: _{' FOR'.I.-: -1 ' STEP 1'UNTII'.Ñ'DO....XU(/4/) :=X( / I ,LL/ ) ; S.I:M04p00}

00123

NEWSIMP(X,XU,.N);

_S!MO4QOO

0.0.1.24 SY-SAC T(1 3-);O U T ST RJNG( I. ,' ( 'NEW S H1PLEX l

SlMO4ROO

00126

_{SYSACT(i,14,1);'GoTQ'Sl;}

.

SIMO4SOO

00.128 PRINT:

SYSACT(l,14,3;

.

SiM04U0O

00129

_{'IF'IND=i'THENOUTSTRING(l,'('EXPANSIUN')'j;}

_SIMO4VOO

00.1.30 I IF ....INO-2THNi.OUT. ( 'REFLECT') ) ;

S1M04W0O

00131

_SIMO4XOO

0.0 .1..3.3 F0.R.!..I ; - 1 '.S.T.EP '-1 .UNT 1L .'.N...DO.' XU( 1.1.1..).:

_SiMO4YOO

00134

FIX(1,10,O,IH);

_SIMO4ZOO

OO.135 a FOR'l( : -4 'STEP '.-1 UNTIL '-N'D O 'FIX (1. , 5...4,XU..../K/)) ;

_SIMO5AL0

00136

_{BLANK(1,1O);OUTREAL(1,Fx(/LH/)J;}

_SIMO5A2O

00....38 GOTO.' S2;

_S1MO5BOO

00139

STOP:

_{SYSACT(1,14,5);OUTSTR1NG(1,1(1CONvERGEa');}

_SIMO5COO

00-141 a END.' ; V