"CODE" 34432;
"PROCEDURE" PRAXIS(N, X, FUNCT, IN, OUT);
"VALUE" N; "INTEGER" N;
"ARRAY" X, IN, OUT;
"REAL" "PROCEDURE" FUNCT;
"BEGIN"
"COMMENT"THIS PROCEDURE MINIMIZES FUNCT(N,X),WITH THE
PRINCIPAL AXIS METHOD (SEE BRENT,R.P, 1973, ALGORITHMS
FOR MINIMIZATION WITHOUT DERIVATIVES,CH.7);
"PROCEDURE" SORT;
"BEGIN" "INTEGER" I, J, K; "REAL" S;
"FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO"
"BEGIN" K:= I; S:= D[I];
"FOR" J:= I+1 "STEP" 1 "UNTIL" N "DO" "IF" D[J]>S "THEN"
"BEGIN" K:= J; S:= D[J] "END";
"IF" K>I "THEN"
"BEGIN" D[K]:= D[I]; D[I]:= S;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" S:=V[J,I]; V[J,I]:= V[J,K]; V[J,K]:= S
"END"
"END"
"END"
"END" SORT
"PROCEDURE" MIN(J, NITS, D2, X1, F1, FK); "VALUE" J, NITS, FK;
"INTEGER" J, NITS; "REAL" D2, X1, F1; "BOOLEAN" FK;
"BEGIN"
"REAL" "PROCEDURE" FLIN(L); "VALUE" L; "REAL" L;
"BEGIN" "INTEGER" I; "ARRAY" T[1:N];
"IF" J > 0 "THEN"
"BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
T[I]:= X[I] + L * V[I,J]
"END" "ELSE"
"BEGIN" "COMMENT" SEARCH ALONG PARABOLIC SPACE CURVE;
QA:= L * (L - QD1) / (QD0 * (QD0 + QD1));
QB:= (L + QD0) * (QD1 - L) /(QD0 * QD1);
QC:= L * (L + QD0) / (QD1 * (QD0 + QD1));
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
T[I]:= QA * Q0[I] +QB * X[I] + QC * Q1[I]
"END";
NF:= NF + 1; FLIN:= FUNCT(N, T)
"END" FLIN;
"INTEGER" K; "BOOLEAN" DZ;
"REAL" X2, XM, F0, F2, FM, D1, T2, S, SF1, SX1;
SF1:= F1; SX1:= X1;
K:= 0; XM:= 0; F0:= FM:= FX; DZ:= D2 < RELTOL;
S:= SQRT(VECVEC(1,N,0,X,X));
T2:= M4 * SQRT(ABS(FX) / ("IF" DZ "THEN" DMIN "ELSE" D2)
+ S * LDT) + M2 * LDT; S:= S * M4 + ABSTOL;
"IF" DZ "AND" T2 > S "THEN" T2:= S;
"IF"T2<SMALL"THEN"T2:= SMALL;
"IF"T2>0.01*H "THEN"T2:= 0.01*H;
"IF"FK"AND"F1<=FM "THEN"
"BEGIN"XM:=X1; FM:= F1 "END";
"IF" ^ FK"OR"ABS(X1)<T2"THEN"
"BEGIN"X1:="IF"X1>0 "THEN"T2"ELSE"-T2;
F1:= FLIN(X1)
"END";
"IF"F1<= FM"THEN"
"BEGIN"XM:= X1; FM:= F1 "END";
L0: "IF" DZ "THEN"
"BEGIN" "COMMENT"EVALUATE FLIN AT ANOTHER POINT
AND ESTIMATE THE SECOND DERIVATIVE;
X2:= "IF" F0 < F1 "THEN" -X1 "ELSE" X1 * 2;
F2:= FLIN(X2); "IF"F2 <= FM "THEN"
"BEGIN" XM:= X2; FM:= F2 "END";
D2:=(X2*(F1-F0)-X1*(F2-F0))/(X1*X2*(X1-X2))
"END";
"COMMENT"ESTIMATE FIRST DERIVATIVE AT 0;
D1:=(F1-F0)/X1-X1*D2; DZ:="TRUE";
X2:= "IF"D2<=SMALL"THEN"
("IF"D1<0"THEN"H"ELSE"-H)
"ELSE"-0.5*D1/D2;
"IF"ABS(X2)>H"THEN"X2:="IF"X2>0"THEN"H"ELSE"-H;
L1: F2:=FLIN(X2);
"IF"K<NITS"AND"F2>F0"THEN"
"BEGIN"K:=K+1;
"IF"F0<F1"AND"X1*X2>0"THEN" "GOTO"L0;
X2:= 0.5*X2; "GOTO"L1
"END";
NL:= NL+1;
"IF"F2>FM"THEN"X2:=XM"ELSE"FM:=F2;
D2:="IF"ABS(X2*(X2-X1))>SMALL"THEN"
(X2*(F1-F0)-X1*(FM-F0))/(X1*X2*(X1-X2))
"ELSE" "IF"K>0"THEN"0"ELSE"D2;
"IF"D2<=SMALL"THEN"D2:=SMALL;
X1:=X2; FX:=FM;
"IF"SF1<FX"THEN"
"BEGIN" FX:=SF1; X1:=SX1 "END";
"IF"J>0"THEN"ELMVECCOL(1,N,J,X,V,X1)
"END" MIN;
"PROCEDURE"QUAD;
"BEGIN" "INTEGER" I; "REAL" L, S;
S:= FX; FX:= QF1; QF1:= S; QD1:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN"S:=X[I]; X[I]:= L:= Q1[I]; Q1[I]:= S;
QD1:= QD1 + (S - L) ** 2
"END";
L:=QD1:=SQRT(QD1); S:= 0;
"IF"(QD0*QD1>DWARF)"AND"NL>=3*N*N"THEN"
"BEGIN"MIN(0,2,S,L,QF1,"TRUE");
QA:= L*(L-QD1)/(QD0*(QD0+QD1));
QB:=(L+QD0)*(QD1-L)/(QD0*QD1);
QC:= L*(L+QD0)/(QD1*(QD0+QD1))
"END" "ELSE"
"BEGIN" FX:= QF1; QA:= QB:= 0; QC:= 1 "END";
QD0:= QD1;"FOR"I:= 1"STEP"1"UNTIL"N"DO"
"BEGIN"S:=Q0[I]; Q0[I]:=X[I];
X[I]:= QA*S + QB*X[I]+QC*Q1[I]
"END"
"END" QUAD;
"BOOLEAN" ILLC;
"INTEGER" I, J, K, K2, NL, MAXF, NF, KL, KT, KTM;
"REAL" S, SL, DN, DMIN, FX, F1, LDS, LDT, SF, DF, QF1, QD0,
QD1, QA, QB, QC, M2, M4, SMALL, VSMALL, LARGE, VLARGE, SCBD,
LDFAC,T2, MACHEPS, RELTOL, ABSTOL, H;
"ARRAY" V[1:N,1:N], D, Y, Z, Q0, Q1[1:N];
MACHEPS:= IN[0]; RELTOL:= IN[1]; ABSTOL:= IN[2]; MAXF:= IN[5];
H:= IN[6]; SCBD:= IN[7]; KTM:= IN[8]; ILLC:= IN[9] < 0;
SMALL:= MACHEPS ** 2; VSMALL:= SMALL ** 2;
LARGE:= 1/SMALL; VLARGE:= 1/VSMALL;
M2:= RELTOL; M4:= SQRT(M2); SETRANDOM(0.5);
LDFAC:= "IF" ILLC "THEN" 0.1 "ELSE" 0.01;
KT:=NL:=0; NF:=1; OUT[3]:= QF1:=FX:=FUNCT(N,X);
ABSTOL:=T2:= SMALL+ABS(ABSTOL); DMIN:= SMALL;
"IF" H<ABSTOL*100"THEN"H:=ABSTOL*100; LDT:=H;
INIMAT(1,N,1,N,V,0);
"FOR"I:=1"STEP"1"UNTIL"N"DO"V[I,I]:= 1;
D[1]:= QD0:= 0; DUPVEC(1,N,0,Q1,X);
INIVEC(1,N,Q0,0);
"COMMENT"MAIN LOOP;
L0: SF:=D[1]; D[1]:= S:= 0;
MIN(1,2,D[1],S,FX,"FALSE");
"IF" S <= 0 "THEN" MULCOL(1, N, 1, 1, V, V, -1);
"IF" SF <= 0.9 * D[1] "OR" 0.9 * SF >= D[1] "THEN"
INIVEC(2,N,D,0);
"FOR" K:= 2"STEP"1"UNTIL"N"DO"
"BEGIN" DUPVEC(1,N,0,Y,X); SF:=FX;
ILLC:= ILLC "OR" KT>0;
L1: KL:=K; DF:= 0; "IF" ILLC "THEN"
"BEGIN" "COMMENT"RANDOM STOP TO GET OFF
RESULTION VALLEY;
"FOR"I:= 1 "STEP"1"UNTIL"N"DO"
"BEGIN"S:=Z[I]:=(0.1*LDT+T2*10**KT)
*(RANDOM-0.5);
ELMVECCOL(1,N,I,X,V,S)
"END";
FX:= FUNCT(N,X); NF:= NF+1
"END";
"FOR"K2:= K "STEP" 1 "UNTIL" N "DO"
"BEGIN" SL:=FX; S:= 0;
MIN (K2, 2, D[K2], S, FX, "FALSE");
S:="IF" ILLC "THEN" D[K2] * (S + Z[K2]) ** 2
"ELSE"SL-FX;"IF"DF<S"THEN"
"BEGIN"DF:=S;KL:= K2"END";
"END";
"IF" ^ILLC "AND" DF < ABS(100 * MACHEPS * FX) "THEN"
"BEGIN" ILLC:= "TRUE"; "GOTO" L1 "END";
"FOR" K2:= 1"STEP" 1"UNTIL"K-1"DO"
"BEGIN" S:= 0; MIN(K2, 2, D[K2], S, FX, "FALSE") "END";
F1:= FX; FX:= SF; LDS:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" SL:= X[I]; X[I]:= Y[I]; SL:= Y[I]:= SL - Y[I];
LDS:= LDS + SL * SL
"END"; LDS:= SQRT(LDS);
"IF" LDS > SMALL "THEN"
"BEGIN" "FOR" I:= KL - 1 "STEP" -1 "UNTIL" K "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
V[J, I + 1]:= V[J,I]; D[I + 1]:= D[I]
"END";
D[K]:= 0; DUPCOLVEC(1, N, K, V, Y);
MULCOL(1, N, K, K, V, V, 1 / LDS);
MIN(K, 4, D[K], LDS, F1, "TRUE"); "IF" LDS <= 0 "THEN"
"BEGIN" LDS:= LDS; MULCOL(1, N, K, K, V, V, -1) "END"
"END";
LDT:= LDFAC * LDT; "IF" LDT < LDS "THEN" LDT:= LDS;
T2:= M2 * SQRT(VECVEC(1, N, 0, X, X)) + ABSTOL;
KT:= "IF" LDT > 0.5 * T2 "THEN" 0 "ELSE" KT + 1;
"IF" KT > KTM "THEN" "BEGIN" OUT[1]:= 0; "GOTO" L2 "END"
"END";
QUAD;
DN:= 0;"FOR"I:= 1"STEP"1"UNTIL"N"DO"
"BEGIN"D[I]:= 1/SQRT(D[I]);
"IF"DN<D[I]"THEN"DN:=D[I]
"END";
"FOR"J:= 1"STEP"1"UNTIL"N"DO"
"BEGIN"S:= D[J]/DN; MULCOL(1,N,J,J,V,V,S)"END";
"IF"SCBD>1"THEN"
"BEGIN"S:=VLARGE; "FOR"I:=1 "STEP"1"UNTIL"N"DO"
"BEGIN" SL:= Z[I]:= SQRT(MATTAM(1, N, I, I, V, V));
"IF"SL<M4"THEN"Z[I]:= M4;
"IF" S>SL "THEN" S:= SL
"END";
"FOR"I:=1"STEP"1"UNTIL"N"DO"
"BEGIN"SL:=S/Z[I];Z[I]:= 1/SL;
"IF"Z[I]>SCBD"THEN"
"BEGIN"SL:=1/SCBD; Z[I]:= SCBD"END";
MULROW(1, N, I, I, V, V, SL)
"END"
"END";
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
ICHROWCOL(I + 1, N, I, I, V);
"BEGIN" "ARRAY" A[1:N,1:N], EM[0:7];
EM[0]:= EM[2]:= MACHEPS;
EM[4]:= 10 * N; EM[6]:= VSMALL;
DUPMAT(1, N, 1, N, A, V);
"IF" QRISNGVALDEC(A, N, N, D, V, EM) ^= 0 "THEN"
"BEGIN" OUT[1]:= 2; "GOTO" L2 "END";
"END";
"IF"SCBD>1"THEN"
"BEGIN" "FOR"I:=1"STEP"1"UNTIL"N"DO"
MULROW(1,N,I,I,V,V,Z[I]);
"FOR"I:= 1"STEP"1"UNTIL"N"DO"
"BEGIN"S:= SQRT(TAMMAT(1,N,I,I,V,V));
D[I]:= S*D[I]; S:= 1/S;
MULCOL(1,N,I,I,V,V,S)
"END"
"END";
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" S:= DN * D[I];
D[I]:= "IF" S > LARGE "THEN" VSMALL "ELSE"
"IF" S < SMALL "THEN" VLARGE "ELSE" S ** (-2)
"END";
SORT;
DMIN:= D[N]; "IF" DMIN < SMALL "THEN" DMIN:= SMALL;
ILLC:= (M2 * D[1]) > DMIN;
"IF" NF < MAXF "THEN" "GOTO" L0 "ELSE" OUT[1]:= 1;
L2: OUT[2]:= FX;
OUT[4]:= NF; OUT[5]:= NL; OUT[6]:= LDT
"END"PRAXIS;
"EOP"