SUBROUTINE RNSGB(A, ALF, B, C, DA, IN, IV, L, L1, LA, LIV, LV, 1 N, NDA, P, V, Y) C C *** ITERATION DRIVER FOR SEPARABLE NONLINEAR LEAST SQUARES, C *** WITH SIMPLE BOUNDS ON THE NONLINEAR VARIABLES. C C *** PARAMETER DECLARATIONS *** C INTEGER L, L1, LA, LIV, LV, N, NDA, P INTEGER IN(2,NDA), IV(LIV) C DIMENSION UIPARM(*) REAL A(LA,L1), ALF(P), B(2,P), C(L), DA(LA,NDA), 1 V(LV), Y(N) C C *** PURPOSE *** C C GIVEN A SET OF N OBSERVATIONS Y(1)....Y(N) OF A DEPENDENT VARIABLE C T(1)...T(N), RNSGB ATTEMPTS TO COMPUTE A LEAST SQUARES FIT C TO A FUNCTION ETA (THE MODEL) WHICH IS A LINEAR COMBINATION C C L C ETA(C,ALF,T) = SUM C * PHI(ALF,T) +PHI (ALF,T) C J=1 J J L+1 C C OF NONLINEAR FUNCTIONS PHI(J) DEPENDENT ON T AND ALF(1),...,ALF(P) C (.E.G. A SUM OF EXPONENTIALS OR GAUSSIANS). THAT IS, IT DETERMINES C NONLINEAR PARAMETERS ALF WHICH MINIMIZE C C 2 N 2 C NORM(RESIDUAL) = SUM (Y - ETA(C,ALF,T )) , C I=1 I I C C SUBJECT TO THE SIMPLE BOUND CONSTRAINTS C B(1,I) .LE. ALF(I) .LE. B(2,I), I = 1(1)P. C C THE (L+1)ST TERM IS OPTIONAL. C C C *** PARAMETERS *** C C A (IN) MATRIX PHI(ALF,T) OF THE MODEL. C ALF (I/O) NONLINEAR PARAMETERS. C INPUT = INITIAL GUESS, C OUTPUT = BEST ESTIMATE FOUND. C C (OUT) LINEAR PARAMETERS (ESTIMATED). C DA (IN) DERIVATIVES OF COLUMNS OF A WITH RESPECT TO COMPONENTS C OF ALF, AS SPECIFIED BY THE IN ARRAY... C IN (IN) WHEN RNSGB IS CALLED WITH IV(1) = 2 OR -2, THEN FOR C I = 1(1)NDA, COLUMN I OF DA IS THE PARTIAL C DERIVATIVE WITH RESPECT TO ALF(IN(1,I)) OF COLUMN C IN(2,I) OF A, UNLESS IV(1,I) IS NOT POSITIVE (IN C WHICH CASE COLUMN I OF DA IS IGNORED. IV(1) = -2 C MEANS THERE ARE MORE COLUMNS OF DA TO COME AND C RNSGB SHOULD RETURN FOR THEM. C IV (I/O) INTEGER PARAMETER AND SCRATCH VECTOR. RNSGB RETURNS C WITH IV(1) = 1 WHEN IT WANTS A TO BE EVALUATED AT C ALF AND WITH IV(1) = 2 WHEN IT WANTS DA TO BE C EVALUATED AT ALF. WHEN CALLED WITH IV(1) = -2 C (AFTER A RETURN WITH IV(1) = 2), RNSGB RETURNS C WITH IV(1) = -2 TO GET MORE COLUMNS OF DA. C L (IN) NUMBER OF LINEAR PARAMETERS TO BE ESTIMATED. C L1 (IN) L+1 IF PHI(L+1) IS IN THE MODEL, L IF NOT. C LA (IN) LEAD DIMENSION OF A. MUST BE AT LEAST N. C LIV (IN) LENGTH OF IV. MUST BE AT LEAST 110 + L + 4*P. C LV (IN) LENGTH OF V. MUST BE AT LEAST C 105 + 2*N + L*(L+3)/2 + P*(2*P + 21 + N). C N (IN) NUMBER OF OBSERVATIONS. C NDA (IN) NUMBER OF COLUMNS IN DA AND IN. C P (IN) NUMBER OF NONLINEAR PARAMETERS TO BE ESTIMATED. C V (I/O) FLOATING-POINT PARAMETER AND SCRATCH VECTOR. C Y (IN) RIGHT-HAND SIDE VECTOR. C C C *** EXTERNAL SUBROUTINES *** C REAL L7SVX, L7SVN, R7MDC EXTERNAL IVSET, ITSUM, L7ITV, L7SVX, L7SVN, RN2GB, Q7APL, 1 Q7RFH, R7MDC, S7CPR, V2AXY, V7CPY, V7PRM, V7SCP C C IVSET.... SUPPLIES DEFAULT PARAMETER VALUES. C ITSUM.... PRINTS ITERATION SUMMARY, INITIAL AND FINAL ALF. C L7ITV... APPLIES INVERSE-TRANSPOSE OF COMPACT LOWER TRIANG. MATRIX. C L7SVX... ESTIMATES LARGEST SING. VALUE OF LOWER TRIANG. MATRIX. C L7SVN... ESTIMATES SMALLEST SING. VALUE OF LOWER TRIANG. MATRIX. C RN2GB... UNDERLYING NONLINEAR LEAST-SQUARES SOLVER. C Q7APL... APPLIES HOUSEHOLDER TRANSFORMS STORED BY Q7RFH. C Q7RFH.... COMPUTES QR FACT. VIA HOUSEHOLDER TRANSFORMS WITH PIVOTING. C R7MDC... RETURNS MACHINE-DEP. CONSTANTS. C S7CPR... PRINTS LINEAR PARAMETERS AT SOLUTION. C V2AXY.... ADDS MULTIPLE OF ONE VECTOR TO ANOTHER. C V7CPY.... COPIES ONE VECTOR TO ANOTHER. C V7PRM.... PERMUTES VECTOR. C V7SCL... SCALES AND COPIES ONE VECTOR TO ANOTHER. C C *** LOCAL VARIABLES *** C INTEGER AR1, CSAVE1, D1, DR1, DR1L, I, I1, 1 IPIV1, IER, IV1, J1, JLEN, K, LL1O2, MD, N1, N2, 2 NML, NRAN, R1, R1L, RD1 REAL SINGTL, T REAL MACHEP, NEGONE, SNGFAC, ZERO C C *** SUBSCRIPTS FOR IV AND V *** C INTEGER AR, CSAVE, D, IERS, IPIVS, IV1SAV, 2 IVNEED, J, MODE, NEXTIV, NEXTV, 2 NFCALL, NFGCAL, PERM, R, 3 REGD, REGD0, RESTOR, TOOBIG, VNEED C C *** IV SUBSCRIPT VALUES *** C C/6 C DATA AR/110/, CSAVE/105/, D/27/, IERS/108/, IPIVS/109/, C 1 IV1SAV/104/, IVNEED/3/, J/70/, MODE/35/, NEXTIV/46/, C 2 NEXTV/47/, NFCALL/6/, NFGCAL/7/, PERM/58/, R/61/, REGD/67/, C 3 REGD0/82/, RESTOR/9/, TOOBIG/2/, VNEED/4/ C/7 PARAMETER (AR=110, CSAVE=105, D=27, IERS=108, IPIVS=109, 1 IV1SAV=104, IVNEED=3, J=70, MODE=35, NEXTIV=46, 2 NEXTV=47, NFCALL=6, NFGCAL=7, PERM=58, R=61, REGD=67, 3 REGD0=82, RESTOR=9, TOOBIG=2, VNEED=4) C/ DATA MACHEP/-1.E+0/, NEGONE/-1.E+0/, SNGFAC/1.E+2/, ZERO/0.E+0/ C C++++++++++++++++++++++++++++++++ BODY ++++++++++++++++++++++++++++++ C C IF (IV(1) .EQ. 0) CALL IVSET(1, IV, LIV, LV, V) N1 = 1 NML = N IV1 = IV(1) IF (IV1 .LE. 2) GO TO 20 C C *** CHECK INPUT INTEGERS *** C IF (P .LE. 0) GO TO 240 IF (L .LT. 0) GO TO 240 IF (N .LE. L) GO TO 240 IF (LA .LT. N) GO TO 240 IF (IV1 .LT. 12) GO TO 20 IF (IV1 .EQ. 14) GO TO 20 IF (IV1 .EQ. 12) IV(1) = 13 C C *** FRESH START -- COMPUTE STORAGE REQUIREMENTS *** C IF (IV(1) .GT. 16) GO TO 240 LL1O2 = L*(L+1)/2 JLEN = N*P I = L + P IF (IV(1) .NE. 13) GO TO 10 IV(IVNEED) = IV(IVNEED) + L IV(VNEED) = IV(VNEED) + P + 2*N + JLEN + LL1O2 + L 10 IF (IV(PERM) .LE. AR) IV(PERM) = AR + 1 CALL RN2GB(B, V, V, IV, LIV, LV, N, N, N1, NML, P, V, V, V, ALF) IF (IV(1) .NE. 14) GO TO 999 C C *** STORAGE ALLOCATION *** C IV(IPIVS) = IV(NEXTIV) IV(NEXTIV) = IV(NEXTIV) + L IV(D) = IV(NEXTV) IV(REGD0) = IV(D) + P IV(AR) = IV(REGD0) + N IV(CSAVE) = IV(AR) + LL1O2 IV(J) = IV(CSAVE) + L IV(R) = IV(J) + JLEN IV(NEXTV) = IV(R) + N IV(IERS) = 0 IF (IV1 .EQ. 13) GO TO 999 C C *** SET POINTERS INTO IV AND V *** C 20 AR1 = IV(AR) D1 = IV(D) DR1 = IV(J) DR1L = DR1 + L R1 = IV(R) R1L = R1 + L RD1 = IV(REGD0) CSAVE1 = IV(CSAVE) NML = N - L IF (IV1 .LE. 2) GO TO 50 C 30 N2 = NML CALL RN2GB(B, V(D1), V(DR1L), IV, LIV, LV, NML, N, N1, N2, P, 1 V(R1L), V(RD1), V, ALF) IF (IABS(IV(RESTOR)-2) .EQ. 1 .AND. L .GT. 0) 1 CALL V7CPY(L, C, V(CSAVE1)) IV1 = IV(1) IF (IV1-2) 40, 150, 230 C C *** NEW FUNCTION VALUE (RESIDUAL) NEEDED *** C 40 IV(IV1SAV) = IV(1) IV(1) = IABS(IV1) IF (IV(RESTOR) .EQ. 2 .AND. L .GT. 0) CALL V7CPY(L, V(CSAVE1), C) GO TO 999 C C *** COMPUTE NEW RESIDUAL OR GRADIENT *** C 50 IV(1) = IV(IV1SAV) MD = IV(MODE) IF (MD .LE. 0) GO TO 60 NML = N DR1L = DR1 R1L = R1 60 IF (IV(TOOBIG) .NE. 0) GO TO 30 IF (IABS(IV1) .EQ. 2) GO TO 170 C C *** COMPUTE NEW RESIDUAL *** C IF (L1 .LE. L) CALL V7CPY(N, V(R1), Y) IF (L1 .GT. L) CALL V2AXY(N, V(R1), NEGONE, A(1,L1), Y) IF (MD .GT. 0) GO TO 120 IER = 0 IF (L .LE. 0) GO TO 110 LL1O2 = L * (L + 1) / 2 IPIV1 = IV(IPIVS) CALL Q7RFH(IER, IV(IPIV1), N, LA, 0, L, A, V(AR1), LL1O2, C) C C *** DETERMINE NUMERICAL RANK OF A *** C IF (MACHEP .LE. ZERO) MACHEP = R7MDC(3) SINGTL = SNGFAC * FLOAT(MAX0(L,N)) * MACHEP K = L IF (IER .NE. 0) K = IER - 1 70 IF (K .LE. 0) GO TO 90 T = L7SVX(K, V(AR1), C, C) IF (T .GT. ZERO) T = L7SVN(K, V(AR1), C, C) / T IF (T .GT. SINGTL) GO TO 80 K = K - 1 GO TO 70 C C *** RECORD RANK IN IV(IERS)... IV(IERS) = 0 MEANS FULL RANK, C *** IV(IERS) .GT. 0 MEANS RANK IV(IERS) - 1. C 80 IF (K .GE. L) GO TO 100 90 IER = K + 1 CALL V7SCP(L-K, C(K+1), ZERO) 100 IV(IERS) = IER IF (K .LE. 0) GO TO 110 C C *** APPLY HOUSEHOLDER TRANSFORMATONS TO RESIDUALS... C CALL Q7APL(LA, N, K, A, V(R1), IER) C C *** COMPUTING C NOW MAY SAVE A FUNCTION EVALUATION AT C *** THE LAST ITERATION. C CALL L7ITV(K, C, V(AR1), V(R1)) CALL V7PRM(L, IV(IPIV1), C) C 110 IF(IV(1) .LT. 2) GO TO 220 GO TO 999 C C C *** RESIDUAL COMPUTATION FOR F.D. HESSIAN *** C 120 IF (L .LE. 0) GO TO 140 DO 130 I = 1, L 130 CALL V2AXY(N, V(R1), -C(I), A(1,I), V(R1)) 140 IF (IV(1) .GT. 0) GO TO 30 IV(1) = 2 GO TO 160 C C *** NEW GRADIENT (JACOBIAN) NEEDED *** C 150 IV(IV1SAV) = IV1 IF (IV(NFGCAL) .NE. IV(NFCALL)) IV(1) = 1 160 CALL V7SCP(N*P, V(DR1), ZERO) GO TO 999 C C *** COMPUTE NEW JACOBIAN *** C 170 IF (NDA .LE. 0) GO TO 240 DO 180 I = 1, NDA I1 = IN(1,I) - 1 IF (I1 .LT. 0) GO TO 180 J1 = IN(2,I) K = DR1 + I1*N T = NEGONE IF (J1 .LE. L) T = -C(J1) CALL V2AXY(N, V(K), T, DA(1,I), V(K)) 180 CONTINUE IF (IV1 .EQ. 2) GO TO 190 IV(1) = IV1 GO TO 999 190 IF (L .LE. 0) GO TO 30 IF (MD .GT. 0) GO TO 30 K = DR1 IER = IV(IERS) NRAN = L IF (IER .GT. 0) NRAN = IER - 1 IF (NRAN .LE. 0) GO TO 210 DO 200 I = 1, P CALL Q7APL(LA, N, NRAN, A, V(K), IER) K = K + N 200 CONTINUE 210 CALL V7CPY(L, V(CSAVE1), C) 220 IF (IER .EQ. 0) GO TO 30 C C *** ADJUST SUBSCRIPTS DESCRIBING R AND DR... C NRAN = IER - 1 DR1L = DR1 + NRAN NML = N - NRAN R1L = R1 + NRAN GO TO 30 C C *** CONVERGENCE OR LIMIT REACHED *** C 230 IF (IV(REGD) .EQ. 1) IV(REGD) = RD1 IF (IV(1) .LE. 11) CALL S7CPR(C, IV, L, LIV) GO TO 999 C 240 IV(1) = 66 CALL ITSUM(V, V, IV, LIV, LV, P, V, ALF) C 999 RETURN C C *** LAST CARD OF RNSGB FOLLOWS *** END