SUBROUTINE MNG(N, D, X, CALCF, CALCG, IV, LIV, LV, V, 1 UIPARM, URPARM, UFPARM) C C *** MINIMIZE GENERAL UNCONSTRAINED OBJECTIVE FUNCTION USING *** C *** ANALYTIC GRADIENT AND HESSIAN APPROX. FROM SECANT UPDATE *** C INTEGER N, LIV, LV INTEGER IV(LIV), UIPARM(1) REAL D(N), X(N), V(LV), URPARM(1) C DIMENSION V(71 + N*(N+15)/2), UIPARM(*), URPARM(*) EXTERNAL CALCF, CALCG, UFPARM C C *** PURPOSE *** C C THIS ROUTINE INTERACTS WITH SUBROUTINE RMNG IN AN ATTEMPT C TO FIND AN N-VECTOR X* THAT MINIMIZES THE (UNCONSTRAINED) C OBJECTIVE FUNCTION COMPUTED BY CALCF. (OFTEN THE X* FOUND IS C A LOCAL MINIMIZER RATHER THAN A GLOBAL ONE.) C C-------------------------- PARAMETER USAGE -------------------------- C C N........ (INPUT) THE NUMBER OF VARIABLES ON WHICH F DEPENDS, I.E., C THE NUMBER OF COMPONENTS IN X. C D........ (INPUT/OUTPUT) A SCALE VECTOR SUCH THAT D(I)*X(I), C I = 1,2,...,N, ARE ALL IN COMPARABLE UNITS. C D CAN STRONGLY AFFECT THE BEHAVIOR OF MNG. C FINDING THE BEST CHOICE OF D IS GENERALLY A TRIAL- C AND-ERROR PROCESS. CHOOSING D SO THAT D(I)*X(I) C HAS ABOUT THE SAME VALUE FOR ALL I OFTEN WORKS WELL. C THE DEFAULTS PROVIDED BY SUBROUTINE IVSET (SEE IV C BELOW) REQUIRE THE CALLER TO SUPPLY D. C X........ (INPUT/OUTPUT) BEFORE (INITIALLY) CALLING MNG, THE CALL- C ER SHOULD SET X TO AN INITIAL GUESS AT X*. WHEN C MNG RETURNS, X CONTAINS THE BEST POINT SO FAR C FOUND, I.E., THE ONE THAT GIVES THE LEAST VALUE SO C FAR SEEN FOR F(X). C CALCF.... (INPUT) A SUBROUTINE THAT, GIVEN X, COMPUTES F(X). CALCF C MUST BE DECLARED EXTERNAL IN THE CALLING PROGRAM. C IT IS INVOKED BY C CALL CALCF(N, X, NF, F, UIPARM, URPARM, UFPARM) C WHEN CALCF IS CALLED, NF IS THE INVOCATION C COUNT FOR CALCF. NF IS INCLUDED FOR POSSIBLE USE C WITH CALCG. IF X IS OUT OF BOUNDS (E.G., IF IT C WOULD CAUSE OVERFLOW IN COMPUTING F(X)), THEN CALCF C SHOULD SET NF TO 0. THIS WILL CAUSE A SHORTER STEP C TO BE ATTEMPTED. (IF X IS IN BOUNDS, THEN CALCF C SHOULD NOT CHANGE NF.) THE OTHER PARAMETERS ARE AS C DESCRIBED ABOVE AND BELOW. CALCF SHOULD NOT CHANGE C N, P, OR X. C CALCG.... (INPUT) A SUBROUTINE THAT, GIVEN X, COMPUTES G(X), THE GRA- C DIENT OF F AT X. CALCG MUST BE DECLARED EXTERNAL IN C THE CALLING PROGRAM. IT IS INVOKED BY C CALL CALCG(N, X, NF, G, UIPARM, URPARM, UFAPRM) C WHEN CALCG IS CALLED, NF IS THE INVOCATION C COUNT FOR CALCF AT THE TIME F(X) WAS EVALUATED. THE C X PASSED TO CALCG IS USUALLY THE ONE PASSED TO CALCF C ON EITHER ITS MOST RECENT INVOCATION OR THE ONE C PRIOR TO IT. IF CALCF SAVES INTERMEDIATE RESULTS C FOR USE BY CALCG, THEN IT IS POSSIBLE TO TELL FROM C NF WHETHER THEY ARE VALID FOR THE CURRENT X (OR C WHICH COPY IS VALID IF TWO COPIES ARE KEPT). IF G C CANNOT BE COMPUTED AT X, THEN CALCG SHOULD SET NF TO C 0. IN THIS CASE, MNG WILL RETURN WITH IV(1) = 65. C (IF G CAN BE COMPUTED AT X, THEN CALCG SHOULD NOT C CHANGED NF.) THE OTHER PARAMETERS TO CALCG ARE AS C DESCRIBED ABOVE AND BELOW. CALCG SHOULD NOT CHANGE C N OR X. C IV....... (INPUT/OUTPUT) AN INTEGER VALUE ARRAY OF LENGTH LIV (SEE C BELOW) THAT HELPS CONTROL THE MNG ALGORITHM AND C THAT IS USED TO STORE VARIOUS INTERMEDIATE QUANTI- C TIES. OF PARTICULAR INTEREST ARE THE INITIALIZATION/ C RETURN CODE IV(1) AND THE ENTRIES IN IV THAT CONTROL C PRINTING AND LIMIT THE NUMBER OF ITERATIONS AND FUNC- C TION EVALUATIONS. SEE THE SECTION ON IV INPUT C VALUES BELOW. C LIV...... (INPUT) LENGTH OF IV ARRAY. MUST BE AT LEAST 60. IF LIV C IS TOO SMALL, THEN MNG RETURNS WITH IV(1) = 15. C WHEN MNG RETURNS, THE SMALLEST ALLOWED VALUE OF C LIV IS STORED IN IV(LASTIV) -- SEE THE SECTION ON C IV OUTPUT VALUES BELOW. (THIS IS INTENDED FOR USE C WITH EXTENSIONS OF MNG THAT HANDLE CONSTRAINTS.) C LV....... (INPUT) LENGTH OF V ARRAY. MUST BE AT LEAST 71+N*(N+15)/2. C (AT LEAST 77+N*(N+17)/2 FOR MNF, AT LEAST C 78+N*(N+12) FOR MNH). IF LV IS TOO SMALL, THEN C MNG RETURNS WITH IV(1) = 16. WHEN MNG RETURNS, C THE SMALLEST ALLOWED VALUE OF LV IS STORED IN C IV(LASTV) -- SEE THE SECTION ON IV OUTPUT VALUES C BELOW. C V........ (INPUT/OUTPUT) A FLOATING-POINT VALUE ARRAY OF LENGTH LV C (SEE BELOW) THAT HELPS CONTROL THE MNG ALGORITHM C AND THAT IS USED TO STORE VARIOUS INTERMEDIATE C QUANTITIES. OF PARTICULAR INTEREST ARE THE ENTRIES C IN V THAT LIMIT THE LENGTH OF THE FIRST STEP C ATTEMPTED (LMAX0) AND SPECIFY CONVERGENCE TOLERANCES C (AFCTOL, LMAXS, RFCTOL, SCTOL, XCTOL, XFTOL). C UIPARM... (INPUT) USER INTEGER PARAMETER ARRAY PASSED WITHOUT CHANGE C TO CALCF AND CALCG. C URPARM... (INPUT) USER FLOATING-POINT PARAMETER ARRAY PASSED WITHOUT C CHANGE TO CALCF AND CALCG. C UFPARM... (INPUT) USER EXTERNAL SUBROUTINE OR FUNCTION PASSED WITHOUT C CHANGE TO CALCF AND CALCG. C C *** IV INPUT VALUES (FROM SUBROUTINE IVSET) *** C C IV(1)... ON INPUT, IV(1) SHOULD HAVE A VALUE BETWEEN 0 AND 14...... C 0 AND 12 MEAN THIS IS A FRESH START. 0 MEANS THAT C IVSET(2, IV, LIV, LV, V) C IS TO BE CALLED TO PROVIDE ALL DEFAULT VALUES TO IV AND C V. 12 (THE VALUE THAT IVSET ASSIGNS TO IV(1)) MEANS THE C CALLER HAS ALREADY CALLED IVSET AND HAS POSSIBLY CHANGED C SOME IV AND/OR V ENTRIES TO NON-DEFAULT VALUES. C 13 MEANS IVSET HAS BEEN CALLED AND THAT MNG (AND C RMNG) SHOULD ONLY DO THEIR STORAGE ALLOCATION. THAT IS, C THEY SHOULD SET THE OUTPUT COMPONENTS OF IV THAT TELL C WHERE VARIOUS SUBARRAYS ARRAYS OF V BEGIN, SUCH AS IV(G) C (AND, FOR MNH AND RMNH ONLY, IV(DTOL)), AND RETURN. C 14 MEANS THAT A STORAGE HAS BEEN ALLOCATED (BY A CALL C WITH IV(1) = 13) AND THAT THE ALGORITHM SHOULD BE C STARTED. WHEN CALLED WITH IV(1) = 13, MNG RETURNS C IV(1) = 14 UNLESS LIV OR LV IS TOO SMALL (OR N IS NOT C POSITIVE). DEFAULT = 12. C IV(INITH).... IV(25) TELLS WHETHER THE HESSIAN APPROXIMATION H SHOULD C BE INITIALIZED. 1 (THE DEFAULT) MEANS RMNG SHOULD C INITIALIZE H TO THE DIAGONAL MATRIX WHOSE I-TH DIAGONAL C ELEMENT IS D(I)**2. 0 MEANS THE CALLER HAS SUPPLIED A C CHOLESKY FACTOR L OF THE INITIAL HESSIAN APPROXIMATION C H = L*(L**T) IN V, STARTING AT V(IV(LMAT)) = V(IV(42)) C (AND STORED COMPACTLY BY ROWS). NOTE THAT IV(LMAT) MAY C BE INITIALIZED BY CALLING MNG WITH IV(1) = 13 (SEE C THE IV(1) DISCUSSION ABOVE). DEFAULT = 1. C IV(MXFCAL)... IV(17) GIVES THE MAXIMUM NUMBER OF FUNCTION EVALUATIONS C (CALLS ON CALCF) ALLOWED. IF THIS NUMBER DOES NOT SUF- C FICE, THEN MNG RETURNS WITH IV(1) = 9. DEFAULT = 200. C IV(MXITER)... IV(18) GIVES THE MAXIMUM NUMBER OF ITERATIONS ALLOWED. C IT ALSO INDIRECTLY LIMITS THE NUMBER OF GRADIENT EVALUA- C TIONS (CALLS ON CALCG) TO IV(MXITER) + 1. IF IV(MXITER) C ITERATIONS DO NOT SUFFICE, THEN MNG RETURNS WITH C IV(1) = 10. DEFAULT = 150. C IV(OUTLEV)... IV(19) CONTROLS THE NUMBER AND LENGTH OF ITERATION SUM- C MARY LINES PRINTED (BY ITSUM). IV(OUTLEV) = 0 MEANS DO C NOT PRINT ANY SUMMARY LINES. OTHERWISE, PRINT A SUMMARY C LINE AFTER EACH ABS(IV(OUTLEV)) ITERATIONS. IF IV(OUTLEV) C IS POSITIVE, THEN SUMMARY LINES OF LENGTH 78 (PLUS CARRI- C AGE CONTROL) ARE PRINTED, INCLUDING THE FOLLOWING... THE C ITERATION AND FUNCTION EVALUATION COUNTS, F = THE CURRENT C FUNCTION VALUE, RELATIVE DIFFERENCE IN FUNCTION VALUES C ACHIEVED BY THE LATEST STEP (I.E., RELDF = (F0-V(F))/F01, C WHERE F01 IS THE MAXIMUM OF ABS(V(F)) AND ABS(V(F0)) AND C V(F0) IS THE FUNCTION VALUE FROM THE PREVIOUS ITERA- C TION), THE RELATIVE FUNCTION REDUCTION PREDICTED FOR THE C STEP JUST TAKEN (I.E., PRELDF = V(PREDUC) / F01, WHERE C V(PREDUC) IS DESCRIBED BELOW), THE SCALED RELATIVE CHANGE C IN X (SEE V(RELDX) BELOW), THE STEP PARAMETER FOR THE C STEP JUST TAKEN (STPPAR = 0 MEANS A FULL NEWTON STEP, C BETWEEN 0 AND 1 MEANS A RELAXED NEWTON STEP, BETWEEN 1 C AND 2 MEANS A DOUBLE DOGLEG STEP, GREATER THAN 2 MEANS C A SCALED DOWN CAUCHY STEP -- SEE SUBROUTINE DBLDOG), THE C 2-NORM OF THE SCALE VECTOR D TIMES THE STEP JUST TAKEN C (SEE V(DSTNRM) BELOW), AND NPRELDF, I.E., C V(NREDUC)/F01, WHERE V(NREDUC) IS DESCRIBED BELOW -- IF C NPRELDF IS POSITIVE, THEN IT IS THE RELATIVE FUNCTION C REDUCTION PREDICTED FOR A NEWTON STEP (ONE WITH C STPPAR = 0). IF NPRELDF IS NEGATIVE, THEN IT IS THE C NEGATIVE OF THE RELATIVE FUNCTION REDUCTION PREDICTED C FOR A STEP COMPUTED WITH STEP BOUND V(LMAXS) FOR USE IN C TESTING FOR SINGULAR CONVERGENCE. C IF IV(OUTLEV) IS NEGATIVE, THEN LINES OF LENGTH 50 C ARE PRINTED, INCLUDING ONLY THE FIRST 6 ITEMS LISTED C ABOVE (THROUGH RELDX). C DEFAULT = 1. C IV(PARPRT)... IV(20) = 1 MEANS PRINT ANY NONDEFAULT V VALUES ON A C FRESH START OR ANY CHANGED V VALUES ON A RESTART. C IV(PARPRT) = 0 MEANS SKIP THIS PRINTING. DEFAULT = 1. C IV(PRUNIT)... IV(21) IS THE OUTPUT UNIT NUMBER ON WHICH ALL PRINTING C IS DONE. IV(PRUNIT) = 0 MEANS SUPPRESS ALL PRINTING. C DEFAULT = STANDARD OUTPUT UNIT (UNIT 6 ON MOST SYSTEMS). C IV(SOLPRT)... IV(22) = 1 MEANS PRINT OUT THE VALUE OF X RETURNED (AS C WELL AS THE GRADIENT AND THE SCALE VECTOR D). C IV(SOLPRT) = 0 MEANS SKIP THIS PRINTING. DEFAULT = 1. C IV(STATPR)... IV(23) = 1 MEANS PRINT SUMMARY STATISTICS UPON RETURN- C ING. THESE CONSIST OF THE FUNCTION VALUE, THE SCALED C RELATIVE CHANGE IN X CAUSED BY THE MOST RECENT STEP (SEE C V(RELDX) BELOW), THE NUMBER OF FUNCTION AND GRADIENT C EVALUATIONS (CALLS ON CALCF AND CALCG), AND THE RELATIVE C FUNCTION REDUCTIONS PREDICTED FOR THE LAST STEP TAKEN AND C FOR A NEWTON STEP (OR PERHAPS A STEP BOUNDED BY V(LMAXS) C -- SEE THE DESCRIPTIONS OF PRELDF AND NPRELDF UNDER C IV(OUTLEV) ABOVE). C IV(STATPR) = 0 MEANS SKIP THIS PRINTING. C IV(STATPR) = -1 MEANS SKIP THIS PRINTING AS WELL AS THAT C OF THE ONE-LINE TERMINATION REASON MESSAGE. DEFAULT = 1. C IV(X0PRT).... IV(24) = 1 MEANS PRINT THE INITIAL X AND SCALE VECTOR D C (ON A FRESH START ONLY). IV(X0PRT) = 0 MEANS SKIP THIS C PRINTING. DEFAULT = 1. C C *** (SELECTED) IV OUTPUT VALUES *** C C IV(1)........ ON OUTPUT, IV(1) IS A RETURN CODE.... C 3 = X-CONVERGENCE. THE SCALED RELATIVE DIFFERENCE (SEE C V(RELDX)) BETWEEN THE CURRENT PARAMETER VECTOR X AND C A LOCALLY OPTIMAL PARAMETER VECTOR IS VERY LIKELY AT C MOST V(XCTOL). C 4 = RELATIVE FUNCTION CONVERGENCE. THE RELATIVE DIFFER- C ENCE BETWEEN THE CURRENT FUNCTION VALUE AND ITS LO- C CALLY OPTIMAL VALUE IS VERY LIKELY AT MOST V(RFCTOL). C 5 = BOTH X- AND RELATIVE FUNCTION CONVERGENCE (I.E., THE C CONDITIONS FOR IV(1) = 3 AND IV(1) = 4 BOTH HOLD). C 6 = ABSOLUTE FUNCTION CONVERGENCE. THE CURRENT FUNCTION C VALUE IS AT MOST V(AFCTOL) IN ABSOLUTE VALUE. C 7 = SINGULAR CONVERGENCE. THE HESSIAN NEAR THE CURRENT C ITERATE APPEARS TO BE SINGULAR OR NEARLY SO, AND A C STEP OF LENGTH AT MOST V(LMAXS) IS UNLIKELY TO YIELD C A RELATIVE FUNCTION DECREASE OF MORE THAN V(SCTOL). C 8 = FALSE CONVERGENCE. THE ITERATES APPEAR TO BE CONVERG- C ING TO A NONCRITICAL POINT. THIS MAY MEAN THAT THE C CONVERGENCE TOLERANCES (V(AFCTOL), V(RFCTOL), C V(XCTOL)) ARE TOO SMALL FOR THE ACCURACY TO WHICH C THE FUNCTION AND GRADIENT ARE BEING COMPUTED, THAT C THERE IS AN ERROR IN COMPUTING THE GRADIENT, OR THAT C THE FUNCTION OR GRADIENT IS DISCONTINUOUS NEAR X. C 9 = FUNCTION EVALUATION LIMIT REACHED WITHOUT OTHER CON- C VERGENCE (SEE IV(MXFCAL)). C 10 = ITERATION LIMIT REACHED WITHOUT OTHER CONVERGENCE C (SEE IV(MXITER)). C 11 = STOPX RETURNED .TRUE. (EXTERNAL INTERRUPT). SEE THE C USAGE NOTES BELOW. C 14 = STORAGE HAS BEEN ALLOCATED (AFTER A CALL WITH C IV(1) = 13). C 17 = RESTART ATTEMPTED WITH N CHANGED. C 18 = D HAS A NEGATIVE COMPONENT AND IV(DTYPE) .LE. 0. C 19...43 = V(IV(1)) IS OUT OF RANGE. C 63 = F(X) CANNOT BE COMPUTED AT THE INITIAL X. C 64 = BAD PARAMETERS PASSED TO ASSESS (WHICH SHOULD NOT C OCCUR). C 65 = THE GRADIENT COULD NOT BE COMPUTED AT X (SEE CALCG C ABOVE). C 67 = BAD FIRST PARAMETER TO IVSET. C 80 = IV(1) WAS OUT OF RANGE. C 81 = N IS NOT POSITIVE. C IV(G)........ IV(28) IS THE STARTING SUBSCRIPT IN V OF THE CURRENT C GRADIENT VECTOR (THE ONE CORRESPONDING TO X). C IV(LASTIV)... IV(44) IS THE LEAST ACCEPTABLE VALUE OF LIV. (IT IS C ONLY SET IF LIV IS AT LEAST 44.) C IV(LASTV).... IV(45) IS THE LEAST ACCEPTABLE VALUE OF LV. (IT IS C ONLY SET IF LIV IS LARGE ENOUGH, AT LEAST IV(LASTIV).) C IV(NFCALL)... IV(6) IS THE NUMBER OF CALLS SO FAR MADE ON CALCF (I.E., C FUNCTION EVALUATIONS). C IV(NGCALL)... IV(30) IS THE NUMBER OF GRADIENT EVALUATIONS (CALLS ON C CALCG). C IV(NITER).... IV(31) IS THE NUMBER OF ITERATIONS PERFORMED. C C *** (SELECTED) V INPUT VALUES (FROM SUBROUTINE IVSET) *** C C V(BIAS)..... V(43) IS THE BIAS PARAMETER USED IN SUBROUTINE DBLDOG -- C SEE THAT SUBROUTINE FOR DETAILS. DEFAULT = 0.8. C V(AFCTOL)... V(31) IS THE ABSOLUTE FUNCTION CONVERGENCE TOLERANCE. C IF MNG FINDS A POINT WHERE THE FUNCTION VALUE IS LESS C THAN V(AFCTOL) IN ABSOLUTE VALUE, AND IF MNG DOES NOT C RETURN WITH IV(1) = 3, 4, OR 5, THEN IT RETURNS WITH C IV(1) = 6. THIS TEST CAN BE TURNED OFF BY SETTING C V(AFCTOL) TO ZERO. DEFAULT = MAX(10**-20, MACHEP**2), C WHERE MACHEP IS THE UNIT ROUNDOFF. C V(DINIT).... V(38), IF NONNEGATIVE, IS THE VALUE TO WHICH THE SCALE C VECTOR D IS INITIALIZED. DEFAULT = -1. C V(LMAX0).... V(35) GIVES THE MAXIMUM 2-NORM ALLOWED FOR D TIMES THE C VERY FIRST STEP THAT MNG ATTEMPTS. THIS PARAMETER CAN C MARKEDLY AFFECT THE PERFORMANCE OF MNG. C V(LMAXS).... V(36) IS USED IN TESTING FOR SINGULAR CONVERGENCE -- IF C THE FUNCTION REDUCTION PREDICTED FOR A STEP OF LENGTH C BOUNDED BY V(LMAXS) IS AT MOST V(SCTOL) * ABS(F0), WHERE C F0 IS THE FUNCTION VALUE AT THE START OF THE CURRENT C ITERATION, AND IF MNG DOES NOT RETURN WITH IV(1) = 3, C 4, 5, OR 6, THEN IT RETURNS WITH IV(1) = 7. DEFAULT = 1. C V(RFCTOL)... V(32) IS THE RELATIVE FUNCTION CONVERGENCE TOLERANCE. C IF THE CURRENT MODEL PREDICTS A MAXIMUM POSSIBLE FUNCTION C REDUCTION (SEE V(NREDUC)) OF AT MOST V(RFCTOL)*ABS(F0) C AT THE START OF THE CURRENT ITERATION, WHERE F0 IS THE C THEN CURRENT FUNCTION VALUE, AND IF THE LAST STEP ATTEMPT- C ED ACHIEVED NO MORE THAN TWICE THE PREDICTED FUNCTION C DECREASE, THEN MNG RETURNS WITH IV(1) = 4 (OR 5). C DEFAULT = MAX(10**-10, MACHEP**(2/3)), WHERE MACHEP IS C THE UNIT ROUNDOFF. C V(SCTOL).... V(37) IS THE SINGULAR CONVERGENCE TOLERANCE -- SEE THE C DESCRIPTION OF V(LMAXS) ABOVE. C V(TUNER1)... V(26) HELPS DECIDE WHEN TO CHECK FOR FALSE CONVERGENCE. C THIS IS DONE IF THE ACTUAL FUNCTION DECREASE FROM THE C CURRENT STEP IS NO MORE THAN V(TUNER1) TIMES ITS PREDICT- C ED VALUE. DEFAULT = 0.1. C V(XCTOL).... V(33) IS THE X-CONVERGENCE TOLERANCE. IF A NEWTON STEP C (SEE V(NREDUC)) IS TRIED THAT HAS V(RELDX) .LE. V(XCTOL) C AND IF THIS STEP YIELDS AT MOST TWICE THE PREDICTED FUNC- C TION DECREASE, THEN MNG RETURNS WITH IV(1) = 3 (OR 5). C (SEE THE DESCRIPTION OF V(RELDX) BELOW.) C DEFAULT = MACHEP**0.5, WHERE MACHEP IS THE UNIT ROUNDOFF. C V(XFTOL).... V(34) IS THE FALSE CONVERGENCE TOLERANCE. IF A STEP IS C TRIED THAT GIVES NO MORE THAN V(TUNER1) TIMES THE PREDICT- C ED FUNCTION DECREASE AND THAT HAS V(RELDX) .LE. V(XFTOL), C AND IF MNG DOES NOT RETURN WITH IV(1) = 3, 4, 5, 6, OR C 7, THEN IT RETURNS WITH IV(1) = 8. (SEE THE DESCRIPTION C OF V(RELDX) BELOW.) DEFAULT = 100*MACHEP, WHERE C MACHEP IS THE UNIT ROUNDOFF. C V(*)........ IVSET SUPPLIES TO V A NUMBER OF TUNING CONSTANTS, WITH C WHICH IT SHOULD ORDINARILY BE UNNECESSARY TO TINKER. SEE C SECTION 17 OF VERSION 2.2 OF THE NL2SOL USAGE SUMMARY C (I.E., THE APPENDIX TO REF. 1) FOR DETAILS ON V(I), C I = DECFAC, INCFAC, PHMNFC, PHMXFC, RDFCMN, RDFCMX, C TUNER2, TUNER3, TUNER4, TUNER5. C C *** (SELECTED) V OUTPUT VALUES *** C C V(DGNORM)... V(1) IS THE 2-NORM OF (DIAG(D)**-1)*G, WHERE G IS THE C MOST RECENTLY COMPUTED GRADIENT. C V(DSTNRM)... V(2) IS THE 2-NORM OF DIAG(D)*STEP, WHERE STEP IS THE C CURRENT STEP. C V(F)........ V(10) IS THE CURRENT FUNCTION VALUE. C V(F0)....... V(13) IS THE FUNCTION VALUE AT THE START OF THE CURRENT C ITERATION. C V(NREDUC)... V(6), IF POSITIVE, IS THE MAXIMUM FUNCTION REDUCTION C POSSIBLE ACCORDING TO THE CURRENT MODEL, I.E., THE FUNC- C TION REDUCTION PREDICTED FOR A NEWTON STEP (I.E., C STEP = -H**-1 * G, WHERE G IS THE CURRENT GRADIENT AND C H IS THE CURRENT HESSIAN APPROXIMATION). C IF V(NREDUC) IS NEGATIVE, THEN IT IS THE NEGATIVE OF C THE FUNCTION REDUCTION PREDICTED FOR A STEP COMPUTED WITH C A STEP BOUND OF V(LMAXS) FOR USE IN TESTING FOR SINGULAR C CONVERGENCE. C V(PREDUC)... V(7) IS THE FUNCTION REDUCTION PREDICTED (BY THE CURRENT C QUADRATIC MODEL) FOR THE CURRENT STEP. THIS (DIVIDED BY C V(F0)) IS USED IN TESTING FOR RELATIVE FUNCTION C CONVERGENCE. C V(RELDX).... V(17) IS THE SCALED RELATIVE CHANGE IN X CAUSED BY THE C CURRENT STEP, COMPUTED AS C MAX(ABS(D(I)*(X(I)-X0(I)), 1 .LE. I .LE. P) / C MAX(D(I)*(ABS(X(I))+ABS(X0(I))), 1 .LE. I .LE. P), C WHERE X = X0 + STEP. C C------------------------------- NOTES ------------------------------- C C *** ALGORITHM NOTES *** C C THIS ROUTINE USES A HESSIAN APPROXIMATION COMPUTED FROM THE C BFGS UPDATE (SEE REF 3). ONLY A CHOLESKY FACTOR OF THE HESSIAN C APPROXIMATION IS STORED, AND THIS IS UPDATED USING IDEAS FROM C REF. 4. STEPS ARE COMPUTED BY THE DOUBLE DOGLEG SCHEME DESCRIBED C IN REF. 2. THE STEPS ARE ASSESSED AS IN REF. 1. C C *** USAGE NOTES *** C C AFTER A RETURN WITH IV(1) .LE. 11, IT IS POSSIBLE TO RESTART, C I.E., TO CHANGE SOME OF THE IV AND V INPUT VALUES DESCRIBED ABOVE C AND CONTINUE THE ALGORITHM FROM THE POINT WHERE IT WAS INTERRUPT- C ED. IV(1) SHOULD NOT BE CHANGED, NOR SHOULD ANY ENTRIES OF IV C AND V OTHER THAN THE INPUT VALUES (THOSE SUPPLIED BY IVSET). C THOSE WHO DO NOT WISH TO WRITE A CALCG WHICH COMPUTES THE C GRADIENT ANALYTICALLY SHOULD CALL MNF RATHER THAN MNG. C MNF USES FINITE DIFFERENCES TO COMPUTE AN APPROXIMATE GRADIENT. C THOSE WHO WOULD PREFER TO PROVIDE F AND G (THE FUNCTION AND C GRADIENT) BY REVERSE COMMUNICATION RATHER THAN BY WRITING SUBROU- C TINES CALCF AND CALCG MAY CALL ON RMNG DIRECTLY. SEE THE COM- C MENTS AT THE BEGINNING OF RMNG. C THOSE WHO USE MNG INTERACTIVELY MAY WISH TO SUPPLY THEIR C OWN STOPX FUNCTION, WHICH SHOULD RETURN .TRUE. IF THE BREAK KEY C HAS BEEN PRESSED SINCE STOPX WAS LAST INVOKED. THIS MAKES IT C POSSIBLE TO EXTERNALLY INTERRUPT MNG (WHICH WILL RETURN WITH C IV(1) = 11 IF STOPX RETURNS .TRUE.). C STORAGE FOR G IS ALLOCATED AT THE END OF V. THUS THE CALLER C MAY MAKE V LONGER THAN SPECIFIED ABOVE AND MAY ALLOW CALCG TO USE C ELEMENTS OF G BEYOND THE FIRST N AS SCRATCH STORAGE. C C *** PORTABILITY NOTES *** C C THE MNG DISTRIBUTION TAPE CONTAINS BOTH SINGLE- AND DOUBLE- C PRECISION VERSIONS OF THE MNG SOURCE CODE, SO IT SHOULD BE UN- C NECESSARY TO CHANGE PRECISIONS. C ONLY THE FUNCTIONS I7MDCN AND R7MDC CONTAIN MACHINE-DEPENDENT C CONSTANTS. TO CHANGE FROM ONE MACHINE TO ANOTHER, IT SHOULD C SUFFICE TO CHANGE THE (FEW) RELEVANT LINES IN THESE FUNCTIONS. C INTRINSIC FUNCTIONS ARE EXPLICITLY DECLARED. ON CERTAIN COM- C PUTERS (E.G. UNIVAC), IT MAY BE NECESSARY TO COMMENT OUT THESE C DECLARATIONS. SO THAT THIS MAY BE DONE AUTOMATICALLY BY A SIMPLE C PROGRAM, SUCH DECLARATIONS ARE PRECEDED BY A COMMENT HAVING C/+ C IN COLUMNS 1-3 AND BLANKS IN COLUMNS 4-72 AND ARE FOLLOWED BY C A COMMENT HAVING C/ IN COLUMNS 1 AND 2 AND BLANKS IN COLUMNS 3-72. C THE MNG SOURCE CODE IS EXPRESSED IN 1966 ANSI STANDARD C FORTRAN. IT MAY BE CONVERTED TO FORTRAN 77 BY COMMENTING OUT ALL C LINES THAT FALL BETWEEN A LINE HAVING C/6 IN COLUMNS 1-3 AND A C LINE HAVING C/7 IN COLUMNS 1-3 AND BY REMOVING (I.E., REPLACING C BY A BLANK) THE C IN COLUMN 1 OF THE LINES THAT FOLLOW THE C/7 C LINE AND PRECEDE A LINE HAVING C/ IN COLUMNS 1-2 AND BLANKS IN C COLUMNS 3-72. THESE CHANGES CONVERT SOME DATA STATEMENTS INTO C PARAMETER STATEMENTS, CONVERT SOME VARIABLES FROM REAL TO C CHARACTER*4, AND MAKE THE DATA STATEMENTS THAT INITIALIZE THESE C VARIABLES USE CHARACTER STRINGS DELIMITED BY PRIMES INSTEAD C OF HOLLERITH CONSTANTS. (SUCH VARIABLES AND DATA STATEMENTS C APPEAR ONLY IN MODULES ITSUM AND PARCK. PARAMETER STATEMENTS C APPEAR NEARLY EVERYWHERE.) THESE CHANGES ALSO ADD SAVE STATE- C MENTS FOR VARIABLES GIVEN MACHINE-DEPENDENT CONSTANTS BY R7MDC. C C *** REFERENCES *** C C 1. DENNIS, J.E., GAY, D.M., AND WELSCH, R.E. (1981), ALGORITHM 573 -- C AN ADAPTIVE NONLINEAR LEAST-SQUARES ALGORITHM, ACM TRANS. C MATH. SOFTWARE 7, PP. 369-383. C C 2. DENNIS, J.E., AND MEI, H.H.W. (1979), TWO NEW UNCONSTRAINED OPTI- C MIZATION ALGORITHMS WHICH USE FUNCTION AND GRADIENT C VALUES, J. OPTIM. THEORY APPLIC. 28, PP. 453-482. C C 3. DENNIS, J.E., AND MORE, J.J. (1977), QUASI-NEWTON METHODS, MOTIVA- C TION AND THEORY, SIAM REV. 19, PP. 46-89. C C 4. GOLDFARB, D. (1976), FACTORIZED VARIABLE METRIC METHODS FOR UNCON- C STRAINED OPTIMIZATION, MATH. COMPUT. 30, PP. 796-811. C C *** GENERAL *** C C CODED BY DAVID M. GAY (WINTER 1980). REVISED SUMMER 1982. C THIS SUBROUTINE WAS WRITTEN IN CONNECTION WITH RESEARCH C SUPPORTED IN PART BY THE NATIONAL SCIENCE FOUNDATION UNDER C GRANTS MCS-7600324, DCR75-10143, 76-14311DSS, MCS76-11989, C AND MCS-7906671. C. C C---------------------------- DECLARATIONS --------------------------- C EXTERNAL IVSET, RMNG C C IVSET... SUPPLIES DEFAULT IV AND V INPUT COMPONENTS. C RMNG... REVERSE-COMMUNICATION ROUTINE THAT CARRIES OUT MNG ALGO- C RITHM. C INTEGER G1, IV1, NF REAL F C C *** SUBSCRIPTS FOR IV *** C INTEGER NEXTV, NFCALL, NFGCAL, G, TOOBIG, VNEED C C/6 C DATA NEXTV/47/, NFCALL/6/, NFGCAL/7/, G/28/, TOOBIG/2/, VNEED/4/ C/7 PARAMETER (NEXTV=47, NFCALL=6, NFGCAL=7, G=28, TOOBIG=2, VNEED=4) C/ C C+++++++++++++++++++++++++++++++ BODY ++++++++++++++++++++++++++++++++ C IF (IV(1) .EQ. 0) CALL IVSET(2, IV, LIV, LV, V) IV1 = IV(1) IF (IV1 .EQ. 12 .OR. IV1 .EQ. 13) IV(VNEED) = IV(VNEED) + N IF (IV1 .EQ. 14) GO TO 10 IF (IV1 .GT. 2 .AND. IV1 .LT. 12) GO TO 10 G1 = 1 IF (IV1 .EQ. 12) IV(1) = 13 GO TO 20 C 10 G1 = IV(G) C 20 CALL RMNG(D, F, V(G1), IV, LIV, LV, N, V, X) IF (IV(1) - 2) 30, 40, 50 C 30 NF = IV(NFCALL) CALL CALCF(N, X, NF, F, UIPARM, URPARM, UFPARM) IF (NF .LE. 0) IV(TOOBIG) = 1 GO TO 20 C 40 NF = IV(NFGCAL) CALL CALCG(N, X, NF, V(G1), UIPARM, URPARM, UFPARM) IF (NF .LE. 0) IV(TOOBIG) = 1 GO TO 20 C 50 IF (IV(1) .NE. 14) GO TO 999 C C *** STORAGE ALLOCATION C IV(G) = IV(NEXTV) IV(NEXTV) = IV(G) + N IF (IV1 .NE. 13) GO TO 10 C 999 RETURN C *** LAST CARD OF MNG FOLLOWS *** END