*********************************************************************** * * * PEQN - DISCRETE NEWTON METHOD FOR LARGE-SCALE SYSTEMS OF * * NONLINEAR EQUATIONS WITH SPARSE JACOBIAN MATRICES. * * * *********************************************************************** 1. Introduction: ---------------- The double-precision FORTRAN 77 basic subroutine PEQN is designed to find a close approximation to a solution of nonlinear equations FA_1(X) = 0, FA_2(X) = 0, ..., FA_N(X)=0. Here X is a vector of N variables and FA_I(X), 1 <= I <= N, are twice continuously differentiable functions. We assume that N is large, but partial functions FA_I(X), 1 <= I <= N depend on a small number of variables. Thus the mapping AF(X) = [FA_1(X), FA_2(X), ..., FA_N(X)] has a sparse Jacobian matrix, which will be denoted by AG(X) (it has N rows and N columns). The sparsity pattern of the Jacobian matrix is stored in the coordinate form if ISPAS=1 or in the standard compressed row format if ISPAS=2 using arrays IAG and JAG. For example, if the Jacobian matrix has the following pattern AG = | * * 0 * | | * * * 0 | | * 0 0 * | | 0 * * 0 | (asterisks denote nonzero elements) then arrays IAG and JAG contain elements IAG(1)=1, IAG(2)=1, IAG(3)=1, IAG(4)=2, IAG(5)=2, IAG(6)=2, IAG(7)=3, IAG(8)=3, IAG(9)=4, IAG(10)=4, JAG(1)=1, JAG(2)=2, JAG(3)=4, JAG(4)=1, JAG(5)=2, JAG(6)=3, JAG(7)=1, JAG(8)=4, JAG(9)=2, JAG(10)=3 if ISPAS=1 or IAG(1)=1, IAG(2)=4, IAG(3)=7, IAG(4)=9, IAG(5)=11, JAG(1)=1, JAG(2)=2, JAG(3)=4, JAG(4)=1, JAG(5)=2, JAG(6)=3, JAG(7)=1, JAG(8)=4, JAG(9)=2, JAG(10)=3 if ISPAS=2. In the first case, nonzero elements can be sorted in an arbitrary order (not only by rows as in the above example). Arrays IAG and JAG have to be declared with lengths N+MA and MA at least, respectively, where MA is the number of nonzero elements. In the second case, nonzero elements can be sorted only by rows. Components of IAG contain total numbers of nonzero elements in all previous rows increased by 1 and elements of JAG contain corresponding column indices (note that IAG has N+1 elements and the last element is equal to MA+1). Arrays IAG and JAG have to be declared with length N+1 and MA at least, respectively. To simplify user's work, an additional easy to use subroutine PEQNU is added. It calls the basic general subroutine PEQN. All subroutines contain a description of formal parameters and extensive comments. Furthermore, test program TEQNU is included, which contains several test problems (see e.g. [2]). This test program serves as an example for using the subroutine PEQNU, verifies its correctness and demonstrates its efficiency. In this short guide, we describe all subroutines which can be called from the user's program. A detailed description of the method is given in [1]. In the description of formal parameters, we introduce a type of the argument that specifies whether the argument must have a value defined on entry to the subroutine (I), whether it is a value which will be returned (O), or both (U), or whether it is an auxiliary value (A). Besides formal parameters, we can use a COMMON /STAT/ block containing statistical information. This block, used in each subroutine has the following form: COMMON /STAT/ NRES,NDEC,NIN,NIT,NFV,NFG,NFH The arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- NRES O Positive INTEGER variable that indicates the number of restarts. NDEC O Positive INTEGER variable that indicates the number of matrix decompositions. NIN O Positive INTEGER variable that indicates the number of inner iterations (for solving linear systems). NIT O Positive INTEGER variable that indicates the number of iterations. NFV O Positive INTEGER variable that indicates the number of function evaluations. NFG O Positive INTEGER variable that indicates the number of gradient evaluations. NFH O Positive INTEGER variable that indicates the number of Hessian evaluations. 2. Subroutine PEQNU: -------------------- The calling sequence is CALL PEQNU(N,MA,X,AF,IAG,JAG,IPAR,RPAR,F,GMAX,IDER,ISPAS,IPRNT,ITERM) The arguments have the following meaning. Argument Type Significance ---------------------------------------------------------------------- N I Positive INTEGER variable that specifies the number of variables of the partially separable function. MA I Number of nonzero elements in the Jacobian matrix. This parameter is used as input only if ISPAS=1 (it defines dimensions of arrays IAG and JAG in this case). X(N) U On input, DOUBLE PRECISION vector with the initial estimate to the solution. On output, the approximation to the minimum. AF(N) O DOUBLE PRECISION vector which contains values of partial functions. IAG(N+1) I INTEGER array which contains pointers of the first elements in rows of the Jacobian matrix. JAG(MA) I INTEGER array which contains column indices of the nonzero elements. IPAR(7) U INTEGER parameters: IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)-unused, IPAR(4)-unused, IPAR(5)=MOS1, IPAR(6)=MOS2, IPAR(7)-unused. Parameters MIT, MFV, MOS1, MOS2 are described in Section 3 together with other parameters of the subroutine PEQN. RPAR(9) U DOUBLE PRECISION parameters: RPAR(1)=XMAX, RPAR(2)=TOLX, RPAR(3)=TOLF, RPAR(4)=TOLB, RPAR(5)=TOLG, RPAR(6)-unused, RPAR(7)-unused, RPAR(8)=ETA2, RPAR(9)-unused. Parameters XMAX, TOLX, TOLF, TOLB, TOLG, ETA2 are described in Section 3 together with other parameters of the subroutine PEQN. F O DOUBLE PRECISION value of the objective function at the solution X. GMAX O DOUBLE PRECISION maximum absolute value of a partial derivative of the squared norm. IDER I INGEGER variable that specifies the degree of analytically computed derivatives (0 OR 1). ISPAS I INTEGER variable that specifies sparse structure of the Jacobian matrix: ISPAS= 1 - the coordinate form is used, ISPAS= 2 - the standard row compresed format is used. IPRNT I INTEGER variable that specifies PRINT: IPRNT= 0 - print is suppressed, IPRNT= 1 - basic print of final results, IPRNT=-1 - extended print of final results, IPRNT= 2 - basic print of intermediate and final results, IPRNT=-2 - extended print of intermediate and final results. ITERM O INTEGER variable that indicates the cause of termination: ITERM= 1 - if |X - XO| was less than or equal to TOLX in two subsequent iterations, ITERM= 2 - if |F - FO| was less than or equal to TOLF in two subsequent iterations, ITERM= 3 - if F is less than or equal to TOLB, ITERM= 4 - if GMAX is less than or equal to TOLG, ITERM= 6 - if termination criterion was not satisfied, but the solution is probably acceptable, ITERM=11 - if NIT exceeded MIT, ITERM=12 - if NFV exceeded MFV, ITERM< 0 - if the method failed. The subroutines PEQNU requires the user supplied subroutines FUN and DFUN that define partial functions and their gradients and have the form SUBROUTINE FUN(NF,KA,X,FA) SUBROUTINE DFUN(NF,KA,X,GA) If IDER=0, the subroutine DFUN can be empty. The arguments of the user supplied subroutines have the following meaning. Argument Type Significance ---------------------------------------------------------------------- N I Positive INTEGER variable that specifies the number of variables of the objective function. KA I INTEGER index of the partial function. X(N) I DOUBLE PRECISION an estimate to the solution. FA O DOUBLE PRECISION value of the KA-th partial function at the point X. 3. Subroutine PEQN: ------------------- This general subroutine is called from all subroutines described in Section 2. The calling sequence is CALL PEQN(N,X,GA,AG,IAG,JAG,IB,IW1,IW2,IW3,IW4,G,S,XO,GO,XS,GS, & XP,GP,AF,AFO,AFD,XMAX,TOLX,TOLF,TOLB,TOLG,ETA2,GMAX,F,MIT,MFV, & MOS1,MOS2,IDER,IPRNT,ITERM) The arguments N, X, IAG, JAG, AF, GMAX, F, IDER, IPRNT, ITERM have the same meaning as in Section 2. Other arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- GA(N) A DOUBLE PRECISION gradient of the partial function. AG(MA) A DOUBLE PRECISION nonzero elements of the Jacobian matrix. IB(N) A INTEGER permutation vector. IW1(N) A INTEGER auxiliary array. IW2(N) A INTEGER auxiliary array. IW3(N) A INTEGER auxiliary array. IW4(N) A INTEGER auxiliary array. G(N) A DOUBLE PRECISION gradient of the objective function. S(N) A DOUBLE PRECISION direction vector. XO(N) A DOUBLE PRECISION array which contains increments of variables. GO(N) A DOUBLE PRECISION array which contains increments of gradients. XS(N) A DOUBLE PRECISION auxiliary array. GS(N) A DOUBLE PRECISION auxiliary array. XP(N) A DOUBLE PRECISION auxiliary array. GP(N) A DOUBLE PRECISION auxiliary array. AFO(N) A DOUBLE PRECISION vector which contains old values of partial functions. AFD(N) A DOUBLE PRECISION auxiliary array. XMAX U DOUBLE PRECISION maximum stepsize; the choice XMAX=0 causes that the default value 1.0D+16 will be taken. TOLX U DOUBLE PRECISION tolerance for the change of the coordinate vector X; the choice TOLX=0 causes that the default value TOLX=1.0D-16 will be taken. TOLF U DOUBLE PRECISION tolerance for the change of function values; the choice TOLF=0 causes that the default value TOLF=1.0D-16 will be taken. TOLB U DOUBLE PRECISION minimum acceptable function value; the choice TOLB=0 causes that the default value TOLB=1.0D-16 will be taken. TOLG U DOUBLE PRECISION tolerance for the Lagrangian function gradient; the choice TOLG=0 causes that the default value TOLG=1.0D-6 will be taken. ETA2 U DOUBLE PRECISION damping parametr for an incomplete LU preconditioner; the choice ETA2<0 or ETA2>1 causes that the default value ETA2=0 will be chosen. MIT U INTEGER variable that specifies the maximum number of iterations; the choice MIT=0 causes that the default value 1000 will be taken. MFV U INTEGER variable that specifies the maximum number of function evaluations; the choice MFV=0 causes that the default value 1000 will be taken. MOS1 U INTEGER variable that specifies the smoothing strategy for the CGS method: MOS1=1 - smoothing is not used. MOS1=2 - single smoothing strategy is used. MOS1=3 - double smoothing strategy is used. The choice MOS1=0 causes that the default value MOS1=3 will be taken. MOS2 I INTEGER choice of preconditioning strategy: MOS2=1 - preconditioning is not used. MOS2=2 - preconditioning by the incomplete LU decomposition. MOS2=3 - preconditioning by the incomplete LU decomposition combined with preliminary solution of the preconditioned system. The choice of parameter XMAX can be sensitive in many cases. First, partial functions can be evaluated only in a relatively small region (if it contains exponentials) so that the maximum stepsize is necessary. Secondly, the problem can be very ill-conditioned far from the solution point so that large steps can be unsuitable. Finally, if the problem has more local solutions, a suitably chosen maximum stepsize can lead to obtaining a better local solution. The subroutine PEQN requires the user supplied subroutine FUN which is described in Section 2. 4. Verification of the subroutines: ----------------------------------- Subroutine PEQNU can be verified and tested using the program TEQNU. This program calls the subroutines TIUB18 (initiation), TAFU18 (function evaluation) and TAGU18 (gradient evaluation) containing 30 unconstrained test problems with at most 5000 variables [2]. The results obtained by the program TEQNU on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 10 NFV= 41 NFG= 0 F= 0.224531E-22 G= 0.168207E-07 ITERM= 3 NIT= 9 NFV= 46 NFG= 0 F= 0.106897E-22 G= 0.163517E-06 ITERM= 3 NIT= 3 NFV= 19 NFG= 0 F= 0.333989E-19 G= 0.223053E-06 ITERM= 3 NIT= 7 NFV= 23 NFG= 0 F= 0.348196E-17 G= 0.177085E-02 ITERM= 3 NIT= 12 NFV= 63 NFG= 0 F= 0.117206E-16 G= 0.694210E-06 ITERM= 3 NIT= 17 NFV= 52 NFG= 0 F= 0.110919E-16 G= 0.167579E-11 ITERM= 3 NIT= 13 NFV= 41 NFG= 0 F= 0.339913E-19 G= 0.457009E-03 ITERM= 3 NIT= 13 NFV= 73 NFG= 0 F= 0.125748E-25 G= 0.193922E-04 ITERM= 3 NIT= 13 NFV= 99 NFG= 0 F= 0.432936E-21 G= 0.201706E-03 ITERM= 3 NIT= 5 NFV= 41 NFG= 0 F= 0.803846E-25 G= 0.415983E-03 ITERM= 3 NIT= 12 NFV= 37 NFG= 0 F= 0.189327E-25 G= 0.423583E-05 ITERM= 3 NIT= 18 NFV= 55 NFG= 0 F= 0.129272E-16 G= 0.713317E-13 ITERM= 3 NIT= 18 NFV= 39 NFG= 0 F= 0.105290E-16 G= 0.341327E-13 ITERM= 3 NIT= 4 NFV= 13 NFG= 0 F= 0.774783E-20 G= 0.441968E-05 ITERM= 3 NIT= 5 NFV= 36 NFG= 0 F= 0.182567E-17 G= 0.471251E-03 ITERM= 3 NIT= 53 NFV= 319 NFG= 0 F= 0.462169E-17 G= 0.153957 ITERM= 3 NIT= 14 NFV= 48 NFG= 0 F= 0.449140E-22 G= 0.105525E-03 ITERM= 3 NIT= 27 NFV= 82 NFG= 0 F= 0.249708E-20 G= 0.571681E-05 ITERM= 3 NIT= 2 NFV= 7 NFG= 0 F= 0.309324E-21 G= 0.370062E-09 ITERM= 3 NIT= 13 NFV= 43 NFG= 0 F= 0.428279E-20 G= 0.203421E-07 ITERM= 3 NIT= 12 NFV= 37 NFG= 0 F= 0.200623E-20 G= 0.255404E-10 ITERM= 3 NIT= 7 NFV= 50 NFG= 0 F= 0.195350E-19 G= 0.106707E-05 ITERM= 3 NIT= 29 NFV= 262 NFG= 0 F= 0.390327E-17 G= 0.200697E-10 ITERM= 3 NIT= 6 NFV= 31 NFG= 0 F= 0.822526E-23 G= 0.812457E-09 ITERM= 3 NIT= 9 NFV= 46 NFG= 0 F= 0.147127E-23 G= 0.395357E-09 ITERM= 3 NIT= 12 NFV= 61 NFG= 0 F= 0.608837E-17 G= 0.420862E-07 ITERM= 3 NIT= 10 NFV= 51 NFG= 0 F= 0.275078E-20 G= 0.121824E-06 ITERM= 3 NIT= 10 NFV= 60 NFG= 0 F= 0.229532E-16 G= 0.213811E-05 ITERM= 3 NIT= 4 NFV= 53 NFG= 0 F= 0.124549E-19 G= 0.130673E-05 ITERM= 3 NIT= 12 NFV= 162 NFG= 0 F= 0.222959E-21 G= 0.107876E-07 ITERM= 3 NITER = 379 NFVAL = 1990 NSUCC = 30 TIME= 0:00:04.77 The rows corresponding to individual test problems contain the number of iterations NIT, the number of function evaluations NFV, the number of gradient evaluations NFG, the final value of the objective function F (sum of squares of the partial functions), the value of the criterion for the termination G and the cause of termination ITERM. References: ----------- [1] Luksan L., Matonoha C., Vlcek J.: LSA: Algorithms for large-scale unconstrained and box constrained optimization. Research Report V-896, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2004. [2] Luksan L., Vlcek J.: Sparse and partially separable test problems for unconstrained and equality constrained optimization. Research Report V-767, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 1998. 