***********************************************************************
* *
* PEQL - LIMITED-MEMORY INVERSE COLUMN-UPDATE METHOD FOR *
* LARGE-SCALE SYSTEMS OF NONLINEAR EQUATIONS WITH *
* SPARSE JACOBIAN MATRICES. *
* *
***********************************************************************
1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PEQL 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
PEQLU is added. It calls the basic general subroutine PEQL. All
subroutines contain a description of formal parameters and extensive
comments. Furthermore, test program TEQLU is included, which contains
several test problems (see e.g. [2]). This test program serves as an
example for using the subroutine PEQLU, 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 PEQLU:
--------------------
The calling sequence is
CALL PEQLU(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)=MF.
Parameters MIT, MFV, MOS1, MOS2, MF are described in
Section 3 together with other parameters of the
subroutine PEQL.
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 PEQL.
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 PEQLU 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 PEQL:
-------------------
This general subroutine is called from all subroutines described
in Section 2. The calling sequence is
CALL PEQL(N,X,GA,AG,IAG,JAG,IB,IW1,IW2,IW3,IW4,XM,GM,IM,G,S,XO,
& GO,XS,GS,XP,GP,AF,AFO,AFD,XMAX,TOLX,TOLF,TOLB,TOLG,ETA2,GMAX,
& F,MIT,MFV,MOS1,MOS2,MF,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.
XM(N*MF) A DOUBLE PRECISION array which contains vectors for
inverse column-update.
GM(MF) A DOUBLE PRECISION array which contains values for
inverse column-update.
IM(MF) A INTEGER array which contains indices for inverse
column-update.
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.
MF U The number of limited-memory variable metric updates
in each iteration (they use 2*MF stored vectors).
The choice MF=0 causes that the default value MF=6
will be taken.
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 PEQL requires the user supplied subroutine FUN
which is described in Section 2.
4. Verification of the subroutines:
-----------------------------------
Subroutine PEQLU can be verified and tested using the program
TEQLU. 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 TEQLU on a PC computer with Microsoft
Power Station Fortran compiler have the following form.
NIT= 30 NFV= 64 NFG= 0 F= 0.326079E-18 G= 0.154142E-03 ITERM= 3
NIT= 17 NFV= 57 NFG= 0 F= 0.720058E-19 G= 0.261551E-07 ITERM= 3
NIT= 5 NFV= 11 NFG= 0 F= 0.861220E-16 G= 0.366389E-03 ITERM= 3
NIT= 11 NFV= 19 NFG= 0 F= 0.115060E-18 G= 0.358897E-01 ITERM= 3
NIT= 20 NFV= 56 NFG= 0 F= 0.335602E-16 G= 0.121910E-06 ITERM= 3
NIT= 22 NFV= 31 NFG= 0 F= 0.167377E-16 G= 0.898624E-08 ITERM= 3
NIT= 25 NFV= 42 NFG= 0 F= 0.137004E-20 G= 0.185851E-05 ITERM= 3
NIT= 21 NFV= 60 NFG= 0 F= 0.496243E-28 G= 0.183782E-07 ITERM= 3
NIT= 32 NFV= 71 NFG= 0 F= 0.220876E-21 G= 0.800603E-05 ITERM= 3
NIT= 9 NFV= 24 NFG= 0 F= 0.202316E-20 G= 0.162996E-03 ITERM= 3
NIT= 16 NFV= 23 NFG= 0 F= 0.116022E-21 G= 0.130018E-02 ITERM= 3
NIT= 23 NFV= 40 NFG= 0 F= 0.861690E-16 G= 0.190460E-08 ITERM= 3
NIT= 24 NFV= 32 NFG= 0 F= 0.234892E-16 G= 0.204525E-08 ITERM= 3
NIT= 8 NFV= 13 NFG= 0 F= 0.596974E-21 G= 0.811563E-05 ITERM= 3
NIT= 12 NFV= 28 NFG= 0 F= 0.124901E-17 G= 0.305897 ITERM= 3
NIT= 22 NFV= 78 NFG= 0 F= 0.984840E-20 G= 0.125407E-03 ITERM= 3
NIT= 17 NFV= 43 NFG= 0 F= 0.130235E-20 G= 0.154659E-04 ITERM= 3
NIT= 46 NFV= 61 NFG= 0 F= 0.224793E-17 G= 0.116353E-01 ITERM= 3
NIT= 2 NFV= 5 NFG= 0 F= 0.704403E-18 G= 0.221630E-06 ITERM= 3
NIT= 18 NFV= 30 NFG= 0 F= 0.158787E-16 G= 0.312477E-03 ITERM= 3
NIT= 25 NFV= 34 NFG= 0 F= 0.233925E-16 G= 0.135133E-05 ITERM= 3
NIT= 14 NFV= 45 NFG= 0 F= 0.189862E-17 G= 0.128826E-01 ITERM= 3
NIT= 23 NFV= 106 NFG= 0 F= 0.194742E-18 G= 0.550497E-08 ITERM= 3
NIT= 20 NFV= 53 NFG= 0 F= 0.737500E-17 G= 0.611156E-08 ITERM= 3
NIT= 29 NFV= 50 NFG= 0 F= 0.208794E-17 G= 0.413643E-08 ITERM= 3
NIT= 36 NFV= 67 NFG= 0 F= 0.132055E-17 G= 0.481013E-08 ITERM= 3
NIT= 40 NFV= 75 NFG= 0 F= 0.659356E-17 G= 0.862034E-08 ITERM= 3
NIT= 27 NFV= 83 NFG= 0 F= 0.461856E-18 G= 0.268680E-08 ITERM= 3
NIT= 12 NFV= 95 NFG= 0 F= 0.206962E-16 G= 0.754042E-08 ITERM= 3
NIT= 18 NFV= 145 NFG= 0 F= 0.740533E-16 G= 0.167985E-07 ITERM= 3
NITER = 624 NFVAL = 1541 NSUCC = 30
TIME= 0:00:04.13
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.