C
C HOMPACK is a suite of FORTRAN 77 subroutines for solving nonlinear
C systems of equations by homotopy methods. There are subroutines for
C fixed point, zero finding, and general homotopy curve tracking problems,
C utilizing both dense and sparse Jacobian matrices, and implementing
C three different algorithms: ODE-based, normal flow, and augmented
C Jacobian. The (driver) subroutines called by the user are given in the
C table below, and are well documented internally. The user need not
C be concerned with any other subroutines in HOMPACK.
C
C
C Problem type
C --------|--------|--------|--------|--------|--------|
C x = f(x) | F(x) = 0 |rho(a,lambda,x)=0|
C --------|--------|--------|--------|--------|--------|
C dense | sparse | dense | sparse | dense | sparse | Algorithm
C --------|--------|--------|--------|--------|--------|---------------------
C FIXPDF | FIXPDS | FIXPDF | FIXPDS | FIXPDF | FIXPDS | ODE based
C --------|--------|--------|--------|--------|--------|---------------------
C FIXPNF | FIXPNS | FIXPNF | FIXPNS | FIXPNF | FIXPNS | normal flow
C --------|--------|--------|--------|--------|--------|---------------------
C FIXPQF | FIXPQS | FIXPQF | FIXPQS | FIXPQF | FIXPQS | augmented Jacobian
C --------|--------|--------|--------|--------|--------|---------------------
C
C
C The sparse subroutines use the packed skyline storage scheme standard in
C structural mechanics, but any sparse storage scheme can be used by
C replacing some of the low-level HOMPACK routines with user-written
C routines. The stepping subroutines STEP?? may be of interest to some
C users with special curve tracking needs.
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
ORGANIZATIONAL DETAILS. HOMPACK is organized in two
different ways: by algorithm/problem type and by subroutine level. There
are three levels of subroutines. The top level consists of drivers, one for
each problem type and algorithm type. Normally these drivers are called by
the user, and the user need know nothing beyond them. They allocate storage
for the lower level routines, and all the arrays are variable dimension, so
there is no limit on problem size. The second subroutine level implements
the major components of the algorithms such as stepping along the homotopy
zero curve, computing tangents, and the end game for the solution at
lambda = 1 . A sophisticated user might call these routines directly to
have complete control of the algorithm, or for some other task such as
tracking an arbitrary parametrized curve over an arbitrary parameter range.
The lowest subroutine level handles the numerical linear algebra, and
includes some BLAS routines. All the linear algebra and associated data
structure handling are concentrated in these routines, so a user could
incorporate his own data structures by writing his own versions of these
low level routines.
The organization of HOMPACK by algorithm/problem type is shown in the
above table, which lists the driver name for each algorithm and problem type.
Using brackets to indicate the three subroutine levels described above,
the natural grouping of the HOMPACK routines is:
[FIXPDF] [FODE, ROOT, SINTRP, STEPS] [DCPOSE]
[FIXPDS] [FODEDS, ROOT, SINTRP, STEPDS] [GMFADS,
MFACDS, MULTDS, PCGDS, QIMUDS, SOLVDS]
[FIXPNF] [ROOTNF, STEPNF, [TANGNF]] [ROOT]
[FIXPNS] [ROOTNS, STEPNS, TANGNS] [GMFADS,
MFACDS, MULTDS, PCGDS, PCGNS, QIMUDS, ROOT, SOLVDS]
[FIXPQF] [ROOTQF, STEPQF, TANGQF] [QRFAQF, QRSLQF, R1UPQF, UPQRQF]
[FIXPQS] [ROOTQS, STEPQS, TANGQS] [GMFADS, MULTDS, PCGQS, SOLVDS]
[POLSYS] [POLYNF, POLYP, ROOTNF, STEPNF, TANGNF]
[DIVP, FFUNP, GFUNP, HFUNP, HFUN1P, INITP, MULP, OTPUTP, POWP,
RHO, RHOJAC, ROOT, SCLGNP, STRPTP]
The BLAS subroutines used by HOMPACK are DAXPY, DCOPY, DDOT, DNRM2, DSCAL,
D1MACH, IDAMAX.
The user written subroutines, of which exactly two must be supplied
depending on the driver chosen, are F, FJAC, FJACS, RHO, RHOA, RHOJAC,
RHOJS.
For testing, there are three main test programs MAINF, MAINP, and MAINS,
and one data file INNHP.DAT (read by MAINP).