HOMPACK90

HOMPACK90 is a suite of FORTRAN 90 subroutines for solving nonlinear systems of equations by homotopy methods. There are subroutines for fixed point, zero finding, and general homotopy curve tracking problems, utilizing both dense and sparse Jacobian matrices, and implementing three different algorithms: ODE-based, normal flow, and augmented Jacobian. The (driver) subroutines called by the user are given in the table below, and are well documented internally. The user need not be concerned with any other subroutines in HOMPACK90.

```                  Problem type
--------|--------|--------|--------|--------|--------|
x = f(x)    |    F(x) = 0     |rho(a,lambda,x)=0|
--------|--------|--------|--------|--------|--------|
dense  | sparse | dense  | sparse | dense  | sparse |  Algorithm
--------|--------|--------|--------|--------|--------|---------------------
FIXPDF | FIXPDS | FIXPDF | FIXPDS | FIXPDF | FIXPDS | ODE based
--------|--------|--------|--------|--------|--------|---------------------
FIXPNF | FIXPNS | FIXPNF | FIXPNS | FIXPNF | FIXPNS | normal flow
--------|--------|--------|--------|--------|--------|---------------------
FIXPQF | FIXPQS | FIXPQF | FIXPQS | FIXPQF | FIXPQS | augmented Jacobian
--------|--------|--------|--------|--------|--------|---------------------

```

The sparse subroutines use either the packed skyline storage scheme standard in structural mechanics or the compressed sparse row storage format, but any sparse storage scheme can be used by replacing some of the low-level HOMPACK90 routines with user-written routines. The stepping subroutines `STEP??` or the reverse call subroutines `STEPNX` and `ROOTNX` may be of interest to some users with special curve tracking needs.

Organizational Details

HOMPACK90 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 LAPACK and 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 HOMPACK90 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 HOMPACK90 routines is:

``` [FIXPDF] [FODE, ROOT, SINTRP, STEPS] [DGEQPF]

[FIXPDS] [FODEDS, ROOT, SINTRP, STEPDS] [GMFADS, GMRES,
GMRILUDS, ILUFDS, ILUSOLVDS, MULTDS, MULT2DS, PCGDS, SOLVDS]

[FIXPNF] [ROOTNF, STEPNF, TANGNF] [DGEQPF, DORMQR, ROOT]

[FIXPNS] [ROOTNS, STEPNS, TANGNS] [GMFADS, GMRES, GMRILUDS,
ILUFDS, ILUSOLVDS,  MULTDS, MULT2DS, PCGDS, ROOT, SOLVDS]

[FIXPQF] [ROOTQF, STEPQF, TANGQF] [DGEQRF, DORGQR, UPQRQF]

[FIXPQS] [ROOTNS, STEPQS, TANGNS] [GMFADS, GMRES, GMRILUDS,
ILUFDS, ILUSOLVDS, MULTDS, MULT2DS, PCGDS, ROOT, SOLVDS]

[POLSYS1H] [FIXPNF, ROOTNF, STEPNF, TANGNF]
[DGEQPF, DGEQRF, DORMQR, DIVP, FFUNP, GFUNP, HFUNP, HFUN1P,
INITP, MULP, OTPUTP, POWP, RHO, RHOJAC, ROOT, SCLGNP, STRPTP]
```

The `LAPACK` and `BLAS` subroutines used by HOMPACK90 are ```DCOPY, DDOT, DGEMM, DGEMV, DGEQPF, DGEQR2, DGEQRF, DGER, DLAIC1, DLAMCH, DLAPY2, DLARF, DLARFB, DLARFG, DLARFT, DNRM2, DORG2R, DORGQR, DORM2R, DORMQR, DSCAL, DSWAP, DTPMV, DTPSV, DTRMM, DTRMV, DTRSV, IDAMAX, ILAENV, LSAME, XERBLA. ```

The user written subroutines, of which exactly two must be supplied depending on the driver chosen, are ```F, FJAC, FJACS, RHO, RHOA, RHOJAC,``` and `RHOJS,`. These external subroutines must conform to the interfaces contained in the module `HOMOTOPY`. The module `REAL_PRECISION` contains machine dependent constants, which must be changed appropriately before compilation. The module `HOMPACK90_GLOBAL` contains global storage, and must be used by the user written subroutines.

Testing and Installation

`HOMPACK90` consists of 4 modules--- `HOMOTOPY` (contains interfaces for the user written external subroutines), `HOMPACK90` (encapsulates all the drivers), `HOMPACK90_GLOBAL` (global dynamic storage), `REAL_PRECISION` (defines precision of all reals)---and external subroutines, all contained in two files `HOMPACK90.f` and `LAPACK.f`. The file sample.f contains templates for the user written subroutines. There are three main programs `MAIN[FPS].f` for testing, with sample output given in the files `MAIN[FPS].out`. `MAINF.f` and `MAINS.f` have no input files; `MAINP.f` reads a data file `INNHP.DAT` and writes the solution in a file `OUTHP.DAT` (for post-processing), since this is normally how the polynomial system driver `POLSYS1H` would be used.

To test the dense (F), sparse (S), polynomial system (P) algorithms respectively in `HOMPACK90`, compile and link in order the files `HOMPACK90.f LAPACK.f MAINF.f` (`MAINS.f, MAINP.f` respectively). The modules and external subroutines in `HOMPACK90.f` and `LAPACK.f` (`BLAS` and `LAPACK` routines) can be kept in module and object libraries and need not be recompiled.