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D .dll, Silverfrost FTN95, and Numerical Recipes:
The Art of Scientific Computing1
(Simulated Annealing2) - FORTRAN Files

(alternatively, Mathematical Procedures - FORTRAN Files)

This is the second part of D .dll (dnrprocs.dll) calling a FORTRAN .dll (mathproc.dll).  This page focuses on mathproc.dll Simulated Annealing
related subprograms.

The FORTRAN Simulated Annealing related subprograms called by dnrprocs.dll are contained in mathproc.for (compiled and linked into mathproc.dll; mathproc.for also contains other engineering, scientific, and mathematical subprograms)3.

mathproc.for was developed using the Silverfrost Plato IDE on an AMD Athlon II X2 running WinXP Pro SP3.  Its various subprograms have been successfully called by FORTRAN, D, Object Pascal, and Lazarus (using an Object Pascal proxy) programs4.

The Simulated Annealing subroutines (original FORTRAN source code published in Numerical Recipes: The Art of Scientific Computing 1986) were modified by breaking DE (difference in cost due to some change) into path cost change DP and extended cost change DE (this also required adding μ, an extended cost associated with each node; this concept was discussed in Numerical Recipes: The Art of Scientific Computing but not implemented).
DP = √ ( (Xi - Xi+1)2 + (Yi - Yi+1)2 )              DE = (μi - μi+1)2
Total Cost Change = DP + DE

(The array IORDER(i) is used as an index into the X(i), Y(i), and MU(i) arrays; only the elements in the IORDER(i) array are modified. X(i) elements are referred to as X(IORDER(i)); the same applies to the Y(i) and MU(i) arrays.)

Also added another variable, IERR, to the parameter list; IERR is a returned error code. IERR = 0 indicates successful call.

Each of these subroutines are called by the D procedure dANNEAL (...) (or the FORTRAN subroutine ANNEAL (...)5, also contained in mathproc.for; used for testing). dANNEAL (...) is a FORTRAN to D port.  Function RLEN (...) (computes DP) is also called by TRNCST (...) and REVCST (...).
Only the subprogram statement and variable declarations are shown to prevent a possible intellectual property violation, since the original FORTRAN source code is published.  The unmodified Numerical Recipes Simulated Annealing FORTRAN source code is available for purchase at Older Numerical Recipes book editions6 (the 1986 edition is not available online). A student version (in English) of Numerical Recipes In 'C': The Art Of Scientific Computing, Second Edition, © 1988-1992 is available from the University of Trieste at no cost.

Function RAN3 (...) is a Portable Random Number Generator (machine independent random number generator for consistent results on different platforms) from Numerical Recipes: The Art of Scientific Computing (1986).  Used by dMETROP (...) (Metropolis Algorithm) and dANNEAL (...);
Also called from Free Pascal Library nrlazrs function fRandom (...) .

SAVE store variables statically. Static variables retain their values until main program execution terminates.  Necessary to retain values of INEXT, INEXTP, and array MA(i) between calls.

 Simulates a coin toss by randomly returning 0 or 1.

 Also called from Free Pascal Library nrlazrs function fRandom (...) .
Function to find the distance between two points (X1, Y1, Z1) and (X2, Y2, Z2) is common enough to be considered public domain.
FORTRAN Function RLEN (...)
Function subprogram RLEN (...) replaces the Arithmetic Function (or Statement Function) ALEN (...) in the original 1986 TRNCST (...) and REVCST (...)
Node Transport Cost Function Subroutine Header
Note that the function RLEN (...) is declared as a variable. This is required since mathproc.for is compiled with IMPLICIT NONE
(Project, Properties, Compiler Options, Language, set global IMPLICIT NONE).
dnrprocs dANNEAL (...) (or mathproc ANNEAL (...) in an all FORTRAN solution) initializes the path cost using RLEN (...), then begins its cycle:

using RAN3 (...), select a segment of 3 or more nodes;

simulates a coin flip with IRBIT1 (...) to decide whether to call TRNCST (...) (difference in cost if the segment is moved) or REVCST (...) (difference in cost if the segment is reversed);

dnrprocs dMETROP (...) (or mathproc METROP (...) in an all FORTRAN solution) (both of which also calls RAN3 (...)) is then called to decide whether to perform the segment move or reversal. If yes, either TRNSPT (...) or REVERS (...) is called to actually move or reverse the segment.

Path Reversal Cost Function Subroutine Header

Calculating DP using RLEN (...), calculating DE:
      DP = 0.0 - RLEN(XX(1),XX(3),YY(1),YY(3),0.0,0.0) - . . .
	     + RLEN(XX(1),XX(4),YY(1),YY(4),0.0,0.0) + . . .
      DE = 0.0 - (MMU(3) - MMU(1))**2 - . . .
	      + (MMU(4) - MMU(1))**2 . . .
(subtract connection, add reverse connection; similar calculation in TRNCST (...) but with different subscript values)
Path Reversal Subroutine Header

The original Numerical Recipes (1986) FORTRAN source code did not have elaborate error checking.  Added IERR to the original parameter list.  If the subroutine detects an error condition (array subscript out of bounds,
etc.), IERR is set to a non-zero value and subroutine immediately exits.  As an example, the code snippet above right is from REVCST (...) and checks that the number of nodes to be swapped exceeds 0.

The FORTRAN .dlls (sttstcs.dll and mathproc.dll) called by Fujitsu COBOL, D, Free Pascal, and Lazarus programs were compiled with a CheckMate Win32 configuration.

With a CheckMate Win32 configuration compiler includes code to perform run-time checking for any undefined variables or array elements in + - / * or ** arithmetic expressions; .NE. .EQ. etc. relational expressions; .AND. .OR. etc. logical expressions; array subscripts; etc.

Project Properties (mathproc.ftn95p), Compiling, Linking:

Use the same Compiler Options, Miscellaneous as in compiling the sttstcs.dll (Output filename, Output filetype, and Alternative compiler options)

Use the same Linker Options as in sttstcs.dll (Export all).

When passing numeric constants from one FORTRAN subroutine to another, may need to set the default INTEGER and REAL kind using Project, Properties, Compiler Options, Numerical.

INTEGER kind 2 corresponds to INTEGER*2.

REAL kind 1 corresponds to REAL (or REAL*4).

Required for mathproc.for since subroutines TRNCST (...) and REVCST (...) both pass constant zero to RLEN (...)
Project, Properties, Compiler Options, Numerical

Project, Properties, Compiler Options, Numerical
Compile and link normally using the Silverfrost FTN95 compiler and SLINK (Project, Set Target, DLL;
Build, Build) to create mathproc.dll.
Plato3 Build Menu
   Plato3 Project Menu, setting compile target     Project Target (.exe, .dll, .lib, .mdl)

.LIB file (MATHPROC.LIB import library):

Depending on the calling program language, may need to use either Silverfrost Plato SLINK or DMD utility implib to build mathproc.lib.

(Free Pascal and Lazarus linking does not require a user generated mathproc.lib)

To use the Silverfrost Plato SLINK to build a Microsoft 32 bit Linker compatible (Microsoft's own variant of the COFF format) MATHPROC.LIB:

click Project, Set Target

click the LIB radio button
click OK

click Build, Build
Plato3 Project Menu, Project Target  


Digital Mars compilers use a different object and import library file format (Intel 32 bit OMF object and library file format) than Microsoft's 32 bit tools (the .dll's produced are compatible). Therefore, for Digital Mars D main programs and .dll's, use the Digital Mars implib utility (file bup.zip) to create a DMD compatible mathproc.lib.
implib mathproc.lib mathproc.dll
in a MS-DOS command prompt

(batch file rrr.bat uses short file names to change to directory
C:\Documents and Settings\Rodney Roberts\My Documents\progProjects)

(Be sure to use the D2 32-bit Command Prompt for compatibility with the CheckMate Win32 configuration, Free Pascal, and Lazarus)
using implib

1. Press, William H., Brian P. Flannery, Saul A Teukolsky, and William T. Vetterling (1986). Numerical Recipes: The Art of Scientific
.  New York:Press Syndicate of the University of Cambridge.
More recent editions available at Numerical Recipes

2. Simulated Annealing, also known as the traveling salesman problem, is used for combinatorial minimization.  The traveling salesman
problem involves a salesman who must travel to a large number of cities (or nodes, represented by X Y coordinates).  What is the shortest
path the salesman can traverse visiting these cities?  In a very large 'solution space', there are some solutions that are 'low cost'
(global minima) relative to other solutions.  Simulated annealing searches for these 'low cost' solutions.  Simulated annealing has also been
applied to designing integrated circuits.  The traveling salesman problem is a NP-Complete problem (nondeterministic polynomial), where
N equals the number of nodes (or cities), and the computation time is proportional to eN.

3. Mathematical Procedures - FORTRAN Files (or more simply, mathproc.for) contains over 60 engineering, mathematical, and scientific
application subprograms (both functions and subroutines), many from Numerical Recipes: The Art of Scientific Computing, others are
from FORTRAN for Engineering 1972,  FORTRAN For Scientists & Engineers 2nd Edition 1995,  algorithms used in
Calculating the Center of Pressure, and others.  This page focuses on the FORTRAN Simulated Annealing subprograms from
Numerical Recipes.  Four root finding subprograms (see note 6 below) are from published algorithms, source available below.

4. As mentioned on other pages, when calling a FORTRAN subprogram from a FORTRAN, Fujitsu COBOL, or Free Pascal program,
the arguments are passed in normal order.  When called from a D program, the arguments are passed in reverse order.

5. Not shown above are subroutine ANNEAL (...) (p. 331; implemented as dANNEAL (...) in D module dnrprocs) and subroutine METROP (...) (p. 333; implemented as function dMETROP (...) in D module dnrprocs).  ANNEAL (...) can be used in an all FORTRAN solution (done for initial testing) or called by an Object Pascal
program.  Added extended cost array μ, total path cost PATH, total extended cost ECOST, input/output flag IOFLG, and IERR to the
argument list.  Used different names for  ANNEAL (...) / dANNEAL (...)  and  METROP (...) / dMETROP (...)  to avoid linking
and/or calling conflicts.

6. Source code for point rotation, triangular segment area, and four root finding subprograms in mathproc.dll not published in
Numerical Recipes 1986, FORTRAN for Engineering 1972, or FORTRAN For Scientists & Engineers, 2nd Edition, 1995 available.
The root finding subprograms focus on Polynomial Root finding.  Includes Cubic Polynomial Roots - X3 + A1*X2 + A2*X + A3 = 0
(algorithm only published) from p. 146 in Numerical Recipes (1986).  Source code includes source of algorithms.

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Feel free to use any of the above in your project (without violating any intellectual property rights); please give credit (same idea as Copyleft).

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