ProfitPress Mega CDROM2 …eeware (MSDOS)(1992)(Eng)/ProfitPress-MegaCDROM2.B6I/PROG/MISC/LIBRY51.ZIP/LIBRY4.DOC

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                                 MATHEMATICAL

These mathematical procedures provide a wide variety  of  functions  that  are
useful for engineering applications.  I began developing this library after my
first experience with SLIM (name changed to protect the guilty) back in  1976.
So  I guess I have them to thank.  If you've ever used SLIM perhaps you've had
a similar experience...

Let's  say  you  want  to solve what one would think to be a simple problem of
simultaneous linear equations, and due to some unfortunate turn of events, you
are given SLIM with which to do this.  You are confronted with no less than 60
subroutines from which to choose.  As it turns out,  none  will  do  what  you
want.   Anyway,  you  narrow  it  down  to  the Smith-Jones-Simpson-Wilkerson-
Feinstein-Gidley  method  for  partially  symmetric  matrices   with   strange
sparseness  stored  in  convoluted-compacted  form  (not  to  be confused with
space-saving   form)   and   the   Kleidowitz-Yonkers-Lakeville-Masseykins
modification of the Orwell-Gauss-Newton-Schmeltz algorithm for ill-conditioned
Hilbert matrices  stored  in  space-saving  form  (not  to  be  confused  with
convoluted-compacted  form).  An obvious choice indeed (any fool with 6 Ph.D.s
in math could see that the latter is the preferred  method).   After  numerous
attempts to link your program with all 347 of the necessary SLIM libraries you
are finally ready to run your program.  Of course, it bombs  out  because  you
were reading tuesday afternoon's manual and it's now wednesday morning.  Tough
luck...

Actually,  SLIM is so bad that even I can't achieve hyperbole.  Language fails
me.  The above could easily be a random selection from one of the six  volumes
that are supposed to be a "user's guide."

Anyway,  I have a completely different philosophy of programming.  I hope that
it works well for you:  if there  is  a  clearly  preferred  method  of  doing
anything, then use it and discard the others.

An  excellent example of this is Gauss quadrature.  There are only two methods
of numerical integration that will (at least  in  theory)  converge  given  an
infinite  order:   trapezoidal rule and Gauss quadrature.  Gauss quadrature is
so far superior in accuracy to any  other  method  (e.g.   Newton-Cotes)  it's
practically  incredible.   Therefore,  why have 50 other methods from which to
choose?  An interesting note about this is that, contrary to  some  rumors  10
subdivisions  of  20-point  Gauss  quadrature (200 point evaluation) is not as
accurate as 96-point  Gauss  quadrature  with  less  than  half  the  function
evaluations.

Another  example  is  Gauss-Jordan  elimination.   SLIM  has  subroutines that
provide a "high accuracy solution"  (If  Andy  Rooney  understood  this,  he'd
probably  ask,  "Who  would  want  a  low accuracy solution anyway?").  And of
course, there's the iterative improvement type  solutions.   What  they  don't
tell  you  is that the residual must be computed in greater precision than the
matrix is reduced in (say single to double precision) which is  OK;   but  why
not  just  use double precision all the way through?  If you are already using
double precision and that's not good enough for you...  well, you're just  out
of  luck.   If  your matrix is ill- conditioned (as is typically the case with
Vandermondes - that's what you get when you're curve-fitting) about  the  best
thing  you  can use is Gauss-Jordan elimination with full column pivoting.  It
is fast enough and I have used vector emulation throughout.  I have tried  all
sorts of things like QR decompositions, Givens rotations, and a whole bunch of
others;  but it all boils down to the precision of the  math  coprocessor.   I
only  use  one  algorithm for solving full matrices.  I have tested it using a
series of standard Hilbert matrices against SLIM's  "high  accuracy  solution"
and found mine to be both faster and more accurate.

One  last  example is the solution of ordinary differential equations.  If you
read numerical mathematics texts, they always give examples of  the  error  in
such-and-such a method as compared to the exact solution.  If you look closely
at the microscopic print in the margin of the legend under the fly  wing  that
got  stuck  on  the  copier, you will probably find the pithy little statement
"exact solution determined using 4-th order Runge-Kutta with 0.0000000001 step
size."  Runge-Kutta  is self-starting (which is a bigger pain in the neck than
you might think if you use a method that isn't), convenient to use, and  works
very  well.   So  why not use it, I ask?  Of course there are certain (unknown
before the fact) cases where other methods  are  much  faster;   but  I  would
rather  spend  less time experimenting with zillions of algorithms and let the
machine sweat a little more.  I've been through it once;  and once is  enough.
Runge-Kutta  is  the one for me.  A word about automatic stepsize control.  It
has been my experience that this takes so long  at  each  step  that  you  are
better  off  to use a smaller step from the start.  I had two such subroutines
that I spent a lot of time on;  but  I  purged  them  one  day  in  a  fit  of
frustration.

One  final  note:   WATCH  YOUR  VARIABLE TYPES!  ESPECIALLY DOUBLE PRECISION!
I've seen many a programmer bewildered by a program that crashes because  they
have  passed  "0."  instead  of  "0.D0"  to  a  subroutine  expecting a double
precision value or forgotten to put the "REAL*8" out in front of a function.

          REAL*8 FUNCTION DOFX(D)

It is really best to use IMPLICIT statements like

          IMPLICIT INTEGER*2 (I-N),REAL*8 (A-H,O-Z)

than to individually type each variable.   You're  bound  to  miss  one.

Another biggy is "O" instead of "0" or vise versa.
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