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Re: Compiled Functions and Mathgroup Archive

• To: mathgroup at smc.vnet.net
• Subject: [mg8935] Re: Compiled Functions and Mathgroup Archive
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 4 Oct 1997 22:08:12 -0400
• Organization: University of Western Australia
• Sender: owner-wri-mathgroup at wolfram.com

Peter Jay Salzman wrote:

> 1
> -
> I'm writing a numerical Schroedinger equation solver and stumbled on the
> 'Compiled' function.  The MMA book mentioned that sometimes functions run
> faster when you don't compile them.

It can be a lot faster to use MathLink to link to external compiled
code.  See, e.g., our second year quantum mechanics courseware at

	http://www.pd.uwa.edu.au/Physics/Courses/Second_Year/Quantum.html

In addition you can use the InterCall Mathematica package to easily link
in compiled code without needing to write MathLink template files.  On
MathSource <http://www.wolfram.com/mathsource/> you will find the
following two listings:

0204-657: Schroedinger's Equation (March 1993)
          Author: Terry Robb

          Schroedinger's equation is numerically solved by calling an
          external subroutine from within Mathematica. By importing the
          Fortran subroutine (named SCHROED, which implements a
          Crank-Nicholson scheme) using InterCall, it is possible to do
          various experiments -- for example shooting a wave-packet at a
          potential barrier and watching an animation of the packet
being
          mostly reflected but also partially tunnelling through the
wall.
          The potential function V[x,t], required by the Fortran SCHROED
          routine, can be written as a Mathematica function which gives
a
          lot of flexibility for interactive experimentation. This
notebook
          demonstrates one such experiment.

          0011:  schroed.ma (772 Kb)

0204-679: Animating Schroedinger's Equation in Two Dimensions (April
1993)
          Author: Terry Robb

          An efficient method for solving parabolic partial differential
          equations is implemented in Mathematica using InterCall and an
          external C routine. As an application, the two-dimensional
          time-dependent Schroedinger's equation is solved for various
          initial conditions and potential functions. Four different
          numerical experiments are given: scattering of a particle from
a
          cylindrical potential barrier; a double slit experiment;
          interaction of wave-packets; and stirring a wave-packet with a
          potential "stick". The resulting animations make excellant
          demonstrations of the properties of Schroedinger's equation. 
The
          technique used can also be applied to other similar parabolic
          partial differential equations.

          0011:  Schroed2D.ma (1.7 Mb)
 
> I know that people must have asked this question before, but when should
> I use Compile to generate faster code?  My program obviously involes a
> tremendous number of repetitive arithmetic operations, so it seems like I
> should compile things like
> 
> wavefunction[r_] = (alpha/Pi)^.75  E^(-.5 alpha r^2)
> 
> or
> 
> norm[wave] = Sqrt[ (4 Pi (dr)^3 Sum[ (Abs[ wave[[i+1]] ]*i)^2,
>      {i, 0, arsize-1} ]
> 
> where arsize is the gridsize and will be on the order of thousands.

I assume that you can use matrix (or vector) method instead of
element-by-element computation?  Using version 3.0 you can get very
considerable speed-ups by compiling matrix operations.

> 2
> -
> Also, is there an archive of Mathgroup questions so I can look up
> questions like this in the future?  It's only useful if it's searchable.

See http://smc.vnet.net/MathGroup.html and
http://www.wolfram.com/cgi-bin/mathgroup/  However it is not
searchable.   I have Cc:d this to suggestions at wolfram.com so that they
can consider making the archive searchable (which I agree would be
_very_ useful).

> 3
> -
> How can I have MMA print something like:
>     The gridsize is arsize
> but replace arsize with the contents of the variable arsize

You can, of course, use Print.  A better way is to use the built-in
message command.  After defining a string such as

   In[1]:= GridSize::size = "The gridsize is `1`."; 

then, for

   In[2]:= arsize = 0.2;

get your code to call message, e.g.,

   In[2]:= Message[GridSize::size,arsize]
   GridSize::"size": "The gridsize is 0.2."

Have a look at Messages.m in 
	
	SystemFiles
	    Kernel
		TextResources
		    English

to complete see a list of the built-in messages.
 
> 4
> -
> I print a bunch of parameters for the current 'run', and would like to
> prompt myself "If this is OK, press enter to continue but if it's not OK,
> press any other key to end the program".

You can use Input to do this:

  In[3]:= ?Input

  "Input[ ] interactively reads in one Mathematica expression. 
   Input["prompt"] requests input, using the specified string as a
prompt."

Cheers,
	Paul 

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia           
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au 
AUSTRALIA                             http://www.pd.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________