Re: [BLACKBOX] BlackBox 1.6 final version - COMPLEX datatypes?

From: [at]} <Robert>
Date: Sun, 5 Jul 2009 09:01:19 +0100

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Hi

There has been little discussion of 'What (computer) Complex numbers are?'
and 'What (computer) Complex numbers do/should do?' in this thread.


WHAT ARE Complex NUMBERS?

The Oakwood guidelines http://www.oberon2005.ru/paper/bk1995-01e.pdf suggest
'Polar format' or 'Cartesian Format' as options.
I think most people would favour Cartesian format.

I guess the first suggestions for the type Complex would be:

TYPE
   Complex* : RECORD re*, im* : REAL END;
   Complex* : POINTER TO RECORD re*, im* : REAL END;
   Complex* : POINTER TO RECORD re-, im- : REAL END;


The first is the simplest, but does not support Complex valued
functions in CP (as it is currently defined).

The next option I think is 'dangerous' as the simple code

a := b;

can be lead to unintended and unnoticed side effects. So my libraries use the
'immutable' third option.

The third option generates a lot of garbage, and may be too inefficient for
some applications.

My current libraries (they have moved on since I published them on CPC) are not
so short (I will explain some of the reasons below):

CONST
   isReal* = 1;
   isImag* = 2;
   isComp* = 3;

TYPE
   Complex* = POINTER TO ABSTRACT RECORD
                  re-, im- : REAL;
                  is- : INTEGER
                END;

   StdComplex = POINTER TO RECORD (Complex) END;
   AuxComplex = POINTER TO RECORD (Complex) END;


Conclusion? Maybe there is not a universal agreement on what is a right
or good design, and so it would be better for the compiler NOT to implement
Complex, but to implement facilities (Overloading, Structured returns) that
facilitate external Complex libraries that can we written as required.


WHAT SHOULD Complex ARITHMETIC DO?

This is actually the most interesting point (to me). Look at what
Kahan has to say:

(Kahan was the principal architect of the IEEE 754 standard, and knows a thing
or two about floating point arithmnetic.)

http://www.eecs.berkeley.edu/~wkahan/JAVAhurt.pdf

Look at the pictures at the top of page 13, then (maybe) read from page 11.

My published Complex library produces the wrong picture on the right. My current
Complex library produces (with the same application code) the correct picture on
the left.

The reason, Kahan says, is that one needs separate classes for Real, (general)
Complex, and Imaginary. The FORTRAN (and OberonX, and many others) style of
representing Complex as a (real, imaginary) pair of REALs is incorrect. I
simulate this with the 'is' field above that can take the values isReal, isImag,
or isComplex.

The reason that REAL pairs are inadequate is that computer complex numbers are
different to mathematical ones. Computer arithmetic (as I have said on these
pages before !) distinguishes between +0 & -0, and this can be useful (in this
case it enables you to plot the correct graph).

As a flavour of the kind of code necessary I show my Square Root and
Reciprocal functions below.


Is this kind of library 'Simple'?

The library itself is NOT simple.
The 'correct' application code that uses it is.

So is this approach in the 'As simple as possible' tradition?


Regards

Robert


PS1 - Please read this as a contribution to the discussion, not as a
conclusion to 'do this' or 'do that'.


PS2 - If anyone wants to see my current Complex library in detail I
will post it and its Docu file.


PS3 - While this library is +/- zero friendly, it is not NaN friendly. BlackBox
support for NaNs is so bad that writing application code that exploits them is
next to impossible - sadly.)



PROCEDURE IsNeg* (x : REAL) : BOOLEAN; (* Detects -0. also *)
   BEGIN
     RETURN (x < 0.) OR (SYSTEM.VAL (LONGINT, x) = MIN (LONGINT))
   END IsNeg;


PROCEDURE Sqrt* (a : Complex) : Complex;
   VAR
     re, im, q, w : REAL;
     sqrt : StdComplex;
   BEGIN
ASSERT (a.is # 0, 54);
     NEW (sqrt);
     IF a.is = isReal THEN
       IF a.re >= 0. THEN sqrt.is := isReal; sqrt.re := M.Sqrt ( a.re)
                       ELSE sqrt.is := isImag; sqrt.im := M.Sqrt (-a.re) END
     ELSE
       sqrt.is := isComp;
       im := ABS (a.im);
       IF a.is = isImag THEN
         sqrt.re := M.Sqrt (im / 2.);
         IF R.IsNeg (a.im) THEN sqrt.im := -sqrt.re ELSE sqrt.im :=
sqrt.re END
       ELSE
         re := ABS (a.re);
         IF re > im THEN
           q := im / re; w := M.Sqrt (re) * M.Sqrt ((1. + M.Sqrt (1. + q *
q)) / 2.)
         ELSE
           IF im = 0. THEN sqrt.re := 0.; sqrt.im := a.im; RETURN sqrt
  END;
           q := re / im; w := M.Sqrt (im) * M.Sqrt ((q + M.Sqrt (1. + q *
q)) / 2.)
         END;
         IF a.re >= 0. THEN
           sqrt.re := w; sqrt.im := a.im / (2. * w)
         ELSE
           sqrt.re := im / (2. * w);
           IF IsNeg (a.im) THEN sqrt.im := -w ELSE sqrt.im := w END
         END
       END
     END;
     RETURN sqrt
   END Sqrt;


PROCEDURE Recip* (a : Complex) : Complex;
   VAR
     fac : REAL;
     res : StdComplex;
   BEGIN
ASSERT (a.is # 0, 54);
     NEW (res);
     CASE a.is OF
       isReal : res.re := 1. / a.re; res.is := isReal
     | isImag : res.im := -1. / a.im; res.is := isImag
     | isComp :
         res.is := isComp;
         IF ABS (a.re) >= ABS (a.im) THEN
           ASSERT (a.re # 0., 20);
           fac := a.im / a.re;
           res.re := 1. / (a.re + a.im * fac); res.im := -res.re * fac
         ELSE
           fac := a.re / a.im;
           res.im := -1. / (a.re * fac + a.im); res.re := -res.im * fac
         END
     END;
     RETURN res
   END Recip;




Georgy Jikia wrote:
> Let me try to ask a slightly different question. Suppose I do need
> COMPLEX data types in BlackBox and I am ready to get my hands dirty and
> implement COMPLEX as a new basic type. So a new COMPLEX basic type
> should be defined, mathematical operations should be implemented and the
> new type should be available in modules importing COMPLEX. Without
> procedures returning structured types and without operator
> overloading the implementation could only be done by extending system
> modules of the BlackBox. Although implementation using standard BlackBox
> modules is not possible, it seems COMPLEX could be implemented similar
> to SYSTEM or COM (pseudo-) modules, which do not exist as usual BlackBox
> SYSTEM.odc or COM.odc, but are hard coded in Dev subsystem. The problem
> is that the documentation for BlackBox modules intended for internal use
> only is absent. So, can somebody who knows inner workings of BlackBox
> make a comment on how such an extension of the core of BlackBox could be
> done? Can such an extension be localized or would it have to be spread
> all over the system modules? I guess the task is much more difficult,
> than writing overloaded operations in active oberon or C++, but it is
> interesting to know how much more difficult?
>
> Georgy
>
>
>
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Received on Sun Jul 05 2009 - 10:01:19 UTC