🚀 go-pugleaf

Checksum:  662440219 (verify with brik -cv)
Submitted-by: John Meissner <meissner%triton.unm.edu@lynx.unm.edu>
Posting-number: Volume 6, Issue 21
Archive-name: laplace_jm/part01


BEGIN_DOC laplace.doc
     These routines are an upgrade to my previous Laplace
transform utility.  Many of the new programs have been rewritten
in SYSTEM-RPL.  As always, system RPL can cause some unpredictable
things to happen -- including MEMORY LOST errors.
     I have strived to do my best to make the routines safe, but I 
am new to syystem-RPL programming and there are undoubtedly some bugs
still left in the routines.  I am posting these routine to comp.sys.hp48
so that these bugs can be tested and corrected over the next few weeks
so that I can release some kind of a final product.
     I have had no reports of memory lost errors or and serious mishaps
of any kind since the pre beta-test release about a month ago..  So,
hopefully, these routines are safe.  As alaways, report any bugs/sugestions
to me as soon as you see them.

	Summary of changes from beta release:
	1) You can now define your own independant variable.
	2) RT-> gives better results.
	3) A few other quirks have been fixed.
 
	Summary of changes in these routines:

	1) system-RPL, these routines run 5 to 10 times faster.
	2) Remove flags [0-5] usage.
	3) Fixed bugs in & and RT-> that caused incorrect transforms.
	4) install a patch that allows 't' to exists elsewhere in the PATH.
	5) Remove 'e^t' notation.
	6) Recognizes and handles correctly the 'SQ()' function.
	7) Many changes in how the routines work that speed up or 
	   otherwise enhance their performance.

     The origional routines used Wanye Scotts polinomial routines for most
of the grunt work.  I don't think any of Wayne's routines are still in this 
package, but he deserves credit for the notation and untimately these 
routines since I never would have bother without his work.
     Also, Bill Wickes' port of a polinomial root finder is included in 
these routines.  Thank you Bill for the efforts.
	Thank you also to Detlef Mueller for is compiler/decompiler that
I used to write the majority of code in these routines.

	The remainder of this article is an edited version of my origional 
post. 

     The following programs are released to the public domain
provided that no money exchange hands (unless of course my hands
are involved).

     A few disclaimer are in order.

     1) The routines in this directory use Wayne Scott's polynomial
root finder routines extensively.  Beware -- if you already have
these routines and use them, then you need to read all of these
documents to determine what effects might surface.  For the most part,
these routines are entirely different even though they do accomplish
some of the same tasks. 

     3) If you are unfamiliar with the format of the polynomials
used by Wayne's routines, here is a brief refresher:

the polynomial   :       x^4 - 3*x^3 - 2*x -1
is represented as:       { 1 -3 0 -2 -1 }

     This is to say the first element of the list is the
coefficient for the x^4 term and so on through the equation.  It is
customary to use 's' for the Laplace variable, so from here on, I
will use 's' in my examples.   
        Some of the routines require two polynomials on the stack.
For example, the ratio of the two polynomials representing this
expression:
                           3*s^5 - 3*s^3 + 5*s^2 - 2
                   ---------------------------------------
                   s^5 - 7*s^4 + 15*s^3 - 5*s^2 -16*s + 12 

would be represented on the stack as:
                   
                   2:  {3 0 -3 5 0 -2}
                   1:  {1 -7 15 -5 -16 12}

Another method of representing the same ratio is:

             2:  {3 0 -3 5 0 -2)
             1:  {3 2 2 1 -1}

     where level 2 contains the same list as the previous example. 
However, level 1 contains the roots of the denominator.

     Hopefully, these examples are clear enough for you to get the
picture.

     The following are descriptions and examples of each of the
functions in the LAPLACE directory:

t

     This variable returns an unevaluated copy of itself.  I allows the
routines to operate when ther is another 't' defined in the current path.
It is also useful for keying is formulae.  If you want the independant 
variable to be something besides 't' then simply store your vaiable in 't.'

->L
     This program takes an algebraic object and returns its Laplace
Transform.  The input function must be a function of the variable 
stored in 't.'  that is to say that the only independent variable 
in the equasion on level one of the stack must be stored in the variable
lower case t.  The functions that it recognizes are constants,
exponentials, sines, cosines, and polynomials.  The program should
allow any combinations of multiplication, addition, subtraction,
and integer exponents.  Division is not supported, however.  
For example:

1: 'COS(2*t)^2'
->L

after a pause will return :

2: { 1 0 2 }              ; numerator 
1: { (0,4) (0,0) (0,-4) } ; roots of the denominator 
 
  
     This represent the transform.  With this as an input, the
inverse transform function returns :

L->

1: '1/2 + 1/2*COS(4*t)'

     This doesn't look the same as the original function; but, the
answer is correct.  COS(t)^2 = 1/2 + 1/2*COS(2+t) is a
trigonometric identity.  Kinda nifty eh?  

     Another interesting result of the transformation is
integration. Integration in the time domain is division by s in the
s domain.
So, take the function:

1: 'EXP(-(2*t))*t*COS(t)^2'

->L

Returns:

2: {1 8 26 40 32 }
1: { (-2,2) (-2,2) (-2,0) (-2,0) (-2,-2) (-2,-2) }

     The list on level one is the roots of the denominator.  If you
add a zero to the list (edit the list or just put a zero on the
stack and press the + key) you are effectively multiplying the
denominator by s.  This is the same as dividing the fraction by s.
The result in the time domain, after the inverse transform is done,
is the integration of the original function between the limits of
0 and 't.' For example:

2: { 1 8 26 40 32 }
1: { (-2,2) (-2,2) (-2,0) (-2,0) (-2,2) (-2,-2) 0 }

->L

Returns:

1: '1/8-1/8*EXP(-(2*t))*COS(2*t)*t+1/8*EXP(-(2*t))*SIN(2*t)*t+1/16*
(EXP(-(2*t))*sin(2*t)-1/4*EXP(2*t)*t-1/8*EXP(-(2*t)'

     This is the integral of the original function evaluated
between 0 and t.  In many cases the general solution can be
obtained by replacing the constant (in this case 1/8) with a
general constant of integration.  Needless to say, this method of
integration was a little easier than looking in the integral
tables.  The description of the inverse Laplace Transformer
follows. 

L->
     This program takes the numerator, level 2, and the
denominator, level 1 (factored), and returns its inverse Laplace
Transform.  The factors of the denominator need not be grouped
together as Wayne's routines require since they are sorted before
anything is done to them.  I originally wrote this routine and
uploaded it to hpcvbbs, but I did not include much for
documentation.  Since then I have seen it posted a few times and
have heard of some bugs that I am unable to duplicate.  This may be
because the users are using Wayne's routines with this one, or, I
fixed the bugs and don't remember.  Either way here is the latest
version.
     The routine should correctly transform the ratio of any two
polynomials with real coefficients.  The only limitation i know of
is that the numerator should be of lower order than the
denominator.  You may still get the correct answer if this
condition is not met but there are no guarantees. 
     I have found absolutely no problems with this routine.  In
fact, I have found 2 errors in the transform tables of the "CRC
Handbook of Mathematical Formulas.'  I am not willing to say that
there are no bugs because I am sure they exist.  Please let me know
as they become apparent.  I have been truly amazed with the results
this program returns;  Wayne did a good job with his routines.  He
did almost all of the work.  Here is and example:

2: { 2 0 4 }
1: { 1 1 0 4 }
L->

Returns: 

1:'-1 - 2*t*EXP(t) + EXP(4*t)'

RT->
     The results of this routine are different depending on the 
number of lists on the stack.  If there are 2 lists of numbers on
the stack, the routine will assume that the represent a fraction and
the numerator is divided by the coefficient of the highest order term
in the denominator.  This is necessary to keep the ratio correct 
since this term is lost in the expasion.  In the event that the stack
contains only one list this program returns the roots of that list.  

1: { 1 -6 1 24 -20 }

RT->

returns:

1: { 1 2 -2 5 }

or:

2: { 1 2 4 }
1: { 2 0 -2 }

RT->

returns:

2: {.5 1 2 }
1: { -1 1 } 

FADD
     This routine adds two partial freactions.  They most be
in the form:
4: list    (numerator)
3: list    (denominator roots)
2: list    (second nemerator)
1: list    (second denominator roots)

&
     This routine convolves (sp?) two partial fractions.  It is
used by the routine ->L.  It either does a partial fraction
expansion of one of the two terms and then does an 's' shifted
summation of the terms, or is does a straight multiplication of the
terms -- depending on which is appropriate.  It is still written
in user-RPL since there was no real speed increase when I rewrote
it.  I believe that user-RPL should be used whenever possible to
avoid as many headaches as possible.


SORT

     Fairly self explanatory.  Not the bubble variety.  It is much
smaller than the bubble sorts that I tried and is a little faster
for lists that are shorter than about 20 elements.  The sort order
is largest to smalles with wieghting on the real value.  This means 
that the unstable roots will be at the begining of the list if they
exist.

RED
     This routine uses PDIV to remove any common roots in the
numerator and denominator.  It is slow but necessary if you are
working with complex transforms.

Q  
     This routine takes an object from the stack and attempts to 
rationalize it to within 4 digits of accuracy. In other words, if
you have a nomber like 5.200004454 and want to round it to the nearest
fraction then Q is your program.  It is necessary because the roots
returned by the root finder are not exact.  And the calculator 
drops digits here and there when dividing or finding roots.  The 
routine will also accept lists, algebraics, complex numbers, and
combinations of any of the above.

PMUL
     This routine take two polynomials and returns their products. 
I have completely rewritten this program to speed it up and
slightly change the function.  The objects do not have to be lists. 
The routine will now multiply a list by a constant.  It should
function identically to Wayne's routines other than this change.

IRT
     This routine take the roots of the polynomial and multiplies
them to find the polynomial. 

PADD
     This routine takes two polynomials and adds them.  In addition,
if the object on level 1 is a number (real or complex), the routine
adds the number to each of the elements in the list in the other 
stack level.  These changes were made in the interest of speed. 

TRIM 
     This routine takes a list and performs Q and then strips the 
leading zeros.  It works identically to the original I believe.  

PDIV
     This routine is NOT the same a Wayne's.  This routines accepts
a list, representing a polinomial, and a number, representing a root
of the polinomial, and returns the list with that root divided out,
and the remainder which is also the value of the polinomial evaluated 
at that root.  Example

2:  { 1 -2 1 }
1:  1

PDIV  (returns)

2: 0
1: { 1 -1 }

RDER 
     This routine takes two lists (numerator,denominator) and
returns the numerator of the derivative of the ratio. 

PDER 
     Accepts a list representing a polinomial and returns a list 
representing  the derivative of that polynomial. 

PF
     Does partial fraction expansion of the ratio (numerator, 
roots) on the stack.   



     The ASC encoded directory follows. Use ASC-> to decode.  The 
total size is 6127 bytes. 

Please let me know id this program is in any way useful to amy of
you.  Also please let me know of any suggestions or bugs that
occur.

Good luck,

John Meissner
END_DOC

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end
sum -r/size 14415/8478 section (from "begin" to "end")
sum -r/size 41242/6135 entire input file
END_UU