POLY, by Wayne Scott  [upgrade to POLY on Goodies Disk #1.  -jkh-]

Here it is, my polynomial routines version 3.2

Changes from version 3.1:
          - Faster PMUL
          - RT now works with complex cooeffients
          - various bug fixes

These routines are in the public domain, but I ask that if you use any of them
in one of your programs you give me credit.  I am also not responsible for any
damage caused by these programs.

This package include the following programs.

TRIM     Strip leading zeros from polynomial object.
IRT      Invert root program. Given n roots it return a nth degree polynomial.
PDER     Derivative of a polynomial.
RDER     Derivative of a rational function.
PF       Partial Fractions.  (Handles multiple roots!)
FCTP     Factor polynomial
RT       Find roots of any polynomial
L2A      Convert a list to an array and back.
PADD     Add two polynomials
PMUL     Multiply two polynomials.
PDIV     Divide two polynomials.
EVPLY    Evaluate a polynomial at a point.
COEF     Given an equation return a polynomial list.

These programs are for the HP-48sx.  I have a version of them that works
correctly on the HP-28.  Send me mail if you want it.

I think people will find these very useful and work as I say, but if you find
any bugs please send me E-Mail.  Comments are also welcome.

Some of these routines could be faster (PF, PMUL, ...) tell me if you know how
to speed them up.

Wayne Scott             |  INTERNET:   wscott@en.ecn.purdue.edu
Electrical Engineering  |  BITNET:     wscott%ea.ecn.purdue.edu@purccvm
Purdue University       |  UUCP:      {purdue,pur-ee}!en.ecn.purdue.edu!wscott


These programs all work on polynomials in the following form:

3*X^3-7*X^2+5 is entered as  { 3 -7 0 5 }


Reasons why I use lists instead of arrays include:

        * lists look better on the stack.  (multi-line)
        * The interactive stack can quickly put serveral items in a
          list.
        * Hitting EVAL will quickly return the elements on a list.
        * the '+' key will add a element to the end or beginning of a list
          or concatenate two lists
        * Internally the main program that needs to be fast (BAIRS)
          does 100% stack maniplations so speed of arrays vs. lists was
          not a major issue.
        * It would be easier for later releases to include symbolic
          polynomials.

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³ FCTP ³ (FaCTor Polynomial)
ÀÄÄÄÄÄÄÙ
When it is passed the cooeficients of a polynomial in a list it returns the
factor of that polynomial.  Example:
 
1: { 1 -17.8 99.41 -261.218 352.611 -134.106 }
FCTP
3: { 1 -4.2 2.1 }
2: { 1 -3.3 6.2 }
1: { 1 -10.3 }

This tells us that X^5-17.8*X^4+99.41*X^3-261.218*X^2+352.611*X-134.106
factors to (X^2-4.2*X+2.1)*(X^2-3.3*X+6.2)*(X-10.3)

ÚÄÄÄÄ¿
³ RT ³ (RooTs)
ÀÄÄÄÄÙ
If given a polynomial it return its roots.  Example:

1: { 1 -17.8 99.41 -261.218 352.611 -134.106 }
RT
5: 3.61986841536
4: .58013158464
3: (1.65, 1.8648056199)
2: (1.65, -1.8648056199)
1: 10.3

RT will work with complex cooeffients in the polynomial.

These programs use the BAIRS program which performs Bairstow's method of
quadratic factors and QUD which does the quadratic equation.

ÚÄÄÄÄÄÄ¿
³ TRIM ³is used to strip the leading zeros from a polynomial list.
ÀÄÄÄÄÄÄÙ
Example: { 0 0 3 0 7 } TRIM => { 3 0 7 }

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³ RDER ³will give the DERivative of a Rational function.  Example:
ÀÄÄÄÄÄÄÙ
    d        x + 4                   -X^2 - 8*x + 31
    --   -------------  =   --------------------------------
    dx   x^2 - 7*x + 3      x^4 - 14*x^3 + 55*x^2 - 42*x + 9

2: { 1 4 }
1: { 1 -7 3 }
RDER
2: { -1 -8 31 }
1: { 1 -14 55 -42 9 }

ÚÄÄÄÄÄ¿
³ IRT ³(Inverse RT) will return a polynomial whose roots you specify.
ÀÄÄÄÄÄÙ
Example:  If a transfer function has zeros at -1, 1 and 5 the function is
          x^3 - 5*x^2 - x + 5

1: { -1 1 5 }
IRT
1: { 1 -5 -1 5 }

ÚÄÄÄÄÄÄ¿
³ PDER ³will return the DERivative of a Polynomial.  Example:
ÀÄÄÄÄÄÄÙ
   The  d/dx (x^3 - 5*x^2 - x + 5) = 3*x^2 - 10*x - 1

1: { 1 -5 -1 5 }
PDER
1: { 3 -10 -1 }

ÚÄÄÄÄ¿
³ PF ³will do a Partial Fraction expansion on a transfer function.
ÀÄÄÄÄÙ
Example:
           s + 5         1/18    5/270     2/3       1/9     2/27
     ----------------- = ----- + ----- - ------- - ------- - -----
     (s-4)(s+2)(s-1)^3   (s-4)   (s+2)   (s-1)^3   (s-1)^2   (s-1) 

2: { 1 5 }
1: { 4 -2 1 1 1 }
PF
1: { 5.5555e-2 1.85185e-2 -.6666 -.11111 -.074074 }

This program expects the polynomial of the numerator to be on level 2 and
a list with the poles to be on level 1.  Repeated poles are suported but
they must be listed in order.  The output is a list of the values of the 
fraction in the same order as the poles were entered.

ÚÄÄÄÄÄÄÂÄÄÄÄÄÄÂÄÄÄÄÄÄ¿
³ PADD ³ PMUL ³ PDIV ³are all obvious, they take two polynomial lists off
ÀÄÄÄÄÄÄÁÄÄÄÄÄÄÁÄÄÄÄÄÄÙ
the stack and perform the operation on them.

PDIV returns the quotient polynomial and the remainder polynomial.

ÚÄÄÄÄÄ¿
³ L2A ³converts a list to an array. (and back)
ÀÄÄÄÄÄÙ
1: { 1 2 3 }
L2A
1: [ 1 2 3 ]
L2A
1: { 1 2 3 }

ÚÄÄÄÄÄÄÄ¿
³ EVPLY ³evaluates any polynomial at a point.
ÀÄÄÄÄÄÄÄÙ
x^3 - 3*x^2 +10*x - 5 | x=2.5   = 16.875

2: { 1 -3 10 -5 }
1: 2.5
EVPLY
1: 16.875

P.S. Many thanks to Mattias Dahl & Henrik Johansson for fixs they have
     made.