Multivariate Polynomials  for the HP-41

Overview
 

 1°)  Two Variables

  1-a)  Evaluation of Polynomial Functions
  1-b)  Degree of a Polynomial
  1-c)  Product of 2 Polynomials

 2°)  N Variables

  2-a)  Evaluation of Polynomial Functions
  2-b)  Product of 2 Polynomials
  2-c)  Deleting the Zeros
  2-d)  Simplifications
  2-e)  Partial Derivatives
 
 

1°)  Two Variables
 

-In this paragraph, the coefficients  a , b , c , d , e , f , g , h , i , j , ....  of a polynomial p(x,y) are to be stored into consecutive registers as follows:

   p(x,y) = a + b x + c y + c x² + d x.y + e y² + f x³ + g x²y + h x.y² + i y³ + j x⁴ + k x³y + l x²y² + m x.y³ + n y⁴ +......

-Thus, we start with the constant term, then the monomials of degree 1 , the monomials of degree 2 , ... etc ...
-The polynomial is identified by the control number of its block of registers:  bbb.eee  that begins with Rbb and ends with Ree.
-All the coefficients must be stored ( including the zeros )

-So, we use:

     1 register for a polynomial of degree  0
     3 registers for a polynomial of degree  1
     6 registers for a polynomial of degree  2
    10 registers for a polynomial of degree  3   and, more generally,  (p+1)(p+2)/2  registers for a polynomial of degree p

-Inversely,  the degree of the polynomial = (1/2)(-3+(8n+1)^1/2)  where n is the number of coefficients in the block of registers ( cf §1-b) ).
 

     a) Evaluation of Polynomial Functions
 
 

Data Registers:              R00 = bbb.eee .....  1+eee.eee                            ( Registers Rbb thru Ree are to be initialized before executing "PXY" )

                                      •  Rbb  ......  •  Ree = coefficients of the polynomial
Flags: /
Subroutines: /

-Synthetic registers M N O P may be replaced by R01  R02  R03  R04
-In this case, delete line 42 and replace line 02 by   0   STO 04   RDN
 
 

 
   ( 75 bytes )
 
 

   with  bbb > 000

Example:      p(x,y) = 1 + 2 x + 3 y + 3 x² + 2 x.y - 4 y² + 2 x³ + 4 x²y - x.y² + 3 y³      >>>>   Evaluate  p(3,4)

-Suppose we choose the first register = R11

-Store    1       2      3       3      2     -4      2       4     -1      3
   into   R11  R12  R13  R14  R15  R16  R17  R18  R19  R20   respectively    ( control number = 11.020 )

   11.020   ENTER^
       4        ENTER^
       3        XEQ "PXY"   >>>>   p(3;4)  =  348    ( in 4 seconds )
 

     b)  Degree of a Polynomial
 

-This short routine calculates the degree of a polynomial from its control number and it produces DATA ERROR ( line 20 ) if this control number is invalid.
-It is used to compute the product of 2 polynomials in paragraph 1-c)
 

Data Registers: /
Flags: /
Subroutines: /
 
 

 
  ( 34 bytes )
 

Examples:

    11.020   XEQ "DEGXY"  yields  3
     1.028           R/S              yields   6
     1.029           R/S              yields   DATA ERROR  1.029 is not a valid control-number
 

     c) Product of 2 Polynomials
 

-Storing the coefficients as explained above allows to write a relatively fast program to multiplicate 2 polynomials of 2 variables:
 

Data Registers:              R00 to R07: temp               ( Registers Rbb thru Ree & Rbb' thru Ree' are to be initialized before executing "PROXY" )

                                      •  Rbb  ......  •  Ree  = coefficients of the 1st polynomial     bb > 07
                                      •  Rbb'  .....  •  Ree' = coefficients of the 2nd polynomial    bb' > 07
Flags: /
Subroutine:  "DEGXY"

-If you don't have an HP-41CX, replace lines 27-28-29  by

   SIGN
   CLX
   STO 00
   LBL 00
   STO IND L
   ISG L
   GTO 00
 
 

 
   ( 113 bytes )
 
 

   with  bbb and bbb' > 007

Example:

    p(x,y) = 1 + 2 x + 3 y + 3 x² + 2 x.y - 4 y² + 2 x³ + 4 x²y - x.y² + 3 y³
    q(x,y) = 2 + 3 x + 4 y + 5 x² + 6 x.y + 7 y²

-If we choose Rbb = R11 for p(x,y)   and   Rbb' = R21 for q(x,y)

-Store    1       2      3       3      2     -4      2       4     -1      3
   into   R11  R12  R13  R14  R15  R16  R17  R18  R19  R20   respectively    ( control number = 11.020 )

   and     2       3       4       5        6       7
   into   R21  R22   R23   R24   R25   R26   respectively   ( control number = 21.026 )

-We can take for instance  Rbb" = R31

    11.020  ENTER^
    21.026  ENTER^
        31     XEQ "PROXY"  >>>>   bbb.eee'' =  31.051   ( in 47 seconds )   and we have the coefficients of the product in registers R31 thru R51, namely:
 

 p(x,y) q(x,y) =  2 + 7 x + 10 y + 17 x² + 27 x.y + 11 y² + 23 x³ + 53 x²y + 26 x.y² + 11 y³ + 21 x⁴ + 48 x³y + 26 x²y² - 5 x.y³ -16 y⁴
                            + 10 x⁵ + 32 x⁴y + 33 x³y² + 37 x²y³ + 11 x.y⁴ + 21 y⁵
 
 

2°)  N Variables
 

-In the following programs, each monomial  µ x^a y^b z^c t^d u^e  is stored into 2 consecutive registers
-The coefficient µ is stored in one data register  and the exponents are coded in the next register under the form  aa.bbccddee

-For instance, to store  12 x² y³ z⁴ t¹⁵ u⁶  in registers  R14-R15 ,

          12          STO 14
   2.03041506  STO 15

-Thus, we assume that all the ( partial ) degrees are smaller than 99 and that the number of variables does not exceed 5.
-However, if we are sure that all the degrees will be smaller than 10, we could deal with polynomials of up to 10 variables ( x² y³ z⁴ t⁵ u⁶ v⁷ is coded 2.34567 )

-On the other hand, if you want to handle degrees up to 999, the maximum number of variables will be reduced to 3 and each exponent will be coded with 3 digits.
 ( x²¹ y³⁴ z⁴⁷¹ will be coded 21.034471 )
 

     a) Evaluation of Polynomial Functions
 

Data Registers:              R00: temp              ( Registers R01 thru Rnn  &  Rbb thru Ree are to be initialized before executing "PEVAL" )

                                      •  R01 = x    •  R02 = y   ......
                                      •  Rbb  ......  •  Ree = coefficients of the polynomial
Flags: /
Subroutines: /

-If you don't want to use the synthetic registers  M N O , replace them by R06 R07 R08
-In this case, delete line 32 and replace line 01 by  0  STO 06  RDN

-Replace line 21 by  10  if you code the exponents with one digit - provided they are smaller than 10
  or  by  E3  if you code the exponents with  3  digits - provided they are smaller than 1000
 
 

 
   ( 58 bytes )
 
 

  with bbb > nnn  ( n = number of variables )

Example:      p(x,y,z) = 3 x y⁴ z⁷ + 4 x² y² z³ - 7 y⁵ z + 2      Evaluate  p(2,3,4)

-If you choose to store the first coefficient into R11

   3   STO 11    1.0407  STO 12
   4   STO 13    2.0203  STO 14
  -7  STO 15    0.0501  STO 16
   2   STO 17        0       STO 18            ( control number = 11.018 )

   2  STO 01
   3  STO 02
   4  STO 03
       11.018   XEQ "PEVAL"  >>>>   p(2,3,4) = 7965038   ( in 8 seconds )

Note:

-There will be a DATA ERROR line 18 if a variable and a corresponding exponent equal zero because the HP-41 refuses to compute 0⁰
-So, you can add after line 17    X#0?  GTO 02  SIGN  STO Y  LBL 02     and you'll get for instance the correct result  p(0,3,4) = -6802

-You can also replace line 18 by a M-Code routine  Y^X  that does produce 0⁰ = 1 for example:

098   "X"
01E   "^"
019   "Y"
0F8   READ3(X)
10E   A=C  ALL
0B8   READ2(Y)
355   ?NCXQ          this subroutine checks A and C for alphadata
050   14D5              executes A<>C  and  SETDEC
128   WRIT4(L)      saves X in L-register
2EE   ?C#0?  ALL
02F   JC+ 05
04E   C=0  ALL      C
35C   PT= 12          =
050    LD@PT-1     1
023   JNC+04
044   CLRF 4          C
045   ?NCXQ          =
06C   1B11            A^C
0E8   WRIT3(X)
078   READ1(Z)
0A8   WRIT2(Y)
046   C=0 S&X
270   RAMSLCT
038   READDATA
068   WRIT1(Z)
3E0   RTN

   XEQ "Y^X"  will work like the built-in function Y^X but it yields 0⁰ = 1

-However, this M-Code routine does not check for overflows and underflows.
 

     b) Product of 2 Polynomials
 

Data Registers:              R00 to R05: temp              ( Registers Rbb thru Ree  &  Rbb' thru Ree' are to be initialized before executing "PMUL" )

                                      •  Rbb  ......  •  Ree = coefficients of the 1st polynomial           bb > 05
                                      •  Rbb'  .....  •  Ree' = coefficients of the 2nd polynomial        bb' > 05
Flag:  F07 is cleared
Subroutine:  "DEL0"  ( cf next paragraph )
 
 

 
  ( 113 bytes )
 
 

   with  bbb and bbb' > 005

Example:            p(x,y,z) = 2 x + 3 x.y²z + 4 x³y.z⁴                                          q(x,y,z) = 4 - 6 y²z  + x²y.z⁴

-With  bb = 07 & bb' = 13

    2  STO 07         3      STO 09          4      STO 11     control number             4  STO 13         -6     STO 15          1      STO 17          control number
    1  STO 08    1.0201  STO 10     3.0104  STO 12          = 7.012                   0  STO 14     0.0201  STO 16    2.0104  STO 18             = 13.018

-If you choose  bb'' = 21

    7.012   ENTER^
   13.018  ENTER^
      21      XEQ "PMUL"  >>>>   21.030   ( execution time = 15 seconds )    and we have:

      R21 = 8      R23 = 4              R25 = 18             R27 = -18           R29 = -21
      R22 = 1      R24 = 5.0208     R26 = 3.0104      R28 =  1.0402     R30 =  3.0305

-So,    p(x,y,z) q(x,y,z) =  8 x + 4 x⁵y²z⁸ + 18 x³y.z⁴ - 18 x.y⁴z² - 21 x³y³z⁵

Notes:

-Each new monomial is compared to the other ones to collect like terms as soon as possible ( lines 38 to 44 & 54 to 59 )
-It reduces the number of registers during the calculations.
-A shorter program may be written if the HP-41 simplifies the polynomial at the end
  but sometimes, ( too? ) many registers are used, especially if there are many like-terms.

-"SIM" is listed in paragraph 2-d)
 
 

 
  ( 69 bytes )
 
 

   with  bbb and bbb' > 004

-The same example is solved in 21 seconds ( instead of 15 )
-Note that the monomials are stored in a different order.
-But if you are sure that there is no like-term in the product, delete line 38 and "PMUL2" will be much faster than "PMUL"
 

     c) Deleting the Zeros
 

-This subroutine is called by "PMUL" and "SIM" to eliminate the monomials that equal zero.
 

Data Registers:              R00 to R02: temp              ( Registers Rbb thru Ree  are to be initialized before executing "DEL0" )

                                      •  Rbb  ......  •  Ree = coefficients of the polynomial           bb > 03
Flags:  /
Subroutines:  /
 
 

 
  ( 71 bytes )
 
 

  with bbb > 003

-  bbb.eee = bbb.eee'  if all the coefficients are different from zero
-If all the coefficients equal zero  eee' = bbb+1 &  Rbb = Ree' = 0
 

     d) Simplifications
 

-This subroutine collects like terms to simplify a polynomial expression.
 

Data Registers:              R00 to R04: temp              ( Registers Rbb thru Ree  are to be initialized before executing "SIM" )

                                      •  Rbb  ......  •  Ree = coefficients of the polynomial           bb > 04
Flags:  /
Subroutine:  "DEL0"  ( cf § 2-c) )

-Line 51 may be replaced by  GTO "DEL0"
 
 

 
  ( 96 bytes )
 
 

  with bbb > 004

Example:      p(x,y) = 2 x + 3 x²y + 4 x.y + 5 x²y² + 8 x.y + 7 x²y

-If we store this polynomial into R05 R06 .....

    2  STO 05       3    STO 07       4    STO 09       5    STO 11       8     STO 13       7    STO 15
    1  STO 06    2.01  STO 08    1.01  STO 10    2.02  STO 12     1.01  STO 14    2.01  STO 16

    5.016  XEQ "SIM"  >>>>   5.012      ( in 8 seconds )      and we find

    R05 = 2    R07 = 10      R09 = 12      R11 = 5
    R06 = 1    R08 = 2.01   R10 = 1.01   R12 = 2.02

-So   p(x,y) = 2 x + 10 x²y + 12 x.y + 5 x²y²

Note:

-This program may be used to add 2 polynomials:
-Store the 2 polynomials into consecutive registers so that their control numbers (bbb.eee)₁ and (bbb.eee)₂ verify  bbb₂ = 1+eee₁
-Then,  key in  bbb₁.eee₂  XEQ "SIM"
 

     e) Partial Derivatives
 

-This program calculates  ¶ ^a+b+c+...p / ¶x^a¶y^b¶z^c....       [  d ^a+b+c+...p / dx^ady^bdz^c....  if the symbol  ¶ doesn't appear correctly ]
 

Data Registers:              R00 to R04: temp              ( Registers Rbb thru Ree  are to be initialized before executing "PDRV" )

                                      •  Rbb  ......  •  Ree = coefficients of the polynomial           bb > 04
Flags:  /
Subroutine:  "SIM"  ( cf § 2-d) )

-Replace line 31 by  10  if you code the exponents with one digit - provided they are smaller than 10
  or  by  E3  if you code the exponents with  3  digits - provided they are smaller than 1000
 
 

 
  ( 74 bytes )
 
 

   with bbb > 004

Example:      p(x,y,z) = 2 x³ y⁴ z⁵ +  x² y z³ + x y z³ + 3 x¹² y⁷ z¹⁰ + 4       Calculate   ¶⁶p / ¶x²¶y ¶z³   ( aa.bbcc = 2.0103 )

-If p(x,y,z) is stored in R05 R06 ....

        2      STO 05         1        STO 07         1       STO 09          3        STO 11     4   STO 13
   3.0405  STO 06     2.0103   STO 08    1.0103   STO 10     12.0710  STO 12     0   STO 14      control number = 5.014

   5.014     ENTER^
   2.0103   XEQ "PDRV"  >>>>   5.010   ( in 24 seconds )  and we have

   R05 = 2880       R07 = 12    R09 = 1995840
   R06 = 1.0302    R08 =  0     R10 = 10.0607

-Thus,    ¶⁶p / ¶x²¶y ¶z³ = 2880 x y³ z² + 12 + 1995840 x¹⁰ y⁶ z⁷

Note:

-This routine replaces the polynomial by its derivative.
-So you'll have to store it in a second block of registers if you want to save it.