wscott@en.ecn.purdue.edu (Wayne H Scott) (09/03/90)
Alright everyone,
When I first released my polynomial routines I was suprised at the
number of people who responded and said that I was just what they
were looking for.
Well...
Here it is, my polynomial routines version 3.
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 Evalulate a polynomial at a point.
COEF Given an equation return a polynomial list.
These programs should work on a 28s, but I might have use the HP-48 only
routines. When someone gets it working on a 28 send me any changes you
have to make.
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 follows form:
3*X^3-7*X^2+5 is entered as { 3 -7 0 5 }
so going down the list...
The first program is FCTP. (factor polynomial)
When it is passed the cooeficients of a polynomial in a list it returns the
factor of that polynomal. ex:
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)
Neat!
The next program is RT. (Roots)
If given a polynmoial it return its roots. ex:
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
Very Useful!
These programs use the BAIRS program which performs Bairstow's method of
quadratic factors and QUD with does the quadratic equation.
TRIM is used to strip the leading zeros from a polynomial list.
{0 0 3 0 7 } TRIM => { 3 0 7 }
RDER will give the derivative of a rational function.
ie. 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 }
I don't know about you but I think it's useful.
IRT will return a polynomial whose roots you specify.
ie. 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 }
PDER will return the derivtive of a polynomial.
.ie 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 }
PF will do a partial fraction expansion on a transfer function.
.ie 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.
PADD, PMUL, and PDIV are all obvious, they take two polynomial lists off
the stack and perform the operation on them.
PDIV returns the polynomial and the remainder polynomial.
L2A converts a list to and array. (and back)
1: { 1 2 3 }
L2A
1: [ 1 2 3 ]
L2A
1: { 1 2 3 }
EVPLY evalutates and polynomial at a point.
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
%%HP: T(3)A(R)F(.);
DIR
TRIM
\<< OBJ\-> \-> n
\<< n
WHILE
ROLL DUP 0 ==
REPEAT
DROP 'n' DECR
END n
ROLLD
IF n 0
==
THEN {
0 }
ELSE n
\->LIST
END
\>>
\>>
RDER
\<< \-> F G
\<< G F
PDER PMUL G PDER {
-1 } PMUL F PMUL
PADD G G PMUL
\>>
\>>
IRT
\<< OBJ\-> \-> n
\<<
IF n 0
>
THEN 1
n
START
n ROLL { 1 } SWAP
NEG +
NEXT
ELSE {
1 }
END
IF n 1
>
THEN 2
n
START
PMUL
NEXT
END
\>>
\>>
PDER
\<< OBJ\-> \-> n
\<< 1 n
FOR i n
ROLL n i - *
NEXT
DROP
IF n 1
==
THEN {
0 }
ELSE n
1 - \->LIST
END
\>>
\>>
PF
\<< MAXR { }
\-> Z P OLD LAST
\<< 1 P
SIZE
FOR I P
I GET \-> p1
\<<
IF p1 OLD \=/
THEN Z p1 EVPLY 1 P
SIZE
FOR J
IF P J GET P I
GET \=/
THEN p1 P J GET
- /
END
NEXT p1 'OLD' STO
{ } 'LAST' STO
ELSE
IF { } LAST SAME
THEN 1 { } 1 P
SIZE
FOR K P K GET
IF DUP p1 ==
THEN DROP
ELSE +
END
NEXT IRT Z SWAP
ELSE LAST OBJ\->
DROP
END RDER DUP2 5
PICK 1 + 3 ROLLD 3
\->LIST 'LAST' STO p1
EVPLY SWAP p1 EVPLY
SWAP / SWAP ! /
END
\>>
NEXT P
SIZE \->LIST
\>>
\>>
FCTP
\<<
IF DUP
SIZE 3 >
THEN DUP
BAIRS SWAP OVER
PDIV DROP FCTP
END
\>>
RT
\<< TRIM DUP
SIZE \-> n
\<<
IF n 3
>
THEN
DUP BAIRS SWAP OVER
PDIV DROP \-> A B
\<< A
RT B RT
\>>
ELSE
IF n
2 >
THEN
QUD
ELSE
LIST\-> DROP NEG SWAP
/
END
END
\>>
\>>
L\178A
\<<
IF DUP
TYPE 5 ==
THEN OBJ\->
\->ARRY
ELSE OBJ\->
1 GET \->LIST
END
\>>
PADD
\<< DUP2 SIZE
SWAP SIZE \-> A B nB
nA
\<< A L\178A B
L\178A
IF nA
nB <
THEN
SWAP
END
IF nA
nB \=/
THEN 1
nA nB - ABS
START
0
NEXT
END nA
nB - ABS 1 + ROLL
OBJ\-> 1 GET nA nB -
ABS + \->ARRY + L\178A
\>>
\>>
PMUL
\<< DUP2 SIZE
SWAP SIZE \-> A B nB
nA
\<< { }
IF nB 1
>
THEN 2
nB
START
0 +
NEXT
END DUP
'A' STO+ 'A' SWAP
STO+ A OBJ\-> \->ARRY B
OBJ\-> DROP
IF nB 1
>
THEN 2
nB
FOR J
J ROLL
NEXT
END
IF 3 nA
nB + \<=
THEN 3
nA nB +
START
0
NEXT
END nA
nB 1 - 2 * + \->ARRY
2 nA nB +
START
DUP2 DOT 3 ROLLD
OBJ\-> SWAP nA nB 1 -
2 * + 1 + ROLLD
\->ARRY
NEXT
DROP2 nA nB + 1 -
\->LIST
\>>
\>>
PDIV
\<< DUP SIZE
3 ROLLD OBJ\-> \->ARRY
SWAP OBJ\-> \->ARRY \-> c
b a
\<< a b
WHILE
OVER SIZE 1 GET c \>=
REPEAT
DIVV
END
DROP \-> d
\<< a
SIZE 1 GET c 1 - -
\->LIST d OBJ\-> OBJ\->
DROP \->LIST
\>>
\>>
\>>
EVPLY
\<< OVER
IF DUP
TYPE 5 ==
THEN SIZE
ELSE SIZE
1 GET
END \-> a r
n
\<< a 1 GET
IF n 1
>
THEN 2
n
FOR i
r * a i GET +
NEXT
END
\>>
\>>
COEF
\<< \-> E n
\<< 0 n
FOR I 0
'X' STO E EVAL 'X'
PURGE E 'X' \.d 'E'
STO I ! /
NEXT 2
n 1 +
FOR I I
ROLL
NEXT n
1 + \->LIST
\>>
\>>
EQ 1
DIVV
\<< DUP 1 GET \-> a
b c
\<< a 1 GET c /
DUP b * a SIZE RDM
a SWAP - OBJ\-> 1
GETI 1 - PUT \->ARRY
SWAP DROP b
\>>
\>>
QUD
\<< LIST\-> \->ARRY
DUP 1 GET / ARRY\->
DROP ROT DROP SWAP
2 / NEG DUP SQ ROT
- \v/ DUP2 + 3 ROLLD
-
\>>
BAIRS
\<< OBJ\-> 1 1 \-> n
R S
\<<
DO 0 n 1 +
PICK 0 0 0 4 PICK 5
n + 7
FOR J J
PICK R 7 PICK * + S
8 PICK * + 7 ROLL
DROP DUP 6 ROLLD R
3 PICK * + S 4 PICK
* + 5 ROLL DROP -1
STEP 3
PICK SQ 3 PICK 6
PICK * -
IF DUP 0
==
THEN DROP
1 1
ELSE 6
PICK 6 PICK * 5
PICK 9 PICK * -
OVER / 4 PICK 9
PICK * 8 PICK 7
PICK * - ROT /
END DUP
'S' STO+ SWAP DUP
'R' STO+
UNTIL R\->C
ABS .000000001 < 7
ROLLD 6 DROPN
END n DROPN
1 R NEG S NEG 3
\->LIST
\>>
\>>
END