Thursday, August 21, 2014

Pythonista 2.7 and HP 35S: Given roots of a polynomial, find the coefficients of a polynomial

General

Variables:
Number of Roots: N
Roots: R, S, T, U
Coefficients: A, B, C, D, E

N = 2, roots R and S:
(x - R) * (x - S) → A * x^2 + B * x + C

Formulas:
A = 1
B = -(R + S)
C = R * S

N = 3, roots R, S, and T:
(x - R) * (x - S) * (x - T) → A * x^3 + B * x^2 + C * x + D

Formulas:
A = 1
B = -(R + S + T)
C = R * S + R * T + S * T

N = 4, roots R, S, T, and U:
(x - R) * (x - S) * (x - T) * (x - U) → A * x^4 + B * x^3 + C * x^2 + D * x + E

Formulas:
A = 1
B = -(R + S + T + U)
C = R * S + R * T + R * U + S * T + S * U + T * U
D = -(R * S * T + R * S * U + R * T * U + S * T * U)
E = R * S * T * U


HP-35S: Coefficients To Roots

Program:
C001 LBL C
C002 SF 10 // SF, decimal point, 0
C003 NO OF ROOTS // enter message as an equation
C004 CF 10 // CF, decimal point, 0
C005 INPUT N
C006 4 // error checking
C007 xC008 GTO C113
C009 R-down
C010 2
C011 x>y?
C012 GTO C113
C013 1 // main routine
C014 STO A
C015 INPUT R
C016 INPUT S
C017 RCL N
C018 3
C019 x=y?
C020 GTO C033
C021 R-down
C022 4
C023 x=y?
C024 GTO C054
C025 RCL R // two roots
C026 RCL+ S
C027 +/-
C028 STO B
C029 RCL R
C030 RCLx S
C031 STO C
C032 GTO C101
C033 INPUT T // three roots
C034 RCL R
C035 RCL+ S
C036 RCL+ T
C037 +/-
C038 STO B
C039 RCL R
C040 RCLx S
C041 RCL R
C042 RCLx T
C043 +
C044 RCL S
C045 RCLx T
C046 +
C047 STO C
C048 RCL R
C049 RCLx S
C050 RCLx T
C051 +/-
C052 STO D
C053 GTO C101
C054 INPUT T // four roots
C055 INPUT U
C056 +
C057 RCL+ S
C058 RCL+ R
C059 +/-
C060 STO B
C061 RCL R
C062 RCLx S
C063 RCL R
C064 RCLx T
C065 +
C066 RCL R
C067 RCLx U
C068 +
C069 RCL S
C070 RCLx T
C071 +
C072 RCL S
C073 RCLx U
C074 +
C075 RCL T
C076 RCLx U
C077 +
C078 STO C
C079 RCL R
C080 RCLx S
C081 RCLx T
C082 RCL R
C083 RCLx S
C084 RCLx U
C085 +
C086 RCL R
C087 RCLx T
C088 RCLx U
C089 +
C090 RCL S
C091 RCLx T
C092 RCLx U
C093 +
C094 +/-
C095 STO D
C096 RCL R
C097 RCLx S
C098 RCLx T
C099 RCLx U
C100 STO E
C101 VIEW A // results
C102 VIEW B
C103 VIEW C
C104 RCL N
C105 3
C106 x≤y?
C107 VIEW D
C108 RCL N
C109 4
C110 x=y?
C111 VIEW E
C112 RTN
C113 0 // invoking the error condition
C114 1/x





Pythonista

Input: Enter a vector of coefficients, up to 4 roots
Output: A list of coefficients, in descending order

Note: the triple periods indicate a tab (...)

# let roots be the list of roots, up to 4
# EWS 2014-08-20
import math
roots=input('List of Roots (up to 4):')
n=len(roots)
poly=[1]
# check for order
if n==2:
...# quadratic
...poly.append(-(roots[0]+roots[1]))
...poly.append(roots[0]*roots[1])
...print('List of coefficients: ',poly)
elif n==3:
...# cubic
...temp=-(roots[0]+roots[1]+roots[2])
...poly.append(temp)
...temp=roots[0]*roots[1]+roots[0]*roots[2]+roots[1]*roots[2]
...poly.append(temp)
...temp=-roots[0]*roots[1]*roots[2]
...poly.append(temp)
...print('List of coefficients: ',poly)
elif n==4:
...# quartic
...temp=-(roots[0]+roots[1]+roots[2]+roots[3])
...poly.append(temp)
...temp=roots[0]*roots[1]+roots[0]*roots[2]+roots[0]*roots[3]+roots[1]*roots[2]+roots[1]*roots[3]+roots[2]*roots[3]
...poly.append(temp)
...temp=-(roots[0]*roots[1]*roots[2]+roots[0]*roots[1]*roots[3]+roots[0]*roots[2]*roots[3]+roots[1]*roots[2]*roots[3])
...poly.append(temp)
...temp=roots[0]*roots[1]*roots[2]*roots[3]
...poly.append(temp)
...print('List of Coefficients: ',poly)
else:
...print('Error: not a valid list')


Examples:

Quadratic:
R = 2, S = -4
A = 1, B = 2, C = -8

Cubic:
R = 3, S = -1, T = -3
A = 1, B = 1, C = -9, D = -9

Quartic:
R = 3, S = -1, T = -3, U = 4
A = 1, B = -3, C = -13, D = 27, E = 36

Eddie


This blog is property of Edward Shore. 2014

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