Kedjebråk

Ändliga kedjebråk

# Kedjebråksutveckling av rationellt tal
r=1347
s=7341
print(r/s)
kb1 = continued_fraction(r/s)
kb1

449/2447

[0; 5, 2, 4, 2, 22]

show(kb1)

kb1.quotients()

[0, 5, 2, 4, 2, 22]

kb1.convergents()

[0, 1/5, 2/11, 9/49, 20/109, 449/2447]

list_plot(kb1.convergents()[1:])

kb2 = continued_fraction([-6,7,8,9,114,137,2])
show(kb2)

kb2.quotients()

[-6, 7, 8, 9, 114, 137, 2]

kb2.convergents()

[-6,
 -41/7,
 -334/57,
 -3047/520,
 -347692/59337,
 -47636851/8129689,
 -95621394/16318715]

kb2.value()

-95621394/16318715

Symboliska kvoter, Eulers regel

def KB(L):
   
    n = len(L)
  #  print(L,n)
    if n == 0:
        return []
    elif n == 1:
        return L
    elif n == 2:
        return [(L[0]*L[1] +1)/L[1]]
    else:
        FL = L[:-2]
        LL = L[-2:]
        EL = KB(LL)
      #  print(f'EL={EL}')
        return FL + EL

def KBfromback(L):
    LC = L.copy()
    LU=[]
    while (len(LC) >1):
        LC = KB(LC)
        LU = LU + [LC]
    return LU

L1 = [3,4,5,2]
KBfromback(L1)

[[3, 4, 11/2], [3, 46/11], [149/46]]

continued_fraction(L1)

[3; 4, 5, 2]

continued_fraction(L1).value()

149/46

# symboliska argument
xl =list(var('x',n=5))
KBfromback(xl)

[[x0, x1, x2, (x3*x4 + 1)/x4],
 [x0, x1, ((x3*x4 + 1)*x2/x4 + 1)*x4/(x3*x4 + 1)],
 [x0,
  (x3*x4 + 1)*(x1*((x3*x4 + 1)*x2/x4 + 1)*x4/(x3*x4 + 1) + 1)/(((x3*x4 + 1)*x2/x4 + 1)*x4)],
 [((x3*x4 + 1)*x2/x4 + 1)*((x3*x4 + 1)*(x1*((x3*x4 + 1)*x2/x4 + 1)*x4/(x3*x4 + 1) + 1)*x0/(((x3*x4 + 1)*x2/x4 + 1)*x4) + 1)*x4/((x3*x4 + 1)*(x1*((x3*x4 + 1)*x2/x4 + 1)*x4/(x3*x4 + 1) + 1))]]

ut=KBfromback(xl)[-1:][0][0]
ut

((x3*x4 + 1)*x2/x4 + 1)*((x3*x4 + 1)*(x1*((x3*x4 + 1)*x2/x4 + 1)*x4/(x3*x4 + 1) + 1)*x0/(((x3*x4 + 1)*x2/x4 + 1)*x4) + 1)*x4/((x3*x4 + 1)*(x1*((x3*x4 + 1)*x2/x4 + 1)*x4/(x3*x4 + 1) + 1))

ut.expand()

x0*x1*x2^2*x3^3*x4^2/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + 3*x0*x1*x2^2*x3^2*x4/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + 2*x0*x1*x2*x3^2*x4^2/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + x0*x2*x3^2*x4/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)) + 3*x0*x1*x2^2*x3/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + x2*x3*x4/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x3*x4 + 1)) + 4*x0*x1*x2*x3*x4/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + x0*x1*x3*x4^2/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + 2*x0*x2*x3/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)) + x0*x3*x4/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)) + x2/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x3*x4 + 1)) + 2*x0*x1*x2/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + x0*x1*x2^2/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2*x4) + x4/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x3*x4 + 1)) + x0*x1*x4/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)^2) + x0/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)) + x0*x2/((x1*x2*x3*x4/(x3*x4 + 1) + x1*x2/(x3*x4 + 1) + x1*x4/(x3*x4 + 1) + 1)*(x2*x3 + x2/x4 + 1)*(x3*x4 + 1)*x4)

ut.numerator()

x0*x1*x2*x3*x4 + x0*x1*x2 + x0*x1*x4 + x0*x3*x4 + x2*x3*x4 + x0 + x2 + x4

ut.denominator()

x1*x2*x3*x4 + x1*x2 + x1*x4 + x3*x4 + 1