[This worksheet contains sample solutions for a computer algebra class in the winter semester 2011/12 at Leibniz Universität Hannover]
1.a)　Komplexe Zahlen
x=CC(0,-2)
print x
y=CC(1,1)
print y
z=CC(0,1)
print z
-2.00000000000000*I 1.00000000000000 + 1.00000000000000*I 1.00000000000000*I -2.00000000000000*I 1.00000000000000 + 1.00000000000000*I 1.00000000000000*I |
1.b) Betrag und Argument
print abs(x),arg(x)
print abs(y),arg(y)
print abs(z),arg(z)
2.00000000000000 -1.57079632679490 1.41421356237310 0.785398163397448 1.00000000000000 1.57079632679490 2.00000000000000 -1.57079632679490 1.41421356237310 0.785398163397448 1.00000000000000 1.57079632679490 |
1.c) Multiplikation in Polardarstellung
print abs(x*y),arg(x*y)
print abs(x*z),arg(x*z)
print abs(z^4),arg(z^4)
print abs(y^8),arg(y^8)
2.82842712474619 -0.785398163397448 2.00000000000000 0.000000000000000 1.00000000000000 -0.000000000000000 16.0000000000000 -0.000000000000000 2.82842712474619 -0.785398163397448 2.00000000000000 0.000000000000000 1.00000000000000 -0.000000000000000 16.0000000000000 -0.000000000000000 |
2.a) Die Folge x_n
def folge(c,n):
if n==0: return c
return folge(c,n-1)^2 + c
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2.b) Erste Folgenglieder
for n in range(10):
print folge(-3.,n)
-3.00000000000000 6.00000000000000 33.0000000000000 1086.00000000000 1.17939300000000e6 1.39096784844600e12 1.93479155541049e24 3.74341836288776e48 1.40131810396053e97 1.96369242848753e194 -3.00000000000000 6.00000000000000 33.0000000000000 1086.00000000000 1.17939300000000e6 1.39096784844600e12 1.93479155541049e24 3.74341836288776e48 1.40131810396053e97 1.96369242848753e194 |
for n in range(10):
print folge(.1,n)
0.100000000000000 0.110000000000000 0.112100000000000 0.112566410000000 0.112671196660288 0.112694798556861 0.112700117621772 0.112701316511961 0.112701586743529 0.112701647654509 0.100000000000000 0.110000000000000 0.112100000000000 0.112566410000000 0.112671196660288 0.112694798556861 0.112700117621772 0.112701316511961 0.112701586743529 0.112701647654509 |
2.c) Komplexe c
for n in range(10):
print folge(CC(0,3),n)
3.00000000000000*I -9.00000000000000 + 3.00000000000000*I 72.0000000000000 - 51.0000000000000*I 2583.00000000000 - 7341.00000000000*I -4.72183920000000e7 - 3.79236030000000e7*I 7.91376878564055e14 + 3.58138310501276e15*I -1.22000275809450e31 + 5.66844756517407e30*I 1.16709375176691e62 - 1.38310433272529e62*I -5.50869769790094e123 - 3.22842484953085e124*I -1.01192695057997e249 + 3.55688330729136e248*I 3.00000000000000*I -9.00000000000000 + 3.00000000000000*I 72.0000000000000 - 51.0000000000000*I 2583.00000000000 - 7341.00000000000*I -4.72183920000000e7 - 3.79236030000000e7*I 7.91376878564055e14 + 3.58138310501276e15*I -1.22000275809450e31 + 5.66844756517407e30*I 1.16709375176691e62 - 1.38310433272529e62*I -5.50869769790094e123 - 3.22842484953085e124*I -1.01192695057997e249 + 3.55688330729136e248*I |
for n in range(10):
print n,folge(CC(0,.1),n)
0 0.100000000000000*I 1 -0.0100000000000000 + 0.100000000000000*I 2 -0.00990000000000000 + 0.0980000000000000*I 3 -0.00950599000000000 + 0.0980596000000000*I 4 -0.00952532130627990 + 0.0981356928459920*I 5 -0.00953988246437502 + 0.0981304519880551*I 6 -0.00953857624994589 + 0.0981276940437159*I 7 -0.00953805990146109 + 0.0981280030162653*I 8 -0.00953813038927631 + 0.0981280984584402*I 9 -0.00953814777574650 + 0.0981280828041033*I 0 0.100000000000000*I 1 -0.0100000000000000 + 0.100000000000000*I 2 -0.00990000000000000 + 0.0980000000000000*I 3 -0.00950599000000000 + 0.0980596000000000*I 4 -0.00952532130627990 + 0.0981356928459920*I 5 -0.00953988246437502 + 0.0981304519880551*I 6 -0.00953857624994589 + 0.0981276940437159*I 7 -0.00953805990146109 + 0.0981280030162653*I 8 -0.00953813038927631 + 0.0981280984584402*I 9 -0.00953814777574650 + 0.0981280828041033*I |
2.d) Kriterium für Unbeschränktheit
N = 10
def test(c):
for n in range(N):
if abs(folge(c,n))>2: return 1
return 0
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2.e) Experimente mit test()
print test(-3)
print test(.1)
print test(CC(0,3))
print test(CC(0,.1))
1 0 1 0 1 0 1 0 |
3.a)
for y in range(-9,10):
for x in range(-9,10):
print max(abs(x),y),
print
9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 2 2 2 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 3 3 3 3 3 3 4 5 6 7 8 9 9 8 7 6 5 4 4 4 4 4 4 4 4 4 5 6 7 8 9 9 8 7 6 5 5 5 5 5 5 5 5 5 5 5 6 7 8 9 9 8 7 6 6 6 6 6 6 6 6 6 6 6 6 6 7 8 9 9 8 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 2 2 2 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 3 3 3 3 3 3 4 5 6 7 8 9 9 8 7 6 5 4 4 4 4 4 4 4 4 4 5 6 7 8 9 9 8 7 6 5 5 5 5 5 5 5 5 5 5 5 6 7 8 9 9 8 7 6 6 6 6 6 6 6 6 6 6 6 6 6 7 8 9 9 8 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 |
3.b) Ein Ausschnitt aus dem Apfelmännchen in 0-1
for y in range(-9,10):
for x in range(-9,10):
print test(CC(x*.1,y*.1)),
print
1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 |
3.e) Plots mit Sage
complex_plot(test,(-1,1),(-1,1),plot_points=20)
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complex_plot(test,(-2,1),(-2,2),plot_points=400)
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3.f) Das eigentliche Apfelmännchen
N = 20
def test2(c):
for n in range(N+1):
if abs(folge(c,n))>2:
return (n+1)^2
return 0
complex_plot(test2,(-2,1),(-1.5,1.5),plot_points=400)
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