numerik 6 -- Sage

def sumTrap(f, a, b, N): h = (b-a)/N s=1/2*f(a)+1/2*f(b) for i in range(1,N): x = a+i*h s+=f(x) return (h*s)

f(x) = 1/(sqrt(1-1/2*sin(x-1)))

#a.) list = [1,2,4,8,16,32] for i in list: print "N = %d" %i a = sumTrap(f,0,2*pi,i) print a.n() print a g = f.integrate(x,0,2*pi) fehler = abs(g - a) print "Fehler: %f" %fehler

N = 1
5.27136699864820
(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1))*pi
Fehler: 1.355186
N = 2
6.76341474604790
1/2*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) +
2/sqrt(-1/2*sin(pi - 1) + 1))*pi
Fehler: 0.136862
N = 4
6.61414918603767
1/4*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) +
2/sqrt(-1/2*sin(3/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) +
2/sqrt(-1/2*sin(1/2*pi - 1) + 1))*pi
Fehler: 0.012403
N = 8
6.62654244910166
1/8*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) +
2/sqrt(-1/2*sin(7/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 1) + 1) +
2/sqrt(-1/2*sin(5/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) +
2/sqrt(-1/2*sin(3/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1) + 1) +
2/sqrt(-1/2*sin(1/4*pi - 1) + 1))*pi
Fehler: 0.000010
N = 16
6.62655267967001
1/16*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) +
2/sqrt(-1/2*sin(15/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(13/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 1) + 1) +
2/sqrt(-1/2*sin(11/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(9/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) +
2/sqrt(-1/2*sin(7/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(5/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1) + 1) +
2/sqrt(-1/2*sin(3/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(1/8*pi - 1) + 1))*pi
Fehler: 0.000000
N = 32
6.62655268094638
1/32*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) +
2/sqrt(-1/2*sin(31/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(15/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(29/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(27/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(13/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(25/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 1) + 1) +
2/sqrt(-1/2*sin(23/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(11/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(21/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(19/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(9/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(17/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) +
2/sqrt(-1/2*sin(15/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(13/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(11/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(9/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1) + 1) +
2/sqrt(-1/2*sin(7/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(5/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1) + 1) +
2/sqrt(-1/2*sin(3/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/8*pi - 1) + 1) +
2/sqrt(-1/2*sin(1/16*pi - 1) + 1))*pi
Fehler: 0.000000

N = 1
5.27136699864820
(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1))*pi
Fehler: 1.355186
N = 2
6.76341474604790
1/2*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1))*pi
Fehler: 0.136862
N = 4
6.61414918603767
1/4*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1) + 1))*pi
Fehler: 0.012403
N = 8
6.62654244910166
1/8*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1) + 1))*pi
Fehler: 0.000010
N = 16
6.62655267967001
1/16*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(15/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(13/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(11/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(9/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) + 2/sqrt(-1/2*sin(7/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/8*pi - 1) + 1))*pi
Fehler: 0.000000
N = 32
6.62655268094638
1/32*(1/sqrt(1/2*sin(1) + 1) + 1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(31/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(15/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(29/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(27/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(13/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(25/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(23/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(11/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(21/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(19/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(9/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(17/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1) + 1) + 2/sqrt(-1/2*sin(15/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(13/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(11/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(9/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(5/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/16*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/8*pi - 1) + 1) + 2/sqrt(-1/2*sin(1/16*pi - 1) + 1))*pi
Fehler: 0.000000

#b.) list = [1,2,4,8,16,32] for i in list: print "N = %d" %i a = sumTrap(f,1,2*pi,i) print a.n() print a g = f.integrate(x,1,2*pi) fehler = abs(g - a) print "Fehler: %f" %fehler

N = 1
4.85779409557072
1/2*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 1)*(2*pi - 1)
Fehler: 0.863503
N = 2
5.45843920271536
1/4*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi -
1/2) + 1) + 1)
Fehler: 0.262858
N = 4
5.69888127071675
1/8*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi
- 3/4) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 2/sqrt(-1/2*sin(1/2*pi -
1/4) + 1) + 1)
Fehler: 0.022416
N = 8
5.71509258986420
1/16*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/4*pi
- 7/8) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 3/4) + 1) +
2/sqrt(-1/2*sin(5/4*pi - 5/8) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) +
2/sqrt(-1/2*sin(3/4*pi - 3/8) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1/4) + 1)
+ 2/sqrt(-1/2*sin(1/4*pi - 1/8) + 1) + 1)
Fehler: 0.006204
N = 16
5.71974941218166
1/32*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) +
2/sqrt(-1/2*sin(15/8*pi - 15/16) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 7/8) +
1) + 2/sqrt(-1/2*sin(13/8*pi - 13/16) + 1) + 2/sqrt(-1/2*sin(3/2*pi -
3/4) + 1) + 2/sqrt(-1/2*sin(11/8*pi - 11/16) + 1) +
2/sqrt(-1/2*sin(5/4*pi - 5/8) + 1) + 2/sqrt(-1/2*sin(9/8*pi - 9/16) + 1)
+ 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 2/sqrt(-1/2*sin(7/8*pi - 7/16) + 1) +
2/sqrt(-1/2*sin(3/4*pi - 3/8) + 1) + 2/sqrt(-1/2*sin(5/8*pi - 5/16) + 1)
+ 2/sqrt(-1/2*sin(1/2*pi - 1/4) + 1) + 2/sqrt(-1/2*sin(3/8*pi - 3/16) +
1) + 2/sqrt(-1/2*sin(1/4*pi - 1/8) + 1) + 2/sqrt(-1/2*sin(1/8*pi - 1/16)
+ 1) + 1)
Fehler: 0.001548
N = 32
5.72091025218410
1/64*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) +
2/sqrt(-1/2*sin(31/16*pi - 31/32) + 1) + 2/sqrt(-1/2*sin(15/8*pi -
15/16) + 1) + 2/sqrt(-1/2*sin(29/16*pi - 29/32) + 1) +
2/sqrt(-1/2*sin(7/4*pi - 7/8) + 1) + 2/sqrt(-1/2*sin(27/16*pi - 27/32) +
1) + 2/sqrt(-1/2*sin(13/8*pi - 13/16) + 1) + 2/sqrt(-1/2*sin(25/16*pi -
25/32) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 3/4) + 1) +
2/sqrt(-1/2*sin(23/16*pi - 23/32) + 1) + 2/sqrt(-1/2*sin(11/8*pi -
11/16) + 1) + 2/sqrt(-1/2*sin(21/16*pi - 21/32) + 1) +
2/sqrt(-1/2*sin(5/4*pi - 5/8) + 1) + 2/sqrt(-1/2*sin(19/16*pi - 19/32) +
1) + 2/sqrt(-1/2*sin(9/8*pi - 9/16) + 1) + 2/sqrt(-1/2*sin(17/16*pi -
17/32) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 2/sqrt(-1/2*sin(15/16*pi
- 15/32) + 1) + 2/sqrt(-1/2*sin(7/8*pi - 7/16) + 1) +
2/sqrt(-1/2*sin(13/16*pi - 13/32) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 3/8) +
1) + 2/sqrt(-1/2*sin(11/16*pi - 11/32) + 1) + 2/sqrt(-1/2*sin(5/8*pi -
5/16) + 1) + 2/sqrt(-1/2*sin(9/16*pi - 9/32) + 1) +
2/sqrt(-1/2*sin(1/2*pi - 1/4) + 1) + 2/sqrt(-1/2*sin(7/16*pi - 7/32) +
1) + 2/sqrt(-1/2*sin(3/8*pi - 3/16) + 1) + 2/sqrt(-1/2*sin(5/16*pi -
5/32) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1/8) + 1) +
2/sqrt(-1/2*sin(3/16*pi - 3/32) + 1) + 2/sqrt(-1/2*sin(1/8*pi - 1/16) +
1) + 2/sqrt(-1/2*sin(1/16*pi - 1/32) + 1) + 1)
Fehler: 0.000387

N = 1
4.85779409557072
1/2*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 1)*(2*pi - 1)
Fehler: 0.863503
N = 2
5.45843920271536
1/4*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 1)
Fehler: 0.262858
N = 4
5.69888127071675
1/8*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 3/4) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1/4) + 1) + 1)
Fehler: 0.022416
N = 8
5.71509258986420
1/16*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 7/8) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 3/4) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 5/8) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 3/8) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1/4) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1/8) + 1) + 1)
Fehler: 0.006204
N = 16
5.71974941218166
1/32*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(15/8*pi - 15/16) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 7/8) + 1) + 2/sqrt(-1/2*sin(13/8*pi - 13/16) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 3/4) + 1) + 2/sqrt(-1/2*sin(11/8*pi - 11/16) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 5/8) + 1) + 2/sqrt(-1/2*sin(9/8*pi - 9/16) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 2/sqrt(-1/2*sin(7/8*pi - 7/16) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 3/8) + 1) + 2/sqrt(-1/2*sin(5/8*pi - 5/16) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1/4) + 1) + 2/sqrt(-1/2*sin(3/8*pi - 3/16) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1/8) + 1) + 2/sqrt(-1/2*sin(1/8*pi - 1/16) + 1) + 1)
Fehler: 0.001548
N = 32
5.72091025218410
1/64*(2*pi - 1)*(1/sqrt(-1/2*sin(2*pi - 1) + 1) + 2/sqrt(-1/2*sin(31/16*pi - 31/32) + 1) + 2/sqrt(-1/2*sin(15/8*pi - 15/16) + 1) + 2/sqrt(-1/2*sin(29/16*pi - 29/32) + 1) + 2/sqrt(-1/2*sin(7/4*pi - 7/8) + 1) + 2/sqrt(-1/2*sin(27/16*pi - 27/32) + 1) + 2/sqrt(-1/2*sin(13/8*pi - 13/16) + 1) + 2/sqrt(-1/2*sin(25/16*pi - 25/32) + 1) + 2/sqrt(-1/2*sin(3/2*pi - 3/4) + 1) + 2/sqrt(-1/2*sin(23/16*pi - 23/32) + 1) + 2/sqrt(-1/2*sin(11/8*pi - 11/16) + 1) + 2/sqrt(-1/2*sin(21/16*pi - 21/32) + 1) + 2/sqrt(-1/2*sin(5/4*pi - 5/8) + 1) + 2/sqrt(-1/2*sin(19/16*pi - 19/32) + 1) + 2/sqrt(-1/2*sin(9/8*pi - 9/16) + 1) + 2/sqrt(-1/2*sin(17/16*pi - 17/32) + 1) + 2/sqrt(-1/2*sin(pi - 1/2) + 1) + 2/sqrt(-1/2*sin(15/16*pi - 15/32) + 1) + 2/sqrt(-1/2*sin(7/8*pi - 7/16) + 1) + 2/sqrt(-1/2*sin(13/16*pi - 13/32) + 1) + 2/sqrt(-1/2*sin(3/4*pi - 3/8) + 1) + 2/sqrt(-1/2*sin(11/16*pi - 11/32) + 1) + 2/sqrt(-1/2*sin(5/8*pi - 5/16) + 1) + 2/sqrt(-1/2*sin(9/16*pi - 9/32) + 1) + 2/sqrt(-1/2*sin(1/2*pi - 1/4) + 1) + 2/sqrt(-1/2*sin(7/16*pi - 7/32) + 1) + 2/sqrt(-1/2*sin(3/8*pi - 3/16) + 1) + 2/sqrt(-1/2*sin(5/16*pi - 5/32) + 1) + 2/sqrt(-1/2*sin(1/4*pi - 1/8) + 1) + 2/sqrt(-1/2*sin(3/16*pi - 3/32) + 1) + 2/sqrt(-1/2*sin(1/8*pi - 1/16) + 1) + 2/sqrt(-1/2*sin(1/16*pi - 1/32) + 1) + 1)
Fehler: 0.000387

numerik 6

175 days ago by Sandra_Peterl