Numerieke Analyse PC-Oefening 1.5
x = var('x');
plot( (x-1)^4, (-10,10))
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plot( x^4 - 4*x^3 + 6*x^2 - 4*x + 1 , ( -10 , 10))
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plot((((x-4)*x+6)*x-4)*x+1 , (-10,10))
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plot( (x-1)^4, (0.9997,1.0003))
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plot( x^4 - 4*x^3 + 6*x^2 - 4*x + 1 , (0.9997,1.0003))
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plot( (((x-4)*x+6)*x-4)*x+1, (0.9997,1.0003))
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Numerieke Analyse PC-Oefening 2.3
n=15;
Digits = 10;
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#Lijst van recursief berekende waarden (symbolisch)
jlist_sym = [ ln(6/5) ];
for i in range(n):
jlist_sym.append( (1/(i+1)) - 5*jlist_sym[i] )
show(jlist_sym)
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#Lijst van recursief berekende waarden (FP-voorstelling door omzetting van symbolische waarden)
jlist_symnum = [];
for i in range(n+1):
jlist_symnum.append(N(jlist_sym[i] , digits=Digits))
jlist_symnum
[0.1823215568, 0.08839221604, 0.05803891976, 0.04313873430, 0.03430632781, 0.02846836299, 0.02432486415, 0.02123284340, 0.01883602142, 0.01693344116, 0.01533508301, 0.01428222656, 0.01269531250, 0.01562500000, 0.0000000000, 0.06250000000] [0.1823215568, 0.08839221604, 0.05803891976, 0.04313873430, 0.03430632781, 0.02846836299, 0.02432486415, 0.02123284340, 0.01883602142, 0.01693344116, 0.01533508301, 0.01428222656, 0.01269531250, 0.01562500000, 0.0000000000, 0.06250000000] |
parent(jlist_symnum[0])
Real Field with 37 bits of precision Real Field with 37 bits of precision |
#Lijst van recursief berekende waarden (Simulatie van FP arithmetic)
jlist_num = [N(ln(6/5) , digits=Digits)];
for i in range(n):
jlist_num.append( (1/(i+1)) - 5*jlist_num[i] )
jlist_num
[0.1823215568, 0.08839221604, 0.05803891979, 0.04313873440, 0.03430632798, 0.02846836008, 0.02432486625, 0.02123281162, 0.01883594188, 0.01693140173, 0.01534299133, 0.01419413428, 0.01236266194, 0.01510976721, -0.004120264624, 0.08726798979] [0.1823215568, 0.08839221604, 0.05803891979, 0.04313873440, 0.03430632798, 0.02846836008, 0.02432486625, 0.02123281162, 0.01883594188, 0.01693140173, 0.01534299133, 0.01419413428, 0.01236266194, 0.01510976721, -0.004120264624, 0.08726798979] |
#controle op type:
parent(jlist_num[3])
Real Field with 37 bits of precision Real Field with 37 bits of precision |
#Lijst van berekende waarden van integralen. (Symbolisch berekend, omgezet naar FP)
jlist_i = [];
x = var('x')
for i in range(n+1):
jlist_i.append( N(integral((x^i) / (x+5) , x, 0 , 1), digits=Digits))
jlist_i
[0.1823215568, 0.08839221613, 0.05803891970, 0.04313873500, 0.03430631757, 0.02846837044, 0.02432465553, 0.02123260498, 0.01882934570, 0.01696777344, 0.01538085938, 0.01464843750, 0.01171875000, 0.0000000000, 0.0000000000, 0.0000000000] [0.1823215568, 0.08839221613, 0.05803891970, 0.04313873500, 0.03430631757, 0.02846837044, 0.02432465553, 0.02123260498, 0.01882934570, 0.01696777344, 0.01538085938, 0.01464843750, 0.01171875000, 0.0000000000, 0.0000000000, 0.0000000000] |
#Voor display: plaats de lijsten als (rij)vectoren in matrix, gebruik de show() functie voor matrices.
m = matrix(vector(jlist_symnum));
m = m.stack(vector(jlist_num));
m = m.stack(vector(jlist_i));
#Kolom1: symbolisch berekend, omgezet naar FP
#Kolom2: berekend in FP-arithmetic
#Komom3: integraal berekend, omgezet naar FP
show(m.transpose())
m.transpose()
[ 0.1823215568 0.1823215568 0.1823215568] [ 0.08839221604 0.08839221604 0.08839221613] [ 0.05803891976 0.05803891979 0.05803891970] [ 0.04313873430 0.04313873440 0.04313873500] [ 0.03430632781 0.03430632798 0.03430631757] [ 0.02846836299 0.02846836008 0.02846837044] [ 0.02432486415 0.02432486625 0.02432465553] [ 0.02123284340 0.02123281162 0.02123260498] [ 0.01883602142 0.01883594188 0.01882934570] [ 0.01693344116 0.01693140173 0.01696777344] [ 0.01533508301 0.01534299133 0.01538085938] [ 0.01428222656 0.01419413428 0.01464843750] [ 0.01269531250 0.01236266194 0.01171875000] [ 0.01562500000 0.01510976721 0.0000000000] [ 0.0000000000 -0.004120264624 0.0000000000] [ 0.06250000000 0.08726798979 0.0000000000] [ 0.1823215568 0.1823215568 0.1823215568] [ 0.08839221604 0.08839221604 0.08839221613] [ 0.05803891976 0.05803891979 0.05803891970] [ 0.04313873430 0.04313873440 0.04313873500] [ 0.03430632781 0.03430632798 0.03430631757] [ 0.02846836299 0.02846836008 0.02846837044] [ 0.02432486415 0.02432486625 0.02432465553] [ 0.02123284340 0.02123281162 0.02123260498] [ 0.01883602142 0.01883594188 0.01882934570] [ 0.01693344116 0.01693140173 0.01696777344] [ 0.01533508301 0.01534299133 0.01538085938] [ 0.01428222656 0.01419413428 0.01464843750] [ 0.01269531250 0.01236266194 0.01171875000] [ 0.01562500000 0.01510976721 0.0000000000] [ 0.0000000000 -0.004120264624 0.0000000000] [ 0.06250000000 0.08726798979 0.0000000000] |
#Bij n=15, Digits=10
#Merk op:
show(jlist_sym[15])
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#Bij n=15, Digits=10
#Bij het omzetten naar een FP getal (zie jlist_symnum[15]) treden er fouten indien men vergelijkt met de
#berekening van de integraal jlist_i[15].
show([N(2005047491650199/360360) , N(-30517578125 * log(6/5))])
#Er is een aftrekking van bijna gelijke grootheden (cancellation).
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