Numerical Integration

192 days ago by tazzalenghe

def left(i,n): if i==n: return 0 else: return 1 def right(i,n): if i==0: return 0 else: return 1 def mid(i,n): if i%2==1: return 2 else: return 0 def trap(i,n): if i==0 or i==n: return 1/2 else: return 1 def simp(i,n): if i==0 or i==n: return 1/3 elif i%2==1: return 4/3 else: return 2/3 
       
RRR = RealField(256) ln3= RRR(ln(3)) leftapp = vector(RRR,[0,0,0,0,0]) rightapp = vector(RRR,[0,0,0,0,0]) midapp = vector(RRR,[0,0,0,0,0]) trapapp = vector(RRR,[0,0,0,0,0]) simpapp = vector(RRR,[0,0,0,0,0]) lefterr = vector(RRR,[0,0,0,0,0]) righterr = vector(RRR,[0,0,0,0,0]) miderr = vector(RRR,[0,0,0,0,0]) traperr = vector(RRR,[0,0,0,0,0]) simperr = vector(RRR,[0,0,0,0,0]) 
       
ln3 
        
1.0986122886681096913952452369225257046474905578227494517346943336374942\
93219
1.098612288668109691395245236922525704647490557822749451734694333637494293219
for j in range(0,5): n = 2*10^j dx = RRR(2/n) for i in range(0,n+1): fi = RRR(1./(1.+i*dx)) leftapp[j]+=fi*left(i,n) rightapp[j]+=fi*right(i,n) midapp[j]+=fi*mid(i,n) trapapp[j]+=fi*trap(i,n) simpapp[j]+=fi*simp(i,n) leftapp[j]*=dx rightapp[j]*=dx midapp[j]*=dx trapapp[j]*=dx simpapp[j]*=dx lefterr[j]=abs(ln(3.)-leftapp[j]) righterr[j]=abs(ln(3.)-rightapp[j]) miderr[j]=abs(ln(3.)-midapp[j]) traperr[j]=abs(ln(3.)-trapapp[j]) simperr[j]=abs(ln(3.)-simpapp[j]) 
       
leftapp, lefterr 
        
((1.50000000000000000000000000000000000000000000000000000000000000000000\
0000000,
1.1326855436188037640210524159556371159851551055908203125000000000000000\
00000,
1.1019530293265498671617663717370305676013231277465820312500000000000000\
00000,
1.0989456960755088699710491084715613396838307380676269531250000000000000\
00000,
1.0986456227421837644087077023868914693593978881835937500000000000000000\
00000),
(0.401387711331890217891782413062173873186111450195312500000000000000000\
0000000,
0.0340732549506939097483382283826358616352081298828125000000000000000000\
0000000,
0.0033407406584400334281781397294253110885620117187500000000000000000000\
00000000,
0.0003334074073990667130829024245031177997589111328125000000000000000000\
000000000,
0.0000333340740739096474953839788213372230529785156250000000000000000000\
0000000000))
((1.500000000000000000000000000000000000000000000000000000000000000000000000000, 1.132685543618803764021052415955637115985155105590820312500000000000000000000, 1.101953029326549867161766371737030567601323127746582031250000000000000000000, 1.098945696075508869971049108471561339683830738067626953125000000000000000000, 1.098645622742183764408707702386891469359397888183593750000000000000000000000), (0.4013877113318902178917824130621738731861114501953125000000000000000000000000, 0.03407325495069390974833822838263586163520812988281250000000000000000000000000, 0.003340740658440033428178139729425311088562011718750000000000000000000000000000, 0.0003334074073990667130829024245031177997589111328125000000000000000000000000000, 0.00003333407407390964749538397882133722305297851562500000000000000000000000000000))
rightapp, righterr 
        
((0.83333333333333331482961625624739099293947219848632812500000000000000\
00000000,
1.0660188769521370955040140415803762152791023254394531250000000000000000\
00000,
1.0952863626598832003100625342995044775307178497314453125000000000000000\
00000,
1.0982790294088422032858787247278087306767702102661132812500000000000000\
00000,
1.0985789560755170977401906640125162084586918354034423828125000000000000\
00000),
(0.265278955334776522789752561948262155055999755859375000000000000000000\
0000000,
0.0325934117159727421153547766152769327163696289062500000000000000000000\
0000000,
0.0033259260082265651448096832609735429286956787109375000000000000000000\
00000000,
0.0003332592592675265308344023651443421840667724609375000000000000000000\
000000000,
0.0000333325925927496768963465001434087753295898437500000000000000000000\
0000000000))
((0.8333333333333333148296162562473909929394721984863281250000000000000000000000, 1.066018876952137095504014041580376215279102325439453125000000000000000000000, 1.095286362659883200310062534299504477530717849731445312500000000000000000000, 1.098279029408842203285878724727808730676770210266113281250000000000000000000, 1.098578956075517097740190664012516208458691835403442382812500000000000000000), (0.2652789553347765227897525619482621550559997558593750000000000000000000000000, 0.03259341171597274211535477661527693271636962890625000000000000000000000000000, 0.003325926008226565144809683260973542928695678710937500000000000000000000000000, 0.0003332592592675265308344023651443421840667724609375000000000000000000000000000, 0.00003333259259274967689634650014340877532958984375000000000000000000000000000000))
midapp, miderr 
        
((1.00000000000000000000000000000000000000000000000000000000000000000000\
0000000,
1.0971420940086142881142450278275646269321441650390625000000000000000000\
00000,
1.0985974750053126636206712873899959959089756011962890625000000000000000\
00000,
1.0986121405200767744414491744464612565934658050537109375000000000000000\
00000,
1.0986122871866282208030973777113104006275534629821777343750000000000000\
00000),
(0.098612288668109782108217586937826126813888549804687500000000000000000\
00000000,
0.0014701946594954939939725591102614998817443847656250000000000000000000\
00000000,
0.0000148136627970973933088316698558628559112548828125000000000000000000\
0000000000,
1.4814803295237766178615856915712356567382812500000000000000000000000000\
00000e-7,
1.4814816040598088875412940979003906250000000000000000000000000000000000\
00000e-9))
((1.000000000000000000000000000000000000000000000000000000000000000000000000000, 1.097142094008614288114245027827564626932144165039062500000000000000000000000, 1.098597475005312663620671287389995995908975601196289062500000000000000000000, 1.098612140520076774441449174446461256593465805053710937500000000000000000000, 1.098612287186628220803097377711310400627553462982177734375000000000000000000), (0.09861228866810978210821758693782612681388854980468750000000000000000000000000, 0.001470194659495493993972559110261499881744384765625000000000000000000000000000, 0.00001481366279709739330883166985586285591125488281250000000000000000000000000000, 1.481480329523776617861585691571235656738281250000000000000000000000000000000e-7, 1.481481604059808887541294097900390625000000000000000000000000000000000000000e-9))
trapapp, traperr 
        
((1.16666666666666665741480812812369549646973609924316406250000000000000\
0000000,
1.0993522102854704297625332287680066656321287155151367187500000000000000\
00000,
1.0986196959932165337359144530182675225660204887390136718750000000000000\
00000,
1.0986123627421755366284639165996850351803004741668701171875000000000000\
00000,
1.0986122894088504310744491831997038389090448617935180664062500000000000\
00000),
(0.068054377998556958573317388072609901428222656250000000000000000000000\
00000000,
0.0007399216173606948387941883993335068225860595703125000000000000000000\
000000000,
7.4073251068451639866907498799264430999755859375000000000000000000000000\
00000e-6,
7.4074065770091124250029679387807846069335937500000000000000000000000000\
00000e-8,
7.4074057998529951873933896422386169433593750000000000000000000000000000\
00000e-10))
((1.166666666666666657414808128123695496469736099243164062500000000000000000000, 1.099352210285470429762533228768006665632128715515136718750000000000000000000, 1.098619695993216533735914453018267522566020488739013671875000000000000000000, 1.098612362742175536628463916599685035180300474166870117187500000000000000000, 1.098612289408850431074449183199703838909044861793518066406250000000000000000), (0.06805437799855695857331738807260990142822265625000000000000000000000000000000, 0.0007399216173606948387941883993335068225860595703125000000000000000000000000000, 7.407325106845163986690749879926443099975585937500000000000000000000000000000e-6, 7.407406577009112425002967938780784606933593750000000000000000000000000000000e-8, 7.407405799852995187393389642238616943359375000000000000000000000000000000000e-10))
simpapp, simperr 
        
((1.11111111111111110494320541874913033097982406616210937500000000000000\
0000000,
1.0986155048598517158797704951211926527321338653564453125000000000000000\
00000,
1.0986122889972485770308333978088436803470055262247721354166666666666666\
66667,
1.0986122886681426158994590025486104423180222511291503906250000000000000\
00000,
1.0986122886681096943173319147035726928152143955230712890625000000000000\
00000),
(0.012498822443001378346139063069131225347518920898437500000000000000000\
00000000,
3.2161917420392427402475732378661632537841796875000000000000000000000000\
00000e-6,
3.2913871628181823325576260685920715332031250000000000000000000000000000\
00000e-10,
3.2862601528904633596539497375488281250000000000000000000000000000000000\
00000e-14,
0.0000000000000000000000000000000000000000000000000000000000000000000000\
000000))
((1.111111111111111104943205418749130330979824066162109375000000000000000000000, 1.098615504859851715879770495121192652732133865356445312500000000000000000000, 1.098612288997248577030833397808843680347005526224772135416666666666666666667, 1.098612288668142615899459002548610442318022251129150390625000000000000000000, 1.098612288668109694317331914703572692815214395523071289062500000000000000000), (0.01249882244300137834613906306913122534751892089843750000000000000000000000000, 3.216191742039242740247573237866163253784179687500000000000000000000000000000e-6, 3.291387162818182332557626068592071533203125000000000000000000000000000000000e-10, 3.286260152890463359653949737548828125000000000000000000000000000000000000000e-14, 0.0000000000000000000000000000000000000000000000000000000000000000000000000000))