Re: Does the order matter to add a sequence of floating numbers

On Oct 30, 11:10 am, JosephLee <shoupe...@xxxxxxxxx> wrote:
Let's say float arr[100], we want to add them up, does the order
matter? what is the different between arr[0] +..+ arr[99], and
arr[99]+...+arr[0]?

One way around this (if applicable) is to use rationals as
terms instead of floats.

# Python
import gmpy
print 'The Basel problem: sum of square inverses'
print
terms = [gmpy.mpq(1,i**2) for i in xrange(1,101)]
print 'first 100 terms forwards:'
for i in terms:
print i,
print
print
print 'sum of forwards terms'
forward_sum = sum(terms)
print forward_sum
print
print 'first 100 terms backwards:'
for i in terms[::-1]:
print i,
print
print
print 'sum of backwards terms'
backward_sum = sum(terms[::-1])
print backward_sum
print
print 'forwards same as backwards?',
print forward_sum==backward_sum
print
print """Convert to float AFTER summing the rationals to
eliminate accumulating rounding errors."""
nf = gmpy.mpf(gmpy.numer(forward_sum))
df = gmpy.mpf(gmpy.denom(forward_sum))
print
ff = nf/df
print 'forwards float\n',ff
nb = gmpy.mpf(gmpy.numer(backward_sum))
db = gmpy.mpf(gmpy.denom(backward_sum))
print
bf = nb/db
print 'backwards float\n',bf
print
print 'forwards same as backwards?',
print ff==bf
print
print
print 'The Basel problem: the stupid way'
print
terms = [gmpy.mpf(1.0/i**2) for i in xrange(1,101)]
print 'first 100 terms forwards:'
for i in terms:
print i,
print
print
print 'sum of forwards terms'
forward_sum = sum(terms)
print forward_sum
print
print 'first 100 terms backwards:'
for i in terms[::-1]:
print i,
print
print
print 'sum of backwards terms'
backward_sum = sum(terms[::-1])
print backward_sum
print
print 'forwards same as backwards?',
print forward_sum==backward_sum

## The Basel problem: sum of square inverses
##
## first 100 terms forwards:
## 1 1/4 1/9 1/16 1/25 1/36 1/49 1/64 1/81 1/100 1/121
## 1/144 1/169 1/196 1/225 1/256 1/289 1/324 1/361
## 1/400 1/441 1/484 1/529 1/576 1/625 1/676 1/729
## 1/784 1/841 1/900 1/961 1/1024 1/1089 1/1156 1/1225
## 1/1296 1/1369 1/1444 1/1521 1/1600 1/1681 1/1764
## 1/1849 1/1936 1/2025 1/2116 1/2209 1/2304 1/2401
## 1/2500 1/2601 1/2704 1/2809 1/2916 1/3025 1/3136
## 1/3249 1/3364 1/3481 1/3600 1/3721 1/3844 1/3969
## 1/4096 1/4225 1/4356 1/4489 1/4624 1/4761 1/4900
## 1/5041 1/5184 1/5329 1/5476 1/5625 1/5776 1/5929
## 1/6084 1/6241 1/6400 1/6561 1/6724 1/6889 1/7056
## 1/7225 1/7396 1/7569 1/7744 1/7921 1/8100 1/8281
## 1/8464 1/8649 1/8836 1/9025 1/9216 1/9409 1/9604
## 1/9801 1/10000
##
## sum of forwards terms
## 158950869413303787311229792851755385970238349854
## 3709859889432834803818131090369901/9721861444343
## 810305896579766726231441619755839957462417827203
## 54705517986165248000
##
## first 100 terms backwards:
## 1/10000 1/9801 1/9604 1/9409 1/9216 1/9025 1/8836
## 1/8649 1/8464 1/8281 1/8100 1/7921 1/7744 1/7569
## 1/7396 1/7225 1/7056 1/6889 1/6724 1/6561 1/6400
## 1/6241 1/6084 1/5929 1/5776 1/5625 1/5476 1/5329
## 1/5184 1/5041 1/4900 1/4761 1/4624 1/4489 1/4356
## 1/4225 1/4096 1/3969 1/3844 1/3721 1/3600 1/3481
## 1/3364 1/3249 1/3136 1/3025 1/2916 1/2809 1/2704
## 1/2601 1/2500 1/2401 1/2304 1/2209 1/2116 1/2025
## 1/1936 1/1849 1/1764 1/1681 1/1600 1/1521 1/1444
## 1/1369 1/1296 1/1225 1/1156 1/1089 1/1024 1/961
## 1/900 1/841 1/784 1/729 1/676 1/625 1/576 1/529
## 1/484 1/441 1/400 1/361 1/324 1/289 1/256 1/225
## 1/196 1/169 1/144 1/121 1/100 1/81 1/64 1/49 1/36
## 1/25 1/16 1/9 1/4 1
##
## sum of backwards terms
## 158950869413303787311229792851755385970238349854
## 3709859889432834803818131090369901/9721861444343
## 810305896579766726231441619755839957462417827203
## 54705517986165248000
##
## forwards same as backwards? True
##
## Convert to float AFTER summing the rationals to
## eliminate accumulating rounding errors.
##
## forwards float
## 1.6349839001848928650771694981803237666833217000312
## 63813853110980485631588881260641124619
##
## backwards float
## 1.6349839001848928650771694981803237666833217000312
## 63813853110980485631588881260641124619
##
## forwards same as backwards? True
##
##
## The Basel problem: the stupid way
##
## first 100 terms forwards:
## 1.0 0.25 0.111111111111111104943 0.0625
## 0.0400000000000000008327 0.0277777777777777762358
## 0.0204081632653061208205 0.015625 0.012345679012345678327
## 0.0100000000000000002082 0.00826446280991735560129
## 0.00694444444444444405895 0.00591715976331360932888
## 0.00510204081632653020512 0.00444444444444444444059
## 0.00390625 0.00346020761245674750989
## 0.00308641975308641958176 0.00277008310249307479825
## 0.00250000000000000005204 0.00226757369614512477654
## 0.00206611570247933890032 0.00189035916824196602477
## 0.00173611111111111101474 0.00160000000000000007667
## 0.00147928994082840233222 0.00137174211248285312335
## 0.00127551020408163255128 0.00118906064209274662385
## 0.00111111111111111111015 0.00104058272632674307309
## 9.765625e-4 9.18273645546372808447e-4
## 8.65051903114186877472e-4 8.16326530612244928228e-4
## 7.71604938271604895439e-4 7.30460189919649393285e-4
## 6.92520775623268699563e-4 6.57462195923734418063e-4
## 6.2500000000000001301e-4 5.94883997620464038091e-4
## 5.66893424036281194135e-4 5.40832882639264507681e-4
## 5.1652892561983472508e-4 4.93827160493827172112e-4
## 4.72589792060491506193e-4 4.52693526482571295157e-4
## 4.34027777777777753684e-4 4.16493127863390270611e-4
## 4.00000000000000019169e-4 3.84467512495194167765e-4
## 3.69822485207100583055e-4 3.5599857600569599636e-4
## 3.42935528120713280838e-4 3.3057851239669423923e-4
## 3.1887755102040813782e-4 3.07787011388119409981e-4
## 2.97265160523186655964e-4 2.87273771904625124188e-4
## 2.77777777777777777537e-4 2.68744961031980626135e-4
## 2.60145681581685768273e-4 2.5195263290501388213e-4
## 2.44140625e-4 2.36686390532544376408e-4
## 2.29568411386593202112e-4 2.22766763198930731063e-4
## 2.16262975778546719368e-4 2.10039907582440651349e-4
## 2.04081632653061232057e-4 1.98373338623289029083e-4
## 1.9290123456790122386e-4 1.87652467629949342353e-4
## 1.82615047479912348321e-4 1.77777777777777786297e-4
## 1.73130193905817174891e-4 1.68662506324843982459e-4
## 1.64365548980933604516e-4 1.6023073225444639826e-4
## 1.56250000000000003253e-4 1.5241579027587258432e-4
## 1.48720999405116009523e-4 1.45158949049208871361e-4
## 1.41723356009070298534e-4 1.38408304498269907985e-4
## 1.3520822065981612692e-4 1.32117849121416309609e-4
## 1.2913223140495868127e-4 1.26246686024491849626e-4
## 1.23456790123456793028e-4 1.20758362516604271465e-4
## 1.18147448015122876548e-4 1.15620302925193658224e-4
## 1.13173381620642823789e-4 1.10803324099722987051e-4
## 1.08506944444444438421e-4 1.06281220108406850586e-4
## 1.04123281965847567653e-4 1.02030405060708089827e-4
## 1.00000000000000004792e-4
##
## sum of forwards terms
## 1.63498390018489285491
##
## first 100 terms backwards:
## 1.00000000000000004792e-4 1.02030405060708089827e-4
## 1.04123281965847567653e-4 1.06281220108406850586e-4
## 1.08506944444444438421e-4 1.10803324099722987051e-4
## 1.13173381620642823789e-4 1.15620302925193658224e-4
## 1.18147448015122876548e-4 1.20758362516604271465e-4
## 1.23456790123456793028e-4 1.26246686024491849626e-4
## 1.2913223140495868127e-4 1.32117849121416309609e-4
## 1.3520822065981612692e-4 1.38408304498269907985e-4
## 1.41723356009070298534e-4 1.45158949049208871361e-4
## 1.48720999405116009523e-4 1.5241579027587258432e-4
## 1.56250000000000003253e-4 1.6023073225444639826e-4
## 1.64365548980933604516e-4 1.68662506324843982459e-4
## 1.73130193905817174891e-4 1.77777777777777786297e-4
## 1.82615047479912348321e-4 1.87652467629949342353e-4
## 1.9290123456790122386e-4 1.98373338623289029083e-4
## 2.04081632653061232057e-4 2.10039907582440651349e-4
## 2.16262975778546719368e-4 2.22766763198930731063e-4
## 2.29568411386593202112e-4 2.36686390532544376408e-4
## 2.44140625e-4 2.5195263290501388213e-4
## 2.60145681581685768273e-4 2.68744961031980626135e-4
## 2.77777777777777777537e-4 2.87273771904625124188e-4
## 2.97265160523186655964e-4 3.07787011388119409981e-4
## 3.1887755102040813782e-4 3.3057851239669423923e-4
## 3.42935528120713280838e-4 3.5599857600569599636e-4
## 3.69822485207100583055e-4 3.84467512495194167765e-4
## 4.00000000000000019169e-4 4.16493127863390270611e-4
## 4.34027777777777753684e-4 4.52693526482571295157e-4
## 4.72589792060491506193e-4 4.93827160493827172112e-4
## 5.1652892561983472508e-4 5.40832882639264507681e-4
## 5.66893424036281194135e-4 5.94883997620464038091e-4
## 6.2500000000000001301e-4 6.57462195923734418063e-4
## 6.92520775623268699563e-4 7.30460189919649393285e-4
## 7.71604938271604895439e-4 8.16326530612244928228e-4
## 8.65051903114186877472e-4 9.18273645546372808447e-4
## 9.765625e-4 0.00104058272632674307309
## 0.00111111111111111111015 0.00118906064209274662385
## 0.00127551020408163255128 0.00137174211248285312335
## 0.00147928994082840233222 0.00160000000000000007667
## 0.00173611111111111101474 0.00189035916824196602477
## 0.00206611570247933890032 0.00226757369614512477654
## 0.00250000000000000005204 0.00277008310249307479825
## 0.00308641975308641958176 0.00346020761245674750989
## 0.00390625 0.00444444444444444444059
## 0.00510204081632653020512 0.00591715976331360932888
## 0.00694444444444444405895 0.00826446280991735560129
## 0.0100000000000000002082 0.012345679012345678327
## 0.015625 0.0204081632653061208205 0.0277777777777777762358
## 0.0400000000000000008327 0.0625 0.111111111111111104943
## 0.25 1.0
##
## sum of backwards terms
## 1.63498390018489285551
##
## forwards same as backwards? False


.

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