Examples of homogeneous functions of two and four variables, which are calculated incorrectly in Excel, OneNote, Word
Definition.
A homogeneous function of degree q is a numerical function f : R^n → R such that for any v ∈ R^n from the domain of the function f and any λ ∈ R the following equality holds: f(λv)=λ^qf(v), where q is called the order of homogeneity.
For simplicity, we take q=1, and consider functions of the form f(a,b)=(a^117-b^117)^(1/117), f(a,b,c,d)=(a^117- b^117)^(1/117)-(c^117-d^117)^(1/117).
After simple numerical calculations in Python and Java, the following equalities were obtained.
f(352,54)=353.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999…
f(357,59)=356.99999999999999999999999999999999999999999999999999999999999999999999999999999999999998972…
f(361,63)=360.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999…
MS Excel:
f(352.54)=352.000000000001
f(357.59)=357.000000000001
f(361.63)=361.000000000001
Python script:
from decimal import *
getcontext().prec = 308
>>> x=(Decimal(352)**Decimal(117)-Decimal(54)**Decimal(117))(Decimal(1)/Decimal(117))
>>> print(x)
351.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998329159190818894696994980506790444791082984001875203468391468411917936881307214870551679958590777317468915163628481716135549589884152531038661217609555941316588708754835741185821790242245778320061791986803024655
>>> x=(Decimal(357)**Decimal(117)-Decimal(59)**Decimal(117))(Decimal(1)/Decimal(117))
>>> print(x)
356.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999989720158196073692153897626133606380322329055537315843030235571925317558982699264816534943225755076027658785035832219023824960424396774347405335963769219643742987625523407845824626821939913224289857224063032843297400
>>> x=(Decimal(361)**Decimal(117)-Decimal(63)**Decimal(117))(Decimal(1)/Decimal(117))
>>> print(x)
360.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999993921127936674583510825707397891980952152159386267417777344776761307280088078727565211233452220813716097076319749507809649466951661213836906215116123352999333771519212777104493376322107139264556222426180333043207111704
The Java script is similar, the only difference is that you need to find a library
to assign the form: RESULT=BigDecimalMath.root(POWER,RESULT);
I advise you to take the library A Java Math.BigDecimal Implementation of Core Mathematical Functions
MS Excel:
f(398,100,398,308)=7.38964445190504E-13
f(398,100,398,309)=7.38964445190504E-13
Python script:
>>> x=(Decimal(398)**Decimal(117)-Decimal(100)Decimal(117))(Decimal(1)/Decimal(117))-(Decimal(398)**Decimal(117)-Decimal(309)Decimal(117))(Decimal(1)/Decimal(117))
>>> print(x)
4.6829537694880400495961208753068636086170585442527635529935481848840914660367426890426968518388082410943344799080628343292984538641902867227063198133880387010206424683985254813925375685020961046102263610649485204384515121661567412189563550051412170861083133399095734629020724759126523504592133E-13
>>> x=(Decimal(398)**Decimal(117)-Decimal(100)Decimal(117))(Decimal(1)/Decimal(117))-(Decimal(398)**Decimal(117)-Decimal(308)Decimal(117))(Decimal(1)/Decimal(117))
>>> print(x)
3.2048787849158093680423522610101807533554641830481900686328630295592641185012838297491718621152607396025746152496522964996997085198842879841026218584337924367377107257472526086034104095639711035849133200650433251604302283626496677147152423702885746810118429098239541285725629711670483877984277E-13
Calculations in Java give a similar result.
At the same time, we get 0 on OneNote.
(398^117−100^117 )^(1/117)−(398^117−308^117 )^(1/117)=0.0
(398^117−100^117 )^(1/117)−(398^117−309^117 )^(1/117)=0.0
Moreover, we will set the greater precision of each member of the difference in OneNote.
f (398,100,398,309) = (15357461407072152272595341560605217489962580453253454782381000698280555150698266871781901372815261064006144084350804830440836948323660592372567223353043470694217100899363125243630508686232896955794063869972410449739272809998037351885263675820458842645745358787039602864806831680779682282811435404889161728-1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) ^ (1/117) – (15357461407072152272595341560605217489962580453253454782381000698280554150698266871781901372815261064006144084350804830440836948323660592372567223353043470694217100899363125243630508686232896955794063869972410449739272809998037351885263675820458842645745358787039602864806831680779682282811435404889161728-21141806454681545366099205930386900017436193168455251825836186523361858890822868376427076038295100051116564154489591346865 40978425229790947408474334697737411374763474080732609023460376667948921254970853067562089178774007996892973246633386570133532092567215418841778357322933607577932579981669) ^ (1/117) = 0.0
Which makes its use unsuitable even for elementary calculations.
However, if we consider this example in Access, we get:
SELECT (15357461407072152272595341560605217489962580453253454782381000698280555150698266871781901372815261064006144084350804830440836948323660592372567223353043470694217100899363125243630508686232896955794063869972410449739272809998037351885263675820458842645745358787039602864806831680779682282811435404889161728 – 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) ^ (1/117) – (15357461407072152272595341560605217489962580453253454782381000698280554150698266871781901372815261064006144084350804830440836948323660592372567223353043470694217100899363125243630508686232896955794063869972410449739272809998037351885263675820458842645745358787039602864806831680779682282811435404889161728-211418064546815453660992059303869000174361931684552518258361865233618588908228683764270760382951000511165641544895913468654097842522 9790947408474334697737411374763474080732609023460376667948921254970853067562089178774007996892973246633386570133532092567215418841778357322933607577932579981669) ^ (1/117);
or
SELECT (398^117-100^117)^(1/117)-(398^117-309^117)^(1/117);
with the same result f(398,100,398,309)=4.54747350886464E-13.
MS Word has not only its own mathematics, but also its own understanding of calculations.
We make a table of one column and calculate.
15357461407072152272595341560605217489962580453253454782381000698280555150698266871781901372815261064006144084350804830440836948323660592372567223353043470694217100899363125243630508686232896955794063869972410449739272809998037351885263675820458842645745358787039602864806831680779682282811435404889161728
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
A1-A2 = 2.16
15357461407072152272595341560605217489962580453253454782381000698280554150698266871781901372815261064006144084350804830440836948323660592372567223353043470694217100899363125243630508686232896955794063869972410449739272809998037351885263675820458842645745358787039602864806831680779682282811435404889161728
2114180645468154536609920593038690001743619316845525182583618652336185889082286837642707603829510005111656415448959134686540978425229790947408474334697737411374763474080732609023460376667948921254970853067562089178774007996892973246633386570133532092567215418841778357322933607577932579981669
A5-A4=2.14
A3-A6=0.02
Which is completely wrong (and the formula A5-A4 u003d 2.14 does not find an explanation anywhere).
The summary is like this. There are many pairs of homogeneous functions of two and four variables that cause difficulties in MS Office 365 Excel, Word, OneNote, but not in Access.
