REVUE D'ANALYSE NUMÉRIQUE
ET DE LA TIIÉORIE Dn rÁppRoxIMATIoN,
Tomc3,
No1,
1974, Pp,ll-zl
ON THE GENERALIZED SUM OF INTERVALS
by
STEFAN N (cluj)
l.
Complex sum and generalized sumof
intervalsIn the
setI of real
closed intervalsA : lar, ør], B : lbt br),
...
the
complexsum (or simply:
sum)of intervals [2] is
defined asA + B : {u + alu e A &a t B} : lø, *
br,ør*
brf.11
ø, e R
(ø,is a real
number)then it is
customaryto
denot'e a',- - lør,
a1], henceR C I'
The
pair (H,
@) is ca11ed. a generøl,ized, sumof intervals, if the
follo-wing
cond.itionsare
satisfied :SJ H is
a set of some ord.ered pairs of intervals'Sr)
@is a rule
which associateswith
each ord.eredpait (A, B) = tI
an unique interval,
denoted bYA
@ B.Sa)
If (A, B) = H n
R'z(i'e' A
anð'B
are numbers anð'(A' B) e H)'
then
A@ B: A +
BWe
can givethe following
examples.1) If H:
I2 and the ruleis (A, B): fu ! alu e A &a e B),
thenwe
havethe
complexsum of
intervals.THE GENERALIZED SUM OF INTERVALS 13
72 STEFAN N. BERTr
2) I,el à be the width of ,4
(i.e.Ã: a,z- a').
11hz ll2
an'dH: {(A, B)
1.4<
B}then by the
ruleø,B)-\W I
rve
have tl¡e
h-q,uas,isuøt,of intervals [1]. The
Þ-quasisumis
denotedby A@hB,
3) If / : (Ír, Ír, Í", /r) is a
s¡zstemof 4
real'umbers, then for
H: {(A, B)lÃ(f, - /,) + Bu, - /,) <
0}the
rule(A, B) * lh I
b,I ÃÍ, * Bfr,
o,-f ó' * Ãf"'l
Bfo)clefines
a
generalized st1m, clenoteclhy A@¡B'
4) If u
andu
are functionsof 4 real
arguments,then
forn : {(A, B)
| u(ar, Q.2, b1, br)- u(øt,
a.t,bt,
b')a(ar,
ar, br,
br)-
a(ar,øt, bt,
bt))the
rule(A, B) - let l-
å,* u(at, ør, br,
br)-
w(ør,ar, br,
br),h I
brl
a(ar, &2, by, br)-
u(ør,at, bt,
bt)lclefines
a
generalized sum denoteclb)' A
@,,,, B.5)
For the generalized. sum we call give thefollowing
representation:1et
a and p be
functionsof 4 reaT
arguments suchthat rot
a.t, Ór=
Rit
follows :a(ø1,
ør, br,
br):
þ(at, a'1,bt,
br): ørl bti
then
for
H: {(A, B)l
a(a.r,ar, br,
br)4
þ(øt, ør,b',
br)}we have the generalized sum given by the mapping
(A, B) ,-,
la(ø1,6r, bt,
br), þ(at,ør, br,
br))'In the
present papef we shal1give
sorne results aboutthe
sums 2),3)
and. 4).2. On thc
quasisumsof
intervalsIn
connectionwith
some interval- qamsioþerations:if HcI3, f:ll+1, 2 rrn
Rsthe real
equationf(A,.B, X X :'A f B, then the interval
solutionÍ(A, B, X) : g(A, B, X) (A, B, X, = l)
is
called. quasisum.It
is immediatethe
getetaTizationfor a
quasioperation generatedby a
realbinary or
n-aÍy operation.In the þaper [1] is
given the proof ofthe
following theorem:tnponirw-l . iti tu":1'-o I (å: l, 2, 3,
.")
the-møþs
d'etined' bysr(A,
B) - A - B,
so(A,B): A - (A - sr-'(/, B))(å:2,3,4' "')
ahereA
- B - {r -rlu e A &u e B\.
Th-e'interua'1,equøtionto(\'4):
: B
høs a.n,inl,eiuø1,-solrttioniJ
and,onty'
i,fA < B.
The sol'ul'ionis
d,eno-ted, by
A
@hB
ønd,is
equø|,with
tke interuø|' 2 3(ø, I ø, -l- ót * br)h - a'- b,
2h-t
A e)þB
-.--.-l
a,I \ +
bz)k-
aÀ-2h-l
This is q.n
interuø\, øl,sofor
rea.l,h
genu.ølized'qw,øsisim
A
@oB (h,-q + B)
thentke h-quøs,isum
is
tke cwe define for å
sumsA@rpB: (-oo, +oo)
anó'AO- B : "fib'-!L'
where A @rpB is the
improperinterval
(theset of real
numbers) and ,4@* B is a real
number.Èor the wid.th of quasisum we have the formula
B--A
2h-l
4 5 THE GENERALIZED Sulvl OF ]NTERVALS 15
74 $TEFAN N. BERTI
In
the case of complex operations the inclusionsACC
andBÇD
(denoted(A,
B)C(C,D)) imply that
AoBÇCoD,
i.e. the interval arithmetic is
inclusion neonotonicl2].
For
the
quasioperations theinterval
arithmeticis not
inclusion mono-tonic. First
we givethe
following theorem'THEoREM 2,.
t¡ çe, Ð Cç, D)
ønd'Ã < E, e <D, then ue
cangiae tke Jol,touing þarørnetricøl' forn+s
for
tkese interuøl's :A:
lar,ql al, B: lbr,lbri
ø+ bl,
C: løt-(b-+cld')sr.2r*o.t +
(ó+
cI
d')(r-
s)rl.D:lbr-c,btlø*b+d),
where
&1,
bre
R, ø, b,c,
de R+
(i.e. are positive numtrers) an!' 0< s, I <
1.eroo¡.
The
above parameiricalforms follow lot A, B
and'C,
hence fromA < B
andB QD.
From -4 CCitfollows:C:lør- x,atløly]
wherex, ye R+.
Frome < D
we havethe
inequationsx<b*c*d, Y<blcld-x,
which has the
solution,:
(b{cld)s, y:
(b-l¿* d')(t-s)t
and the
theoFor the q ypot
ha¡reTHEoREM and'
raal,sA @uB,
C @nDexist, then
for "Í A, B,
C,D
giuenin
theorern
2, ue
høaeA
@e" :lo i
øt*
b,+ #,
a'+ at I h +#rl'
-sAfs)
This form of
quasisumsfollows immediately from the
TheoremI
THEoRÞM 4.
For
tke quasisurns giaenin
theorem3 ue
ha'ae:A@nB:A+Beh:hr:!tb. - 2øIb
ønd'
B :ølb
Ã+E 2ølb
where O
< s'< I is
ørbitrøry.The proof
is
immediately.D alblc*d'
" c+n 2(ø¡b+a+d)-st(btald') alb*cld'
C@nD:C+D+k'-hz:
2(a
t
b-l
a 1- d)-
st(bI
a{
d)We denote s
: I - s
a'nd't :
7-
t.Proof.
We have (for the
parametricalforms given in
theorem 2)Ã:ø, B-:ø+b,
e:a'ibtcld'-st(b *c-l d)' D:alb!c{d',
therefore (theorem 1)
ftl-t-_
The theorem
is
proved.TrrEoREM 5.
For the
quas'isums giaenin
tkeorem3 ae
ka'ue(A
@uB : A+ B
&c @nD:c +
D)' +ä : " b(alb*a+d)+a(otd') !\ølb +
a+d)&h: atb 2ølb
The theorem
is
an immed.iately consequence of theorem 4.THEOR_EM
6. The
quøsisumsA @nB ønd c @nD
øre comþtrex sumsfor
ø common aa,lueof h, if
and' onl'yif
:î b(atb*cl-d) b(alb+a+d)+ø(c+d)
i,e.
for -ald)-b-d+
þ+c+
d\((l -
s)rÞb(ø*btrc*.d\, b(a+b+c+d)+a(ald)s'
" - t(o+b+c+d,l
+øþ¡
d.)s"' b(ølb-l
a-ld)|a(ctd,)
C@nD:lonø'I\*
hþh(b-ctd')
2k-l
-l
c )- (bI
ct
d')((t-
s) th -sÞ-
(1-
s)l)ø*hIb'l
2h-l16 $TEFAN N' BERTI 6
For the following
theoremwe recall the
d.efinitionsof
some bìnaty,"latîJÅ -ã-i
ti,".";"ú;"1.-;T I2;
see [1]): for the intervars A and
B we haveA <
B+
ø21br, A )
B+
øv <bL<
ø,1br'
AC
B+
b'1
ø'1
ø21bz' A> B+ B 1A, Al B+ B )A, Al Bo B CA'
IHDoREM6.Ifp={(,},l,l,C'))then'inttrt'elcyþotkesisof
lheorerns
2
and'3 it
fol'l'ous(A
@oB)p(C @nD)o l-b, o)
plkþ-b -
d+ qs'
hþi
a-
4(1- s)ll'
uhere
þ :
d-
c|
(b+ t+
d)((1-
s)'- s)
øndI :
b*
c*
d'Proof.
By
ad'ditionto the
endpointsof the
intervals[-ó, 0)
and' lhþ-
b-
d'+
qs, hþ+
c- q(l-
s)']oftheconstantnumberk,b,next,bydivisionwiththepositivenumber
2k
- 1 and next ¡î-ä¿ãiti"" of ihe
constant numberI *-9t { ót
weããt"i" th"
""ãpãi"is'otJrtã
quasisumsA @nB
anð' C@nD'The
theoremis
proved.IHEoRÐM
7.
With'the
notøtionsof
tkeorern6
we haae the fol'loaing equiualences :A
@uB;' c
@r D+
hþ<q(l -
s)t-
b-
c'A@*Bl-C@nD+
q(t-s)t,-c iÎ
(1--s)lf s.#*
oq(l-s)r-b-c<kþ< if (1-s)rfsrf-i.,
T 'rHE cENERALIzED suM oF INTERVALS t7
A@nB4C@eDo
A
@nB I
C @nDe
h'þ>
b+
d-
qt'We give only
th
ences relateð't'oA 9., F l--9
9uD' Th"
pioãls -ot itre can
be madesimilarly. We
have r i.,-r,rih"otem 6) :,4 t-ó,01 l-
lhþ-
b-
d'I
qs'hþl
+*;l;(i''iít* --b <hþ l' - c(l -
s)ú{
0+
o q(l -
s),-
b-
ct
hþ<min
(e(t- s)t-c, d-q.t) "
(q(1-
s)l-
ó- -;'< hþ < q(r -
s)I -
c &q(r -
s)ú-
c<
d-
qs) V (g(1-
s)t-
b- -
c<
hþ< d-
4s & d-
qs< q(l -
s)l-
c)'Since
q(l -
s)t-
c<
d-qs e
(1-
s),+ t <ry.:L q btald'
and
q(r- s)r- c>d,- qsë(1 -s)rf s> #h
the
equivalence fo11orvs.fn the
endof ifrir
putograph wegive an
exarnple.We
consider the iutervalsA : lør, ør* 31, B : lh *6, a, * 101,
c: lør-2' øtl 4l' D:lh*5, qtlltu)
where
¿r € R
and. ue
R+.these
intervals satisfy the cond.itions of theorems2
and.3with the
Parameterss
I
aL br a. b C d s t
a,r
at*6 l+u
uo 1
I l*u 3lu l-Fu 3+u r+
d-qs In this way
we haveAe)þB:
and
+
19)A-
2ø'-
l02h-r
(4ørf19)h-2a1-9
2h-l
(4at
A@nBcc @hD.*(1-s)rf
s. #heq$ -
s)t-
c<
hþ<
d- q"
A
@nB)C @hD.'(1 -s)rl-s, # *&'d -
qst
hþ< q(l -
s)t-
c' (4øt I 18J_w)k-2ør- I-tt' (4o, 2h-ll-18+u)h-2øt-
c@þB: 2h-l
2 - Rcvue d'analyse numérique et de la théorie de l'aPproximation' torne 3' no 1' 1974
I I
THE GENERALIZED SUM OF INTERVALS 191B STEFAN N. BERTI
For
thethe
followingcomparison
of the
quasisumsA@hB
anð' C @oD we
haveequivalences :
hence f.or
u :2.
'We havethe
complex sumfor the point
Pwith u:2 A@þB{
c@o D+u> 1 & h> h+
(u,h)e D'
A@hBl
C@o D+u> 1 & I <k' a-.+ (u'k) e D'
A@hBCC@o D+w> 1 & îtu (1e (u'k) eD"
4@uB]
C@e D+u< 1 &
h<7 + (u'k) e
DnA@þ Bl-
c@o Deu< 1 & | <k a¿+(ø'h)eDu A@hBlc@¡ D+u<1& hr¿+@'h) =Dc' In the figure 1
are representedthe
domainsD' - D*
anð,
k: !
7
(Fig.
1.).The
monotonic 1awvalid for P (interval
arithmetic sum)is
valid. alsoin a
neighbourhoodof P, namely in
thei.e.
complexdomain Dr.
3.
Some eonsiderations onthe
generalized surn ,4 @o B($ 1;
3)I1
Í : (lr, Ír, f",
Ín)is
a systemof 4
real numbers,then
forH: {(/, B) lA(f, -,f,) + BU,- /,) <
0}k
we define the generalized. sum
A
@îB
asthe
interval lørl
b,* ÃÍ,I BÍ",
ú,i
b,-l ÃÍ, -f Bf^l
D6 Dl TrrEoREM B.
For
the generølized, sutnA
@rB ue
høae oneof
the follo-uing
3 formes:+\\
.,{r
,
D5 D,
l) Í: (Ír, fr*8, f'+Í, Íù, B <4!;
c
2)
Í : U, * f,
.fr, .fr, .f,* g), B > 4!;
3).f : (f', Í,, h+l, fr] g),
ö tl
7
D3
--¿P___
!
22 u uhere
fr,,f, = n
øndf, g e R+
ørearbitrar
real, nunt'bers, The quasisunoA @þ
B
ha.s theform 2),
nømely witlt'Fig'l
Only
the
points ofthe domain
u> 0 * O' + defi
@uif";1;; B and Ã
CoJ 9, ã
O.""a
C @ÈD the
comPlex sumfor h,
iL:.ft:.fz:
2h-1h-l øndf : s:;-,
1 (B<Ã)
therefore
| +u
u6+u A@hB:A@t h Þ-t h-t h \B
I 2h
-1'
2h=j' 2k-r'
2n-tll+1t
4(3*u)Ð"
3+u
20 $TEFAN
N. BERTI 10
of
11 THE GENERALIZED SUM OF ]NTERVALS 2L
the set of
elementsAB
whereA
isunicuelv
d.etermincd,but is
notichte'Präuss. Akad.
\Miss' Berlin ectedwith inversability of
complex ervals)is the
scopeof the
investi- sums and generally of the quasiop- Ptoof. From the
conditi on.4'(f1- /t) + B(f, - /t) < 0 holds
onethe
folloiving Possibilities :I
Í"_ f,>o & ro-r,<o & B.n(!"-!',)'
Í,-Ín
Í"-l',<0 & Ín-f,>0 & B '++'
or r"-Ír>o & fn-r'>o
and
the
parametrical forms givenin
theoremhold' An
immediately proof sivesthal
the quasisum has the form Z'rHÊoREM
9. Ti,;';r;;;.t;;;'t-i** Àø, B
giaes ø qør,øsisutnif
a'nd, onlyif
fr- l":1,*fn:l
ønd'Ã(f"-f) + BUo-lr) >
0'tine þroof
is
verY simPle'RDFERENCDS
[1] B et!i, S. N., Aritmeti'ca çi analizø interoalel'or' Rev' de anal' nu[r' 9i teot' apr' l' 1'
2t-39
(1e72).[2]
Moor,
E.RaÍl
on, Interual' analysis' Prentice-Hall seriesin
automatic computa-tion, 1966.
Receivetl 28. II. 19?3.
lttsl,itutul d,e ealcul ddn Clu'i aI Aca' dcmiei Reþubl'icii Socialisle Romd'nxa
4. An
example relatedto the
gencralizeil sum/
@¡,eB (s1,
4)¡I\he above sum
is not in the
present papef in general'We give
onlythe following
examPle,If
u(ø7, ø2, br, br)
:
o't6z* brb,
anda(øt'
ø2' b1' b'):
(øtt
bt)(ø'*
b')'then
A
@rop Bis the
intervallør* brl øú, *
brb,- ø!-
b?'ø'+
b''* (ø'ib')(ø'lb') - (ø'+b')'l'
d.efined- for
Ãb,.+Bar<0.
5. A
historical remarhThecomplexsulnshavebeenconsidered(in'1895),bvl'"}.ROEBENIUS