View of On the generalized sum of intervals

REVUE D'ANALYSE NUMÉRIQUE

ET DE LA TIIÉORIE Dn rÁppRoxIMATIoN,

^Tomc

^3,

^No

^1,

^{1974, Pp,}

ll-zl

ON THE GENERALIZED SUM OF INTERVALS

by

STEFAN N (cluj)

l.

Complex sum and generalized sum

of

intervals

In the

set

I ^{of real}

^{closed intervals}

A : _lar, ør], ^B : _lbt br),

^.

..

the

complex

sum (or simply:

^sum)

of intervals [2] is

^{defined as}

A + ^B : _{{u +} alu e A &a t B} : _lø, *

^br,

ør*

^brf.

11

ø, e R

^(ø,

is a real

number)

then it ^is

^customary

to

denot'e a',

- - ^lør,

^{a1], hence}

^{R C I'}

The

pair (H,

@) is ca11ed. a generøl,ized, sum

of intervals, if the

follo-

wing

cond.itions

are

satisfied ^:

SJ H is

a set of some ord.ered pairs of intervals'

Sr)

@

is a rule

which associates

with

each ord.ered

pait (A, B) ₌ tI

an unique interval,

denoted bY

A

@ 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 +

^B

We

can give

the following

examples.

1) If H:

^{I2 and the rule}

^{is (A,} B): _fu ! ^{alu e} A &a e B),

^then

we

have

the

complex

sum 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').

11

hz ll2

^an'd

H: {(A, B)

1.4

<

B}

then by the

rule

ø,B)-\W I

rve

have tl¡e

h-q,uas,isuøt,

of intervals [1]. ^The

Þ-quasisum

is

denoted

by A@hB,

3) If / : ^{(Ír, Ír, Í", /r) is a}

^s¡zstem

^{of 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, clenotecl

hy A@¡B'

4) If u

^and

u

are functions

of 4 real

arguments,

then

for

n : _{(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

^br

l

^{a(ar, &2, by, br)}

-

^u(ør,

^{at, bt,}

^bt)l

clefines

a

generalized sum ^denotecl

b)' A

^@,,,, B.

5)

For the generalized. sum we call ^give the

following

representation:

1et

a and p be

functions

of 4 reaT

arguments such

that rot

^{a.t, Ór}

=

^R

it

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 shal1

give

sorne results about

the

sums ^2),

3)

and. 4).

2. On thc

^quasisums

of

intervals

In

connection

with

some interval- qamsioþerations:

if HcI3, f:ll+1, 2 rrn

^Rs

^{the real}

^equation

f(A,.B, ^X X :'A f ^B, then the interval

solution

Í(A, ^{B, X)} : g(A, B, X) ^{(A, B, X,} _{= l)}

is

called. quasisum.

It

is immediate

the

getetaTization

for a

quasioperation generated

by a

^real

binary or

n-aÍy operation.

In _{the þaper} [1] ^is

^given the proof of

the

following theorem:

tnponirw-l . iti tu":1'-o I (å: l, ^2, 3,

^.

")

^the-

^møþs

^{d'etined' by}

sr(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,-solrttion

iJ

^and,

onty'

i,f

A < B.

The sol'ul'ion

is

^d,eno-

ted, by

A

^@h

B

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,so

for

^rea.l,

h

genu.ølized'

qw,øsisim

A

@o

B (h,-q + B)

^then

tke h-quøs,isum

is

^{tke c}

we define for å

^sums

A@rpB: (-oo, +oo)

^anó'

^{AO- B} : "fib'-!L'

where A @rpB is the

^improper

interval

^(the

^{set 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 inclusions

ACC

^and

BÇD

^(denoted

^(A,

^B)

C(C,D)) imply that

AoBÇCoD,

i.e. the interval arithmetic is

inclusion neonotonic

l2].

For

the

quasioperations the

interval

arithmetic

is not

inclusion mono-

tonic. First

we give

the

following theorem'

THEoREM ^2,.

t¡ _{çe, Ð} Cç, D)

^ønd'

^Ã < ^{E, e} <D, ^{then ue}

^can

giae 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

+

^c

I

^d')(r

-

^s)rl.

D:lbr-c,btlø*b+d),

where

&1,

bre

R, ø, b,

c,

^d

e _R+

(i.e. are positive numtrers) an!' 0

< s, I <

^1.

eroo¡.

The

above parameirical

forms follow lot A, B

and'

C,

hence from

A < B

^and

^B QD.

From -4 CCitfollows:C:lør- x,atløly]

^where

^{x, ye R+.}

^From

e < D

we have

the

inequations

x<b*c*d, Y<blcld-x,

which has the

solution

,:

^(b

{cld)s, y:

^(b

-l¿* d')(t-s)t

and the

theo

For the q ypot

^ha¡re

THEoREM and'

^raal,s

^A @uB,

C @nD

exist, then

for "Í ^{A, B,}

^C,

^D

^giuen

ⁱⁿ

theorern

2, ue

høae

A

@e

" :lo i

^øt

^*

^b,

⁺ #,

^a'

^{+ at} I ^h +#rl'

-sAfs)

This form of

^quasisums

follows immediately from the

^Theorem

I

THEoRÞM 4.

For

tke quasisurns giaen

in

^theorem

^{3 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(b

I

^a

{

^d)

We denote s

: I - ^s

^a'nd'

^t :

7

-

^t.

Proof.

We ^have (for the

parametrical

forms 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 giaen

in

^tkeorem

^{3 ae}

^ka'ue

(A

@u

B : 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

quøsisums

A _@nB ønd c _@nD

øre ^{comþtrex sums}

for

^{ø common aa,lue}

of h, if

and' onl'y

if

:î b(atb*cl-d) b(alb+a+d)+ø(c+d)

i,e.

for -ald)-b-d+

^þ

+c+

^d\(

^(l _-

^s)rÞ

b(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\*

^hþ

h(b-ctd')

2k-l

-l

c )- ^(b

I

^c

t

^d')((t

-

^{s) th -sÞ}

-

⁽¹

-

^s)l)

ø*hIb'l

_2h-l

16 ^{$TEFAN N'} BERTI ⁶

For the following

^theorem

we recall the

d.efinitions

of

some bìnaty

,"latîJÅ -ã-i

ti,".";"ú;"1.-;T ^I2;

^{see [1])}

^{: for the intervars} A and

^B we have

A <

^B

+

^ø2

1br, ^A )

^B

⁺

^{øv <bL}

^<

^ø,

^1br'

^A

^C

^B

⁺

^b'

¹

^ø'

¹

^ø2

^1bz' 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'

^hþ

i

^a

-

⁴⁽¹

- ^s)ll'

uhere

þ :

d

-

^c

|

^(b

+ t+

^d)((1

-

^s)'

- ^s)

^ønd

I :

^b

*

^c

*

^d'

Proof.

By

ad'dition

to the

endpoints

of 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 number

I ^*-9t { ^ót

^we

ããt"i" th"

""ãpãi"is'otJrtã

^quasisums

A _@nB

^{anð' C}

@nD'The

^theorem

is

^proved.

IHEoRÐM

7.

With'

the

notøtions

of

tkeorern

6

^{we haae the} fol'loaing equiualences :

A

@u

B;' 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

@n

B I

C @nD

e

^h'þ

>

^b

+

^d

-

^qt'

We give only

th

^ences relateð't'o

A _9., F l--9

_9u

^D' Th"

^{pioãls -}

ot itre ^can

^{be made}

similarly. 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

-

^c

t

^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

end

of ifrir

putograph we

give an

exarnple.

We

consider the iutervals

A : _lør, ør* 31, ^B : _lh *6, ^a, * 101,

^c

: lør-2' øtl ^4l' D:lh*5, qtlltu)

where

¿r € R

and. u

e

_R+.

these

intervals satisfy the cond.itions of theorems

2

and.3

with the

_Parameters

s

I

aL br ^{a. b C} d s t

a,r

at*6 l+u

^u

o 1

I l*u _{3lu l-Fu 3+u r+}

d-qs ^{In this way}

^{we have}

Ae)þB:

and

+

^19)A

-

^2ø'

-

^l0

2h-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 -

^qs

t

^hþ

^{< q(l} -

^s)t

-

^{c' (4øt} I 18J_w)k-2ør- I-tt' ^(4o, 2h-l

l-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 19

1B ^STEFAN N. BERTI

For

^the

the

^following

comparison

of the

^quasisums

A@hB

^{anð' C @o}

D ^we

^have

equivalences ^:

hence f.or

u :2.

'We have

the

complex sum

for the point

P

with 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

Dn

A@þ Bl-

^{c@o D}

eu< ¹ & | <k a¿+(ø'h)eDu A@hBlc@¡ D+u<1& hr¿+@'h) ^=Dc' In the figure 1

^are represented

the

^domains

D' - ^D*

anð,

k: !

7

(Fig.

1.).

The

monotonic 1aw

valid for P (interval

arithmetic sum)

is

valid. also

in a

neighbourhood

of P, namely in

the

i.e.

complex

domain Dr.

3.

Some eonsiderations on

the

generalized surn ,4 @o B($ 1

;

³⁾

I1

Í : ^(lr, Ír, f",

^Ín)

^is

^{a system}

^{of 4}

^{real numbers,}

^then

^for

H: _{(/, ^B) _{lA(f, -,f,)} + BU,- _/,) <

^0}

k

we define the generalized. sum

A

@î

B

as

the

interval lør

l

^b,

* ÃÍ,I ^BÍ",

^ú,

i

^b,

^{-l ÃÍ, -f Bf^l}

D6 Dl TrrEoREM B.

For

the generølized, sutn

A

@r

B ue

høae one

of

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___

!

2

2 u uhere

fr,,f, = n

ønd

f, ^{g e R+}

^øre

^arbitrar

^real, nunt'bers, The quasisuno

A @þ

B

^{ha.s the}

_form 2),

nømely witlt'

Fig'l

Only

the

points of

the domain

^u

> ⁰ * O' ₊ defi

^@u

if";1;; ^{B and Ã}

^C

^{oJ 9, ã}

^O.

""a

^{C @È}

D the

comPlex sum

for ^h,

^iL:.

ft:.fz:

_2h-1^{h-l ønd}

f : s:;-,

^{1 (B}

^<Ã)

therefore

| +u

^u

^{6+u A@hB:A@t h} Þ-t h-t h \B

I 2h

-1'

^{2h=j' 2k}

-r'

^2n-tl

l+1t

^4(3*u)

Ð"

3+u

20 ^$TEFAN

N. BERTI 10

of

11 ^THE GENERALIZED SUM OF ]NTERVALS 2L

the set of

elements

AB

where

A

^is

unicuelv

d.etermincd,

but is

not

ichte'Präuss. Akad.

\Miss' Berlin ected

with inversability of

complex ervals)

is the

scope

of the

investi- sums and generally of the quasiop- Ptoof

. From the

^{conditi on}

.4'(f1- /t) + ^B(f, - ^{/t) < 0 holds}

^one

the

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 given

in

theorem

hold' An

immediately ^proof sives

thal

^the quasisum has the form ^Z'

rHÊoREM

9. Ti,;';r;;;.t;;;'t-i** ^{Àø, B}

^{giaes ø} qør,øsisutn

if

^{a'nd, only}

if

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 ^series

in

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 related

to the

gencralizeil sum

/

^@¡,e

^{B (s1,}

⁴⁾

¡I\he above sum

is not in ^the

^{present papef in general'}

^{We give}

^only

the following

^examPle,

If

u(ø7, ø2, br, br)

:

o't6z

* ^brb,

^and

^a(øt'

ø2' b1' b')

:

(øt

t

^bt)(ø'

*

^b')'

then

A

^{@rop B}

is the

^interval

lør* brl ^øú, *

^brb,

- ^ø!-

^b?'

^ø'+

^b''

* (ø'ib')(ø'lb') - ^(ø'+b')'l'

d.efined- for

Ãb,.+Bar<0.

5. A

historical remarh

Thecomplexsulnshavebeenconsidered(in'1895),bvl'"}.ROEBENIUS

(1g4g_1917) remarking that. for "ottit"".ìr

^(subsêti.

of a

^{group) we}

òan define Úr"

group'opeìation

; "u-"'ly if

^{Q[ and}

!$ ^are

^{subsets ot}