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~
f
n
~
~
=
f
l
!
-
Parru
b
*h
f i3) ~
C
d a p o # . . ~ ~ h - o i l ~ ~ i .
.
,&
p a ~ ~ o ~ l crrplrsenta.tion
up
~ m the
. data, (4) allow economy in o p A - . ~ u
8
I
i\
tim erymbals, and ( 5 ) provide c
without loss of detail.
Tbe sequence of execution of statements w i l l be specifie
' t g T i a W * ~ ~ ~ ~ i s t ~ - i b y
It8
mamers;aro
Branch poinka, at*iah alternative successors
ohoaern according t o the cxltcmma of a comparison between a pair of
quantities, w i l l be represented by a colon placed between the c m pared quantities, and by a label attached to eaoh a r r o w showing
fhe rdatim under which it i s f d
l
.
.
t h
my
b~ 888plC10gd, e.g.,
Any well-dsf ined rela-
equality, inequality, or set member
ship. The aoaditiozwr at each branch poinf must be exhaustive, and
saacesmr is associated w i t h aW, conditions not included
tke li&&
in the & b d
arrows.
Caaaaxily
-i x j
floor
occurring operations t o be defined include the
(largeat integer mt axceedhg x), the ceiling [x],
and the re-e
of x modulo m, t o be denoted by Ix,ml.
-
-
The camon
logical aplsratirma -and
¶
or, and not w i l l be denoted by A,
V
, and -,
I
and w i l l be augmemea oy zne relatianal statement
as fo-.
( X M y)
defined
If x and y an any quantities and rZ is any binary
relaticm M i n e d upaar than, them (xA y) is a lag;it~~I
variable whose
value is 0 or 1 accord in^, as x does
tion d t o y.
or does not stand i n tlre rela-
For example, the absolute value of x may be defined
To illustrate the use of the floor and ceiling operations, concider
a motangular array of dimension a X b whose cells listed in
by rows (O,l,o,.,b-l,b,b+l
,...,
ab-1) are denoted by x
orasr
in orde.?
by oolmm are denoted by yo Than
and
These a r e the transformations used i n determining a c c e s s i b i l i t y I n
as
~
r
h
l memory
~ 1 of a bands with b' slots per band,
be derived frum the identity
-
-
mq be derived frum the identity
'Ihe description of a
process
can b
I
-
representation by defining 'certab ihckmen
They
T
a r b r e d seb,
1,.2.,. ..,v(B)
mw
l ma-
The elabnt of a finite shp1.y-ordered set B of
-
th al.emnt of the set.
such tkt B~ is the i
I a9 B w i l l then
sf an e l
be denoted by r
~ h kth
s
t
k and
,
daffned as the elenaent B ,, where j r (i+k) (mod v (B) ) and B, = x.
IQ B is f:hs set
sgwbdl B may be elided, and if
of
k denotes the kth successor of the
f
#
denote8
tm
inbm i +
l o
The suocesear operation defined upan an element of B can
be extended to aqy sub&
c;;
D such t
~D~t9-
G
i&i
x :'--I,
,
I1
k denotes the s e t
B as follmsr C
k,, ITc and B are identical, the operation
is called lelYI rotation and i s demoted by C t k.
b t a t i o n i s ex-
tamdad d a g m s l y to mctorao
Two sete A end B are equal (A
but are idemtical (A
,,
d-le
okd6r.
6
I
3 ) if thag contain the stme
B) only if they
also have the same
lnodflications in the 'standarddefbftiaxis of' inter-
$~LR%+. & union provide a c;baied ~ s t e mfor ordered 8e%so TQ
7
..L&
I
-
d.?#JE.<l,d-
.oma r k oz aperatton 'apbils, intersection and union a r e hnoted by A and V, a h a &
noted by A and V, a h a &
- md, and or.
used for the analae;oua logical operations
used for the analogous logical operations
:fGjj;x
potential ambiguity is qmidsd bp using distinctive
1'-
4-
$ 1 .
.,.
symbds far each chs8 of o p g m d ; 'fe9fes For s3.1ig e var bles,
iw\ia*
1
-
lid+f
lmer case boldface i t a l i c s far vectors, upper case boldface i t a l i c s
for ma.t,ricss, and o r d m Rannan characters for l i t e r a l s , i.e., for
charcrctws such as the mmerals and the letters of the alphabet which
-
'I
*En the
Bp.d nraim, b~Pdfacecharac825&,
olsmaubrar are encircled thus:
8
v : >'.,f*
P,\'g'4 9
:
In r;e'd add
:
A second set of indicea, c a l l e d oorttrrmu*rsnt indices
spec5fi.e~C as the infix of B having t e n f n a l elements x and y,
The s p b o l
B
lnay
be elided if B is the set of integers,
FWgmm 1 illnstmtea the cmvention~introduced thus far,
.
the w i t and H~ the denomination of the jUU
card in a
-If&n
3
.I
'7W
of thirteen phgdng cards, and if
ia the set of dgnoninations, then the quantity q determined by the
3wgYiiib
-
b&.
&,--#.
,' I .
, _ .,hL,. , ,,
,,4 8 .
-fie
I .
8,
the lallgth of t h e longsst run in
a q
o m suit. A left-
I
I
m
,qt:
i.
Ypositfoml
n o e i a work,
notab17 i n ewitching t h a w and i n the urn of
rmresentaticmr (e.g., c o l m sorting) there is some advantage in using O-origin indexing, i.e.,
in using an index set
beginning a t wro, Although the presenterpositha i s limited t o
l-arigin indexing, O-origin i n d a d q can be ueed as well,
point*
qwnfi%yon the right of
emw eruem4f.s L b
af thQ ra3gm of t5qiaaav beclpuse i* arl-ters
dUf#d.%~88~Pdbt~bk
u-4
in *(t*-
may.
Utll-*cr
am--
ab htdk
sari
be aatdeved by
&gmf&LbW eaah fBp0mtiaa dsfinrs$ Ztpm sbp3.e VarhbLas to
s t s a m ~ e dm
a
ap&
d q r r ~ s a w
a8 7 ~ w b ~ and
r a aprptx!%aea,
(e.e.,
ssd~h
Far
t is a logic81
w2abb), t
m
Operratione mch aa floor, ceiling, and residue are exteslcbd
-wsjyo
&--&
1
c
r
t
;
(ta
~
~
t h a X qrsratm is -toti*s,
8-
Pnd sinoc y i 2
yf fur
w s t o ~Qi &bmriao!
. . thrse, t h e E h l e p r e s ~ c mI s crqud. to es
s
and tha -titr#rrsa 54 eatirbliuhed.
mgn%budaaf x ;
;y, tb f
~
Further thsareans cmcsm*
~
the
t product,
o r m d tfr, bax praduot
idla bg dmihr3y rlqSie formal maniplhti
rjndiv.Wd c a m p m t s of
wleokd by mbamipte and mpmtmripts
of thb m&m
X,
i
th raw -tar
A for the 1
dw
cmpm
L
of a
1
AL
the
- x , for the ith
o d b m flsotor, and A& for tb f f n
3
element. Hoxw gemxdI.ly,
L
it is ~ g b l l ~ s e rto
y apwi.f'y selected iuubeets of the w q o r - ~ n t e .
mW
-W
jd8 f m (1(10h 0-9
8
bJnrw w a f t o n , it
arn ba macWiad by an asaoaiated logical mcto
@ZB@ W I I L Z R Q W jd8 fm (1(10h C98 b
-
arn ba macifiad by an asaoaiated logical metor of the sam
~~.
Y
T i n r e fen an arbitrary mtar x and ooanpotible loglea
u (thaf h, v(u)
-
v(x)), the
&tg#nt
I
and ~olunmampressicm, de5fned by
mcpanafon of a vectorx bjr
va&aru
lug*
is 66..
and fs defked as foJSarrs,
thet ~ ( u rn) v(x).
Class*
V(Z)
:
-
~(u).
byuv] and d t m m espm~son&\XI
ex-
dsfbd
dagaus~.
T b caapess aad expand operatima pr~videcl pmrefful sx-
tanaium
am5hu.g maLrix ambra. For example,
w
C
Z WUl
be d @ ~ ~ &&C ~ ~ ~ dt6
i r tb
g id9Zltfty
!hfzicar o.n be dm-s68
o t b mnmz-. A
fek:
sbcilarly,
arly
mnerlcal
Hamam, the conventional
of the mare important identities are, fbr
MPdwisatian ovur thaw awponmts of x for uhich u
3
w~
wu
o t . t by
i
urx.
~ o l~. i
s
~
l
y
o
1
-
~
h o r b q t a l line m m a plot of x .m%il,%% .Couohes %he Largest eat&
v t ,
by the
IrfnQ.
'Ptrur if x
v/x = (6,6,6), and
byup.
> 0)[x
J
-
-
x t t ~ ~ c w
-mgnts
(6,3,-8,5,6)3
thsn T =elk
-positin
-
6: > o ) [ x , d f w
vabm o f x mj bs &%noted,
the pmmt ennp*s
(0,1,0,0,0).
ProgPsar 2 filustrah~t b um of tUrr mdation
.in a
1
haription of thr
Tbe veet;or
srP --*Ma-
,I-
:',..r':-
*..+iL-.
*. ;q
'.
si.pb*
y &aterrrrlnsd is
&La .r.----J
the opt-
and rrpecrifbs tb a
t z m
- . -..,
cam-
A:.
4,
mlution of the follawing
@ m - r l L l ~L - 4 - a
4-
4
L r-a~rrl4-m
tcr, be
-tor
C1,0s0,3.,3),
( ~ b
= 6.) ~
Wlddmtton i s d 9 a W rmlcgon8W
TIM
for axumpb,
(2
r
~ wa i
t~hrrn
g i m initially
f ~ l a i b l ebasis; x is ths uorrrssponding
ob nmago w t d a t b s ,
A pawar of
a
lsatrir
i s denoted by
solution unbounded
y
of the ood) x,
An
$t;y obds
I 8 . I u€#mbkds i
.&,
leg%thts
na~~ugaxs
o&mn w s s ,
.
FUIE
.
maclk, column mesh,zid raw mesh
me I s . a i d fo be a pm~taticmof the other, and thsre exists a
mtor p mch that
4#~+&%@)~
~ 4 s
.. #@me pmmtik
vtw~m. ~i B$ = A
y
$
,
i.
-~(BT~
then B may be denoted by A
)
for
pr
P
Parmuhttasl w i l l be axtended anslogaualy t o vectors and
i?muattm ag the eqrrare
.
X, It i s e s s i l y shown that
.
2- x for a q *-a of inverse vectors p
$.A
.$ (&,SF I . ,
.J
&A
eTf"-i
, &*+,;&!
;&%-
l .
wptxix
rT,
&-lr:;*-"~"
k$a 1 .
T-
B to a otwreepandent a of an arbitrary aet A can be raprescnted
b.p a penmbtion mmtar p surrh that B,
.
If, for tmmple,
{apple, booty, b a t ,
A
.
and the
~ v s p si n t o Am
.
.
,, '
k,
57%
-.:
q-
,'
!;b#&' i<-
'. ieaute,
,:
. cT LC,
fa ikan a (leman wrd tcr its €@.iwh
o
o
x
m
~
~
,
. I
. '
.
'-
,:,7aslt\*:
r
..:-
&'b in 33, tba
Obl
tlmk
-8ii
~~~~
of
?hh k j
5.
Qsfinsd &a the act considered 8s
A
a uwtor, 1.e.
whaare m and eolumw are aL1 genmutst%on metors
mu-
~ L be
U o a ~ da:permtation
raplremt
at^.*
A permutation ntrlx un c l e a r ~
tb apozrtfem itl sir &%tract group.
The group is Abelian
i f and an3y if tb 2~ltz3.x21 qmetri~.
A -%or
-
is frequently rapressnted (stored) in a serial-
aumrs iSls in which the aolllpmmta
ars made available only in
their aalaml.-s
algmithns upon vsctora so
~~~8
mall
To der8-
it So omPMIent %o i
61 foXbr~.
-
A fZlQ
n ~ ~ o srP;eitwl
l s
p of 1 . W h n
notation for a
a representation of a
mator x o f d l m z w l ~n ~arranged
~
ae follmm:
X ( ~ ) * X ~ *.,k(n)final(n+ll
I(~I
T h n ~ t ~ o f t z . P r n s cf x~ms p~ o m m a f i l a t o epe~iFg
a qumbiQ jr 18 cwxbd
P,
the f i b and is &maW by y-
The tmmu!!er i s tmmimtad by t
b txmmxw of a pertitian symbol,
an& if U s
A
rLs X(8) th fib i s them araM to bs in positim
-
-
fFI. q b i U r r be read ftmmrd (d.mted b y g $ ) m baclnu'd
forrard it tmmdga th canpmaat xj and stop8 in p i t f a n (*l),
j e
.
{
tb
A
f i l e rwtd baokwiud frm pottittion 1 1 trantsfera
on,,
.topa
posit3 c{(l,n)} *
Tbpasitioa uf a f i b BTUZUbe denatedby n(B,).
Xj
tb s t r s g + d $
w ( 4)-
1
ifba g
s paitfcns t;hs fils to
8.
liikl
the poebtiorr
9
,w b r e a a
In perbimlar, n(4 ).+-I denotes
of tlwa fils, and aitbar n(O )+-(MI)
tb.
2;
Thud
or (us-
oontra-
mmmt bidmclrg on tb ( ~ 1 poaitionrr)
)
n( ;8 ) +~1 demote pasition-
to thd
Qld
of t b f l h , atpr File for w H & the general position-
tha p s i t i a n & ths file b;gr am.
(d8-ta)
raoad.d hulrg a pmaeim
S
I
ulhsd
aosa3 o f%&~
wh&zh is o J y mad is a # b d
u! e
en
E.Oh partition eyrihl ny abarns
mbpmpa w i t h i n the f o e .
A W e w Mch is
m%
fl.1~~
on o f 1
values,
m x .ere ltmlf
a
Thus 3f'&h
t m e of th. pro-
3
uf eah
reaeooia~itam in the fm sccor-iae
9.8
ussd
wtral an
.
ftmadhg. W h
- .-
m
i
t
i
c
n
r
,rprabal
bl)* brwsh in tlru pmgmm
k chbndned by the
The *h
I
-
e
the recording is fcemrd
I
roMoh ZesnainoliXt8 the 1-d.
DlfYemnt
fib8
oummbg
11)a ~PWNBCI& OriU be
dist5mguishsd
l
q rQYlltW s u b d p t a ~a d rraps~edqtis, the lottier being gemrally
re-d
t o d.mt. -jar
.
clam. of if;las (a*&., lnplt a d m t p t ) .
An .nV of ifis. i.tihaat!-
b-4
1 ~ . mtcix and ths not.-
tim fur ampresaioll lnrfr be applied tq s
p
e
m
w mbssts d'the w a y .