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titute
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C++C++C++C++Programming
ProgrammingProgrammingProgramming
Chapter Chapter Chapter Chapter 5 Functions and an
5 Functions and an 5 Functions and an 5 Functions and an
Introduction to RecursionIntroduction to RecursionIntroduction to RecursionIntroduction to Recursion
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(02) 3366(02) 3366(02) 3366(02) 3366--- -3603
360336033603
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2
ObjectivesObjectivesObjectivesObjectives
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3
OutlineOutlineOutlineOutline
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OutlineOutlineOutlineOutline
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�C
on
struct p
rog
rams fro
m sm
all, simp
le pieces, o
r com
po
nen
ts.
�d
ivid
e and
con
qu
er
�D
eclare and
use fu
nctio
ns to
facilitate the d
esign
, imp
lemen
tation
, o
peratio
n an
d m
ainten
ance o
f large p
rog
rams
�O
verv
iew sev
eral C+
+ S
tand
ard L
ibrary
math
fun
ction
s
�L
earn h
ow
to d
eclare yo
ur o
wn
fun
ction
s
�R
and
om
nu
mb
er gen
eration
and
a versio
n o
f the casin
o d
ice gam
e
�sto
rage classes an
d sco
pe ru
les
�im
pro
ve p
rog
ram p
erform
ance
�in
line fu
nctio
ns an
d referen
ce param
eters
�F
un
ction
ov
erload
ing
-o
ne fu
nctio
n o
f the sam
e nam
e
�p
erform
similar task
s for arg
um
ents o
f differen
t typ
es or p
ossib
ly
for d
ifferent n
um
bers o
f argu
men
ts
�F
un
ction
temp
lates -a m
echan
ism fo
r defin
ing
a family
of o
verlo
aded
fu
nctio
ns
�R
ecursio
n -
fun
ction
s that call th
emselv
es
5
5.1Introduction
IntroductionIntroductionIntroduction
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�C
++
pro
gram
s are typ
ically w
ritten b
y co
mb
inin
g
�n
ew fu
nctio
ns an
d classes y
ou
write w
ith “p
repack
aged
” fun
ction
s
�classes av
ailable in
the C
++
Stan
dard
Lib
rary
�C
++
Stan
dard
Lib
rary p
rov
ides
�co
mm
on
math
ematical calcu
lation
s
�strin
g m
anip
ulatio
ns
character m
anip
ulatio
ns
�in
pu
t/ou
tpu
terro
r check
ing
�m
any
oth
er usefu
l op
eration
s
�F
un
ction
s
�m
od
ularize a p
rog
ram b
y sep
arating its task
s into
self-contain
ed u
nits
�b
y u
ser-
user-d
efined
fun
ctions
or p
rog
ramm
er-defin
ed fu
nctio
ns
�M
otiv
ation
s for “fu
nctio
nalizin
g” a p
rog
ram
�d
ivid
e-and-co
nq
uer m
akes p
rog
ram d
evelo
pm
ent m
ore m
anag
eable
�so
ftware reu
sability
–
�usin
g ex
isting fu
nctio
ns as b
uild
ing b
lock
s to create n
ew p
rogram
s
�av
oid
repeatin
g co
de in
a pro
gram
6
5.2
Pro
gra
m C
om
po
ne
nts
in C
++
��
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�A
fun
ction
is
inv
ok
ed b
y a
fun
ction
call, and
wh
en th
e called
fun
ction
com
pletes
its task, it eith
er
return
s a result o
r
simp
ly retu
rns
con
trol to
the
caller
7
5.2
Pro
gra
m C
om
po
ne
nts
in C
++
YP
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�F
un
ction
s typ
es
�m
emb
ers of a class
�g
lob
al fun
ction
s -n
ot m
emb
ers of a class
�<cmath>
head
er file -
perfo
rm co
mm
on
math
ematical calcu
lation
s
�G
lob
al fun
ctions (all) –
type fx_name(arguments)�
Arg
um
ents: co
nstan
ts, variab
les or m
ore co
mp
lex ex
pressio
ns
8
5.3
Ma
th L
ibra
ry F
un
ctio
ns
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�http://msdn.microsoft.com/en-us/library/7wsh95e5.aspx
9
5.3
Ma
th L
ibra
ry F
un
ctio
ns
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5.4
Fu
nc
tion
De
finitio
ns
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�fo
rmat o
f a fun
ction
defin
ition
retu
rn-v
alu
e-ty
pe
functio
n-n
am
e(para
mete
r-list)
{decla
ratio
ns a
nd sta
tem
ents
}
�retu
rn-va
lue-typ
evoid -
a fun
ction
do
es no
t return
a valu
e�
All v
ariables d
efined
in fu
nctio
n d
efinitio
ns are lo
cal variab
les—th
ey’re k
no
wn
on
ly in
the fu
nctio
n in
wh
ich th
ey’re d
efined
�A
fun
ction
’s param
eters are also lo
cal variab
les of th
at fun
ction
.�
Fu
nctio
n square
(lines 1
9–
22
) receives a co
py
of th
e valu
e of arg
um
ent x
from
line 1
3 an
d sto
res it in th
e param
eter y.
�L
ine 6
is a fun
ction
pro
toty
pe.
�N
ot req
uired
if the d
efinitio
n ap
pears b
efo
re th
e functio
n’s first u
se in th
e
pro
gram
-th
e functio
n h
eader also
serves as th
e functio
n p
roto
type
11
5.4
Fu
nc
tion
De
finitio
ns
expects to
receive an
integ
er valu
e from
the caller
info
rms th
e com
piler to
return
s an in
teger resu
lt to th
e callerC
heck
by
com
piler
If data ty
pe n
ot m
atch =
> co
nvert
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�p
ara
meter-list : a
com
ma
-sepa
rated
list con
tainin
g th
e declaratio
ns
of th
e param
eters received
by
the fu
nctio
n w
hen
it’s called
�If n
o v
alue receiv
ed, p
ara
meter-list is v
oid
or sim
ply
left emp
ty
�A
typ
e mu
st be listed
for each
param
eter in th
e param
eter list
�d
eclara
tion
s an
d sta
temen
ts : fun
ction
bo
dy
, also called
a blo
ck
or co
mp
ou
nd
statemen
t.
�V
ariables can
be d
eclared in
any b
lock
, and
blo
cks can
be n
ested.
�T
hree w
ays to
return
con
trol as a fu
nctio
n w
as inv
ok
ed
�n
ot retu
rnin
g a resu
lt: con
trol retu
rns w
hen
the p
rog
ram reach
es the
fun
ction
-end
ing
righ
t brace, o
r by ex
ecutin
g th
e statemen
treturn;
�retu
rnin
g a resu
lt, the statem
ent
return
expre
ssion;
evalu
ates exp
ressio
nan
d retu
rns th
e valu
e of ex
pressio
n to
the
caller.
12
5.4
Fu
nc
tion
De
finitio
ns
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14
5.5
Fu
nc
tion
s
with
Mu
ltiple
P
ara
me
ters
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5.5
Fu
nc
tion
s w
ith M
ultip
le P
ara
me
ters
�W
hen
maximum
is called (lin
e 20
), the p
arameters x
(y,z) is in
itialized
with
the v
alue o
f the arg
um
ent n
um
ber1
(nu
mb
er2, n
um
ber3
)
�T
here m
ust b
e on
e argu
men
t in th
e fun
ction
call for each
param
eter (also
called a fo
rmal p
arameter) in
the fu
nctio
n d
efinitio
n
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5.5
Fu
nc
tion
s w
ith M
ultip
le P
ara
me
ters
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5.5
Fu
nc
tion
s w
ith M
ultip
le P
ara
me
ters
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�A
fun
ction
pro
toty
pe (also
called a fu
nctio
n d
eclaration
) tells the
com
piler th
e nam
e of a fu
nctio
n, th
e typ
e of d
ata return
ed b
y th
e
fun
ction
, the n
um
ber o
f param
eters the fu
nctio
n ex
pects to
receive,
the ty
pes o
f tho
se param
eters and
the o
rder in
wh
ich th
e param
eters
of th
ose ty
pes are ex
pected
.
�C
om
piler ch
ecks calls to
a fun
ction
with
the fu
nctio
n p
roto
typ
e
�n
um
ber an
d ty
pes o
f argu
men
ts, argu
men
ts in th
e correct o
rder
�v
alue retu
rned
by
the fu
nctio
n can
be u
sed co
rrectly
18
5.6
Fu
nc
tion
Pro
toty
pe
s a
nd
Arg
um
en
t Co
erc
ion
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�T
he p
ortio
n o
f a fun
ction
pro
toty
pe th
at inclu
des th
e nam
e of th
e
fun
ction
and
the ty
pes o
f its argu
men
ts (no
t inclu
din
g th
e return
typ
e)
is called th
e fun
ction
sign
ature
or sim
ply
the sig
natu
re
�F
un
ction
s in th
e sam
e scop
e mu
st ha
ve u
niq
ue
sign
atures.
�T
he sco
pe is w
here th
e fun
ction is k
no
wn
and
accessible.
19
5.6
Fu
nc
tion
Pro
toty
pe
s a
nd
Arg
um
en
t Co
erc
ion
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�A
rgu
men
t coercio
n-
forcin
g arg
um
ents to
the ap
pro
priate ty
pes
specified
by
the p
arameter d
eclaration
s
�e.g
. intin
call bu
t double in p
roto
typ
e
�C
on
versio
n b
y p
rom
otio
n ru
les -If arg
um
ent v
alues th
at do
no
t co
rrespond
precisely
to th
e param
eter typ
es in th
e fun
ction p
roto
typ
e can b
e co
nv
erted b
y th
e com
piler to
the p
rop
er typ
e befo
re the fu
nctio
n is called
.
�w
itho
ut lo
sing
data, e.g
., inttodouble
�T
he ty
pe o
f each v
alue in
a mix
ed-ty
pe ex
pressio
n is p
rom
oted
to th
e “h
igh
est” typ
e in th
e exp
ression (actu
ally a tem
po
rary v
ersion
of each
valu
e is created
and
used
for th
e exp
ression
—th
e orig
inal v
alues rem
ain
un
chan
ged
).
20
5.6
Fu
nc
tion
Pro
toty
pe
s a
nd
Arg
um
en
t Co
erc
ion
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�T
he C
++
Stan
dard
Lib
rary is d
ivid
ed in
to m
any
po
rtion
s, each w
ith
its ow
n h
eader file.
�T
he h
eader files co
ntain
the fu
nctio
n p
roto
typ
es for th
e related
fun
ction
s that fo
rm each
po
rtion
of th
e library
.
�T
he h
eader files also
con
tain d
efinitio
ns o
f vario
us class ty
pes an
d
fun
ction
s, as well as co
nstan
ts need
ed b
y th
ose fu
nctio
ns.
�A
head
er file “instru
cts” the co
mp
iler on
ho
w to
interface w
ith
library
and
user-w
ritten co
mp
on
ents.
22
5.7
C+
+ S
tan
da
rd L
ibra
ry H
ea
de
r File
s
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5.7
C+
+ S
tan
da
rd L
ibra
ry H
ea
de
r File
s
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5.7
C+
+ S
tan
da
rd L
ibra
ry H
ea
de
r File
s
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5.7
C+
+ S
tan
da
rd L
ibra
ry H
ea
de
r File
s
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5.7
C+
+ S
tan
da
rd L
ibra
ry H
ea
de
r File
s
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�D
evelo
p a g
ame-p
layin
g p
rog
ram th
at inclu
des m
ultip
le fun
ction
s
�rand
and
RAND_MAX
in th
e <cstdlib>
head
er file
i= rand();
gen
erates an u
nsig
ned
integ
er betw
een 0
and
RAND_MAX
�V
isual S
tud
io: RAND_MAX
= 3
27
67
= 2
15-1
�G
NU
C+
+: RAND_MAX
= 2
14
748
364
7 =
23
1-1
�ev
ery n
um
ber b
etween
0 an
d RAND_MAX
has an
equ
al cha
nce (o
r
pro
bab
ility) o
f bein
g ch
osen
each tim
e rand
is called
�S
caling
1 + rand() % 6
�6
: scaling
factor, sh
ift the ran
ge to
1-6
from
0-5
by ad
din
g 1
�F
un
ction
rand
actually
gen
erates pseu
do
rand
om
nu
mb
ers
�R
epeatab
ility is an
imp
ortan
t characteristic o
f rand
�R
epeated
ly callin
g rand
pro
du
ces a sequ
ence o
f nu
mb
ers that ap
pears to
be ran
do
m, b
ut th
e sequ
ence rep
eats itself each tim
e the p
rog
ram ex
ecutes.
27
5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
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5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
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�T
o sh
ow
that th
e nu
mb
ers pro
duced
by rand
occu
r with
app
rox
imately
equ
al likelih
oo
d, F
ig.5
.8 sim
ulates 6
,00
0,0
00 ro
lls of a d
ie
29
5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
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5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
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5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
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�srand-
in h
eader file <
cstdlib>
, it takes an
unsigned
integ
er
argu
men
t and
seeds
the r
and
fun
ction
to p
rod
uce a d
ifferent seq
uen
ce of
rand
om
nu
mb
ers for each
execu
tion
.
�unsignedint:
�tw
o-b
ytes: 0
–6
553
5 (2
16-1
) fou
r-bytes: 0
–4
294
967
295
(23
2-1)
32
5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
33
5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�T
o ran
do
mize w
itho
ut h
avin
g to
enter a seed
each tim
esrand( time( 0
) );
�time
in <ctime>
�ty
pically
return
s the cu
rrent tim
e as the n
um
ber o
f seconds sin
ce January
1,
1970, at m
idnig
ht G
reenw
ich M
ean T
ime (G
MT
)
�R
ollin
g a six
-sided
die w
ith th
e statemen
tface = 1+ rand() % 6;
�g
eneralize th
is result as
nu
mb
er=
shiftin
gV
alu
e+rand() %
scalin
gF
acto
r;
34
5.8
Ca
se
Stu
dy
: Ra
nd
om
Nu
mb
er G
en
era
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�C
raps:
On
e of th
e mo
st po
pu
lar gam
es of ch
ance p
layed
in casin
os an
d
back
alleys w
orld
wid
e.
�A
play
er rolls tw
o d
ice. Each
die h
as six faces. T
hese faces co
ntain
1, 2
, 3, 4
, 5
and 6
spots. A
fter the d
ice hav
e com
e to rest, th
e sum
of th
e spots o
n th
e two
upw
ard faces is calcu
lated. If th
e sum
is 7 o
r 11 o
n th
e first roll, th
e play
er win
s. If th
e sum
is 2, 3
or 1
2 o
n th
e first roll (called
“craps”), th
e play
er loses (i.e.,
the “h
ouse” w
ins). If th
e sum
is 4, 5
, 6, 8
, 9 o
r 10 o
n th
e first roll, th
en th
at sum
beco
mes th
e play
er’s “poin
t.” To w
in, y
ou m
ust co
ntin
ue ro
lling th
e dice u
ntil
you “m
ake y
our p
oin
t.” The p
layer lo
ses by
rollin
g a 7
befo
re mak
ing th
e poin
t.
�E
nu
mera
tion
: declares a u
ser-defin
ed ty
pe
�gameStatus
is declared
to b
e of n
ew ty
pe S
tatus
�set o
f integ
er con
stants rep
resented
by id
entifiers
�T
he v
alues o
f these en
um
eration
con
stants
start at 0, u
nless sp
ecified
oth
erwise, an
d in
cremen
t by 1
�enumMonths {
JAN
= 1, FEB, MAR, APR, MAY, JUN, JUL, AUG,
SEP, OCT, NOV, DEC };
�resu
lting in
the v
alues 1
thro
ugh
12
35
5.9
Ca
se
Stu
dy: G
am
e o
f Ch
an
ce
; Intro
du
cin
g enum
enum
enum
enum
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
36
5.9
Ca
se
Stu
dy: G
am
e o
f Ch
an
ce
; Intro
du
cin
g enum
enum
enum
enum
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
37
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
38
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
39
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
40
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�T
he attrib
utes o
f variab
les inclu
de n
ame, ty
pe, size an
d v
alue
�E
ach id
entifier in
a pro
gram
has o
ther attrib
utes, in
clud
ing
storag
e class,
scop
ean
d lin
kag
e
�C
++
pro
vid
es five sto
rage-class sp
ecifiers: auto
, register
,
extern
, mutable
and
static
�A
n id
entifier’s sto
rag
e class d
etermin
esth
e perio
d d
urin
g w
hich
tha
t
iden
tifier exists in
mem
ory
.
�A
n id
entifier’s sco
pe
is wh
ere the id
entifier can
be referen
ced in
a
pro
gram
�th
roug
hou
t a pro
gram
; oth
ers from
limited
po
rtions o
f a pro
gram
. (5.1
1)
�A
n id
entifier’s lin
ka
ge
determ
ines w
heth
er it’s kn
ow
n
�o
nly
in th
e sou
rce file wh
ere it’s declared
or
�acro
ss mu
ltiple files th
at are com
piled
, then
link
ed to
geth
er.
�A
n id
entifier’s sto
rage-class sp
ecifierh
elps d
etermin
e its storag
e class
and
link
age.
41
5.1
0S
tora
ge
Cla
ss
es
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�S
torag
e-classsp
ecifierscan
be sp
lit into
two
: auto
matic
and
static
�A
uto
ma
ticsto
rage class: k
eyw
ord
s auto
and
register
�v
ariables are created
wh
en p
rog
ram ex
ecutio
n en
ters the b
lock
in
wh
ich th
ey’re d
efined
�th
ey ex
ist wh
ile the b
lock
is active an
d th
ey’re d
estroy
ed w
hen
the p
rog
ram ex
its the b
lock
.
�S
tatic
storag
e class: key
wo
rds e
xtern
and
static
�v
ariables ex
ist from
the p
oin
t at wh
ich th
e pro
gram
beg
ins
execu
tion
and
last for th
e du
ration
of th
e pro
gram
�allo
cated w
hen
the p
rog
ram b
egin
s execu
tion
�in
itialized o
nce w
hen
its declaratio
n is en
cou
ntered
�fo
r fun
ction
s, the n
ame o
f the fu
nctio
n ex
ists wh
en th
e pro
gram
b
egin
s execu
tion
, just as fo
r all oth
er fun
ction
s
�n
ot n
ecessary u
sed th
rou
gh
ou
t the p
rog
ram ev
en th
ou
gh
they
ex
ist from
the start o
f pro
gram
execu
tion
42
5.1
0S
tora
ge
Cla
ss
es
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�A
uto
matic sto
rage class
�o
nly
local v
ariables o
f a fun
ction
can b
e�
E.g
. auto doublex;
exists o
nly
in th
e nearest en
closin
g p
air of cu
rly b
races with
in th
e b
od
y o
f the fu
nctio
n in
wh
ich th
e defin
ition ap
pears:
�L
ocal v
ariables are o
f auto
matic sto
rage class b
y d
efau
lt, so k
eyw
ord
auto
rarely
is used
.
43
5.1
0S
tora
ge
Cla
ss
es
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�register intcounter = 1;
�su
gg
eststh
at the in
teger v
ariable c
ounter
be p
laced in
on
e of th
e
com
pu
ter’s registers (in
mach
ine-lan
gu
age n
orm
ally lo
aded
into
a register)
�T
he co
mp
iler mig
ht ig
no
re register
declaratio
ns, e.g
., insu
fficient
�register
can b
e used
on
ly w
ith lo
cal variab
les and
fun
ction p
arameters.
44
5.1
0S
tora
ge
Cla
ss
es
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�S
tatic
-k
eyw
ord
s extern
and
static
declare id
entifiers fo
r variab
les
�S
tatic-storag
e-class variab
les ( and
the n
ame o
f fun
ctions) ex
ist (allo
cated) fro
m th
e po
int at w
hich
the p
rog
ram b
egin
s execu
tion
and
last fo
r the d
uratio
n o
f the p
rog
ram.
�S
uch
a variab
le is initialized
on
ce wh
en its d
eclaration
is enco
un
tered.
�T
wo
typ
es of id
entifiers
�ex
ternal id
entifiers (su
ch as g
lob
al variab
lesan
d g
lob
al fun
ction
nam
es)
�lo
cal variab
les declared
with
the sto
rage-class sp
ecifierstatic
.
�G
lob
al variab
les
�p
lacing
variab
le declaratio
ns o
utsid
e any class o
r fun
ction d
efinitio
n.
�retain
their v
alues th
rou
ghou
t the ex
ecutio
n o
f the p
rog
ram.
�G
lob
al variab
les and
glo
bal fu
nctio
ns can
be referen
ced b
y an
y fu
nctio
n
that fo
llow
s their d
eclaration
s or d
efinitio
ns in
the so
urce file
45
5.1
0S
tora
ge
Cla
ss
es
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�L
ocal v
ariables d
eclared static
�e.g.
static intcount = 1;
�k
no
wn
on
ly in
the fu
nctio
n in
wh
ich th
ey’re d
eclared,
�b
ut retain
their v
alues w
hen
the fu
nctio
n retu
rns to
its caller.
�T
he n
ext tim
e the fu
nctio
n is called
, the s
tatic
local v
ariables
con
tain th
e valu
es they
had
wh
en th
e fun
ction last co
mp
leted ex
ecutio
n.
�A
ll nu
meric v
ariables o
f the static sto
rage class are in
itialized
to zero
if th
ey’re n
ot ex
plicitly
initialized
by y
ou
46
5.1
0S
tora
ge
Cla
ss
es
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�S
cop
eo
f an id
entifier: p
ortio
n o
f the p
rog
ram w
here it ca
n b
e used
�E
.g., d
eclare a local v
ariable in
a blo
ck: it can
be referen
ced o
nly
in
that b
lock
and
in b
lock
s nested
with
in th
at blo
ck.
�T
his sectio
n—
functio
nsco
pe, g
lob
al nam
espace sco
pe, lo
cal scop
ean
d
fun
ction-p
roto
typ
e scop
e
�tw
o o
ther sco
pes—
class scop
e(C
h.9
) and
nam
espace sco
pe
(Ch
.2
4)
�g
lob
al n
am
espa
ce scop
e
�A
n id
entifier d
eclared o
utsid
e any fu
nctio
n o
r class. it is “kn
ow
n” in
all
fun
ctions fro
m th
e po
int at w
hich
it’s declared
un
til the en
d o
f the file
�G
lob
al variab
les, fun
ction d
efinitio
ns an
d fu
nctio
n p
roto
typ
es placed
ou
tside a fu
nctio
n
�L
abels
(iden
tifiers follo
wed
by a co
lon
such
as start:
) are the o
nly
iden
tifiers with
fun
ction
scop
e.
�can
be u
sed an
yw
here in
the fu
nctio
n in
which
they
appear, b
ut n
ot b
e outsid
e
�used
in goto
statemen
ts (Appen
dix
F).
�im
plem
entatio
n d
etails that fu
nctio
ns h
ide fro
m o
ne an
oth
er.
47
5.1
1S
co
pe
Ru
les
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�L
oca
lsco
pe -
iden
tifiers declared
insid
e a blo
ck
�b
egin
s at the id
entifier’s d
eclaration
and
end
s at the term
inatin
g rig
ht
brace (}
) of th
e blo
ck in
wh
ich th
e iden
tifier is declared
�L
ocal v
ariables (in
cludin
g static)
, fun
ction p
arameters
�A
ny b
lock
can co
ntain
variab
le declaratio
ns.
�W
hen
blo
cks are n
ested an
d an
iden
tifier in an
ou
ter blo
ck h
as the sam
e n
ame as an
iden
tifier in an
inn
er blo
ck, th
e iden
tifier in th
e ou
ter blo
ck is
“hid
den
” un
til the in
ner b
lock
termin
ates.
48
5.1
1S
co
pe
Ru
les
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�F
un
ction
pro
toty
pe sco
pe: o
nly
iden
tifiers in th
e param
eter list of a fu
nctio
n
pro
toty
pe (n
ames n
ot req
uired
, only
types req
uired
)
�N
ames in
the list are ig
nored
by
the co
mpiler
�Id
entifiers u
sed in
a functio
n p
roto
type can
be reu
sed elsew
here in
the p
rogram
w
ithout am
big
uity
�In
a single p
roto
type, a p
articular id
entifier can
be u
sed o
nly
once
49
5.1
1S
co
pe
Ru
les
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
50
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
51
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�S
tack
: a data stru
cture an
alog
ou
s to a p
ile of d
ishes
�p
ush
ing th
e dish
on
to th
e stack, an
d p
op
pin
g th
e dish
off th
e stack (b
oth
from
the to
p)
�last-in
, first-out (L
IFO
) data stru
ctures
�F
un
ction
call stack(p
rogram
execu
tion
stack)
�w
ork
ing “b
ehin
d th
e scenes” -
sup
po
rts the fu
nctio
n call/retu
rn m
echan
ism
�su
pp
orts th
e creation
, main
tenan
ce and
destru
ction o
f each called
fun
ction’s au
tom
atic variab
les
�as each
fun
ction (fx
) is called
, it may
, in tu
rn, call o
ther fx
(s)
�k
eep tra
ck o
f the retu
rn a
dd
resses to retu
rn co
ntro
l to th
e fxcallin
g it
�E
ach tim
e a fxcalls an
oth
er fx, a
n en
try is p
ush
ed o
nto
the sta
ck
�T
his en
try, called
a stack
fram
eo
r an activ
ation
record
, con
tains th
e
return
add
ress and
som
e add
ition
al info
rmatio
n
52
5.1
2F
un
ctio
n C
all S
tac
k a
nd
Ac
tiva
tion
Re
co
rds
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�If th
e called fu
nctio
n retu
rns, in
stead o
f calling
ano
ther fu
nctio
n b
efore
return
ing, th
e stack
fram
e for th
e fun
ction
call is p
op
ped
, and
con
trol
transfers to
the retu
rn ad
dress in
the p
op
ped
stack fram
e.
�called
fun
ction
find
s the in
form
ation
to retu
rn at th
eto
po
f the stack
�If o
ne fu
nctio
n m
akes a call to
ano
ther, a n
ew stack
frame is p
ush
ed
�T
hu
s, the retu
rn ad
dress req
uired
by th
e new
ly called
fun
ction to
retu
rn to
its caller is no
w lo
cated at th
e top
of th
e stack.
�A
no
ther im
po
rtant resp
on
sibility
of stack
frame
�A
uto
ma
tic varia
bles, p
arameters an
d lo
cal variab
les the fu
nctio
n d
eclares
�n
eed to
exist w
hile a fu
nctio
n is ex
ecutin
g
�n
eed to
remain
active if th
e fun
ction m
akes calls to
oth
er fun
ctions
�W
hen
called fu
nctio
n retu
rns—
and
no
lon
ger n
eeds its lo
cal auto
matic
variab
les—its stack
frame is p
op
ped
from
the stack
, and
tho
se local
auto
matic v
ariables are n
o lo
ng
er kn
ow
n to
the p
rog
ram
�O
nly
a certain am
ou
nt o
f mem
ory
can b
e used
to sto
re activatio
n reco
rds o
n
the fu
nctio
n call stack
du
e to fin
ite com
pu
ter mem
ory
�stack
ov
erflow
-if m
ore fu
nctio
n calls o
ccur th
an can
stored
53
5.1
2F
un
ctio
n C
all S
tac
k a
nd
Ac
tiva
tion
Re
co
rds
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
54
5.1
2F
un
ctio
n C
all S
tac
k a
nd
Ac
tiva
tion
Re
co
rds
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�activ
ation
record
�1. OS
calls main: retu
rn ad
dress R
1 an
d au
to-v
ar. a�
2. main
calls square: retu
rn ad
dressR2 an
d au
to-v
ar. x
55
5.1
2F
un
ctio
n C
all S
tac
k a
nd
Ac
tiva
tion
Re
co
rds
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
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rog
ram
min
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TU
BA
56
5.1
2F
un
ctio
n C
all S
tac
k a
nd
Ac
tiva
tion
Re
co
rds
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�square retu
rns to
main: stack
is po
pp
ed, g
ives R2
and
loo
sing
x�
main return
s to O
S: stack is p
op
ped
again
, giv
es R1an
d lo
osin
g a
57
5.1
2F
un
ctio
n C
all S
tac
k a
nd
Ac
tiva
tion
Re
co
rds
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�A
n em
pty
param
eter list: writin
g eith
er void
or n
oth
ing at all in
paren
theses
�T
he p
roto
typ
e
voidprint();
specifies th
at fun
ction
do
es no
t take arg
um
ents an
d d
oes n
ot retu
rn
a valu
e
58
5.1
3F
un
ctio
ns
with
Em
pty
Pa
ram
ete
r Lis
ts
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
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rog
ram
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BA
59
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
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+ P
rog
ram
min
gN
TU
BA
�C
++
pro
vid
es inlin
e fun
ctions
to h
elp red
uce th
e execu
tion-tim
e ov
erhead
of fu
nctio
n calls—
especially
for sm
allfu
nctio
ns.
�q
ualifier i
nline
befo
re a fun
ction’s retu
rn ty
pe in
the fu
nctio
n d
efinitio
n:
“adv
ises” the co
mp
iler to g
enerate a co
py o
f the fu
nctio
n’s co
de in
place
(wh
en ap
pro
priate) to
avo
id a fu
nctio
n call.
�T
he co
mp
iler can ig
no
re the i
nline
qu
alifier and
typ
ically d
oes so
for all
bu
t the sm
allest fun
ction
s
�trad
e-off b
etween
mem
ory
and
time
60
5.1
4In
line
Fu
nc
tion
s
YP
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NT
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& E
EIn
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n to
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rog
ram
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BA
�K
eyw
ord
const
in fu
nctio
n cube
’s param
eter list (line 9
) tells the
com
piler th
at the fu
nctio
n d
oes n
ot m
od
ify v
ariable s
ide
, ensu
ring
side
’s valu
e is no
t chan
ged
by th
e fun
ction d
urin
g th
e calculatio
n.
�N
otice th
at the co
mp
lete defin
ition
of fu
nctio
n cube
app
ears befo
re it’s u
sed in
the p
rog
ram.
�T
his is req
uired
so th
at the co
mp
iler kn
ow
s ho
w to
exp
and
a cube
fun
ction
call into
its inlin
edco
de.
�F
or th
is reason, reu
sable in
line fu
nctio
ns are ty
pically
placed
in h
eader files,
so th
at their d
efinitio
ns can
be in
clud
ed in
each so
urce file th
at uses th
em.
61
5.1
4In
line
Fu
nc
tion
s
YP
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NT
U G
IPO
& E
EIn
trod
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rog
ram
min
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BA
62
5.1
4In
line
Fu
nc
tion
s
YP
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NT
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& E
EIn
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rog
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BA
Tw
o w
ays to
pass arg
um
ents to
fun
ction
s
�p
ass-by
-va
lue
�a co
py
of th
e argu
men
t’s valu
e is mad
e and
passed
(on
the
fun
ction
call stack) to
the called
fun
ction
�ch
ang
es to th
e cop
y d
o n
ot affect th
e orig
inal
�p
reven
ts the accid
ental sid
e effects
�p
ass-by
-reference
�th
e caller giv
es the called
fun
ction
the ab
ility to
access th
e
caller’s d
ata
directly, an
d to
mo
dify
that d
ata
�referen
ce param
eter: an alias fo
r its corresp
on
din
g arg
um
ent in
a
fun
ction
call.
�fu
nctio
n p
roto
typ
e by
an am
persan
d (&
)
�th
e orig
inal v
ariable can
be m
od
ified b
y th
e called fu
nctio
n
�to
specify
a reference to
a con
stant, p
lace the c
onst
qu
alifier
�F
un
ction
s can retu
rn referen
ces, bu
t this can
be d
ang
erous.
63
5.1
5R
efe
ren
ce
s a
nd
Re
fere
nc
e P
ara
me
ters
YP
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NT
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IPO
& E
EIn
trod
uctio
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rog
ram
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BA
64
5.1
5R
efe
ren
ce
s a
nd
Re
fere
nc
e P
ara
me
ters
YP
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NT
U G
IPO
& E
EIn
trod
uctio
n to
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rog
ram
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BA
65
5.1
5R
efe
ren
ce
s a
nd
Re
fere
nc
e P
ara
me
ters
YP
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NT
U G
IPO
& E
EIn
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uctio
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rog
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66
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
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rog
ram
min
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BA
�R
eferences can
also b
e used
as alia
sesfo
r oth
er variab
les
with
in a fu
nctio
n // u
se the sam
e mem
ory
67
5.1
5R
efe
ren
ce
s a
nd
Re
fere
nc
e P
ara
me
ters
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
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rog
ram
min
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TU
BA
68
5.1
5R
efe
ren
ce
s a
nd
Re
fere
nc
e P
ara
me
ters
YP
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NT
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& E
EIn
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uctio
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rog
ram
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TU
BA
�U
sed in
a fu
nctio
n rep
eated
ly w
ith th
e sam
e arg
um
ent v
alu
e for
a p
articu
lar p
ara
meter
�O
mits an
argu
men
t for a p
arameter w
ith a d
efault arg
um
ent in
a fu
nctio
n call, th
e com
piler rew
rites the fu
nctio
n call an
d in
serts the
defau
lt valu
e of th
at argu
men
t.
�D
efault arg
um
ents m
ust b
e the rig
htm
ost (trailin
g) arg
um
ents in
a fu
nctio
n’s p
arameter list.
�D
efault arg
um
ents m
ust b
e specified
with
the first o
ccurren
ce of th
e
fun
ction
nam
e—ty
pically
, in th
e fun
ction
pro
toty
pe.
�D
efault v
alues can
be an
y ex
pressio
n, in
clud
ing
con
stants, g
lob
al v
ariables o
r fun
ction
calls.
�D
efault arg
um
ents also
can b
e used
with
inline
fun
ction
s.
69
5.1
6D
efa
ult A
rgu
me
nts
YP
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NT
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IPO
& E
EIn
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70
YP
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NT
U G
IPO
& E
EIn
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uctio
n to
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rog
ram
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TU
BA
71
5.1
6D
efa
ult A
rgu
me
nts
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
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+ P
rog
ram
min
gN
TU
BA
�u
nary
scop
e resolu
tion
op
erator (:
:)
: to access a g
lob
al
variab
le wh
en a lo
cal variab
le of th
e same n
ame is in
scop
e
72
5.1
7U
na
ry S
co
pe
Re
so
lutio
n O
pe
rato
r
YP
C -
NT
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& E
EIn
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uctio
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rog
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73
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
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+ P
rog
ram
min
gN
TU
BA
�F
un
ction
s of th
e sam
e na
me w
ith d
ifferent sig
na
tures
�U
sed to
create several fu
nctio
ns o
f the sam
e nam
e that p
erform
similar
tasks, b
ut o
n d
ifferent d
ata typ
es
�T
he co
mp
iler uses o
nly
the p
arameter lists to
distin
gu
ish b
etween
ov
erload
ed fu
nctio
ns.
�F
un
ction
main
is no
t man
gled
, becau
se it cann
ot b
e ov
erload
ed.
�U
se cautio
n w
hen
ov
erload
ing
fun
ction
s with
defau
lt param
eters,
becau
se this m
ay cau
se amb
igu
ity,
�e.g
. a fxw
ith 3
par. &
a fx. w
ith 3
par +
1 d
efault p
ar. (com
pilatio
n erro
r)
74
5.1
8F
un
ctio
n O
ve
rloa
din
g
YP
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75
YP
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NT
U G
IPO
& E
EIn
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rog
ram
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gN
TU
BA
�F
un
ction
temp
lates:id
entica
lp
rog
ram lo
gic an
d o
peratio
ns fo
r
�N
ote: O
verlo
aded
fun
ctions p
erform
similar o
peratio
ns th
at inv
olv
e
differen
t pro
gram
log
ic on
differen
t data ty
pes.
�D
efines a w
ho
le family
of o
verlo
aded
fun
ction
s
�G
iven
the arg
um
ent ty
pes =
> au
tom
atically g
enerates sep
arate fun
ction
temp
late specializatio
ns
to h
and
le each ty
pe o
f call app
rop
riately
76
5.1
9F
un
ctio
n T
em
pla
tes
YP
C -
NT
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IPO
& E
EIn
trod
uctio
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77
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
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rog
ram
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BA
78
5.1
9F
un
ctio
n T
em
pla
tes
�E
very
param
eter in th
e temp
late param
eter list (often
referred to
as a form
al typ
e param
eter) is preced
ed b
y k
eyw
ord
typename
or k
eyw
ord
class
(wh
ich are sy
no
ny
ms in
this co
ntex
t).
�T
he fo
rmal ty
pe p
arameters are p
laceho
lders fo
r fun
dam
ental
typ
es or u
ser-defin
ed ty
pes.
�T
he n
ame o
f a typ
e param
eter mu
st be u
niq
ue in
the tem
plate
param
eter list for a p
articular tem
plate d
efinitio
n.
�M
ore th
an o
ne class, e.g
. template <class T1, class T2>
YP
C -
NT
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IPO
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EIn
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�R
ecursiv
e fun
ction: fu
nctio
n callin
g itself, eith
er directly
, or in
directly
(thro
ug
h an
oth
er fun
ction)
�B
ase case:fu
nctio
n so
lvin
g o
nly
the sim
plest case, sim
ply
return
s a result
�m
ore co
mp
lex: d
ivid
ing th
e pro
blem
into
two
con
ceptu
al pieces
�A
: a piece th
at the fu
nctio
n k
no
ws h
ow
to d
o an
d
�B
: a piece th
at it do
es no
t kn
ow
ho
w to
do
�T
o m
ake recu
rsion feasib
le, B m
ust resem
ble th
e orig
inal p
rob
lem, b
ut
be a slig
htly
simp
ler or sm
aller versio
n.
�T
his n
ew p
rob
lem lo
ok
s like th
e orig
inal, so
the fu
nctio
n calls a co
py o
f itself to
wo
rk o
n th
e smaller p
rob
lem—
this is referred
to as a recu
rsive call
and
is also
called th
e recursio
n step
.
�T
he recu
rsion
step o
ften in
clud
es the k
eyw
ord
return -
its result w
ill be
com
bin
ed w
ith th
e po
rtion o
f the p
rob
lem th
e fun
ction k
new
ho
w to
solv
e to
form
the resu
lt passed
back
to th
e orig
inal caller, p
ossib
ly main
�E
ach tim
e the fu
nctio
n calls itself w
ith a slig
htly
simp
ler versio
n o
f the
orig
inal p
rob
lem in
ord
er to ev
entu
ally co
nv
erge o
n th
e base case.
�A
t that p
oin
t, the fu
nctio
n reco
gn
izes the b
ase case and
return
s a result to
the
prev
ious co
py o
f the fu
nctio
n, an
d a seq
uen
ce of retu
rns en
sues u
p th
e line
un
til the o
rigin
al call even
tually
return
s the fin
al result to
the o
rigin
al caller
79
5.2
0R
ec
urs
ion
YP
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NT
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IPO
& E
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�Ex. factorial
80
5.2
0R
ec
urs
ion
YP
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NT
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& E
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81
5.2
0R
ec
urs
ion
YP
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NT
U G
IPO
& E
EIn
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uctio
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rog
ram
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�unsignedlongint:
>=
4 B
ytes: 0
to >
= (2
32-1
) = 4
29
49
67
29
5
�int:
4 B
ytes =
> -
23
1to
(2
31-1
) = 2
14
74
83
647
82
5.2
0R
ec
urs
ion
YP
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NT
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IPO
& E
EIn
trod
uctio
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ram
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�F
ibo
nacci n
um
ber is th
e sum
of th
e prev
iou
s two
Fib
on
acci nu
mb
ers.
�fibonacci(0) = 0
fibonacci(1) = 1
fibonacci(n) = fibonacci(n–1) + fibonacci(n–2)
�T
he F
ibo
nacci series
0, 1, 1, 2, 3, 5, 8, 13, 21, …
�A
series occu
rs in n
ature an
d, in
particu
lar, describ
es a form
of sp
iral.
�T
he ratio
of su
ccessive F
ibo
nacci n
um
bers co
nv
erges o
n a co
nstan
t v
alue o
f 1.6
18
….
�T
his n
um
ber, to
o, freq
uen
tly o
ccurs in
natu
re and
has b
een called
the
go
lden
ratioo
r the g
old
en m
ean.
�H
um
ans ten
d to
find
the g
old
en m
ean aesth
etically p
leasing
.
�A
rchitects o
ften d
esign
win
do
ws, ro
om
s and
bu
ildin
gs w
ho
se leng
th
and
wid
th are in
the ratio
of th
e go
lden
mean
.
�P
ostcard
s are often
desig
ned
with
a go
lden
mean
leng
th/w
idth
ratio.
83
5.2
1E
xa
mp
le U
sin
g R
ec
urs
ion
: Fib
on
ac
ci S
erie
s
YP
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NT
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& E
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84
YP
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NT
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85
5.2
1E
xa
mp
le U
sin
g R
ec
urs
ion
: Fib
on
ac
ci S
erie
s
YP
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NT
U G
IPO
& E
EIn
trod
uctio
n to
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rog
ram
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�C
++
do
es no
t specify
the o
rder in
wh
ich th
e op
erand
s of m
ost
op
erators (in
clud
ing
+) are to
be ev
aluated
.
�D
on
’t assum
e the o
rder in
wh
ich th
ese calls execu
te (It’s OK
in
Fig
.5.2
9.)
�fibonacci(n) = fibonacci(n–1) + fibonacci(n–2)
�In
som
e pro
gram
s the ev
aluatio
n o
f an o
peran
d can
hav
e side
effects(ch
ang
es to d
ata valu
es) that co
uld
affect the fin
al result o
f
the ex
pressio
n
�C
++
specifies th
e ord
er of ev
aluatio
n o
f the o
peran
ds o
f on
ly fo
ur
op
erators—
&&
, ||
, com
ma (,
) and
?:
.
86
5.2
1E
xa
mp
le U
sin
g R
ec
urs
ion
: Fib
on
ac
ci S
erie
s
YP
C -
NT
U G
IPO
& E
EIn
trod
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n to
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�E
ach lev
el of recu
rsion
in fu
nctio
n fibonacci
has a d
ou
blin
g
effect on
the n
um
ber o
f fun
ction
calls; i.e., the n
um
ber o
f recursiv
e
calls that are req
uired
to calcu
late the n
th F
ibo
nacci n
um
ber is o
n
the o
rder o
f 2n.
�C
alculatin
g th
e 20
th F
ibo
nacci n
um
ber: 2
20≈
a millio
n calls
the 3
0th
Fib
on
acci nu
mb
er: 23
0≈a b
illion
calls
�C
om
pu
ter scientists refer to
this as ex
po
nen
tial com
plex
ity.
87
5.2
1E
xa
mp
le U
sin
g R
ec
urs
ion
: Fib
on
ac
ci S
erie
s
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�B
oth
are based
on
a con
trol statem
ent:
iteration
: repetitio
nstru
cture recu
rsion
: a selection
structu
re
�B
oth
inv
olv
e repetitio
n:
exp
licitly a rep
etition
structu
re thro
ug
h rep
eated fu
nctio
n calls
�B
oth
inv
olv
e a termin
ation
test:
loo
p-co
ntin
uatio
n co
nd
ition
fails a base case is reco
gn
ized
�B
oth
grad
ually
app
roach
termin
ation
:
mo
difies a co
un
ter p
rod
uces sim
pler v
ersion
s
�B
oth
can o
ccur in
finitely
:
the lo
op
-con
tinu
ation
test nev
er beco
mes false
the recu
rsion
step d
oes n
ot co
nv
erges o
n th
e base case
88
5.2
2R
ec
urs
ion
vs
. Itera
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
89
5.2
2R
ec
urs
ion
vs
. Itera
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
90
5.2
2R
ec
urs
ion
vs
. Itera
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
�N
egativ
eso
f Recu
rsion
�rep
eatedly
inv
ok
es the m
echan
ism, an
d co
nseq
uen
tly th
e o
verh
ead, o
f fun
ction
calls.
�ex
pen
sive in
bo
th p
rocesso
r time an
d m
emo
ry sp
ace.
�E
ach recu
rsive call cau
ses ano
ther co
py
of th
e fun
ction
(actually
o
nly
the fu
nctio
n’s v
ariables) to
be created
; this can
con
sum
e co
nsid
erable m
emo
ry.
�Iteratio
n n
orm
ally o
ccurs w
ithin
a fun
ction
, so th
e ov
erhead
of
repeated
fun
ction
calls and
extra m
emo
ry assig
nm
ent is o
mitted
.
91
5.2
2R
ec
urs
ion
vs
. Itera
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
92
5.2
2R
ec
urs
ion
vs
. Itera
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
93
5.2
2R
ec
urs
ion
vs
. Itera
tion
YP
C -
NT
U G
IPO
& E
EIn
trod
uctio
n to
C+
+ P
rog
ram
min
gN
TU
BA
94
5.2
2R
ec
urs
ion
vs
. Itera
tion
