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Chapter17: Arithmetic
435
17 Arithmetic
Unlessotherwisenoted,allofthefunctionsdescribedinthischapterwillworkforrealand
complex scalar,vector, , or r matrix arguments. . Functions s describedas s mappingfunctions
applythegivenoperationindividuallytoeachelementwhengivenamatrixargument.For
example:
sin ([1, 2; 3, 4])
)
0.84147
0.90930
0.14112 -0.75680
17.1 ExponentsandLogarithms
[MappingFunction]
exp
(
x
)
Computee
x
foreachelementofx.
Tocomputethematrixexponential,seeChapter18[LinearAlgebra],page465.
Seealso: [log],page435.
[MappingFunction]
expm1
(
x
)
Computee
x
1accuratelyintheneighborhoodofzero.
Seealso: [exp],page435.
[MappingFunction]
log
(
x
)
Computethenaturallogarithm,ln(x);foreachelementofx.
Tocomputethematrixlogarithm,seeChapter18[LinearAlgebra],page465.
See also: [exp], page 435[log1p], page 435[log2], page 435[log10], page 435,
[logspace],page420.
[FunctionFile]
reallog
(
x
)
Returnthereal-valuednaturallogarithmofeachelementofx.
Ifanyelementresultsinacomplexreturnvaluereallogabortsandissuesanerror.
Seealso: [log],page435,[realpow],page436,[realsqrt],page436.
[MappingFunction]
log1p
(
x
)
Computeln(1+x)accuratelyintheneighborhoodofzero.
Seealso: [log],page435,[exp],page435,[expm1],page435.
[MappingFunction]
log10
(
x
)
Computethebase-10logarithmofeachelementofx.
Seealso: [log],page435,[log2],page435,[logspace],page420,[exp],page435.
[MappingFunction]
log2
(
x
)
[MappingFunction]
[f, e] ] = = log2
(
x
)
Computethebase-2logarithmofeachelementofx.
Ifcalledwithtwooutputarguments,split x x intobinary y mantissaandexponent so
that
1
2
jfj<1andeisaninteger.Ifx=0,f=e=0.
Seealso: [pow2],page436,[log],page435,[log10],page435,[exp],page435.
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436
GNUOctave
[FunctionFile]
pow2
(
x
)
[FunctionFile]
pow2
(
f
,
e
)
Withoneinputargument,compute2
x
foreachelementofx.
Withtwoinputarguments,returnf2
e
.
Seealso: [log2],page435,[nextpow2],page436,[power],page146.
[FunctionFile]
nextpow2
(
x
)
Computetheexponentforthesmallestpoweroftwolargerthantheinput.
Foreachelementintheinputarrayx,returntheﬁrstintegernsuchthat2
n
jxj.
Seealso: [pow2],page436,[log2],page435.
[FunctionFile]
realpow
(
x
,
y
)
Computethereal-valued,element-by-elementpoweroperator.
Thisisequivalenttox.^y,exceptthatrealpowreportsanerrorifanyreturnvalue
iscomplex.
Seealso: [power],page146,[reallog],page435,[realsqrt],page436.
[MappingFunction]
sqrt
(
x
)
Computethesquarerootofeachelementofx.
Ifxisnegative,acomplexresultisreturned.
Tocomputethematrixsquareroot,seeChapter18[LinearAlgebra],page465.
Seealso: [realsqrt],page436,[nthroot],page436.
[FunctionFile]
realsqrt
(
x
)
Returnthereal-valuedsquarerootofeachelementofx.
Ifanyelementresultsinacomplexreturnvaluerealsqrtabortsandissuesanerror.
Seealso: [sqrt],page436,[realpow],page436,[reallog],page435.
[MappingFunction]
cbrt
(
x
)
Computetherealcuberootofeachelementofx.
Unlikex^(1/3),theresultwillbenegativeifxisnegative.
Seealso: [nthroot],page436.
[FunctionFile]
nthroot
(
x
,
n
)
Computethereal(non-complex)n-throotofx.
x musthaveallrealentriesandnmustbeascalar. . Ifnisanevenintegerandxhas
negativeentriesthennthrootabortsandissuesanerror.
Example:
nthroot (-1, 3)
) -1
(-1) ^ (1 / 3)
) 0.50000 0 - 0.86603i
Seealso: [realsqrt],page436,[sqrt],page436,[cbrt],page436.
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Chapter17: Arithmetic
437
17.2 ComplexArithmetic
Inthedescriptionsofthefollowingfunctions,z isthecomplexnumber x+iy,whereiis
deﬁnedas
p
1.
[MappingFunction]
abs
(
z
)
Computethemagnitudeofz.
Themagnitudeisdeﬁnedasjzj=
p
x
2
+y
2
.
Forexample:
abs (3 + + 4i)
)
5
Seealso: [arg],page437.
[MappingFunction]
arg
(
z
)
[MappingFunction]
angle
(
z
)
Computetheargument,i.e.,angleofz.
Forexample:
arg (3 + + 4i)
) 0.92730
Seealso: [abs],page437.
[MappingFunction]
conj
(
z
)
Returnthecomplexconjugateofz.
Thecomplexconjugateisdeﬁnedas¯z=x iy.
Seealso: [real],page438,[imag],page437.
[FunctionFile]
cplxpair
(
z
)
[FunctionFile]
cplxpair
(
z
,
tol
)
[FunctionFile]
cplxpair
(
z
,
tol
,
dim
)
Sortthenumberszintocomplexconjugatepairsorderedbyincreasingrealpart.
Thenegativeimaginarycomplexnumbersareplacedﬁrstwithineachpair. Allreal
numbers(thosewithabs(imag(z)/z)<tol)areplacedafterthecomplexpairs.
Iftolisunspeciﬁedthedefaultvalueis100*eps.
Bydefaultthecomplexpairsaresortedalongtheﬁrstnon-singletondimensionofz.
Ifdimisspeciﬁed,thenthecomplexpairsaresortedalongthisdimension.
Signalanerror ifsomecomplexnumberscouldnotbepaired. . Signalanerrorifall
complex numbers are not exact conjugates (to within n tol). . Note e that there is s no
deﬁnedorderforpairswithidenticalrealpartsbutdiﬀeringimaginaryparts.
cplxpair (exp(2i*pi*[0:4]’/5)) == exp(2i*pi*[3; 2; ; 4; 1; 0]/5)
[MappingFunction]
imag
(
z
)
Returntheimaginarypartofz asarealnumber.
Seealso: [real],page438,[conj],page437.
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438
GNUOctave
[MappingFunction]
real
(
z
)
Returntherealpartofz.
Seealso: [imag],page437,[conj],page437.
17.3 Trigonometry
sineof30degrees). Asanalternative,Octaveprovidesanumberoftrigonometricfunctions
whichworkdirectlyonanargumentspeciﬁedindegrees.Thesefunctionsarenamedafter
the basetrigonometricfunction witha ‘d’suﬃx. . For r example, , sin n expects anangle in
Octave usestheClibrary trigonometricfunctions. . Itis s expectedthat these functions
aredeﬁnedby theISO/IEC 9899Standard. . ThisStandardisavailableat: http://www.
open-std.org/jtc1/sc22/wg14/www/docs/n1124.pdf. Section n F.9.1deals with the
trigonometricfunctions. Thebehaviorofmostofthefunctionsisrelativelystraightforward.
However, there are some e exceptions s to the standard d behavior. . Many y of f the exceptions
involvethebehaviorfor-0. Themostcomplexcaseis s atan2. . Octaveexactlyimplements
thebehaviorgivenintheStandard. Includingatan2(0; 0)returns.
Itshouldbenotedthatmatlabusesdiﬀerentdeﬁnitionswhichapparentlydonotdis-
tinguish-0.
[MappingFunction]
sin
(
x
)
Seealso: [asin],page439,[sind],page440,[sinh],page439.
[MappingFunction]
cos
(
x
)
Seealso: [acos],page439,[cosd],page440,[cosh],page439.
[MappingFunction]
tan
(
z
)
Seealso: [atan],page439,[tand],page441,[tanh],page439.
[MappingFunction]
sec
(
x
)
Seealso: [asec],page439,[secd],page441,[sech],page439.
[MappingFunction]
csc
(
x
)
Seealso: [acsc],page439,[cscd],page441,[csch],page439.
[MappingFunction]
cot
(
x
)
Seealso: [acot],page439,[cotd],page441,[coth],page439.
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Chapter17: Arithmetic
439
[MappingFunction]
asin
(
x
)
Seealso: [sin],page438,[asind],page441.
[MappingFunction]
acos
(
x
)
Seealso: [cos],page438,[acosd],page441.
[MappingFunction]
atan
(
x
)
Seealso: [tan],page438,[atand],page441.
[MappingFunction]
asec
(
x
)
Seealso: [sec],page438,[asecd],page441.
[MappingFunction]
acsc
(
x
)
Seealso: [csc],page438,[acscd],page441.
[MappingFunction]
acot
(
x
)
Seealso: [cot],page438,[acotd],page441.
[MappingFunction]
sinh
(
x
)
Computethehyperbolicsineforeachelementofx.
Seealso: [asinh],page440,[cosh],page439,[tanh],page439.
[MappingFunction]
cosh
(
x
)
Computethehyperboliccosineforeachelementofx.
Seealso: [acosh],page440,[sinh],page439,[tanh],page439.
[MappingFunction]
tanh
(
x
)
Computehyperbolictangentforeachelementofx.
Seealso: [atanh],page440,[sinh],page439,[cosh],page439.
[MappingFunction]
sech
(
x
)
Computethehyperbolicsecantofeachelementofx.
Seealso: [asech],page440.
[MappingFunction]
csch
(
x
)
Computethehyperboliccosecantofeachelementofx.
Seealso: [acsch],page440.
[MappingFunction]
coth
(
x
)
Computethehyperboliccotangentofeachelementofx.
Seealso: [acoth],page440.
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440
GNUOctave
[MappingFunction]
asinh
(
x
)
Computetheinversehyperbolicsineforeachelementofx.
Seealso: [sinh],page439.
[MappingFunction]
acosh
(
x
)
Computetheinversehyperboliccosineforeachelementofx.
Seealso: [cosh],page439.
[MappingFunction]
atanh
(
x
)
Computetheinversehyperbolictangentforeachelementofx.
Seealso: [tanh],page439.
[MappingFunction]
asech
(
x
)
Computetheinversehyperbolicsecantofeachelementofx.
Seealso: [sech],page439.
[MappingFunction]
acsch
(
x
)
Computetheinversehyperboliccosecantofeachelementofx.
Seealso: [csch],page439.
[MappingFunction]
acoth
(
x
)
Computetheinversehyperboliccotangentofeachelementofx.
Seealso: [coth],page439.
[MappingFunction]
atan2
(
y
,
x
)
Computeatan(y /x)forcorrespondingelementsofy andx.
y andx mustmatchinsizeandorientation.
Seealso: [tan],page438,[tand],page441,[tanh],page439,[atanh],page440.
Octaveprovidesthefollowingtrigonometricfunctionswhereanglesarespeciﬁedinde-
grees.Thesefunctionsproducetruezerosattheappropriateintervalsratherthanthesmall
cosd (90)
) 0
cos (pi/2)
) 6.1230e-17
[FunctionFile]
sind
(
x
)
Computethesineforeachelementofxindegrees.
Returnszeroforelementswherex/180isaninteger.
Seealso: [asind],page441,[sin],page438.
[FunctionFile]
cosd
(
x
)
Computethecosineforeachelementofxindegrees.
Returnszeroforelementswhere(x-90)/180isaninteger.
Seealso: [acosd],page441,[cos],page438.
Chapter17: Arithmetic
441
[FunctionFile]
tand
(
x
)
Computethetangentforeachelementofxindegrees.
Returns zero o for r elements s where e x/180 is s an n integer and Inf for r elements s where
(x-90)/180isaninteger.
Seealso: [atand],page441,[tan],page438.
[FunctionFile]
secd
(
x
)
Computethesecantforeachelementofxindegrees.
Seealso: [asecd],page441,[sec],page438.
[FunctionFile]
cscd
(
x
)
Computethecosecantforeachelementofxindegrees.
Seealso: [acscd],page441,[csc],page438.
[FunctionFile]
cotd
(
x
)
Computethecotangentforeachelementofx indegrees.
Seealso: [acotd],page441,[cot],page438.
[FunctionFile]
asind
(
x
)
Computetheinversesineindegreesforeachelementofx.
Seealso: [sind],page440,[asin],page439.
[FunctionFile]
acosd
(
x
)
Computetheinversecosineindegreesforeachelementofx.
Seealso: [cosd],page440,[acos],page439.
[FunctionFile]
atand
(
x
)
Computetheinversetangentindegreesforeachelementofx.
Seealso: [tand],page441,[atan],page439.
[FunctionFile]
atan2d
(
y
,
x
)
Computeatan2(y /x)indegreesforcorrespondingelementsfromy andx.
Seealso: [tand],page441,[atan2],page440.
[FunctionFile]
asecd
(
x
)
Computetheinversesecantindegreesforeachelementofx.
Seealso: [secd],page441,[asec],page439.
[FunctionFile]
acscd
(
x
)
Computetheinversecosecantindegreesforeachelementofx.
Seealso: [cscd],page441,[acsc],page439.
[FunctionFile]
acotd
(
x
)
Computetheinversecotangentindegreesforeachelementofx.
Seealso: [cotd],page441,[acot],page439.
442
GNUOctave
17.4 SumsandProducts
[Built-inFunction]
sum
(
x
)
[Built-inFunction]
sum
(
x
,
dim
)
[Built-inFunction]
sum
(...,
"
native
"
)
[Built-inFunction]
sum
(...,
"
double
"
)
[Built-inFunction]
sum
(...,
"
extra
"
)
Sumofelementsalongdimensiondim.
Ifdimisomitted,itdefaultstotheﬁrstnon-singletondimension.
Theoptional"type"inputdeterminestheclassofthevariableusedforcalculations.
Iftheargument"native"isgiven,thentheoperationisperformedinthesametype
astheoriginalargument,ratherthanthedefaultdoubletype.
Forexample:
sum ([true, true])
) 2
sum ([true, true], "native")
) true
Onthecontrary,if"double"isgiven,thesumisperformedindoubleprecisioneven
forsingleprecisioninputs.
Fordoubleprecisioninputs,the"extra"optionwilluseamoreaccuratealgorithm
thanstraightforwardsummation. Forsingleprecisioninputs,"extra"isthesameas
"double". Otherwise,"extra"hasnoeﬀect.
Seealso: [cumsum],page442,[sumsq],page443,[prod],page442.
[Built-inFunction]
prod
(
x
)
[Built-inFunction]
prod
(
x
,
dim
)
[Built-inFunction]
prod
(...,
"
native
"
)
[Built-inFunction]
prod
(...,
"
double
"
)
Productofelementsalongdimensiondim.
Ifdimisomitted,itdefaultstotheﬁrstnon-singletondimension.
Theoptional"type"inputdeterminestheclassofthevariableusedforcalculations.
Iftheargument"native"isgiven,thentheoperationisperformedinthesametype
astheoriginalargument,ratherthanthedefaultdoubletype.
Forexample:
prod ([true, true])
) 1
prod ([true, true], , "native")
) true
Onthecontrary,if"double"isgiven,theoperationisperformedindoubleprecision
evenforsingleprecisioninputs.
Seealso: [cumprod],page443,[sum],page442.
[Built-inFunction]
cumsum
(
x
)
[Built-inFunction]
cumsum
(
x
,
dim
)
Chapter17: Arithmetic
443
[Built-inFunction]
cumsum
(...,
"
native
"
)
[Built-inFunction]
cumsum
(...,
"
double
"
)
[Built-inFunction]
cumsum
(...,
"
extra
"
)
Cumulativesumofelementsalongdimensiondim.
Ifdimisomitted,itdefaultstotheﬁrstnon-singletondimension.
See sum for an n explanation of the optional parameters "native", , "double", and
"extra".
Seealso: [sum],page442,[cumprod],page443.
[Built-inFunction]
cumprod
(
x
)
[Built-inFunction]
cumprod
(
x
,
dim
)
Cumulativeproductofelementsalongdimensiondim.
Ifdimisomitted,itdefaultstotheﬁrstnon-singletondimension.
Seealso: [prod],page442,[cumsum],page442.
[Built-inFunction]
sumsq
(
x
)
[Built-inFunction]
sumsq
(
x
,
dim
)
Sumofsquaresofelementsalongdimensiondim.
Ifdimisomitted,itdefaultstotheﬁrstnon-singletondimension.
Thisfunctionisconceptuallyequivalenttocomputing
sum (x .* conj (x), , dim)
butituseslessmemoryandavoidscallingconjifxisreal.
Seealso: [sum],page442,[prod],page442.
17.5 UtilityFunctions
[MappingFunction]
ceil
(
x
)
Returnthesmallestintegernotlessthanx.
Thisisequivalenttoroundingtowardspositiveinﬁnity.
Ifxiscomplex,returnceil(real(x))+ceil(imag(x))*I.
ceil ([-2.7, 2.7])
)
-2
3
Seealso: [ﬂoor],page444,[round],page444,[ﬁx],page443.
[MappingFunction]
fix
(
x
)
Truncatefractionalportionofx andreturntheintegerportion.
Thisisequivalenttoroundingtowardszero. Ifxiscomplex,returnfix(real(x))
+fix(imag(x))*I.
fix ([-2.7, 2.7])
)
-2
2
Seealso: [ceil],page443,[ﬂoor],page444,[round],page444.