Chapter17: Arithmetic
455
Ellipticintegralsofthefirstkindaredefinedas
K(m)=
Z
1
0
dt
p
(1 t
2
)(1 mt
2
)
Ellipticintegralsofthesecondkindaredefinedas
E(m)=
Z
1
0
p
1 mt
2
p
1 t
2
dt
Reference: Milton n Abramowitz z and Irene A. . Stegun, , Handbook k of f Mathematical
Functions,Chapter17,Dover,1965.
Seealso: [ellipj],page454.
[MappingFunction]
erf
(
z
)
Computetheerrorfunction.
Theerrorfunctionisdefinedas
erf(z)=
2
p
Z
z
0
e
t
2
dt
See also: [erfc], page 455[erfcx], page 455[erfi], page 455[dawson], page e 454,
[erfinv],page456,[erfcinv],page456.
[MappingFunction]
erfc
(
z
)
Computethecomplementaryerrorfunction.
Thecomplementaryerrorfunctionisdefinedas1 erf(z).
See also: [erfcinv],page 456,[erfcx],page455,[erfi],page 455[dawson],page454,
[erf],page455,[erfinv],page456.
[MappingFunction]
erfcx
(
z
)
Computethescaledcomplementaryerrorfunction.
Thescaledcomplementaryerrorfunctionisdefinedas
e
z
2
erfc(z)e
z
2
(1 erf(z))
Seealso:[erfc],page455,[erf],page455,[erfi],page455,[dawson],page454,[erfinv],
page456,[erfcinv],page456.
[MappingFunction]
erfi
(
z
)
Computetheimaginaryerrorfunction.
Theimaginaryerrorfunctionisdefinedas
ierf(iz)
Seealso:[erfc],page455,[erf],page455,[erfcx],page455,[dawson],page454,[erfinv],
page456,[erfcinv],page456.
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456
GNUOctave
[MappingFunction]
erfinv
(
x
)
Computetheinverseerrorfunction.
Theinverseerrorfunctionisdefinedsuchthat
erf (y) == x
Seealso:[erf],page455,[erfc],page455,[erfcx],page455,[erfi],page455,[dawson],
page454,[erfcinv],page456.
[MappingFunction]
erfcinv
(
x
)
Computetheinversecomplementaryerrorfunction.
Theinversecomplementaryerrorfunctionisdefinedsuchthat
erfc (y) ) == = x
Seealso:[erfc],page455,[erf],page455,[erfcx],page455,[erfi],page455,[dawson],
page454,[erfinv],page456.
[FunctionFile]
expint
(
x
)
Computetheexponentialintegral:
E
1
(x)=
Z
1
x
e
t
t
dt
Note:Forcompatibility,thisfunctionsusesthematlabdefinitionoftheexponential
integral. MostothersourcesrefertothisparticularvalueasE
1
(x),andtheexponen-
tialintegralas
Ei(x)= 
Z
1
x
e
t
t
dt:
Thetwodefinitionsarerelated,forpositiverealvaluesofx,byE
1
( x)= Ei(x) i:
[MappingFunction]
gamma
(
z
)
ComputetheGammafunction.
TheGammafunctionisdefinedas
Γ(z)=
Z
1
0
t
z 1
e
t
dt:
ProgrammingNote: Thegammafunctioncangrowquitelargeevenforsmallinput
values.Inmanycasesitmaybepreferabletousethenaturallogarithmofthegamma
function(gammaln) incalculationstominimizelossofprecision. . Thefinalresult t is
thenexp(result_using_gammaln).
Seealso: [gammainc],page456,[gammaln],page458,[factorial],page449.
[MappingFunction]
gammainc
(
x
,
a
)
[MappingFunction]
gammainc
(
x
,
a
,
"
lower
"
)
[MappingFunction]
gammainc
(
x
,
a
,
"
upper
"
)
Computethenormalizedincompletegammafunction.
Thisisdefinedas
(x;a)=
1
Γ(a)
Z
x
0
t
a 1
e
t
dt
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Chapter17: Arithmetic
457
withthelimitingvalueof1asxapproachesinfinity. ThestandardnotationisP(a;x),
e.g.,AbramowitzandStegun(6.5.1).
Ifaisscalar,thengammainc(x,a)isreturnedforeachelementofxandviceversa.
Ifneitherx noraisscalar,thesizesofxandamustagree,andgammaincisapplied
element-by-element.
Bydefaulttheincomplete gamma functionintegratedfrom0tox x iscomputed. . If
"upper" is giventhenthe complementary functionintegratedfrom m x to infinity y is
calculated. Itshouldbenotedthat
gammainc (x, , a) )  1 1 - gammainc (x, , a, "upper")
Seealso: [gamma],page456,[gammaln],page458.
[FunctionFile]
l = = legendre
(
n
,
x
)
[FunctionFile]
l = = legendre
(
n
,
x
,
normalization
)
ComputetheLegendrefunctionofdegreenandorderm=0... n.
Thevaluenmustbearealnon-negativeinteger.
x isavectorwithreal-valuedelementsintherange[-1,1].
The optionalargument normalizationmay be one of "unnorm","sch",or "norm".
Thedefaultifnonormalizationisgivenis"unnorm".
Whentheoptionalargumentnormalizationis"unnorm",computetheLegendrefunc-
tionofdegreenandordermandreturnallvaluesform=0...n. Thereturnvalue
hasonedimensionmorethanx.
TheLegendreFunctionofdegreenandorderm:
P
m
n
(x)=( 1)
m
(1 x
2
)
m=2
d
m
dx
m
P
n
(x)
withLegendrepolynomialofdegreen:
P(x)=
1
2
n
n!
d
n
dx
n
(x
2
1)
n
legendre(3,[-1.0,-0.9,-0.8])returnsthematrix:
x |
-1.0
|
-0.9
|
-0.8
------------------------------------
m=0 | | -1.00000 0 | | -0.47250 0 | | -0.08000
m=1 | | 0.00000 0 | | -1.99420 0 | | -1.98000
m=2 | | 0.00000 0 | | -2.56500 0 | | -4.32000
m=3 | | 0.00000 0 | | -1.24229 9 | | -3.24000
Whentheoptionalargument normalization is"sch",computetheSchmidt semi-
normalizedassociatedLegendre function. . The e Schmidt semi-normalizedassociated
LegendrefunctionisrelatedtotheunnormalizedLegendrefunctionsbythefollowing:
ForLegendrefunctionsofdegreenandorder0:
SP
0
n
(x)=P
0
n
(x)
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458
GNUOctave
ForLegendrefunctionsofdegreenandorderm:
SP
m
n
(x)=P
m
n
(x)( 1)
m
2(n m)!
(n+m)!
0:5
Whentheoptionalargumentnormalizationis"norm",computethefullynormalized
associatedLegendrefunction. ThefullynormalizedassociatedLegendrefunctionis
relatedtotheunnormalizedLegendrefunctionsbythefollowing:
ForLegendrefunctionsofdegreenandorderm
NP
m
n
(x)=P
m
n
(x)( 1)
m
(n+0:5)(n m)!
(n+m)!
0:5
[MappingFunction]
gammaln
(
x
)
[MappingFunction]
lgamma
(
x
)
Returnthenaturallogarithmofthegammafunctionofx.
Seealso: [gamma],page456,[gammainc],page456.
17.7 RationalApproximations
[FunctionFile]
s = = rat
(
x
,
tol
)
[FunctionFile]
[n, d] ] = = rat
(
x
,
tol
)
Findarationalapproximationtox withinthetolerancedefinedbytolusingacon-
tinuedfractionexpansion.
Forexample:
rat (pi) ) = = 3 + 1/(7 7 + + 1/16) ) = = 355/113
rat (e) = 3 + + 1/(-4 + + 1/(2 + 1/(5 + + 1/(-2 + + 1/(-7)))))
= 1457/536
Whencalledwithtwooutputargumentsreturnthenumeratoranddenominatorsep-
aratelyastwomatrices.
Seealso: [rats],page458.
[Built-inFunction]
rats
(
x
,
len
)
Convertxintoarationalapproximationrepresentedasastring.
Thestringcanbeconvertedbackintoamatrixasfollows:
r = rats s (hilb b (4));
x = str2num (r)
Theoptionalsecondargumentdefinesthemaximumlengthofthestringrepresenting
theelementsofx. Bydefaultlenis9.
Ifthelengthofthesmallestpossiblerationalapproximationexceedslen,anasterisk
(*)paddedwithspaceswillbereturnedinstead.
Seealso: [format],page234,[rat],page458.
C# TWAIN - Query & Set Device Abilities in C#
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Chapter17: Arithmetic
459
17.8 CoordinateTransformations
[FunctionFile]
[theta, r] = cart2pol
(
x
,
y
)
[FunctionFile]
[theta, r, z] = cart2pol
(
x
,
y
,
z
)
[FunctionFile]
[theta, r] = cart2pol
(
C
)
[FunctionFile]
[theta, r, z] = cart2pol
(
C
)
[FunctionFile]
P = = cart2pol
(...)
TransformCartesiancoordinatestopolarorcylindricalcoordinates.
Theinputsx,y (,andz)mustbethesameshape,orscalar. . Ifcalledwithasingle
matrixargumenttheneachrowofCrepresentstheCartesiancoordinate(x,y(,z)).
thetadescribestheanglerelativetothepositivex-axis.
r isthedistancetothez-axis(0,0,z).
IfonlyasinglereturnargumentisrequestedthenreturnamatrixPwhereeachrow
representsonepolar/(cylindrical)coordinate(theta,phi(,z)).
Seealso: [pol2cart],page459,[cart2sph],page459,[sph2cart],page460.
[FunctionFile]
[x, y] ] = = pol2cart
(
theta
,
r
)
[FunctionFile]
[x, y, , z] = = pol2cart
(
theta
,
r
,
z
)
[FunctionFile]
[x, y] ] = = pol2cart
(
P
)
[FunctionFile]
[x, y, , z] = = pol2cart
(
P
)
[FunctionFile]
C = = pol2cart
(...)
TransformpolarorcylindricalcoordinatestoCartesiancoordinates.
Theinputstheta,r,(andz)mustbethesameshape,orscalar.Ifcalledwithasingle
matrix argument then n eachrow of P P represents s the e polar/(cylindrical) coordinate
(theta,r (,z)).
thetadescribestheanglerelativetothepositivex-axis.
r isthedistancetothez-axis(0,0,z).
IfonlyasinglereturnargumentisrequestedthenreturnamatrixCwhereeachrow
representsoneCartesiancoordinate(x,y (,z)).
Seealso: [cart2pol],page459,[sph2cart],page460,[cart2sph],page459.
[FunctionFile]
[theta, phi, r] = cart2sph
(
x
,
y
,
z
)
[FunctionFile]
[theta, phi, r] = cart2sph
(
C
)
[FunctionFile]
S = = cart2sph
(...)
TransformCartesiancoordinatestosphericalcoordinates.
The inputs x, y, andz z must t bethe sameshape, , or r scalar. . If f calledwith asingle
matrixargumenttheneachrowofC representstheCartesiancoordinate(x,y,z).
thetadescribestheanglerelativetothepositivex-axis.
phiistheanglerelativetothexy-plane.
r isthedistancetotheorigin(0,0,0).
IfonlyasinglereturnargumentisrequestedthenreturnamatrixSwhereeachrow
representsonesphericalcoordinate(theta,phi,r).
Seealso: [sph2cart],page460,[cart2pol],page459,[pol2cart],page459.
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RasterEdge.XDoc.PDF.dll. acquire image to file using our C#.NET TWAIN Add-On Group4) device.Compression = TwainCompressionMode.Group3; break; } } acq.FileTranfer
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460
GNUOctave
[FunctionFile]
[x, y, , z] = = sph2cart
(
theta
,
phi
,
r
)
[FunctionFile]
[x, y, , z] = = sph2cart
(
S
)
[FunctionFile]
C = = sph2cart
(...)
TransformsphericalcoordinatestoCartesiancoordinates.
Theinputstheta,phi,andrmustbethesameshape,orscalar.Ifcalledwithasingle
matrixargumenttheneachrowofSrepresentsthesphericalcoordinate(theta,phi,
r).
thetadescribestheanglerelativetothepositivex-axis.
phiistheanglerelativetothexy-plane.
r isthedistancetotheorigin(0,0,0).
IfonlyasinglereturnargumentisrequestedthenreturnamatrixCwhereeachrow
representsoneCartesiancoordinate(x,y,z).
Seealso: [cart2sph],page459,[pol2cart],page459,[cart2pol],page459.
17.9 MathematicalConstants
[Built-inFunction]
e
[Built-inFunction]
e
(
n
)
[Built-inFunction]
e
(
n
,
m
)
[Built-inFunction]
e
(
n
,
m
,
k
,...)
[Built-inFunction]
e
(...,
class
)
Returnascalar,matrix,orN-dimensionalarraywhoseelementsareallequaltothe
baseofnaturallogarithms.
Theconstantesatisfiestheequationlog(e)=1.
Whencalledwithnoarguments,returnascalarwiththevaluee.
When called with a single e argument, return n a a square matrix with h the e dimension
specified.
Whencalledwithmorethanonescalarargumentthefirsttwoargumentsaretakenas
thenumberofrowsandcolumnsandanyfurtherargumentsspecifyadditionalmatrix
dimensions.
Theoptionalargumentclassspecifiesthereturntypeandmaybeeither"double"or
"single".
Seealso: [log],page435,[exp],page435,[pi],page460,[I],page461.
[Built-inFunction]
pi
[Built-inFunction]
pi
(
n
)
[Built-inFunction]
pi
(
n
,
m
)
[Built-inFunction]
pi
(
n
,
m
,
k
,...)
[Built-inFunction]
pi
(...,
class
)
Returnascalar,matrix,orN-dimensionalarraywhoseelementsareallequaltothe
ratioofthecircumferenceofacircletoitsdiameter().
Internally,piiscomputedas‘4.0*atan(1.0)’.
Whencalledwithnoarguments,returnascalarwiththevalueof.
Chapter17: Arithmetic
461
When called with a single e argument, return n a a square matrix with h the e dimension
specified.
Whencalledwithmorethanonescalarargumentthefirsttwoargumentsaretakenas
thenumberofrowsandcolumnsandanyfurtherargumentsspecifyadditionalmatrix
dimensions.
Theoptionalargumentclassspecifiesthereturntypeandmaybeeither"double"or
"single".
Seealso: [e],page460,[I],page461.
[Built-inFunction]
I
[Built-inFunction]
I
(
n
)
[Built-inFunction]
I
(
n
,
m
)
[Built-inFunction]
I
(
n
,
m
,
k
,...)
[Built-inFunction]
I
(...,
class
)
Returnascalar,matrix,orN-dimensionalarraywhoseelementsareallequaltothe
pureimaginaryunit,definedas
p
1.
I,anditsequivalentsi,j,andJ,arefunctionssoanyofthenamesmaybereusedfor
otherpurposes(suchasiforacountervariable).
Whencalledwithnoarguments,returnascalarwiththevaluei.
When called with a single e argument, return n a a square matrix with h the e dimension
specified.
Whencalledwithmorethanonescalarargumentthefirsttwoargumentsaretakenas
thenumberofrowsandcolumnsandanyfurtherargumentsspecifyadditionalmatrix
dimensions.
Theoptionalargumentclassspecifiesthereturntypeandmaybeeither"double"or
"single".
Seealso: [e],page460,[pi],page460,[log],page435,[exp],page435.
[Built-inFunction]
Inf
[Built-inFunction]
Inf
(
n
)
[Built-inFunction]
Inf
(
n
,
m
)
[Built-inFunction]
Inf
(
n
,
m
,
k
,...)
[Built-inFunction]
Inf
(...,
class
)
Returnascalar,matrixorN-dimensionalarraywhoseelementsareallequaltothe
IEEErepresentationforpositiveinfinity.
Infinity is s produced d when results are too large to be represented using the e IEEE
floatingpointformatfornumbers. Twocommonexampleswhichproduceinfinityare
divisionbyzeroandoverflow.
[ 1/0 0 e^800 0 ]
) Inf
Inf
Whencalledwithnoarguments,returnascalarwiththevalue‘Inf’.
When called with a single e argument, return n a a square matrix with h the e dimension
specified.
462
GNUOctave
Whencalledwithmorethanonescalarargumentthefirsttwoargumentsaretakenas
thenumberofrowsandcolumnsandanyfurtherargumentsspecifyadditionalmatrix
dimensions.
Theoptionalargumentclassspecifiesthereturntypeandmaybeeither"double"or
"single".
Seealso: [isinf],page406,[NaN],page462.
[Built-inFunction]
NaN
[Built-inFunction]
NaN
(
n
)
[Built-inFunction]
NaN
(
n
,
m
)
[Built-inFunction]
NaN
(
n
,
m
,
k
,...)
[Built-inFunction]
NaN
(...,
class
)
Returnascalar,matrix,orN-dimensionalarraywhoseelementsareallequaltothe
IEEEsymbolNaN(NotaNumber).
NaNistheresultofoperationswhichdonotproduceawelldefinednumericalresult.
CommonoperationswhichproduceaNaNarearithmeticwithinfinity(1 1),zero
dividedbyzero(0=0),andanyoperationinvolvinganotherNaNvalue(5+NaN).
NotethatNaNalwayscomparesnotequaltoNaN(NaN!=NaN).Thisbehavioris
specifiedbytheIEEEstandardforfloatingpointarithmetic.TofindNaNvalues,use
theisnanfunction.
Whencalledwithnoarguments,returnascalarwiththevalue‘NaN’.
When called with a single e argument, return n a a square matrix with h the e dimension
specified.
Whencalledwithmorethanonescalarargumentthefirsttwoargumentsaretakenas
thenumberofrowsandcolumnsandanyfurtherargumentsspecifyadditionalmatrix
dimensions.
Theoptionalargumentclassspecifiesthereturntypeandmaybeeither"double"or
"single".
Seealso:
[isnan],page406,[Inf],page461.
[Built-inFunction]
eps
[Built-inFunction]
eps
(
x
)
[Built-inFunction]
eps
(
n
,
m
)
[Built-inFunction]
eps
(
n
,
m
,
k
,...)
[Built-inFunction]
eps
(...,
class
)
Returnascalar,matrixorN-dimensionalarraywhoseelementsarealleps,thema-
chineprecision.
Moreprecisely,epsistherelativespacingbetweenanytwoadjacentnumbersinthe
machine’sfloatingpointsystem. Thisnumber r isobviously systemdependent. . On
machinesthatsupportIEEEfloatingpointarithmetic,epsisapproximately2:2204
10
16
fordoubleprecisionand1:192110
7
forsingleprecision.
Whencalledwithnoarguments,returnascalarwiththevalueeps(1.0).
Givenasingleargumentx,returnthedistancebetweenxandthenextlargestvalue.
Whencalledwithmorethanoneargumentthefirsttwoargumentsaretakenasthe
number of rowsandcolumns and d anyfurther r arguments specify additionalmatrix
Chapter17: Arithmetic
463
dimensions.Theoptionalargumentclassspecifiesthereturntypeandmaybeeither
"double"or"single".
See also: [realmax], page 463[realmin], page 463[intmax], page 55[bitmax],
page58.
[Built-inFunction]
realmax
[Built-inFunction]
realmax
(
n
)
[Built-inFunction]
realmax
(
n
,
m
)
[Built-inFunction]
realmax
(
n
,
m
,
k
,...)
[Built-inFunction]
realmax
(...,
class
)
Returnascalar,matrix,orN-dimensionalarraywhoseelementsareallequaltothe
largestfloatingpointnumberthatisrepresentable.
The actual value e is system m dependent. . On n machines s that t support IEEE floating
pointarithmetic,realmaxisapproximately1:797710
308
fordoubleprecisionand
3:402810
38
forsingleprecision.
Whencalledwithnoarguments,returnascalarwiththevaluerealmax("double").
When called with a single e argument, return n a a square matrix with h the e dimension
specified.
Whencalledwithmorethanonescalarargumentthefirsttwoargumentsaretakenas
thenumberofrowsandcolumnsandanyfurtherargumentsspecifyadditionalmatrix
dimensions.
Theoptionalargumentclassspecifiesthereturntypeandmaybeeither"double"or
"single".
Seealso: [realmin],page463,[intmax],page55,[bitmax],page58,[eps],page462.
[Built-inFunction]
realmin
[Built-inFunction]
realmin
(
n
)
[Built-inFunction]
realmin
(
n
,
m
)
[Built-inFunction]
realmin
(
n
,
m
,
k
,...)
[Built-inFunction]
realmin
(...,
class
)
Returnascalar,matrix,orN-dimensionalarraywhoseelementsareallequaltothe
smallestnormalizedfloatingpointnumberthatisrepresentable.
The actual value e is system m dependent. . On n machines s that t support IEEE floating
pointarithmetic,realminisapproximately2:225110
308
fordoubleprecisionand
1:175510
38
forsingleprecision.
Whencalledwithnoarguments,returnascalarwiththevaluerealmin("double").
When called with a single e argument, return n a a square matrix with h the e dimension
specified.
Whencalledwithmorethanonescalarargumentthefirsttwoargumentsaretakenas
thenumberofrowsandcolumnsandanyfurtherargumentsspecifyadditionalmatrix
dimensions.
Theoptionalargumentclassspecifiesthereturntypeandmaybeeither"double"or
"single".
Seealso: [realmax],page463,[intmin],page56,[eps],page462.
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