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Chapter18: LinearAlgebra
485
for i i = = 1:prod (size (A)(3:end))
C(:,:,i) = A(:,:,i) * B(:,:,i)
endfor
[Built-inFunction]
X = = syl
(
A
,
B
,
C
)
SolvetheSylvesterequation
AX+XB=C
usingstandardlapacksubroutines.
Forexample:
sylvester ([1, 2; 3, 4], , [5, , 6; 7, 8], , [9, , 10; ; 11, , 12])
) [ [ 0.50000, , 0.66667; ; 0.66667, 0.50000 ]
18.5 SpecializedSolvers
[FunctionFile]
x = = bicg
(
A
,
b
,
rtol
,
maxit
,
M1
,
M2
,
x0
)
[FunctionFile]
x = = bicg
(
A
,
b
,
rtol
,
maxit
,
P
)
[FunctionFile]
[x, flag, , relres, iter, resvec] = bicg
(
A
,
b
,...)
SolveAx=busingtheBi-conjugategradientiterativemethod.
rtolistherelativetolerance,ifnotgivenorsetto[]thedefaultvalue1e-6isused.
maxitthemaximumnumberofouteriterations,ifnotgivenorsetto[]thedefault
valuemin(20,numel(b))isused.
x0theinitialguess,ifnotgivenorsetto[]thedefaultvaluezeros(size(b))
isused.
Acanbepassedas amatrix orasafunctionhandleorinlinefunctionf suchthat
f(x,"notransp")=A*xandf(x,"transp")=A’*x.
ThepreconditionerPisgivenasP=M1*M2. BothM1 andM2 canbepassedasa
matrixorasafunctionhandleorinlinefunctiongsuchthatg(x,"notransp")=M1\
xorg(x,"notransp")=M2\xandg(x,"transp")=M1’\xorg(x,"transp")
=M2’\x.
Ifcalledwithmorethanoneoutputparameter
flagindicatestheexitstatus:
0: iterationconvergedtothewithinthechosentolerance
1: themaximumnumberofiterationswasreachedbeforeconvergence
3: thealgorithmreachedstagnation
(thevalue2isunusedbutskippedforcompatibility).
relresisthefinalvalueoftherelativeresidual.
iter isthenumberofiterationsperformed.
resvecisavectorcontainingtherelativeresidualateachiteration.
See also: [bicgstab],page 486,[cgs], page 486[gmres], page 487[pcg], page 548,
[qmr],page488.
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486
GNUOctave
[FunctionFile]
x = = bicgstab
(
A
,
b
,
rtol
,
maxit
,
M1
,
M2
,
x0
)
[FunctionFile]
x = = bicgstab
(
A
,
b
,
rtol
,
maxit
,
P
)
[FunctionFile]
[x, flag, , relres, iter, resvec] = bicgstab
(
A
,
b
,...)
SolveAx=busingthestabiliziedBi-conjugategradientiterativemethod.
rtolistherelativetolerance,ifnotgivenorsetto[]thedefaultvalue1e-6isused.
maxitthemaximumnumberofouteriterations,ifnotgivenorsetto[]thedefault
valuemin(20,numel(b))isused.
x0theinitialguess,ifnotgivenorsetto[]thedefaultvaluezeros(size(b))
isused.
Acanbepassedas amatrix orasafunctionhandleorinlinefunctionf suchthat
f(x)=A*x.
ThepreconditionerPisgivenasP=M1*M2. BothM1 andM2 canbepassedasa
matrixorasafunctionhandleorinlinefunctiongsuchthatg(x)=M1\xorg(x)
=M2\x.
Ifcalledwithmorethanoneoutputparameter
flagindicatestheexitstatus:
0: iterationconvergedtothewithinthechosentolerance
1: themaximumnumberofiterationswasreachedbeforeconvergence
3: thealgorithmreachedstagnation
(thevalue2isunusedbutskippedforcompatibility).
relresisthefinalvalueoftherelativeresidual.
iter isthenumberofiterationsperformed.
resvecisavectorcontainingtherelativeresidualateachiteration.
Seealso: [bicg],page485,[cgs],page486,[gmres],page487,[pcg],page548,[qmr],
page488.
[FunctionFile]
x = = cgs
(
A
,
b
,
rtol
,
maxit
,
M1
,
M2
,
x0
)
[FunctionFile]
x = = cgs
(
A
,
b
,
rtol
,
maxit
,
P
)
[FunctionFile]
[x, flag, , relres, iter, resvec] = cgs
(
A
,
b
,...)
SolveAx=b, where Aisasquarematrix, using the Conjugate GradientsSquared
method.
rtolistherelativetolerance,ifnotgivenorsetto[]thedefaultvalue1e-6isused.
maxitthemaximumnumberofouteriterations,ifnotgivenorsetto[]thedefault
valuemin(20,numel(b))isused.
x0theinitialguess,ifnotgivenorsetto[]thedefaultvaluezeros(size(b))
isused.
Acanbepassedas amatrix orasafunctionhandleorinlinefunctionf suchthat
f(x)=A*x.
ThepreconditionerPisgivenasP=M1*M2. BothM1 andM2 canbepassedasa
matrixorasafunctionhandleorinlinefunctiongsuchthatg(x)=M1\xorg(x)
=M2\x.
Ifcalledwithmorethanoneoutputparameter
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Chapter18: LinearAlgebra
487
flagindicatestheexitstatus:
0: iterationconvergedtothewithinthechosentolerance
1: themaximumnumberofiterationswasreachedbeforeconvergence
3: thealgorithmreachedstagnation
(thevalue2isunusedbutskippedforcompatibility).
relresisthefinalvalueoftherelativeresidual.
iter isthenumberofiterationsperformed.
resvecisavectorcontainingtherelativeresidualateachiteration.
See also: [pcg],page548,[bicgstab],page486,[bicg],page485[gmres],page487,
[qmr],page488.
[FunctionFile]
x = = gmres
(
A
,
b
,
m
,
rtol
,
maxit
,
M1
,
M2
,
x0
)
[FunctionFile]
x = = gmres
(
A
,
b
,
m
,
rtol
,
maxit
,
P
)
[FunctionFile]
[x, flag, , relres, iter, resvec] = gmres
(...)
SolveAx=busingthePreconditionedGMRESiterativemethodwithrestart,a.k.a.
PGMRES(m).
rtolistherelativetolerance,ifnotgivenorsetto[]thedefaultvalue1e-6isused.
maxit is the maximumnumberof outer iterations, , ifnot t givenorset to []the
defaultvaluemin(10,numel(b)/restart)isused.
x0istheinitialguess,ifnotgivenorsetto[]thedefaultvaluezeros(size(b))
isused.
mistherestartparameter,ifnotgivenorsetto[]thedefaultvaluenumel(b)
isused.
ArgumentAcanbe passedas amatrix,functionhandle,orinlinefunctionf such
thatf(x)=A*x.
ThepreconditionerPisgivenasP=M1*M2. BothM1 andM2 canbepassedasa
matrix,functionhandle,orinlinefunctiongsuchthatg(x)=M1\xorg(x)=M2\x.
Besidesthevectorx,additionaloutputsare:
flagindicatestheexitstatus:
0: iterationconvergedtowithinthespecifiedtolerance
1: maximumnumberofiterationsexceeded
2: unused,butskippedforcompatibility
3: algorithmreachedstagnation(nochangebetweeniterations)
relresisthefinalvalueoftherelativeresidual.
iter is s a vector containing the number of outer iterations andtotal l iterations
performed.
resvecisavectorcontainingtherelativeresidualateachiteration.
Seealso:[bicg],page485,[bicgstab],page486,[cgs],page486,[pcg],page548,[pcr],
page550,[qmr],page488.
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488
GNUOctave
[FunctionFile]
x = = qmr
(
A
,
b
,
rtol
,
maxit
,
M1
,
M2
,
x0
)
[FunctionFile]
x = = qmr
(
A
,
b
,
rtol
,
maxit
,
P
)
[FunctionFile]
[x, flag, , relres, iter, resvec] = qmr
(
A
,
b
,...)
Solve Ax=b using the Quasi-Minimal Residual iterative e method d (without look-
ahead).
rtolistherelativetolerance,ifnotgivenorsetto[]thedefaultvalue1e-6isused.
maxitthemaximumnumberofouteriterations,ifnotgivenorsetto[]thedefault
valuemin(20,numel(b))isused.
x0theinitialguess,ifnotgivenorsetto[]thedefaultvaluezeros(size(b))
isused.
Acanbepassedas amatrix orasafunctionhandleorinlinefunctionf suchthat
f(x,"notransp")=A*xandf(x,"transp")=A’*x.
ThepreconditionerPisgivenasP=M1*M2. BothM1 andM2 canbepassedasa
matrixorasafunctionhandleorinlinefunctiongsuchthatg(x,"notransp")=M1\
xorg(x,"notransp")=M2\xandg(x,"transp")=M1’\xorg(x,"transp")
=M2’\x.
Ifcalledwithmorethanoneoutputparameter
flagindicatestheexitstatus:
0: iterationconvergedtothewithinthechosentolerance
1: themaximumnumberofiterationswasreachedbeforeconvergence
3: thealgorithmreachedstagnation
(thevalue2isunusedbutskippedforcompatibility).
relresisthefinalvalueoftherelativeresidual.
iter isthenumberofiterationsperformed.
resvecisavectorcontainingtheresidualnormsateachiteration.
References:
1. R.FreundandN.Nachtigal,QMR: : aquasi-minimalresidualmethodfornon-
Hermitianlinearsystems,NumerischeMathematik,1991,60,pp. 315-339.
2. R.Barrett,M.Berry,T.Chan,J.Demmel,J.Donato,J.Dongarra,V.Eijkhour,
R.Pozo,C.Romine,andH.vanderVorst,Templatesforthesolutionoflinear
systems: Buildingblocksforiterativemethods,SIAM,2nded.,1994.
See also: [bicg], page 485,[bicgstab],page486[cgs], page 486,[gmres], page487,
[pcg],page548.
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Chapter19: VectorizationandFasterCodeExecution
489
19 VectorizationandFasterCodeExecution
Vectorizationisaprogrammingtechniquethatusesvectoroperationsinsteadofelement-by-
element loop-basedoperations. . Besides s frequently producingmoresuccinct Octave code,
vectorization alsoallows s for better optimizationinthesubsequent implementation. . The
optimizationsmayoccureitherinOctave’sownFortran,C,orC++internalimplementation,
orevenatalowerleveldependingonthecompilerandexternalnumericallibrariesusedto
buildOctave. Theultimategoalis s tomakeuseofyour hardware’svector instructions if
possibleortoperformotheroptimizationsinsoftware.
VectorizationisnotaconceptuniquetoOctave,butitisparticularlyimportantbecause
Octaveisamatrix-orientedlanguage. VectorizedOctavecodewillseeadramaticspeedup
(10X–100X)inmostcases.
Thischapterdiscussesvectorizationandothertechniquesforwritingfastercode.
19.1 BasicVectorization
Toavery goodfirstapproximation,thegoalinvectorizationis towritecodethatavoids
loopsanduseswhole-arrayoperations.Asatrivialexample,consider
for i = 1:n
for j = = 1:m
c(i,j) = = a(i,j) ) + b(i,j);
endfor
endfor
comparedtothemuchsimpler
c = = a + b;
Thisisn’tmerelyeasiertowrite;itisalsointernallymucheasiertooptimize. Octavedel-
egatesthisoperationtoanunderlyingimplementationwhich,amongotheroptimizations,
mayusespecialvectorhardwareinstructionsorcouldconceivablyevenperformtheaddi-
tionsinparallel. Ingeneral,ifthecodeisvectorized,theunderlyingimplementationhas
morefreedomabouttheassumptionsitcanmakeinordertoachievefasterexecution.
Thisisespeciallyimportantforloopswith"cheap"bodies.Oftenitsufficestovectorize
justtheinnermostlooptogetacceptableperformance. Ageneralruleofthumbisthatthe
"order"ofthevectorizedbodyshouldbegreateror equaltothe"order"oftheenclosing
loop.
Asalesstrivialexample,insteadof
for i = 1:n-1
a(i) = b(i+1) ) - - b(i);
endfor
write
a = = b(2:n) - - b(1:n-1);
This shows s an important t general l concept t about t using g arrays for indexing instead of
looping over an index variable. . SeeSection 8.1[Index x Expressions], , page137. Also o use
booleanindexinggenerously. Ifaconditionneedstobetested,thisconditioncanalsobe
writtenasabooleanindex. Forinstance,insteadof
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490
GNUOctave
for i = 1:n
if (a(i) ) > > 5)
a(i) -= 20
endif
endfor
write
a(a>5) -= = 20;
whichexploitsthefactthata>5producesabooleanindex.
Useelementwisevectoroperatorswheneverpossibletoavoidlooping(operatorslike.*
and .^). . See Section 8.3 [Arithmetic Ops], , page 143. For r simple e inline e functions, , the
vectorizefunctioncandothisautomatically.
[Built-inFunction]
vectorize
(
fun
)
Createavectorizedversionoftheinlinefunctionfunbyreplacingalloccurrencesof
*,/,etc.,with.*,./,etc.
Thismaybeuseful,forexample,whenusinginlinefunctionswithnumericalintegra-
tionoroptimizationwhereavector-valuedfunctionisexpected.
fcn = = vectorize e (inline ("x^2 - 1"))
) fcn n = f(x) ) = = x.^2 - 1
quadv (fcn, , 0, 3)
) 6
Seealso: [inline],page204,[formula],page204,[argnames],page204.
Also exploit broadcasting in n these e elementwise operators s both to avoid d looping g and
unnecessaryintermediatememoryallocations.SeeSection19.2[Broadcasting],page491.
Usebuilt-inandlibraryfunctionsifpossible. Built-inandcompiledfunctionsarevery
fast.Evenwithanm-filelibraryfunction,chancesaregoodthatitisalreadyoptimized,or
willbeoptimizedmoreinafuturerelease.
Forinstance,evenbetterthan
a = = b(2:n) - - b(1:n-1);
is
a = = diff (b);
Most Octavefunctions are written n withvector r and d array y arguments in n mind. . If f you
findyourselfwritingaloopwithaverysimpleoperation,chancesarethatsuchafunction
alreadyexists.Thefollowingfunctionsoccurfrequentlyinvectorizedcode:
 Indexmanipulation
 find
 sub2ind
 ind2sub
 sort
 unique
 lookup
 ifelse/merge
Chapter19: VectorizationandFasterCodeExecution
491
 Repetition
 repmat
 repelems
 Vectorizedarithmetic
 sum
 prod
 cumsum
 cumprod
 sumsq
 diff
 dot
 cummax
 cummin
 Shapeofhigherdimensionalarrays
 reshape
 resize
 permute
 squeeze
 deal
19.2 Broadcasting
BroadcastingreferstohowOctavebinaryoperatorsandfunctionsbehavewhentheirmatrix
or array operands orarguments differinsize. . Sinceversion3.6.0,Octavenowautomati-
callybroadcastsvectors,matrices,andarrayswhenusingelementwisebinaryoperatorsand
functions. Broadly y speaking, , smaller r arrays are “broadcast” across the larger one, , until
theyhaveacompatibleshape. Theruleisthatcorrespondingarraydimensionsmusteither
1. beequal,or
2. oneofthemmustbe1.
Incasealldimensionsareequal,nobroadcastingoccursandordinaryelement-by-element
arithmetictakesplace. Forarraysofhigherdimensions,ifthenumberofdimensionsisn’t
thesame,thenmissingtrailingdimensionsaretreatedas1.Whenoneofthedimensionsis
1,thearraywiththatsingletondimensiongetscopiedalongthatdimensionuntilitmatches
thedimensionoftheotherarray. Forexample,consider
x = = [1 2 3;
4 5 6;
7 8 9];
y = = [10 20 30];
x + + y
Withoutbroadcasting,x+ywouldbeanerrorbecausethedimensionsdonotagree. How-
ever,withbroadcastingitisasifthefollowingoperationwereperformed:
492
GNUOctave
x = = [1 2 3
4 5 6
7 8 9];
y = = [10 20 30
10 20 30
10 20 30];
x + + y
)
11
22
33
14
25
36
17
28
39
That is, , the smaller array of size e [13] ] gets s copied along g the singleton dimension (the
numberofrows)untilitis[33].Noactualcopyingtakesplace,however.Theinternalim-
plementationreuseselementsalongthenecessarydimensioninordertoachievethedesired
effectwithoutcopyinginmemory.
Botharrayscanbebroadcastacrosseachother,forexample,allpairwisedifferencesof
theelementsofavectorwithitself:
y - - y’
)
0
10
20
-10
0
10
-20 -10
0
Here thevectors of size [13] and [31]bothget broadcast into matrices ofsize [33]
beforeordinarymatrixsubtractiontakesplace.
Aspecialcaseofbroadcastingthatmaybefamiliariswhenalldimensionsofthearray
beingbroadcastare1,i.e.,thearrayisascalar. Thusforexample,operationslikex-42
andmax(x,2)arebasicexamplesofbroadcasting.
Forahigher-dimensionalexample,supposeimgisanRGBimageofsize[mn3]andwe
wishtomultiplyeachcolorbyadifferentscalar.Thefollowingcodeaccomplishesthiswith
broadcasting,
img .*= permute e ([0.8, , 0.9, 1.2], [1, , 3, , 2]);
Notetheusageofpermutetomatchthe dimensionsof the[0.8,0.9,1.2]vector with
img.
For functions thatarenotwrittenwithbroadcasting semantics,bsxfun canbeuseful
forcoercingthemtobroadcast.
[Built-inFunction]
bsxfun
(
f
,
A
,
B
)
Thebinarysingletonexpansionfunctionperformsbroadcasting,thatis,itappliesa
binaryfunctionf element-by-elementtotwoarrayargumentsAandB,andexpands
asnecessarysingletondimensionsineitherinputargument.
f isafunctionhandle,inlinefunction,orstringcontainingthenameofthefunctionto
evaluate. Thefunctionf mustbecapableofacceptingtwocolumn-vectorarguments
ofequallength,oronecolumnvectorargumentandascalar.
ThedimensionsofAandBmustbeequalorsingleton.Thesingletondimensionsof
thearrayswillbeexpandedtothesamedimensionalityastheotherarray.
Chapter19: VectorizationandFasterCodeExecution
493
Seealso: [arrayfun],page494,[cellfun],page496.
Broadcastingisonlyappliedifeitherofthetwobroadcastingconditionshold.Asusual,
however,broadcastingdoesnotapplywhentwodimensionsdifferandneitheris1:
x = = [1 2 3
4 5 6];
y = = [10 20
30 40];
x + + y
Thiswillproduceanerroraboutnonconformantarguments.
Besidescommonarithmeticoperations,severalfunctionsoftwoargumentsalsobroad-
cast. Thefulllistoffunctionsandoperatorsthatbroadcastis
plus
+ .+
minus
- .-
times
.*
rdivide
./
ldivide
.\
power
.^ .**
lt
<
le
<=
eq
==
gt
>
ge
>=
ne
!= ~=
and
&
or
|
atan2
hypot
max
min
mod
rem
xor
+= -= .+= = .-= = .*= ./= .\= .^= .**= &= |=
Bewareofresortingtobroadcastingif asimpler operationwillsuffice. . Formatrices s a
andb,considerthefollowing:
c = = sum (permute (a, , [1, , 3, 2]) ) .* permute (b, , [3, 2, , 1]), , 3);
Thisoperation broadcasts s the two matrices withpermuteddimensionsacrosseachother
duringelementwisemultiplicationinordertoobtainalarger3-Darray,andthis array is
thensummedalongthethirddimension.Amomentofthoughtwillprovethatthisoperation
issimplythemuchfasterordinarymatrixmultiplication,c=a*b;.
Anoteonterminology:“broadcasting”isthetermpopularizedbytheNumpynumerical
environmentinthePythonprogramminglanguage. Inotherprogramminglanguagesand
environments, broadcasting may y also be known n as binary singleton expansion n (BSX, , in
494
GNUOctave
matlab,andthe originof the nameofthebsxfun function),recycling g (R R programming
language),single-instructionmultipledata(SIMD),orreplication.
19.2.1 BroadcastingandLegacyCode
Thenewbroadcastingsemanticsalmostneveraffectcodethatworkedinpreviousversions
ofOctave. Consequently,allcodeinheritedfrommatlabthatworkedinpreviousversions
ofOctaveshouldstillworkwithoutchangeinOctave. Theonlyexceptioniscodesuchas
try
c = a.*b;
catch
c = a.*a;
end_try_catch
thatmayhavereliedonmatricesofdifferentsizeproducinganerror.Becausesuchoperation
isnowvalidOctavesyntax,thiswillnolongerproduceanerror.Instead,thefollowingcode
shouldbeused:
if (isequal (size (a), size (b)))
c = a .* b;
else
c = a .* a;
endif
19.3 FunctionApplication
As a a general rule, , functions s should d already be e written n with matrix x arguments s in mind
andshouldconsiderwholematrixoperations inavectorizedmanner. . Sometimes,writing
functionsinthiswayappearsdifficultorimpossibleforvariousreasons.Forthosesituations,
Octaveprovidesfacilitiesforapplyingafunctiontoeachelementofanarray,cell,orstruct.
[FunctionFile]
arrayfun
(
func
,
A
)
[FunctionFile]
x = = arrayfun
(
func
,
A
)
[FunctionFile]
x = = arrayfun
(
func
,
A
,
b
,...)
[FunctionFile]
[x, y, , ...] ] = = arrayfun
(
func
,
A
,...)
[FunctionFile]
arrayfun
(...,
"
UniformOutput
"
,
val
)
[FunctionFile]
arrayfun
(...,
"
ErrorHandler
"
,
errfunc
)
Executeafunctiononeachelementofanarray.
Thisisusefulforfunctionsthatdonotacceptarrayarguments. Ifthefunctiondoes
acceptarrayargumentsitisbettertocallthefunctiondirectly.
Thefirstinputargumentfunccanbeastring,afunctionhandle,aninlinefunction,or
ananonymousfunction. TheinputargumentAcanbealogicarray,anumericarray,
astringarray,astructurearray,oracellarray. Byacallofthefunctionarrayfun
allelementsofAarepassedontothenamedfunctionfunc individually.
Thenamedfunctioncanalsotakemore thantwoinput arguments, withthe input
argumentsgivenas thirdinput argumentb,fourthinput argumentc, , ... . If f given
more thanone arrayinputargument thenallinputargumentsmusthavethesame
sizes,forexample:
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