c# adobe pdf reader : Add page numbers to pdf online Library control component asp.net web page winforms mvc SageTutorial8-part1551

SageTutorial,Release7.1
Atthispoint, acommandlikeview(graphs.CompleteGraph(4))shouldproduceagraphicversionofthe
graphpastedintothenotebook, havingusedpdflatextoprocesstkz-graphcommandstorealizethegraph.
NotethatthereisavarietyofoptionstoaffecthowagraphisrenderedinLaTeXviatkz-graph,whichisagain
outsidethescopeofthissection,seethesectionoftheReferencemanualtitled“LaTeXOptionsforGraphs”for
instructionsanddetails.
5.6 AFullyCapableTeXInstallation
ManyofthemoreadvancedfeaturesoftheintegrationofTeXwithSagerequiresasystem-wideinstallationofTeX.
ManyversionsofLinuxhavebaseTeXpackagesbasedonTeX-live,forOSXthereisTeXshopandforWindowsthere
isMikTeX.TheconvertutilityispartoftheImageMagicksuite(whichshouldbeapackageoraneasydownload),
andthethreeprogramsdvipng,ps2pdf,anddvipsmaybeincludedwithyourTeXdistribution. Thefirsttwo
mayalsobeobtained,respectively,fromhttp://sourceforge.net/projects/dvipng/andaspartofGhostscript.
Renderingcombinatorial graphs s requires arecent t version ofthe PGF library, and the files tkz-graph.sty,
tkz-arith.styandperhapstkz-berge.sty,allfromtheAltermundussite.
5.7 ExternalPrograms
TherearethreeprogramsavailabletofurtherintegrateTeXandSage. Thefirstissagetex. Aconcisedescriptionof
sagetexisthatitisacollectionofTeXmacrosthatallowaLaTeXdocumenttoincludeinstructionstohaveSage
computevariousobjectsand/orformatobjectsusingthelatex()supportbuiltintoSage. Soasanintermediate
stepofcompilingaLaTeXdocument,allofthecomputationalandLaTeX-formattingfeaturesofSagecanbehandled
automatically. Asanexample,amathematicsexaminationcanmaintainacorrectcorrespondencebetweenquestions
andanswersbyusingsagetextohaveSagecomputeonefromtheother.SeeUsingSageTeXformoreinformation.
tex2sws begins witha LaTeXdocument, butdefines extraenvironmentsfortheplacementofSagecode. . When
processedwiththerighttools,theresultisaSageworksheet,withcontentproperlyformattedforMathJaxandthe
Sagecodeincorporatedasinputcells.SoatextbookorarticlecanbeauthoredinLaTeX,blocksofSagecodeincluded,
andthewholedocumentcanbetransformedintoaSageworksheetwherethemathematicaltextisnicelyformatted
andtheblocksofSagecodeare“live.”Currentlyindevelopment,seetex2sws@BitBucketformoreinformation.
sws2texreversestheprocessbybeginningwithaSageworksheetandconvertingittolegitimateLaTeXforsubsequent
processingwithallthetoolsavailableforLaTeXdocuments.Currentlyindevelopment,seesws2tex@BitBucketfor
moreinformation.
5.6. AFullyCapableTeXInstallation
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Chapter5. Sage,LaTeXandFriends
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CHAPTER
SIX
PROGRAMMING
6.1 LoadingandAttachingSagefiles
NextweillustratehowtoloadprogramswritteninaseparatefileintoSage.Createafilecalledexample.sagewith
thefollowingcontent:
print "Hello World"
print 2^3
Youcanreadinandexecuteexample.sagefileusingtheloadcommand.
sage: load("example.sage")
Hello World
8
YoucanalsoattachaSagefiletoarunningsessionusingtheattachcommand:
sage: attach("example.sage")
Hello World
8
Nowifyouchangeexample.sageandenteroneblanklineintoSage(i.e.,hitreturn), thenthecontentsof
example.sagewillbeautomaticallyreloadedintoSage.
Inparticular,attachautomaticallyreloadsafilewheneveritchanges,whichishandywhendebuggingcode,whereas
loadonlyloadsafileonce.
When Sageloads example.sage it converts s it toPython, , whichis s thenexecuted d bythePython interpreter.
This conversion is minimal; it t mainly involves wrapping integer literals in Integer() floating point literals
inRealNumber(), replacing ^‘sby
**
‘s, andreplacinge.g., R.2byR.gen(2). . The e convertedversionof
example.sageiscontainedinthesamedirectoryasexample.sageandiscalledexample.sage.py. This
filecontainsthefollowingcode:
print "Hello World"
print Integer(2)
**
Integer(3)
Integerliteralsarewrappedandthe^isreplacedbya
**
. (InPython^means“exclusiveor”and
**
means“expo-
nentiation”.)
Thispreparsingisimplementedinsage/misc/interpreter.py.)
Youcanpastemulti-lineindentedcodeintoSageaslongastherearenewlinestomakenewblocks(thisisnotnecessary
infiles).However,thebestwaytoentersuchcodeintoSageistosaveittoafileanduseattach,asdescribedabove.
79
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SageTutorial,Release7.1
6.2 CreatingCompiledCode
Speediscrucialinmathematicalcomputations. ThoughPythonisaconvenientveryhigh-levellanguage, , certain
calculationscanbeseveralordersofmagnitudefasterthaninPythoniftheyareimplementedusingstatictypesina
compiledlanguage.SomeaspectsofSagewouldhavebeentooslowifithadbeenwrittenentirelyinPython.Todeal
withthis,Sagesupportsacompiled“version”ofPythoncalledCython([Cyt]and[Pyr]). Cythonissimultaneously
similartobothPythonandC.MostPythonconstructions,includinglistcomprehensions,conditionalexpressions,
codelike+=areallowed;youcanalsoimportcodethatyouhavewritteninotherPythonmodules.Moreover,youcan
declarearbitraryCvariables,andarbitraryClibrarycallscanbemadedirectly. TheresultingcodeisconvertedtoC
andcompiledusingaCcompiler.
InordertomakeyourowncompiledSagecode,givethefilean.spyxextension(insteadof.sage). Ifyouare
workingwiththecommand-lineinterface,youcanattachandloadcompiledcodeexactlylikewithinterpretedcode(at
themoment,attachingandloadingCythoncodeisnotsupportedwiththenotebookinterface).Theactualcompilation
isdone“behindthescenes”withoutyourhavingtodoanythingexplicit.Thecompiledsharedobjectlibraryisstored
under$HOME/.sage/temp/hostname/pid/spyx.ThesefilesaredeletedwhenyouexitSage.
NOSagepreparsingisappliedtospyxfiles,e.g.,1/3willresultin0inaspyxfileinsteadoftherationalnumber1/3.
IffooisafunctionintheSagelibrary,touseitfromaspyxfileimportsage.allandusesage.all.foo.
import sage.all
def foo(n):
return sage.all.factorial(n)
6.2.1 AccessingCFunctionsinSeparateFiles
ItisalsoeasytoaccessCfunctionsdefinedinseparate*.cfiles. Here’s s anexample. . Createfilestest.cand
test.spyxinthesamedirectorywithcontents:
ThepureCcode:test.c
int add_one(int n) ) {
return n n + 1;
}
TheCythoncode:test.spyx:
cdef extern from "test.c":
int add_one(int n)
def test(n):
return add_one(n)
Thenthefollowingworks:
sage: attach("test.spyx")
Compiling (...)/test.spyx...
sage: test(10)
11
IfanadditionallibraryfooisneededtocompiletheCcodegeneratedfromaCythonfile,addthelineclib footo
theCythonsource.Similarly,anadditionalCfilebarcanbeincludedinthecompilationwiththedeclarationcfile
bar.
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Chapter6. Programming
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SageTutorial,Release7.1
6.3 StandalonePython/SageScripts
ThefollowingstandaloneSagescriptfactorsintegers,polynomials,etc:
#!/usr/bin/env sage
import sys
from sage.all import
*
if len(sys.argv) != 2:
print "Usage: %s <n>"%sys.argv[0]
print "Outputs the prime factorization of n."
sys.exit(1)
print factor(sage_eval(sys.argv[1]))
Inordertousethisscript,yourSAGE_ROOTmustbeinyourPATH.Iftheabovescriptiscalledfactor,hereisan
exampleusage:
bash $ ./factor r 2006
2
*
17
*
59
bash $ ./factor r "32
*
x^5-1"
(2
*
x - 1)
*
(16
*
x^4 + + 8
*
x^3 + 4
*
x^2 + + 2
*
x + + 1)
6.4 DataTypes
EveryobjectinSagehasawell-definedtype. Pythonhasawiderangeofbasicbuilt-intypes,andtheSagelibrary
addsmanymore.Somebuilt-inPythontypesincludestrings,lists,tuples,intsandfloats,asillustrated:
sage: "sage"type(s)
<type ’str’>
sage: ’sage’type(s)
# you u can use e either r single or r double e quotes
<type ’str’>
sage: [1,2,3,4]; type(s)
<type ’list’>
sage: (1,2,3,4); type(s)
<type ’tuple’>
sage: int(2006); type(s)
<type ’int’>
sage: float(2006); type(s)
<type ’float’>
Tothis,Sageaddsmanyothertypes.E.g.,vectorspaces:
sage: VectorSpace(QQ, 1000000); V
Vector space of f dimension 1000000 0 over r Rational l Field
sage: type(V)
<class ’sage.modules.free_module.FreeModule_ambient_field_with_category’>
OnlycertainfunctionscanbecalledonV.Inothermathsoftwaresystems,thesewouldbecalledusingthe“functional”
notationfoo(V,...). InSage,certainfunctionsareattachedtothetype(orclass)ofV,andarecalledusingan
object-orientedsyntaxlikeinJavaorC++,e.g.,V.foo(...). Thishelpskeeptheglobalnamespacefrombeing
pollutedwithtensofthousandsoffunctions,andmeansthatmanydifferentfunctionswithdifferentbehaviorcanbe
namedfoo,withouthavingtousetype-checkingofarguments(orcasestatements)todecidewhichtocall.Also,ifyou
6.3. StandalonePython/SageScripts
81
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SageTutorial,Release7.1
reusethenameofafunction,thatfunctionisstillavailable(e.g.,ifyoucallsomethingzeta,thenwanttocompute
thevalueoftheRiemann-Zetafunctionat0.5,youcanstilltypes=.5; s.zeta()).
sage: zeta = -1
sage: s=.5; s.zeta()
-1.46035450880959
Insomeverycommoncases,theusualfunctionalnotationisalsosupportedforconvenienceandbecausemathematical
expressionsmightlookconfusingusingobject-orientednotation.Herearesomeexamples.
sage: 2; n.sqrt()
sqrt(2)
sage: sqrt(2)
sqrt(2)
sage: VectorSpace(QQ,2)
sage: V.basis()
[
(1, 0),
(0, 1)
]
sage: basis(V)
[
(1, 0),
(0, 1)
]
sage: MatrixSpace(GF(7), 2); M
Full MatrixSpace of 2 2 by y 2 2 dense matrices s over Finite Field of size e 7
sage: M([1,2,3,4]); A
[1 2]
[3 4]
sage: A.charpoly(’x’)
x^2 + + 2
*
x + + 5
sage: charpoly(A, ’x’)
x^2 + + 2
*
x + + 5
Tolistallmemberfunctionsfor㔴,usetabcompletion. JusttypeA.,thentypethe[tab]keyonyourkeyboard,as
explainedinReverseSearchandTabCompletion.
6.5 Lists,Tuples,andSequences
Thelistdatatypestoreselementsofarbitrarytype.LikeinC,C++,etc. (butunlikemoststandardcomputeralgebra
systems),theelementsofthelistareindexedstartingfrom0:
sage: [235’x’, SymmetricGroup(3)]; v
[2, 3, , 5, , ’x’, Symmetric c group of order 3! as a permutation group]
sage: type(v)
<type ’list’>
sage: v[0]
2
sage: v[2]
5
(Whenindexingintoalist,itisOKiftheindexisnotaPythonint!)ASageInteger(orRational,oranythingwithan
__index__method)willworkjustfine.
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Chapter6. Programming
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SageTutorial,Release7.1
sage: [1,2,3]
sage: v[2]
3
sage: 2
# SAGE Integer
sage: v[n]
# Perfectly OK!
3
sage: v[int(n)]
# Also OK.
3
TherangefunctioncreatesalistofPythonint’s(notSageIntegers):
sage: range(115)
[1, 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, 10, , 11, , 12, 13, , 14]
Thisisusefulwhenusinglistcomprehensionstoconstructlists:
sage: [factor(n) for in range(115)]
sage: print L
[1, 2, , 3, , 2^2, 5, , 2
*
3, 7, 2^3, , 3^2, 2
*
5, 11, , 2^2
*
3, 13, , 2
*
7]
sage: L[12]
13
sage: type(L[12])
<class ’sage.structure.factorization_integer.IntegerFactorization’>
sage: [factor(n) for in range(115if is_odd(n)]
[1, 3, , 5, , 7, , 3^2, 11, , 13]
Formoreabouthowtocreatelistsusinglistcomprehensions,see[PyT].
Listslicingisawonderfulfeature.IfLisalist,thenL[m:n]returnsthesublistofLobtainedbystartingatthe㕚
㕡ℎ
elementandstoppingatthe(㕛−1)
㕠㕡
element,asillustratedbelow.
sage: [factor(n) for in range(120)]
sage: L[4:9]
[5, 2
*
3, 7, 2^3, 3^2]
sage: print L[:4]
[1, 2, , 3, , 2^2]
sage: L[14:4]
[]
sage: L[14:]
[3
*
5, 2^4, 17, 2
*
3^2, 19]
Tuplesaresimilartolists,excepttheyareimmutable,meaningoncetheyarecreatedtheycan’tbechanged.
sage: (1,2,3,4); v
(1, 2, , 3, , 4)
sage: type(v)
<type ’tuple’>
sage: v[15
Traceback (most recent t call last):
...
TypeError: ’tuple’ object t does s not support item assignment
Sequencesareathirdlist-orientedSagetype.Unlikelistsandtuples,Sequenceisnotabuilt-inPythontype.Bydefault,
asequenceismutable,butusingtheSequenceclassmethodset_immutable,itcanbesettobeimmutable,as
thefollowingexampleillustrates.Allelementsofasequencehaveacommonparent,calledthesequencesuniverse.
sage: Sequence([1,2,3,4/5])
sage: v
6.5. Lists,Tuples,andSequences
83
SageTutorial,Release7.1
[1, 2, , 3, , 4/5]
sage: type(v)
<class ’sage.structure.sequence.Sequence_generic’>
sage: type(v[1])
<type ’sage.rings.rational.Rational’>
sage: v.universe()
Rational Field
sage: v.is_immutable()
False
sage: v.set_immutable()
sage: v[03
Traceback (most recent t call last):
...
ValueError: object is s immutable; please e change a copy instead.
Sequencesderivefromlistsandcanbeusedanywherealistcanbeused:
sage: Sequence([1,2,3,4/5])
sage: isinstance(v, list)
True
sage: list(v)
[1, 2, , 3, , 4/5]
sage: type(list(v))
<type ’list’>
Asanotherexample,basisforvectorspacesareimmutablesequences,sinceit’simportantthatyoudon’tchangethem.
sage: QQ^3; B V.basis(); B
[
(1, 0, , 0),
(0, 1, , 0),
(0, 0, , 1)
]
sage: type(B)
<class ’sage.structure.sequence.Sequence_generic’>
sage: B[0B[1]
Traceback (most recent t call last):
...
ValueError: object is s immutable; please e change a copy instead.
sage: B.universe()
Vector space of f dimension 3 3 over r Rational l Field
6.6 Dictionaries
A dictionary (also o sometimes s called d an n associative e array) ) is a a mapping from m ‘hashable’ objects s (e.g.,
strings, numbers, and d tuples of f such; ; see e the Python documentation http://docs.python.org/tut/node7.html and
http://docs.python.org/lib/typesmapping.htmlfordetails)toarbitraryobjects.
sage: {1:5’sage’:17, ZZ:GF(7)}
sage: type(d)
<type ’dict’>
sage: d.keys()
[1, ’sage’, Integer r Ring]
sage: d[’sage’]
17
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Chapter6. Programming
SageTutorial,Release7.1
sage: d[ZZ]
Finite Field of f size 7
sage: d[1]
5
Thethirdkeyillustratesthattheindexesofadictionarycanbecomplicated,e.g.,theringofintegers.
Youcanturntheabovedictionaryintoalistwiththesamedata:
sage: d.items()
[(1, 5), , (’sage’, , 17), (Integer r Ring, Finite Field of f size 7)]
Acommonidiomistoiteratethroughthepairsinadictionary:
sage: {2:43:94:16}
sage: [a
*
for a, b in d.iteritems()]
[8, 27, , 64]
Adictionaryisunordered,asthelastoutputillustrates.
6.7 Sets
Pythonhasabuilt-insettype.Themainfeatureitoffersisveryfastlookupofwhetheranelementisinthesetornot,
alongwithstandardset-theoreticoperations.
sage: set([1,19,’a’]);
set([1,1,12/3])
sage: X
# random sort order
{1, 19, , ’a’}
sage: == set([’a’1119])
True
sage: Y
{2/3, 1}
sage: ’a’ in X
True
sage: ’a’ in Y
False
sage: X.intersection(Y)
{1}
Sagealsohasitsownsettypethatis(insomecases)implementedusingthebuilt-inPythonsettype,buthasalittlebit
ofextraSage-relatedfunctionality.CreateaSagesetusingSet(...).Forexample,
sage: Set([1,19,’a’]);
Set([1,1,12/3])
sage: X
# random sort order
{’a’, 1, , 19}
sage: == Set([’a’1119])
True
sage: Y
{1, 2/3}
sage: X.intersection(Y)
{1}
sage: print latex(Y)
\left\{1, \frac{2}{3}\right\}
sage: Set(ZZ)
Set of f elements s of f Integer r Ring
6.7. Sets
85
SageTutorial,Release7.1
6.8 Iterators
IteratorsarearecentadditiontoPythonthatareparticularlyusefulinmathematicsapplications. Hereareseveral
examples;see[PyT]formoredetails.Wemakeaniteratoroverthesquaresofthenonnegativeintegersupto10000000.
sage: (n^for in xrange(10000000))
sage: next(v)
0
sage: next(v)
1
sage: next(v)
4
Wecreateaniterateovertheprimesoftheform4㕝+1with㕝alsoprime,andlookatthefirstfewvalues.
sage: (4
*
for in Primes() if is_prime(4
*
p+1))
sage: w
# in n the e next line, 0xb0853d6c is s a a random 0x x number
<generator object at t 0xb0853d6c>
sage: next(w)
13
sage: next(w)
29
sage: next(w)
53
Certainrings,e.g.,finitefieldsandtheintegershaveiteratorsassociatedtothem:
sage: [x for in GF(7)]
[0, 1, , 2, , 3, , 4, , 5, , 6]
sage: ((x,y) for in ZZ for in ZZ)
sage: next(W)
(0, 0)
sage: next(W)
(0, 1)
sage: next(W)
(0, -1)
6.9 Loops,Functions,ControlStatements,andComparisons
Wehaveseenafewexamplesalreadyofsomecommonusesofforloops. InPython,aforloophasanindented
structure,suchas
>>> for in range(5):
...
print(i)
...
0
1
2
3
4
Notethecolonattheendoftheforstatement(thereisno“do”or“od”asinGAPorMaple),andtheindentationbefore
the“body”oftheloop,namelyprint(i).Thisindentationisimportant. InSage,theindentationisautomatically
putinforyouwhenyouhitenteraftera”:”,asillustratedbelow.
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Chapter6. Programming
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