This is pdfTeX, Version 3.1415926-1.40.11 (TeX Live 2010) (format=pdflatex 2010.7.21) 28 MAY 2011 21:38 entering extended mode restricted \write18 enabled. %&-line parsing enabled. **sagebook (./sagebook.tex LaTeX2e <2009/09/24> Babel and hyphenation patterns for english, dumylang, nohyphenation, ge rman-x-2009-06-19, ngerman-x-2009-06-19, ancientgreek, ibycus, arabic, armenian , basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danish, dutch, u kenglish, usenglishmax, esperanto, estonian, farsi, finnish, french, galician, german, ngerman, swissgerman, monogreek, greek, hungarian, icelandic, assamese, bengali, gujarati, hindi, kannada, malayalam, marathi, oriya, panjabi, tamil, telugu, indonesian, interlingua, irish, italian, kurmanji, lao, latin, latvian, lithuanian, mongolian, mongolianlmc, bokmal, nynorsk, polish, portuguese, roma nian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, turkish, turkmen, ukrainian, uppersorbian, welsh, loaded. (/usr/local/texlive/2010/texmf-dist/tex/latex/base/report.cls Document Class: report 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/texlive/2010/texmf-dist/tex/latex/base/size10.clo File: size10.clo 2007/10/19 v1.4h Standard LaTeX file (size option) ) \c@part=\count79 \c@chapter=\count80 \c@section=\count81 \c@subsection=\count82 \c@subsubsection=\count83 \c@paragraph=\count84 \c@subparagraph=\count85 \c@figure=\count86 \c@table=\count87 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) \@input{macros.aux} \openout2 = `macros.aux'. (./macros.tex (/usr/local/texlive/2010/texmf-dist/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. (/usr/local/texlive/2010/texmf-dist/tex/latex/amsmath/amstext.sty Package: amstext 2000/06/29 v2.01 (/usr/local/texlive/2010/texmf-dist/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 \@emptytoks=\toks14 \ex@=\dimen103 )) (/usr/local/texlive/2010/texmf-dist/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d \pmbraise@=\dimen104 ) (/usr/local/texlive/2010/texmf-dist/tex/latex/amsmath/amsopn.sty Package: amsopn 1999/12/14 v2.01 operator names ) \inf@bad=\count88 LaTeX Info: Redefining \frac on input line 211. \uproot@=\count89 \leftroot@=\count90 LaTeX Info: Redefining \overline on input line 307. \classnum@=\count91 \DOTSCASE@=\count92 LaTeX Info: Redefining \ldots on input line 379. LaTeX Info: Redefining \dots on input line 382. LaTeX Info: Redefining \cdots on input line 467. \Mathstrutbox@=\box26 \strutbox@=\box27 \big@size=\dimen105 LaTeX Font Info: Redeclaring font encoding OML on input line 567. LaTeX Font Info: Redeclaring font encoding OMS on input line 568. \macc@depth=\count93 \c@MaxMatrixCols=\count94 \dotsspace@=\muskip10 \c@parentequation=\count95 \dspbrk@lvl=\count96 \tag@help=\toks15 \row@=\count97 \column@=\count98 \maxfields@=\count99 \andhelp@=\toks16 \eqnshift@=\dimen106 \alignsep@=\dimen107 \tagshift@=\dimen108 \tagwidth@=\dimen109 \totwidth@=\dimen110 \lineht@=\dimen111 \@envbody=\toks17 \multlinegap=\skip44 \multlinetaggap=\skip45 \mathdisplay@stack=\toks18 LaTeX Info: Redefining \[ on input line 2666. LaTeX Info: Redefining \] on input line 2667. ) (/usr/local/texlive/2010/texmf-dist/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2009/06/22 v3.00 Basic AMSFonts support \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 96. ) (/usr/local/texlive/2010/texmf-dist/tex/latex/amsfonts/amssymb.sty Package: amssymb 2009/06/22 v3.00 ) (/usr/local/texlive/2010/texmf-dist/tex/latex/amscls/amsthm.sty Package: amsthm 2004/08/06 v2.20 \thm@style=\toks19 \thm@bodyfont=\toks20 \thm@headfont=\toks21 \thm@notefont=\toks22 \thm@headpunct=\toks23 \thm@preskip=\skip46 \thm@postskip=\skip47 \thm@headsep=\skip48 \dth@everypar=\toks24 ) (/usr/local/texlive/2010/texmf-dist/tex/latex/url/url.sty \Urlmuskip=\muskip11 Package: url 2006/04/12 ver 3.3 Verb mode for urls, etc. ) (/usr/local/texlive/2010/texmf-dist/tex/latex/tools/xspace.sty Package: xspace 2006/05/08 v1.12 Space after command names (DPC,MH) ) \c@theorem=\count100 \c@listnum=\count101 (/usr/local/texlive/2010/texmf-dist/tex/latex/graphics/color.sty Package: color 2005/11/14 v1.0j Standard LaTeX Color (DPC) (/usr/local/texlive/2010/texmf-dist/tex/latex/latexconfig/color.cfg File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive ) Package color Info: Driver file: pdftex.def on input line 130. (/usr/local/texlive/2010/texmf-dist/tex/latex/pdftex-def/pdftex.def File: pdftex.def 2010/02/14 v0.04n Graphics/color for pdfTeX \Gread@gobject=\count102 )) (/usr/local/texlive/2010/texmf-dist/tex/latex/listings/listings.sty (/usr/local/texlive/2010/texmf-dist/tex/latex/graphics/keyval.sty Package: keyval 1999/03/16 v1.13 key=value parser (DPC) \KV@toks@=\toks25 ) \lst@mode=\count103 \lst@gtempboxa=\box28 \lst@token=\toks26 \lst@length=\count104 \lst@currlwidth=\dimen112 \lst@column=\count105 \lst@pos=\count106 \lst@lostspace=\dimen113 \lst@width=\dimen114 \lst@newlines=\count107 \lst@lineno=\count108 \lst@maxwidth=\dimen115 (/usr/local/texlive/2010/texmf-dist/tex/latex/listings/lstmisc.sty File: lstmisc.sty 2007/02/22 1.4 (Carsten Heinz) \c@lstnumber=\count109 \lst@skipnumbers=\count110 \lst@framebox=\box29 ) (/usr/local/texlive/2010/texmf-dist/tex/latex/listings/listings.cfg File: listings.cfg 2007/02/22 1.4 listings configuration )) Package: listings 2007/02/22 1.4 (Carsten Heinz) (/usr/local/texlive/2010/texmf-dist/tex/latex/listings/lstlang1.sty File: lstlang1.sty 2004/09/05 1.3 listings language file )) (/usr/local/texlive/2010/texmf-dist/tex/latex/pdfsync/pdfsync.sty Package: pdfsync 2008/01/26 v1.1 ) (/usr/local/texlive/2010/texmf-dist/tex/latex/hyperref/hyperref.sty Package: hyperref 2010/06/18 v6.81g Hypertext links for LaTeX (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/ltxcmds.sty Package: ltxcmds 2010/04/26 v1.7 LaTeX kernel commands for general use (HO) ) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty Package: kvsetkeys 2010/03/01 v1.9 Key value parser (HO) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/infwarerr.sty Package: infwarerr 2010/04/08 v1.3 Providing info/warning/message (HO) ) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/etexcmds.sty Package: etexcmds 2010/01/28 v1.3 Prefix for e-TeX command names (HO) Package etexcmds Info: Could not find \expanded. (etexcmds) That can mean that you are not using pdfTeX 1.50 or (etexcmds) that some package has redefined \expanded. (etexcmds) In the latter case, load this package earlier. )) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/pdfescape.sty Package: pdfescape 2010/03/01 v1.9 Provides hex, PDF name and string conversion s (HO) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/pdftexcmds.sty Package: pdftexcmds 2010/04/01 v0.9 Utility functions of pdfTeX for LuaTeX (HO) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/ifluatex.sty Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) Package ifluatex Info: LuaTeX not detected. ) Package pdftexcmds Info: LuaTeX not detected. Package pdftexcmds Info: \pdf@primitive is available. Package pdftexcmds Info: \pdf@ifprimitive is available. )) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/ifpdf.sty Package: ifpdf 2010/01/28 v2.1 Provides the ifpdf switch (HO) Package ifpdf Info: pdfTeX in pdf mode detected. ) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/ifvtex.sty Package: ifvtex 2010/03/01 v1.5 Switches for detecting VTeX and its modes (HO) Package ifvtex Info: VTeX not detected. ) (/usr/local/texlive/2010/texmf-dist/tex/generic/ifxetex/ifxetex.sty Package: ifxetex 2009/01/23 v0.5 Provides ifxetex conditional ) (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/hycolor.sty Package: hycolor 2009/12/12 v1.6 Color options of hyperref/bookmark (HO) (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/xcolor-patch.sty Package: xcolor-patch 2009/12/12 xcolor patch )) (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/letltxmacro.sty Package: letltxmacro 2008/06/24 v1.3 Let assignment for LaTeX macros (HO) ) \@linkdim=\dimen116 \Hy@linkcounter=\count111 \Hy@pagecounter=\count112 (/usr/local/texlive/2010/texmf-dist/tex/latex/hyperref/pd1enc.def File: pd1enc.def 2010/06/18 v6.81g Hyperref: PDFDocEncoding definition (HO) ) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/intcalc.sty Package: intcalc 2007/09/27 v1.1 Expandable integer calculations (HO) ) (/usr/local/texlive/2010/texmf-dist/tex/latex/latexconfig/hyperref.cfg File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive ) (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/kvoptions.sty Package: kvoptions 2010/02/22 v3.7 Keyval support for LaTeX options (HO) ) Package hyperref Info: Hyper figures OFF on input line 3658. Package hyperref Info: Link nesting OFF on input line 3663. Package hyperref Info: Hyper index ON on input line 3666. Package hyperref Info: Plain pages OFF on input line 3673. Package hyperref Info: Backreferencing OFF on input line 3678. Implicit mode ON; LaTeX internals redefined Package hyperref Info: Bookmarks ON on input line 3874. LaTeX Info: Redefining \url on input line 4159. (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/bitset.sty Package: bitset 2007/09/28 v1.0 Data type bit set (HO) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/bigintcalc.sty Package: bigintcalc 2007/11/11 v1.1 Expandable big integer calculations (HO) )) \Fld@menulength=\count113 \Field@Width=\dimen117 \Fld@charsize=\dimen118 \Field@toks=\toks27 Package hyperref Info: Hyper figures OFF on input line 5137. Package hyperref Info: Link nesting OFF on input line 5142. Package hyperref Info: Hyper index ON on input line 5145. Package hyperref Info: backreferencing OFF on input line 5152. Package hyperref Info: Link coloring OFF on input line 5157. Package hyperref Info: Link coloring with OCG OFF on input line 5162. Package hyperref Info: PDF/A mode OFF on input line 5167. LaTeX Info: Redefining \ref on input line 5207. LaTeX Info: Redefining \pageref on input line 5211. (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/atbegshi.sty Package: atbegshi 2010/03/25 v1.12 At begin shipout hook (HO) ) \Hy@abspage=\count114 \c@Item=\count115 \c@Hfootnote=\count116 ) Package hyperref Warning: Wrong driver `hypertex.def'; (hyperref) pdfTeX is running in PDF mode. (hyperref) Forcing driver `hpdftex.def'. * hyperref using driver hpdftex * (/usr/local/texlive/2010/texmf-dist/tex/latex/hyperref/hpdftex.def File: hpdftex.def 2010/06/18 v6.81g Hyperref driver for pdfTeX (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/atveryend.sty Package: atveryend 2010/03/24 v1.5 Hooks at very end of document (HO) Package atveryend Info: \enddocument detected (standard). ) \Fld@listcount=\count117 \c@bookmark@seq@number=\count118 (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty Package: rerunfilecheck 2010/03/16 v1.6 Rerun checks for auxiliary files (HO) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/uniquecounter.sty Package: uniquecounter 2009/12/18 v1.1 Provides unlimited unique counter (HO) ) Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 71. ) \Hy@SectionHShift=\skip49 ) (/usr/local/texlive/2010/texmf-dist/tex/latex/graphics/graphicx.sty Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) (/usr/local/texlive/2010/texmf-dist/tex/latex/graphics/graphics.sty Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) (/usr/local/texlive/2010/texmf-dist/tex/latex/graphics/trig.sty Package: trig 1999/03/16 v1.09 sin cos tan (DPC) ) (/usr/local/texlive/2010/texmf-dist/tex/latex/latexconfig/graphics.cfg File: graphics.cfg 2010/04/23 v1.9 graphics configuration of TeX Live ) Package graphics Info: Driver file: pdftex.def on input line 91. ) \Gin@req@height=\dimen119 \Gin@req@width=\dimen120 ) (./sagebook.aux) \openout1 = `sagebook.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 27. LaTeX Font Info: ... okay on input line 27. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 27. LaTeX Font Info: ... okay on input line 27. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 27. LaTeX Font Info: ... okay on input line 27. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 27. LaTeX Font Info: ... okay on input line 27. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 27. LaTeX Font Info: ... okay on input line 27. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 27. LaTeX Font Info: ... okay on input line 27. LaTeX Font Info: Checking defaults for OT2/cmr/m/n on input line 27. LaTeX Font Info: Try loading font information for OT2+cmr on input line 27. (/usr/local/texlive/2010/texmf-dist/tex/latex/cyrillic/ot2cmr.fd File: ot2cmr.fd 2001/08/11 v3.2c Computer Modern Cyrillic font definitions ) LaTeX Font Info: ... okay on input line 27. LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 27. LaTeX Font Info: ... okay on input line 27. (/usr/local/texlive/2010/texmf-dist/tex/context/base/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] \scratchcounter=\count119 \scratchdimen=\dimen121 \scratchbox=\box30 \nofMPsegments=\count120 \nofMParguments=\count121 \everyMPshowfont=\toks28 \MPscratchCnt=\count122 \MPscratchDim=\dimen122 \MPnumerator=\count123 \everyMPtoPDFconversion=\toks29 ) \c@lstlisting=\count124 Configuring pdfsync \PDFSYNC@ut=\write3 \openout3 = `sagebook.pdfsync'. \PDFSYNCf@li@=\count125 \PDFSYNC@rder=\count126 **** pdfsync.sty: Synchronizing at \everypar \PDFSYNCeveryp@r=\toks30 **** pdfsync.sty: Synchronizing at \everymath \PDFSYNCeverym@th=\toks31 **** pdfsync.sty: Synchronizing at \everyhbox \PDFSYNCeveryhb@x=\toks32 **** pdfsync.sty: Synchronizing at \everyvbox \PDFSYNCeveryvb@x=\toks33 **** pdfsync.sty: started \AtBeginShipoutBox=\box31 Package hyperref Info: Link coloring OFF on input line 27. (/usr/local/texlive/2010/texmf-dist/tex/latex/hyperref/nameref.sty Package: nameref 2010/04/30 v2.40 Cross-referencing by name of section (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/refcount.sty Package: refcount 2008/08/11 v3.1 Data extraction from references (HO) ) (/usr/local/texlive/2010/texmf-dist/tex/generic/oberdiek/gettitlestring.sty Package: gettitlestring 2009/12/18 v1.3 Cleanup title references (HO) ) \c@section@level=\count127 ) LaTeX Info: Redefining \ref on input line 27. LaTeX Info: Redefining \pageref on input line 27. LaTeX Info: Redefining \nameref on input line 27. (./sagebook.out) (./sagebook.out) \@outlinefile=\write4 \openout4 = `sagebook.out'. (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty Package: epstopdf-base 2010/02/09 v2.5 Base part for package epstopdf (/usr/local/texlive/2010/texmf-dist/tex/latex/oberdiek/grfext.sty Package: grfext 2007/09/30 v1.0 Managing graphics extensions (HO) ) Package grfext Info: Graphics extension search list: (grfext) [.png,.pdf,.jpg,.mps,.jpeg,.jbig2,.jb2,.PNG,.PDF,.JPG,.JPE G,.JBIG2,.JB2,.eps] (grfext) \AppendGraphicsExtensions on input line 436. (/usr/local/texlive/2010/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) LaTeX Font Info: Try loading font information for U+msa on input line 31. (/usr/local/texlive/2010/texmf-dist/tex/latex/amsfonts/umsa.fd File: umsa.fd 2009/06/22 v3.00 AMS symbols A ) LaTeX Font Info: Try loading font information for U+msb on input line 31. (/usr/local/texlive/2010/texmf-dist/tex/latex/amsfonts/umsb.fd File: umsb.fd 2009/06/22 v3.00 AMS symbols B ) [1 {/usr/local/texlive/2010/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./sage book.tocpdfTeX warning (ext4): destination with the same identifier (name{page. 1}) has been already used, duplicate ignored \relax l.37 ...Types and Functions}{45}{subsection.3.3.3} [1 ] [2]) \tf@toc=\write5 \openout5 = `sagebook.toc'. [3] Chapter 1. Overfull \hbox (1.13757pt too wide) in paragraph at lines 63--71 [][][]\OT1/cmr/m/n/10 Sage uses a main-stream pro-gram-ming lan-guage, un-like Maple, Math-e-mat-ica, Magma, [] [4 ] LaTeX Font Info: Try loading font information for OMS+cmr on input line 107. (/usr/local/texlive/2010/texmf-dist/tex/latex/base/omscmr.fd File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 107. [5] Overfull \hbox (6.92914pt too wide) in paragraph at lines 156--156 [][][]\OT1/cmr/bx/n/14.4 ``This unique Amer-i-can idea of the en-trepreneurial [] [6] LaTeX Warning: Reference `ch:profdeb' on page 7 undefined on input line 230. LaTeX Warning: Reference `ch:contrib' on page 7 undefined on input line 231. LaTeX Font Info: Font shape `OT1/cmtt/bx/n' in size <9> not available (Font) Font shape `OT1/cmtt/m/n' tried instead on input line 251. [7] File: graphics/plot1.pdf Graphic file (type pdf) [8 <./graphics/plot1.pdf>] Overfull \hbox (0.38675pt too wide) in paragraph at lines 331--332 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][] [] [9] [10 ] Chapter 2. [11 ] [12] Overfull \hbox (17.3967pt too wide) in paragraph at lines 481--482 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 482--483 [][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 483--484 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (17.3967pt too wide) in paragraph at lines 484--485 [][][][][][] [] [13] [14] [15] [16] [17] File: graphics/3dplot1.png Graphic file (type png) [18 <./graphics/3dplot1.png (PNG copy)>] [19] [20] Overfull \hbox (11.72672pt too wide) in paragraph at lines 954--955 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (0.38675pt too wide) in paragraph at lines 961--962 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [21] [22] [23] Overfull \hbox (51.4166pt too wide) in paragraph at lines 1172--1173 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [24] Overfull \hbox (130.79636pt too wide) in paragraph at lines 1230--1231 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][] [] [25] [26] [27] Overfull \hbox (29.99432pt too wide) in paragraph at lines 1463--1469 \OT1/cmtt/m/n/10 ArithmeticError, ValueError, RuntimeError, NotImplementedError , OverflowError, [] [28] Overfull \hbox (57.08658pt too wide) in paragraph at lines 1474--1475 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (34.40665pt too wide) in paragraph at lines 1486--1487 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (40.07663pt too wide) in paragraph at lines 1490--1491 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [29] Overfull \hbox (0.38675pt too wide) in paragraph at lines 1527--1528 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][] [] Overfull \hbox (68.42654pt too wide) in paragraph at lines 1544--1545 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][] [] Overfull \hbox (28.73666pt too wide) in paragraph at lines 1576--1577 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [30] [31] File: graphics/interact.png Graphic file (type png) [32 <./graphics/interact.png (PNG copy)>] [33] [34] [35] [36] Overfull \hbox (1.83188pt too wide) in paragraph at lines 1948--1951 [][][]\OT1/cmr/m/n/10 For more in-for-ma-tion about how the Python source code is laid out, see the README [] [37] Chapter 3. Overfull \hbox (0.38675pt too wide) in paragraph at lines 1970--1971 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][] [] [38 ] Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! [39] [40] [41] Overfull \hbox (28.73666pt too wide) in paragraph at lines 2227--2228 [][][][][][][][][][][][][][] [] Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! [42] [43] Overfull \hbox (10.69199pt too wide) in paragraph at lines 2323--2327 [][][]\OT1/cmr/m/n/10 For the Cython source code of Sage in-te-gers, in the Sag e li-brary see []\OT1/cmtt/m/n/10 rings/integer.pxd [] [44] [45] [46] [47] Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! [48] [49] Overfull \hbox (3.74455pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Lecture 12: Numpy + Cyth on = AWESOME

[] [] Overfull \hbox (155.99323pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

This lecture is about how to efficiently combine Nump y and Cython to write fast numerical code.

[] [] Overfull \hbox (1415.98224pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

We will focus on the problem of computing the sta ndard deviation of a list of floating point numbers.   Let $x_1, \ldots, x_n$ be a list of $n$ real numbers, and let $$\mu = \frac{1}{n}\sum_{i =1}^n x_i$$ be their mean.   We define their standard deviation to be $$\s qrt{\frac{1}{n}\sum_{i=1}^n (x_i - \mu)^2}.$$

[] [] Overfull \hbox (1363.4827pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Note: In statistics it is common to divide by $n-1$ instead of $n$ when computing the standard deviation of a sampl e and using it to estimate the standard deviation of a population.  We wil l not do this, since our goal today is illustrating programming techniques, not learning techniques of statistics.

[] [] Overfull \hbox (917.23659pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Running Example: Compute the standar d deviation of a list of 64-bit floating point numbers.   Our data set is computed using the random.random method, which generates numbers unifor mly between 0 and 1.

[] [] Overfull \hbox (1809.7288pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10 [0.43811732887872634, 0.78344784289564662, 0.79176725313 41533, 0.43546157784257289, 0.99879630143646858, 0.21470214570255253, 0.5281800 2353940696, 0.51667692205628102, 0.67726646422202585, 0.92183464863760212, 0.54 553592061123968, 0.2143866131543859, 0.90130600825854523, 0.71144055233831971, 0.080614713472959898, 0.81024524942438758, 0.8403186842969067, 0.26527690630696 821, 0.9755892062984004, 0.94353224947123771][] [] Overfull \hbox (155.99323pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

First we write a naive straightforward implementation of computation of the standard deviation.

[] [] [50] Overfull \hbox (423.74089pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Next we try the std function in Sage, which was imple mented by UW undergrad Andrew Hou as part of paid work he did on Sage after he took Math 480.

[] [] [51] Overfull \hbox (560.2397pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Sage also has code for working with TimeSeries, which happens to have a method for computing the standard deviation.  It is a c ouple of times faster than Numpy.  

[] [] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (922.48654pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

The TimeSeries code is nearly optimal.  A TimeSe ries is represented by a contiguous array of double's, and the code for computi ng standard deviation is very straightforward Cython that maps directly to C. & nbsp;(I wrote it, by the way.)

[] [] Overfull \hbox (523.49002pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Goal: Write a function that computes the standard deviation of a numpy array as quickly as stats.TimeSeries does, h ence is faster than Numpy itself.

[] [] Overfull \hbox (323.99176pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

First approach: Use numpy "vectorized operations". &n bsp;This doesn't help at all (and is also wasteful of memory, by the way).

[ ] [] [52] Overfull \hbox (255.74236pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Let's see how the time gets spent between each step.  It turns out to be about equally spent among each line.

[] [] [53] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! [54] Overfull \hbox (239.9925pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Next try Cython with no special type declarations. &n bsp;Not surprisingly, this does not help in the least bit.

[] [] Overfull \hbox (134.99341pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Next try Cython with special support for Numpy.   ;This gets powerful... as we will see.

[] [] [55] Overfull \hbox (906.73668pt too wide) in paragraph at lines 3175--3175 [][][] \OT1/cmtt/m/n/10 exec compile(u'open("___code___.py","w").write("# -* - coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("c 3RkX251bXB5MyhOb25lKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code_ __.py"))' + '\n', '', 'single')[] [] Overfull \hbox (1090.48508pt too wide) in paragraph at lines 3175--3175 [][][] \OT1/cmtt/m/n/10 File "_sagenb_flask_sage_notebook_sagenb_home_openidSf mMv1OuVE_44_code_sage70_spyx_0.pyx", line 8, in _sagenb_flask_sage_notebook_sag enb_home_openidSfmMv1OuVE_44_code_sage70_spyx_0.std_numpy3 (_sagenb_flask_sage_ notebook_sagenb_home_openidSfmMv1OuVE_44_code_sage70_spyx_0.c:713)[] [] Overfull \hbox (45.74419pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10 TypeError: Argument 'v' has incorrect type (expected num py.ndarray, got NoneType)[] [] Overfull \hbox (649.48892pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Look at Cython + Numpy documentation (by Googling "cy thon numpy"), and we learn that if we declare v a little more precisely, then w e get fast direct access to the underlying elements in v.

[] [] [56] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! [57] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (281.99213pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Finally, we try again, after disabling bounds checkin g.   This is even better; almost as good as stats.TimeSeries.

[] [] [58] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (19.49442pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

Yeah, we did it!! &nbs p; 

[] [] Overfull \hbox (659.98883pt too wide) in paragraph at lines 3175--3175 [][][]\OT1/cmtt/m/n/10

For smaller input, interestingly we get a massive win over Numpy.   If you were, e.g., computing a sliding window of standard d eviations (say) for a time series, this would be important.

[] [] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! [59] Chapter 4. [60 ] Overfull \hbox (6.0159pt too wide) in paragraph at lines 3239--3242 [][][]\OT1/cmr/m/n/10 To search the def-i-ni-tions of func-tion, use []\OT1/cmt t/m/n/10 search_def\OT1/cmr/m/n/10 . This works just like []\OT1/cmtt/m/n/10 se arch_src [] [61] [62] Chapter 5. Missing character: There is no à in font cmtt9! Missing character: There is no © in font cmtt9! [63 ] Overfull \hbox (17.3967pt too wide) in paragraph at lines 3375--3377 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [64] Overfull \hbox (27.96887pt too wide) in paragraph at lines 3460--3467 [][][]\OT1/cmr/m/n/10 Fortunately, the pro-cess is well doc-u-mented (see []$[] []\OT1/cmtt/m/n/10 http : / / sagemath . org / doc / developer/$[]\OT1/cmr/m/n/ 10 ), [] Overfull \hbox (164.81625pt too wide) in paragraph at lines 3482--3483 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [65] Overfull \hbox (113.7864pt too wide) in paragraph at lines 3484--3485 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 3487--3488 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (51.4166pt too wide) in paragraph at lines 3488--3489 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (17.3967pt too wide) in paragraph at lines 3491--3492 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 3492--3493 [][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 3493--3494 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (17.3967pt too wide) in paragraph at lines 3494--3495 [][][][][][] [] Overfull \hbox (428.99084pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 0. Turn on screen capture using Quicktime (plus I'l l do this all in one terminal and paste the session log below, though editing w on't get recorded).[] [] Overfull \hbox (953.98627pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 1. Login to the math480@sage.math.washington.edu ac count. NOTE: On Windows, install [http://www.chiark.greenend.org.uk/~sgtatham/ putty/ Putty]. On Linux and OS X use the command line (e.g., Terminal) and type "ssh math480@sage.math.washington.edu".[] [] Overfull \hbox (66.744pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 2. Change to directory with my sage install in it a nd type "here" to setup PATH.[] [] Overfull \hbox (140.24336pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 2. Make a change to the Sage library source code us ing pico, based on a suggestion from class.[] [] Overfull \hbox (287.24208pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 4. Type "hg diff" to see the change (your shell cur rent working directory must be a subdirectory of SAGE_ROOT/devel/sage/)[] [] Overfull \hbox (680.98865pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 5. Type "hg commit" to save changes as a commit. T ype "hg revert --all" to instead undo everything you have done. Enter a log me ssage on the first line and save the file, when you do "hg commit".[] [] Overfull \hbox (785.98773pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 7. Type "hg export tip > file.patch" to create a pa tch file. You can get the patch by navigating to the directory in which you t yped this command, starting here: http://sage.math.washington.edu/home/math480/ scratch/[] [] Overfull \hbox (607.48929pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 8. Congrats, we have our patch. If we don't like it and want to start over, do "hg rollback". This will reset things to be exac tly like they were before we typed "hg commit" above.[] [] Overfull \hbox (854.23714pt too wide) in paragraph at lines 3518--3518 [][][] \OT1/cmtt/m/n/10 9. Remark: For much finer control over making patch es, people usually use "Mercurial Queues", as described here in [http://sagemat h.org/doc/developer/ The Sage developers guide]. You do not have to use th em for your homework.[] [] [66] Chapter 6. Overfull \hbox (386.99121pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

The goal of this lecture is to give you a deeper unde rstanding of some of the fundamental and unique architectural issues involved i n Sage.

[] [] Overfull \hbox (397.49112pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

I built Sage partly from other complete mathematical software systems because I wanted to finish Sage (for my use!) in at most 5 yea rs. 

[] [] Overfull \hbox (87.74382pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

"Building the car instea d of reinventing the wheel."

[] [] Overfull \hbox (1337.23293pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

Each of the above is a full standalone project with i ts own custom programming language, history, culture, etc.  And each has u nique, powerful, debugged code that I don't want to have to rewrite from scratc h, since it would take too long, and writing code from scratch is incredibly di fficult and frustrating. 

[] [] Overfull \hbox (554.98975pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

I also wanted to make it easy to call the following s ystems from Sage for the purposes of benchmarking, porting, migration of users and code, optional functionality, etc.:

[] [] Overfull \hbox (113.99359pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

The Big Problem: How can we make use of the above systems from Python?

[] [] Overfull \hbox (444.7407pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

This question is difficult partly because there are s o many answers, each with pros and cons, and I had to choose (then be criticize d for my choices).

[] [] Overfull \hbox (3090.71764pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10 [[10], [9, 1], [8, 2], [8, 1, 1], [7, 3], [7, 2, 1], [7, 1, 1, 1], [6, 4], [6, 3, 1], [6, 2, 2], [6, 2, 1, 1], [6, 1, 1, 1, 1], [5, 5], [5, 4, 1], [5, 3, 2], [5, 3, 1, 1], [5, 2, 2, 1], [5, 2, 1, 1, 1], [5, 1, 1, 1 , 1, 1], [4, 4, 2], [4, 4, 1, 1], [4, 3, 3], [4, 3, 2, 1], [4, 3, 1, 1, 1], [4, 2, 2, 2], [4, 2, 2, 1, 1], [4, 2, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1, 1], [3, 3, 3 , 1], [3, 3, 2, 2], [3, 3, 2, 1, 1], [3, 3, 1, 1, 1, 1], [3, 2, 2, 2, 1], [3, 2 , 2, 1, 1, 1], [3, 2, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 2] , [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1, 1], [2, 1, 1 , 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]][] [] Overfull \hbox (18074.08699pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10 92027175502604546685596278166825605430729405281023979395 3285763517412985262323501978822916547103339332198764311128926694429965192014469 3371805742588542551019656697136927224393688612370494439001184662672422293588388 0949646021554674211449712293631879438242092222979701858787035045131791561718499 0942766797810155029441933075045772129188981041614489343545384206438995186836592 2625931251702234301276800624906634777438422420020049142313578994862871246786261 0060006610227873354093344771970346402912468019502617741296485750068965727678736 5748796835192363570613191348609145244276270764465804777408575949440508551447566 4188114896304641911150453092801316525477317827937471411504849803193674306114639 9094602347281946619566715867818368113040887581799683872172944577575391666322871 2954510481120704923859087275241592392223665876910286300131474621294645735699401 7362846975817551519000164134514089936709319085908026718561179217042946519785796 8117435299141079155705662249617336911859509285578584413441733816964914269258918 7435301742615221458864914330678814703958326474085781955660425664604942849123726 1204850512724398725459766069032380425223714808312168530010647379469068980174750 4661946299472005981442948094926764563853171727831538610914056742150737497538427 5021240249642090800336572787686941682199463463099066866702204042926855402821066 6766158414720115574024352181802062923401192416721410067481998267445932984517612 6920689181532303574746823885977198276651255478126323692431978852519785180336170 5413947686093663891147724061366833494127467087092245969359401102159299184733830 1438906563276480526765759086551359397824424431168085421773553659282321900119956 9770431572718888108452817614295777255622715362408122265800848932276196267545354 4265567233476857358975517683731089773908740330164186182660688614126800469264172 9957436799211806554382447196471158568214661865173011689077327922743213795448845 7335074000412015820488115033194548391813526724286716271750346470118187739995800 4314416087296985163324282563458624844066236399562733354725457411347627828347256 5930336271866741519736879804021922337163713433125642662988743051959019602770323 1191899072357523708159719377437782971846471928181434133277347743728346062720786 9478917449115162489982657581564116298023459732495270075351700956243598913730840 1334549711182650855857066766448864363318347642642193870541917675024873002552707 9093923132020478303851642054200888831310378483811088425548758508373324093873479 7302734376936793420810867020185209718779670927269908884747550412550772822753523 0866798095362455247464529577455482381783027175201590517548765620068098696273250 3598211618667111264793583287147032328796244469851598884862190118775508811358918 5457016501136471407349024158013999313263656227842088073986822774216895311418550 7699828983806874612655437732175065228812853327497124638951381805103781784970476 9088911818484659276794986806465634286299339344423718490667472287384981129496643 9997475566061223014020274110963671727713854372115721224489228469570767118124546 4557214332012897163090638216734538158656594031674831608270687758454094849618839 8233399350019423833682423300814059493159013417591894979385650634324308841947277 1218165592793359389269115106835490432906902871027335713152227618464826154317860 6181345463363441597417941394202470612998672573447155234616773861350947076075833 8637870579921007168514417341548481513953296373455058614174692678013759737246724 6969311252404574068882891540550303875935489428805492623836212595940806996986432 45355453826567378500963781681659096276126857969078217677288980[] [] Overfull \hbox (197.99286pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

Problem 1: Availability of a specifi c known version of a third party software package.

[] [] [67 ] Overfull \hbox (1736.22945pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

Even if we solve the big problem above, a "vendor" wi ll often just release a new version of their software with numerous changes tha t break our solution.  This happens constantly.  And the act ual versions of software that people have installed (under OS X, various Linuxe s, Solaris, etc.) will be widely distributed over versions of software from the last decade.    

[] [] Overfull \hbox (2329.47427pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

Solution: For the free open systems that (1) we really need, and (2) we can build from source easily enough, we shi p and build a very specific version as part of Sage.   This completely sol ves problem 1, at the expense of a lot of (misplaced) criticism from people who don't understand the problem; at the same time, this accidentally creates a so lution to a different problem (easy-to-install distribution of a bunch of usefu l math software), which many people greatly appreciate. 

[] [] Overfull \hbox (1478.98169pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

For the non-free systems or the free systems that are hard to build, the problem just doesn't get solved.  And indeed our inter faces and code that relies on those systems is sadly fairly brittle; often a ne w version of Magma just breaks with Sage.   Fortunately, of course very li ttle functionality in Sage depends on such systems. 

[] [] Overfull \hbox (245.24245pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

Problem 2: Make a specific version o f some mathematics software (call it M) usable from Python.

[] [] Overfull \hbox (932.98645pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10
  • Naive Subprocess. Start up M, tell it to read in a file, and save the results in a file, and terminate.   This doesn't preserve state between calls, and startup time can easily be secon ds, so this is not viable.
    • [] [] Overfull \hbox (3720.71214pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10
  • Create network protocols.  Def ine an openmath/XML based protocol for well-defined communication of "arbitrary mathematics" (whatever that is) between software, e.g., between M and Python.  Design and implement client and server protocols.  The SCIEnce pr oject, started in 2006, and costing many millions of dollars, is an example .   This seems like the right approach, but it it is slow in certain r elevant benchmarks and too challenging (for me) to develop.  It's probably very useful for something (e.g., writing research papers), but is massively to o complicated for what we need for Sage, which is focused on what is practical now.
    • [] [] Overfull \hbox (2502.72276pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10
  • Pseudo-tty's (ptty) = pexpect. &nbs p; Create a simulated command line prompt that is controled by Python.  Ab solutely anything that one can do at the command line with M immediately become s usable from a Python program.    This is relatively easy to impleme nt and extremely flexible -- one can create a useful interface to any math soft ware system out there in a day.  It is slow in some sense, but still much better than (1) and (2).   This approach has been used in Sage for a l ong time.
    • [] [] Overfull \hbox (1646.98022pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10
  • C/C++ library interfaces.  Cre ate a C/C++ library interface and link the other program into Python itself, us ing Cython.   This is extremely difficult, because none of the M's (except PARI) are designed to be used this way.  However, it is extremely fast. & nbsp;For basic arithmetic, it can be several hundred times faster than (3) abov e.  
    • [] [] Overfull \hbox (1200.73412pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    As of now, people have written fairly polished versio ns of both (3) and (4) for all of: PARI, GAP, Singular, Maxima, and R.   & nbsp;In case of (4), these are all hard work, and aren't necessarily used much in Sage yet, or even included in Sage, but they exist, and are on the way in.&n bsp;

    [] [] Overfull \hbox (1662.73009pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    The rest of this worksheet is about how to use (3) ab ove: the pexpect based interfaces.   This is well worth learning, because these interface all work in almost exactly the same way, and there are interfaces to pretty much every math software system out there.   Sage is the only software in existence that can talk to so many other math software sy stems.  

    [] [] Overfull \hbox (528.73997pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    Here are the basic points, which we'll follow with se veral examples illustrating them.  Suppose m is one of math software syste ms, e.g., r or singular or maxima:

    [] [] Overfull \hbox (780.73778pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10
  • x = m.eval(s): sets x equal to the string obtained by typing the string s into M.   The string s can be sever al lines long.  (This is how many % modes in the noteboook are implemented .)
    • [] [] Overfull \hbox (1258.48361pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10
  • x = m(s): creates a new Python obje ct that "wraps" the result of evaluating s in M.  It's like typing "x.name () = s" into M.  You can then do arithmetic with x, and call functions on it, e.g., x.function(...) or m.function(..., x, ...). 
    • [] [] Overfull \hbox (2607.72185pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    WARNING: There is latency.  Any time you call any function involving a pexpect interface, exp ect it to take on the order of at least 1ms (=one millisecond), even if the actual operation in the system M takes almost no time.   &nb sp;For comparison, adding or multiplying most simple objects in Python/Sage tak es about 1 microsecond (i.e., 1/1000 the time of a call involving pexpect), and adding/multiping objects in Cython can take only a few nanoseconds (1/1000000 the time of a pexpect call).  

    [] [] Overfull \hbox (292.49203pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    Another note: the very first time you do m.eval(...) it may take surprisingly long, since another program is starting up.

    [] [] [68] Overfull \hbox (203.24281pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    There is now a separate Maxima subprocess running. &n bsp;Each process has an id number associated to it:

    [] [] Overfull \hbox (82.49387pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    Next will illustrate creating a Python object that wr aps an expression in Maxima.

    [] [] Overfull \hbox (77.24391pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10
    \newcommand{\Bold}[1]{\mathbf{#1 }}\sin x^3\,\tan y
    [] [] Overfull \hbox (365.9914pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    The result is another Python object (which wraps anot her object defined in Maxima).  We can call functions on that object as we ll.

    [] [] Overfull \hbox (974.98608pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    Conclusion: If you understand the ab ove, you are in extremely good shape.  All the other interfaces work the s ame way.   The examples below are just to illustrate some subtle points an d show how interfaces are useful.  

    [] [] [69] Overfull \hbox (2161.47574pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    It is possible in some systems to seriously mess thin gs up and get things "out of sync".  This is nearly impossible with Maxima , since we use it so heavily and have debugged the heck out of it.  Howeve r, with other systems (like Magma) this can happen.  If it does, do, e.g., maxima.quit().  This completely kills the subprocess, in valides any Python objects that wrap variables in that session, and starts a br and new fresh session.  

    [] [] Overfull \hbox (24.74437pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    Here is an example with each of the five big systems included in Sage:

    [] [] Overfull \hbox (570.73961pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    You can follow standard R tutorials and have the comp utations (except graphics at present) to all definitely "just work".  (Unl ike the potentially confusing rpy2.) 

    [] [] Overfull \hbox (82.49387pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    There is also an interface to Octave, which is very s imilar to Matlab (but free).

    [] [] Overfull \hbox (428.99084pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    Bonus: There is even a pexpect inter face to Sage itself.   (Trivia: this is used in the implementation of the Sage notebook.)

    [] [] [70] Overfull \hbox (481.49039pt too wide) in paragraph at lines 3728--3728 [][][]\OT1/cmtt/m/n/10

    Let's get crazy: a pexpect interface inside a pexpect interface.  And of course, this code is going from the notebook to Sage v ia yet another interface.

    [] [] [71] [72 ] Chapter 7. Overfull \hbox (3.74455pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10
  • most 2d plotting that Mathematica has (with a simila r interface)
    • [] [] Overfull \hbox (1169.23439pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    Sage uses the Python library Matplotlib (http://matplotlib.sourceforge. net/) is used under the hood to render 2d graphics; for 3d graphics, Sage c an use a Java applet (jmol), an HTML5 canvas renderer, or a raytracer.  [] [] Overfull \hbox (187.49295pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    In this worksheet, we'll explain how to use the "math ematica-style" 2d plotting capabilities of Sage.

    [] [] Overfull \hbox (1583.98077pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    First, we'll discuss a simple but very powerful plott ing command in Sage called "line".  It takes as input a list of points, an d draws a sequence of line segments connecting them.    The points ar e given as 2-tuples (x,y), which are the x and y coordinates of a point in the plane.   The output of calling the line command is a line object.  [] [] Overfull \hbox (255.74236pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    To see the actual plot of L , just put L by itself on a line or type show(L) or L.show():

    [] [] Overfull \hbox (260.99231pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    Incidentally, there are many, many options that you c an pass to the show command.   The three most useful are:

    [] [] Overfull \hbox (334.49167pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10
  • frame=True:   Make it so the x-y axis are repla ced by a frame, which is much better when looking at certain types of plots
    • [] [] Overfull \hbox (187.49295pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10
  • gridlines=True: Adds a background grid, which makes it easier to understand the plot in some cases.
    • [] [] Overfull \hbox (355.49149pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10
  • figsize=[w,h]:  Allows you to adjust the size o f the output.  Think of w and h as the width and height in "inches". 
    • [] [] Overfull \hbox (3783.7116pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

     In the notebook you can just click and download the default plots displayed above, since they are png images.   However, if you want to include images in a paper you're writing, or use an image in an editor such as Inkscape, it's much better to save the images in other formats.   Thus a critically useful command is L.save('filename.ext'), which e nables you to save a graphics object to a file.  The extension of the file name determines the type of the file.  For example, below we save L to pdf , eps, and svg formats.  Note that the svg image just appears embedded in your web browser, and you can pan around.  In any case, you can always (ri ght or control) click on the link or image to save it as a file on your compute r.     

    [] [] Overfull \hbox (554.98975pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    Lines (and all other graphics objects) have numerous properties that you can adjust, which you find in the documentation.  The most important properties of lines are:

    [] [] [73 ] Overfull \hbox (292.49203pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10
  • color=...: where for the color you can give a string , e.g., 'red'; or an html color, e.g., '#042a99', or an rgb triple.
    • [] [] Overfull \hbox (14.24446pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10
  • linestyle='--':   the style of the line: '--', '-.', '-', ':'
    • [] [] Overfull \hbox (45.74419pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10 sage: line([(-2,-2), (3,8), (5, 5)], color='purple', thi ckness=3, linestyle='--')[] [] Overfull \hbox (56.2441pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10 sage: line([(-2,-2), (3,8), (5, 5)], color='#042a99', th ickness=1.5, linestyle=':')[] [] Overfull \hbox (318.7418pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    Arithmetic: a key unusual idea in Sage graphics is th at you combine together different graphics using +, as illustrated below:

    [] [] Overfull \hbox (155.99323pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10 sage: L2 = line([(1,0), (2,5), (3,0)], color='purple', t hickness=10, alpha=.7) # alpha = transparency[] [] Overfull \hbox (360.74144pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    There are numerous other important plotting commands in Sage, including point, circle, polygon, arrow, and text, as illustrated belo w:

    [] [] Overfull \hbox (35.24428pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10 sage: G += text(r"$\sqrt{\sin(\pi x^2)}$",(1.8,1.35),col or='black',fontsize=20)[] [] Overfull \hbox (1405.48233pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    There are also a function just called "plot" that mak es a plot of a wide range of Sage objects.  It is very useful especially f or plotting functions of one variable.  It is probably the most used Sage plotting function.  The result is a graphics object, which you can use jus t like any graphics object discussed above.

    [] [] Overfull \hbox (124.4935pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    matrix_plot is another similar plotting function, whi ch allows you to visualize a matrix.

    [] [] Overfull \hbox (266.24226pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    Finally, there is a graphics_array function that lets you assemble several independent plots into a single big plot.

    [] [] Overfull \hbox (113.99359pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10 sage: graphics_array([[matrix_plot(A), matrix_plot(A^2)] , [plot(sin), plot(cos,color='red')]])[] [] Overfull \hbox (523.49002pt too wide) in paragraph at lines 3815--3815 [][][]\OT1/cmtt/m/n/10

    Bonus -- you can animate graphics.  Given any li st of graphics objects, the animate command will make a single animated GIF fil e out of them.  For example:

    [] [] Overfull \hbox (481.49039pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    In Sage, just as with 2d graphics, you make 3d graphi cs by creating various primitives and combining them together using addition to create a 3d scene. 

    [] [] Overfull \hbox (3741.71196pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    There are many 3d graphics primitives in Sage.   For example, you can draw platonic solids using tetrahedron, cube, oct ahedron, dodecahedron, icosahedron.  You can plot round objects u sing sphere and point3d.   You can plot 1d arrows, lines, and curves in space using arrow3d, bezier3d, lin e3d.   There are also numerous powerful tools for plotting d ata, functions, and surfaces in 3d, including cylindrical_plot3d, impli cit_plot3d, list_plot3d, parametric_plot3d, plot3d, plot_vector_field3d, polygo n3d, revolution_plot3d, spherical_plot3d, and implicit_plot3d< /strong>.  Finally, you can place text in 3d using the text3d function.

    [] [] Overfull \hbox (833.23732pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    All 3d graphics objects have translate and rotate (and rotateX, rotateY, rotateZ) methods, which allow you to position the object or collection of objects anywh ere you want.

    [] [] Overfull \hbox (192.7429pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Also, you can set the color and opacity of any 3d obj ect when you create it, as an optional parameter.

    [] [] Overfull \hbox (1678.47995pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Finally, you can display a 3d scene G using either&nb sp; jmol (java) via G.show(), the Tachyon raytracer via G.show(viewer='tachyon'), or HTML5 canvas via G.show(viewer= 'canvas3d').  The show command also takes an aspect_ratio option; e.g., sometimes aspect_ratio=1 is useful, in order to make sp here round, etc.

    [] [] [74] Overfull \hbox (1022.23567pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    There are also some rudimentary 3d plotting capabilit ies in matplotlib.  I had once announced an intention to improve those for Sage, but upon closer inspection one finds that matplotlib is very 2d oriented and the 3d stuff just doesn't feel right at all.

    [] [] Overfull \hbox (4796.95276pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    History: William Stein included Tach yon in Sage, then Tom Boothby, Josh Kantor and William Stein wrote some very pr eliminary 3d plotting functionality that relied entirely on tachyon, and could plot functions $z=f(x,y)$.  A year later, during Christmas break, William stumbled on the jmol Java viewer (that only uses Java's 2d render!) for molecul es and he and Robert Bradshaw snuck off and figured out how to make jmol show m ore general mathematical graphics, and also wrote most of the 3d plotting libra ry on top of this, motivated by the upcoming joint math meetings in San Diego ( Jan 2008).   David Joyner then submitted many examples to the documen tation.  Next, William Cauchois (as a UW freshman project) and Carl Witty added an implicit_plot3d function, and Cauchois also added HTML5 canvas renderi ng.  Other people added plotting of vector fields, cylindrical plotting, e tc., driven by Calculus teaching needs.   

    [] [] Overfull \hbox (2560.47226pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Note: The 3d plotting in Sage is mai nly oriented toward mathematical visualization, rather than visualizing large 3d datasets that come up in Scientific computing.   Scientsts are ru mored to make great use of other Python-friendly options, none of which are inc luded with Sage or are easy to install in Sage at present, though all are free, open source, and can be installed if one is "sufficiently motivated":    MyaVI (which uses the VTK C++ library),   ScientificPyth on,   ...? 

    [] [] Overfull \hbox (2051.2267pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Shortcoming: The biggest shortcoming s are that (1) realtime interaction with 3d graphics is not supported in any wa y, (2) there is no easy way to make high quality movie animations of 3d scenes (it is possible, but requires optional tools), and (3) your browser can run out of Java memory if you display too many jmol java-based 3d plots at once, and r efuse to display anymore -- this has been fixed in a patch that has gone into S age yet.

    [] [] Overfull \hbox (119.24355pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    The rest of this worksheet illustrates with examples how to create 3d images using Sage.

    [] [] Overfull \hbox (255.74236pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Problem: Draw all of the platonic so lids next to each other in different colors in a single plot.

    [] [] Overfull \hbox (302.99194pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Problem: Plot 40 semi-transparent ra ndom spheres.   Similarly, plot a few hundred random points.

    [] [] Overfull \hbox (8.9945pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: G = sum( sphere((random(), random(), random()), co lor=hue(random()),[] [] Overfull \hbox (134.99341pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: G = sum( point3d((random(), random(), random()), c olor=hue(random())) for _ in range(1000) )[] [] Overfull \hbox (98.24373pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: G = sum([text3d('%.1f'%n, (cos(n),sin(n),n), color =hue(n/8)) for n in [0,0.3,..,12]])[] [] Overfull \hbox (61.49405pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: plot3d( sin(pi*(x^2+y^2))/2,(x,-B,B),(y,-B,B), plo t_points=100, color='gree' )[] [] Overfull \hbox (129.74345pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Problem: Plot an implicit 3d surface defined by an equation $f(x,y,z)=0$.

    [] [] Overfull \hbox (197.99286pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: p(x,y,z) = 2 - (cos(x + T*y) + cos(x - T*y) + cos( y + T*z) + cos(y - T*z) + cos(z - T*x) + cos(z + T*x))[] [] Overfull \hbox (19.49442pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: implicit_plot3d(p, (x, -r, r), (y, -r, r), (z, -r, r), plot_points=40)[] [] Overfull \hbox (113.99359pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: implicit_plot3d(p==1, (x, -r, r), (y, -r, r), (z, -r, r), plot_points=40, color='green')[] [] Overfull \hbox (1221.73393pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10

    Solution: use a standard mesh one fi nds online as follows, which describes a model of Yoda that has over 50,000 tri angles.   Here we use the scipy module "io" to load the model, then u se the IndexFaceSet 3d primitive to construct the 3d image from the triangulati on data.

    [] [] [75] Overfull \hbox (119.24355pt too wide) in paragraph at lines 3883--3883 [][][]\OT1/cmtt/m/n/10 sage: Y = IndexFaceSet(F3,V,color=Color('#444444')) + In dexFaceSet(F4,V,color=Color('#007700'))[] [] Overfull \hbox (2770.47043pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    Though Sage provides its own functions (e.g,. plot, l ine, point, text, circle, etc.) for drawing 2d graphics, they are all very orie nted toward visualizing the sorts of mathematical objects that come up in more pure mathematics (so more like Mathematica).  For the sort of scientific v isualizing that comes up in applications, the matplotlib library provides funct ionality that is very similar to Matlab for plotting.     Also, matpl otlib can be used on any major operating system without using Sage; it only dep ends on numpy, and has a very open license (BSD-compatible). 

    [] [] Overfull \hbox (1085.23512pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    Also, if you're drawing an image that involves a huge amount of data points, directly using matplotlib can be more efficient than using Sage's plotting, since Sage's plotting is built on top of matplotl ib -- using matplotlib directly gets you closer to the metal.

    [] [] Overfull \hbox (192.7429pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    There are two absolutely critical th ings to remember when using matplotlib from Sage:

    [] [] Overfull \hbox (1851.72844pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10
  • Instead of plt.show() use p lt.savefig('a.png').  Memorize this now.  This will make a n ice smooth antialised png image of the plot appear in the Sage notebook.   Using plt.show() may just do nothing in Sage, depending on your setup (it migh t also popup a window).   You can also do plt.savefig('a.pdf') and plt.savefig('a.svg').
    • [] [] Overfull \hbox (402.74107pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10
  • You might have to put your input in a %pytho n cell or turn off the preparser (by typing preparser(False)).
    • [] [] Overfull \hbox (749.23805pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    With these two hints, you should be able to to try ou t the examples at http://matplotlib.sourceforge.net/gallery.html.   

    [] [] Overfull \hbox (182.243pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    In fact, try it now [in class, go to the above websit e, scroll, and let students choose an example]:

    [] [] Overfull \hbox (308.2419pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    Note: There are some images in the gallery that requi re some external data file (e.g, the brain image), so those won't work.

    [] [] Overfull \hbox (455.24062pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    For example, if students choose the first example, we get:

    [] [] Overfull \hbox (14.24446pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: Several examples of standard matplotlib graphics p rimitives (artists)[] [] [76] Overfull \hbox (24.74437pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: art = mpatches.Rectangle(pos[:,1] - np.array([0.02 5, 0.05]), 0.05, 0.1,[] [] Overfull \hbox (35.24428pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: arrow = mpatches.Arrow(pos[0,5]-0.05, pos[1,5]-0.0 5, 0.1, 0.1, width=0.1)[] [] [77] Overfull \hbox (29.99432pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: collection = PatchCollection(patches, cmap=matplot lib.cm.jet, alpha=0.4)[] [] Overfull \hbox (1909.47794pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    Matplotlib has an interface that works much like Matl ab.  This will be very helpful if you know Matlab, and of some value other wise since there is a lot of Matlab code and documentation out there. This mode is called "pyplot", and there is a now tutorial for it at http://matpl otlib.sourceforge.net/users/pyplot_tutorial.html.  

    [] [] Overfull \hbox (1541.98114pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    Below we replicate several examples from this tutoria l in Sage, and you should read this tutorial.  The main point you should r ealize when looking at these examples is how easily one can express scientific data visualization in terms of this interface; many equivalent plots are possib ly directly with Sage's plotting commands, but they are less natural.

    [] [] [78] Overfull \hbox (24.74437pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: n, bins, patches = plt.hist(x, 50, normed=1, facec olor='g', alpha=0.75)[] [] Overfull \hbox (45.74419pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    Incidentally, you can of course combine matplotlib gr aphics with @interact

    [] [] [79] Overfull \hbox (56.2441pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 ... n, bins, patches = plt.hist(x, bins, normed=1, facecolor='g', alpha=0.75)[] [] Overfull \hbox (822.73741pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    There are tons of other examples of pyplot at the mat plotlib website here: http://matplotlib.sourceforge.net/examples/pylab_ex amples/

    [] [] Overfull \hbox (14.24446pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: # open, close, volume, adj_close from the mpl-data /example directory.[] [] Overfull \hbox (8.9945pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: # The record array stores python datetime.date as an object array in[] [] [80] Overfull \hbox (2171.97565pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    There is more to matplotlib than just a Matlab like i nterface.    Matplotlib has its own library interface, primitives, et c., which are documented here: http://matplotlib.sourceforge.net/contents.html.  Also, there is an strong community with much momentum behind matplotlib d evelopment.  It is the de facto standard for 2d plotting in Pytho n, and it keeps getting better.

    [] [] Overfull \hbox (722.98828pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10

    This is basically this example: htt p://matplotlib.sourceforge.net/examples/mplot3d/surface3d_demo.html

    [] [] Overfull \hbox (56.2441pt too wide) in paragraph at lines 4163--4163 [][][]\OT1/cmtt/m/n/10 sage: from matplotlib.ticker import LinearLocator, Fixed Locator, FormatStrFormatter[] [] [81] Chapter 8. [82 ] [83] [84] Overfull \hbox (0.38675pt too wide) in paragraph at lines 4345--4346 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] File: graphics/primepi100.pdf Graphic file (type pdf) [85] LaTeX Warning: Command \d invalid in math mode on input line 4420. Overfull \hbox (0.80115pt too wide) in paragraph at lines 4434--4438 \OT1/cmr/m/n/10 dol-lar prize prob-lems: []$[][]\OT1/cmtt/m/n/10 http : / / www . claymath . org / millennium / Riemann _ Hypothesis/$[] [] Overfull \hbox (0.38675pt too wide) in paragraph at lines 4446--4447 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][] [] [86 <./graphics/primepi100.pdf>] File: graphics/rhpic.pdf Graphic file (type pdf) [87 <./graphics/rhpic.pdf>] File: graphics/zeta0.pdf Graphic file (type pdf) File: graphics/zeta1.pdf Graphic file (type pdf) [88 <./graphics/zeta0.pdf> <./graphics/zeta1.pdf>] File: graphics/zeta2.png Graphic file (type png) Overfull \hbox (150.74327pt too wide) in paragraph at lines 4601--4601 [][][]\OT1/cmtt/m/n/10 10000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000267[] [] Overfull \hbox (145.49332pt too wide) in paragraph at lines 4601--4601 [][][]\OT1/cmtt/m/n/10 38994629840785861384457662111217990522007745403201488128 25084038333387229965957683578348930338929160[] [] [89 <./graphics/zeta2.png (PNG copy)>] Overfull \hbox (145.49332pt too wide) in paragraph at lines 4601--4601 [][][]\OT1/cmtt/m/n/10 79473887545115165760984304429323579667762572891406147323 45836374084514192683933294095474801360397543[] [] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (1237.4838pt too wide) in paragraph at lines 4601--4601 [][][]\OT1/cmtt/m/n/10 89884656743115795386465259539451236680898848947115328636 7150405788663379027504815663542386612037680105600569399356966788293948844072083 1124642371531973706218888394671243274263815110980062304705972654147604250288441 9075341171231440736956555270413618581675255342293149119973622969239858152417678 164812112069763[] [] Overfull \hbox (234.74254pt too wide) in paragraph at lines 4601--4601 [][][]\OT1/cmtt/m/n/10
  • For math -- see http://wstein.org/ent (chapter 3).
    • [] [] Overfull \hbox (533.98993pt too wide) in paragraph at lines 4601--4601 [][][]\OT1/cmtt/m/n/10
  • More on cryptography using Sage -- see the book by D avid Kohel that is listed here.
    • [] [] Overfull \hbox (392.24117pt too wide) in paragraph at lines 4601--4601 [][][]\OT1/cmtt/m/n/10
  • There is a library called PyCrypto that is included with Sage . 
    • [] [] [90] File: graphics/group_law.pdf Graphic file (type pdf) [91] [92 <./graphics/group_law.pdf>] File: graphics/birch.png Graphic file (type png) [93 <./graphics/birch.png (PNG copy)>] File: graphics/birch_paper.png Graphic file (type png) File: graphics/bsd_plot.pdf Graphic file (type pdf) [94 <./graphics/birch_paper.png (PNG copy)>] Overfull \hbox (7.35771pt too wide) in paragraph at lines 4817--4819 [][][]\OT1/cmr/m/n/10 This is a Clay Mil-lion Dol-lar prize prob-lem: []$[][]\O T1/cmtt/m/n/10 http : / / www . claymath . org / millennium / [] [95 <./graphics/bsd_plot.pdf>] [96] Chapter 9. Overfull \hbox (533.98993pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    See http://rpy.sourceforge.net/rpy2/doc-2.0 /html/introduction.html.  

    [] [] Overfull \hbox (29.99432pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    Note that we have to explicitly use print to see a ni ce representation:

    [] [] Overfull \hbox (801.7376pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    There is a pexpect interface to r called "r" by defau lt when you start Sage.  This tutorial is not about that interface, but in stead about the C library interface called rpy2, which is much faster and more robust.

    [] [] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (19.49442pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    (frankly, I'm shocked at how slow the rpy2 interface actually is...!)

    [] [] [97 ] Overfull \hbox (119.24355pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10 ValueError: Nothing can be done for the type at the moment.[] [] Overfull \hbox (24.74437pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    WARNING:  Python indexing starts at 0 and R inde xing starts at 1.

    [] [] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (166.49313pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    Most R objects have a string representation that can be directly parsed by R, which can be handy.

    [] [] [98] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (40.49423pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    You can also create R matrices, which are R vectors w ith a dim attribute:

    [] [] Overfull \hbox (449.99066pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    The above illustrates how to call an R function. &nbs p;You get it from the R namespace, then call it in the standard way.  Here 's another example:

    [] [] [99] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (1478.98169pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    GOTCHA: In R variable names with dots in them are all owed, but in Python they are not.   The example below illustrates how to d eal with this (use **kwds).  In this example, we make an R vector with a " NA" in it, which means we don't know that entry; the na.rm option to R's sum co mmand controls how it behaves on lists with NA's in them.

    [] [] Overfull \hbox (113.99359pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    Directly in R, we would just type na.rm=TRUE.  I n Python this does not make sense.

    [] [] [100] Overfull \hbox (203.24281pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10
  • Call the R.png function to tell R where the output i mage should be saved (and what size it should be).
    • [] [] Overfull \hbox (140.24336pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10
  • Tell R to turn the plotting device off, which causes the output file to be written.  
    • [] [] Overfull \hbox (649.48892pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    IMPORTANT: This must all happen in the same notebook cell.  Otherwise the output file gets written in temp directory for a cell that was already evaluated, and the plot may not appear.

    [] [] Overfull \hbox (3.74455pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10 ... x = robjects.IntVector(range(points)); y = R.r norm(int(points))[] [] Overfull \hbox (87.74382pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    Warning again -- Do NOT do this: cal l dev.off in a separate cell!

    [] [] Overfull \hbox (355.49149pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    This is how we would do this directly in R, which we can use from Sage by using the "%r" mode in the notebook (or r.eval("""...""")) :

    [] [] [101] Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Overfull \hbox (124.4935pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    Next, we do the same computation, but via rpy2 (which is unfortunately more complicated):

    [] [] Overfull \hbox (66.744pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10 sage: ctl = robjects.FloatVector([4.17,5.58,5.18,6.11,4. 50,4.61,5.17,4.53,5.33,5.14])[] [] Overfull \hbox (66.744pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10 sage: trt = robjects.FloatVector([4.81,4.17,4.41,3.59,5. 87,3.83,6.03,4.89,4.32,4.69])[] [] [102] Overfull \hbox (113.99359pt too wide) in paragraph at lines 5319--5319 [][][] \OT1/cmtt/m/n/10 stop(gettextf("number of offsets is %d, shou ld equal %d (number of observations)",[] [] Overfull \hbox (113.99359pt too wide) in paragraph at lines 5319--5319 [][][] \OT1/cmtt/m/n/10 y, weights = w, rank = 0L, df.residual = if (is.matrix(y)) nrow(y) else length(y))[] [] Overfull \hbox (3.74455pt too wide) in paragraph at lines 5319--5319 [][][] \OT1/cmtt/m/n/10 else lm.wfit(x, y, w, offset = offset, singular. ok = singular.ok,[] [] [103] Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! [104] Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ˜ in font cmtt10! Missing character: There is no â in font cmtt10! Missing character: There is no € in font cmtt10! Missing character: There is no ™ in font cmtt10! Overfull \hbox (801.7376pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    In R a "data frame" is an array of values with labele d rows and columns (like part of a spreadsheet).  Typically one thinks of a data frame as a table where the rows are observations and the columns are var iables.

    [] [] Overfull \hbox (1557.731pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    If you are using rpy2 and Sage together to deal with large real-world data sets, then it is critical that you can quickly move data back and forth.   If you're working with big data in Sage, you're probably using numpy arrays.  Fortunately, there is a way to very quickly convert a big numpy array to an R vector and conversely, as we illustrate below.

    [] [] Overfull \hbox (1006.48581pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    NOTE: The rpy2 documentation suggests doing "import r py2.robjects.numpy2ri" but this is broken (at least with the versions of R, rpy 2, and numpy in Sage), and gives totally wrong results.  So just explicitl y use FloatVector, etc., as illustrated below.

    [] [] Overfull \hbox (50.99414pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10 ValueError: Nothing can be done for the type at the moment.[] [] [105] Overfull \hbox (276.74217pt too wide) in paragraph at lines 5319--5319 [][][]\OT1/cmtt/m/n/10

    ... CRAP, this seems to be just totally broken in rpy 2.  Maybe it is fixed in a newer version.  Sorry folks.

    [] [] [106] Chapter 10. Overfull \hbox (712.48837pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    The first page of "abstract mathematics" that I ever saw, accidentally misfiled in a the computer book section of Bookman's in Flags taff. (Burton W. Jones's "An Introduction to Modern Algebra", 1975.)

    [] [] Overfull \hbox (329.24171pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    A group is a set $G$ equipped with a binary operation $G \times G \to G$ that we write as a dot below that has three properties:

    [] [] Overfull \hbox (66.744pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10
  • Associativity:  $(a\cdot b)\cd ot c = a\cdot(b\cdot c)$
    • [] [] Overfull \hbox (255.74236pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10
  • Existence of identity: There is $1\ in G$ such that $1\cdot a = a\cdot 1 = a$ for all $a \in G$.
    • [] [] Overfull \hbox (339.74162pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10
  • Existence of inverse: For each $a\i n G$ there is $a^{-1} \in G$ such that $a^{-1} \cdot a = a\cdot a^{-1} = 1$.[] [] Overfull \hbox (129.74345pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    We construct objects in Sage that have a binary opera tion satisfying the above properties.

    [] [] Overfull \hbox (35.24428pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    If it is 7am, what time will it be 10 hours from now?  Answer: 5pm.

    [] [] [107 ] Overfull \hbox (134.99341pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field[] [] Overfull \hbox (35.24428pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 Elliptic Curve defined by y^2 + y = x^3 + x^2 + 5*x over Finite Field of size 7[] [] Overfull \hbox (176.99304pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + x^2 + 5*x over Finite Field of size 7[] [] [108] Overfull \hbox (24.74437pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    The Group of orientation preserving symmetries of th e icosahedron...

    [] [] Overfull \hbox (61.49405pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10
    \newcommand{\Bold}[1]{\mathbf{#1 }}\left(\begin{array}{rrrrr}[] [] Overfull \hbox (50.99414pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 3 & -1 & 0 & \zeta_{5}^{3} + \zeta_{5}^{2} + 1 & -\zeta_ {5}^{3} - \zeta_{5}^{2} \\[] [] Overfull \hbox (50.99414pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 3 & -1 & 0 & -\zeta_{5}^{3} - \zeta_{5}^{2} & \zeta_{5}^ {3} + \zeta_{5}^{2} + 1 \\[] [] Overfull \hbox (1767.72917pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10
    \newcommand{\Bold}[1]{\mathbf{#1 }}x = \frac{{\left(-i \, \sqrt{3} + 1\right)} a}{6 \, {\left(\frac{1}{18} \, \s qrt{4 \, a^{3} + 27 \, b^{2}} \sqrt{3} - \frac{1}{2} \, b\right)}^{\left(\frac{ 1}{3}\right)}} - \frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1 }{18} \, \sqrt{4 \, a^{3} + 27 \, b^{2}} \sqrt{3} - \frac{1}{2} \, b\right)}^{\ left(\frac{1}{3}\right)}
    [] [] Overfull \hbox (14.24446pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 sage: G = GL(2, GF(5)); G # 2x2 invertible matrices wi th entries modulo 5[] [] [109] Overfull \hbox (134.99341pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 AttributeError: 'SpecialLinearGroup_finite_field_with_ca tegory' object has no attribute 'subgroup'[] [] Overfull \hbox (1368.73265pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    See the Sage docs and < a href="http://en.wikipedia.org/wiki/Rubik's_cube_group" target="_blank">Wikipe dia.  See also my complaint.

    [] [] Overfull \hbox (1793.97894pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10 ['(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)', '(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40 )', '(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)', '( 9,1 1,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)', '(25,27,32,30)(2 6,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24)', '( 1, 3, 8, 6)( 2, 5, 7, 4 )( 9,33,25,17)(10,34,26,18)(11,35,27,19)'][] [] Overfull \hbox (218.99268pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    An abelian group is a group $G$ wher e for every $a,b \in G$ we have $a\cdot b = b\cdot a$.

    [] [] Overfull \hbox (192.7429pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    An monoid is the same as a group, ex cept we do not require the existence of inverses.

    [] [] Overfull \hbox (134.99341pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    A ring $R$ is a set with two binary operations, $+$ and $\cdot$ such that:

    [] [] Overfull \hbox (119.24355pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10
  • $(R^*,\cdot)$ is an abelian monoid, where $R^*$ is t he set of nonzero elements of $R$,
    • [] [] Overfull \hbox (19.49442pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10
  • For all $a,b,c \in R$ we have $a\cdot (b+c) = a\cdot b + a\cdot c$.
    • [] [] Overfull \hbox (50.99414pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    A field $K$ is a ring such that $(R^ *, \cdot)$ is a group.

    [] [] Overfull \hbox (108.74364pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    Like with groups, Sage (and mathematics!) comes loade d with numerous rings and fields.

    [] [] [110] Overfull \hbox (334.49167pt too wide) in paragraph at lines 5519--5519 [][][]\OT1/cmtt/m/n/10

    Just as for groups, there is much advanced functional ity available for rings (e.g., Groebner basis), but this is another story...

    [] [] [111] Overfull \hbox (0.38675pt too wide) in paragraph at lines 5582--5583 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][] [] Overfull \hbox (0.38675pt too wide) in paragraph at lines 5583--5584 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (0.38675pt too wide) in paragraph at lines 5588--5589 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][] [] Overfull \hbox (0.38675pt too wide) in paragraph at lines 5589--5590 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [112] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5627--5628 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5628--5629 [][][][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5629--5630 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5630--5631 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5631--5632 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5639--5640 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5640--5641 [][][][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5641--5642 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5642--5643 [][][][][][] [] Overfull \hbox (11.72672pt too wide) in paragraph at lines 5643--5644 [][][][][][] [] Overfull \hbox (57.08658pt too wide) in paragraph at lines 5663--5664 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][] [] [113] Overfull \hbox (0.38675pt too wide) in paragraph at lines 5694--5695 [][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [114] [115] [116] Overfull \hbox (6.05673pt too wide) in paragraph at lines 5888--5889 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [117] Chapter 11. Overfull \hbox (0.38675pt too wide) in paragraph at lines 5947--5948 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [118 ] Overfull \hbox (2.02596pt too wide) in paragraph at lines 5977--5984 [][]\OT1/cmr/m/n/10 The \OT1/cmtt/m/n/10 save \OT1/cmr/m/n/10 and \OT1/cmtt/m/n /10 load \OT1/cmr/m/n/10 com-mands from Sec-tion []11.1.1[] above are im-ple-me nted us-ing Python's [] [119] Overfull \hbox (147.8063pt too wide) in paragraph at lines 6041--6042 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (23.06668pt too wide) in paragraph at lines 6043--6044 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][] [] Overfull \hbox (23.06668pt too wide) in paragraph at lines 6068--6069 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][] [] Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! [120] Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! Missing character: There is no  in font cmtt9! Missing character: There is no µ in font cmtt9! Overfull \hbox (539.03513pt too wide) in paragraph at lines 6104--6105 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][] [] [121] [122] Overfull \hbox (119.45639pt too wide) in paragraph at lines 6225--6226 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (816.86429pt too wide) in paragraph at lines 6260--6261 [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [][][][][][][][][][][][][][][][][][][][][][][] [] Overfull \hbox (497.24025pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10

    loads can parse both the compressed and uncompressed pickles (it figures out which is right by assuming compressed, getting an error , then trying uncompressed).

    [] [] [123] Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Missing character: There is no  in font cmtt10! Missing character: There is no µ in font cmtt10! Overfull \hbox (103.49368pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: X = load('http://wiki.wstein.org/11/480a/5-25?acti on=AttachFile&do=get&target=A.sobj')[] [] Overfull \hbox (176.99304pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 Attempting to load remote file: http://wiki.wstein.org/1 1/480a/5-25?action=AttachFile&do=get&target=A.sobj[] [] Overfull \hbox (197.99286pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: X = load('http://wiki.wstein.org/11/480a/5-25?acti on=AttachFile&do=get&target=A.sobj', verbose=False); X[] [] Overfull \hbox (686.2386pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • Understanding object serialization is useful if you do some research computations, and want to record the results in a way that you can later easily recover. As long as later isn't "too late".
    • [] [] Overfull \hbox (334.49167pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • It requires very little thought to use. sav e(obj, 'filename.sobj') and load('filename.sobj')
    • [] [] Overfull \hbox (1011.73576pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • You could make a simple "database" that anybody can easily use over the web by: (1) putting a bunch of sobj's on a webpage, and (2) writing a Python function that uses Sage's load command to remotely grab them off that webpage when requested. Very simple.
    • [] [] Overfull \hbox (1730.97949pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10

    If you want to store a plain string to disk, and load it later, it is critical to master the Python open command. This is very similar to the C library open command, hence to the open command i n most programming languages. You can use this one builtin Python command to b oth read and write files, and also to iterate through the lines of a file, seek to given positions, etc.

    [] [] [124] Overfull \hbox (806.98755pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10

    One can do a lot with a file, or a bunch of files in a directory. Don't use a sophisticated database just because you don't underst and or know how to use files. Now you do. In some cases, they are a great sol ution.

    [] [] Overfull \hbox (103.49368pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10

    Here's a nice decorator (written by Tom Boothby) that combines files with pickling.

    [] [] Overfull \hbox (213.74272pt too wide) in paragraph at lines 6407--6407 [][][] \OT1/cmtt/m/n/10

    File: /sagenb/flask/sage-4.6.2/loc al/lib/python2.6/site-packages/sage/misc/cachefunc.py

    [] [] Overfull \hbox (229.49258pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10

    Decorator for DiskCachedFunction.

    [] [] Overfull \hbox (764.98792pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
    sage: dir = tmp_dir()[] [] Overfull \hbox (407.99103pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: @disk_cac hed_function(dir)[] [] Overfull \hbox (1657.48013pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 ... def < span class="nf">foo(x): return next_prime(2^< span class="n">x)%x[] [] Overfull \hbox (701.98846pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: x < span class="o">= foo(200);x[] [] Overfull \hbox (407.99103pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: @disk_cac hed_function(dir)[] [] Overfull \hbox (980.23604pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 ... def < span class="nf">foo(x): return 1/x[] [] Overfull \hbox (308.2419pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: foo (200)[] [] Overfull \hbox (350.24153pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: foo .clear_cache()[] [] Overfull \hbox (308.2419pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 sage: foo (200)[] [] [125] Overfull \hbox (344.99158pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 ['my_factor-18268770466636286477546060408953537745699156 7873.sobj', 'my_factor-182687704666362864775460604089535377456991567873.key.sob j'][] [] Overfull \hbox (1069.48526pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 ['my_factor-18268770466636286477546060408953537745699156 7875.sobj', 'my_factor-182687704666362864775460604089535377456991567875.key.sob j', 'my_factor-182687704666362864775460604089535377456991567873.sobj', 'my_fact or-182687704666362864775460604089535377456991567873.key.sobj'][] [] Overfull \hbox (229.49258pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10 (((182687704666362864775460604089535377456991567875,), ( )), 5^3 * 557 * 2623880856967509727475197186205175977838299)[] [] Overfull \hbox (560.2397pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • save/load: If you remember nothing else from today's lecture, remember these two commands, which allow you to very easily store and load most any object.
    • [] [] Overfull \hbox (113.99359pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • open: It is easy to open and write to and read from files in Python.
    • [] [] Overfull \hbox (2754.72057pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • disk_cached_function: provides a fu nction decorator that makes a function only ever have to evaluate a given input once, and in the future it just remembers the inputs automatically. It combin es pickling with using open to read and write data to the filesystem. You can m anage the cached inputs to the function outside of Sage, just by adding files t o the cache directory (e.g., if you computed values of a disk_cached_function o n different computers, you could just dump all the directories of files that re sult into a single big directory and have the combined cache).
    • [] [] Overfull \hbox (1720.47958pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • SQL ite: a relational database that is included in Sage. This is not at all Python specific, but it has excellent support for using it from Python. It's an extremely popular database -- according to their website it is the most widely deployed database there is. For example, iPhone apps all use it track their data...
    • [] [] Overfull \hbox (1106.23494pt too wide) in paragraph at lines 6407--6407 [][][]\OT1/cmtt/m/n/10
  • (Maybe) SQLalchemy: an object relational mapper that is included in Sage. SQLalchemy provides much more Python-friendly support on top of some relational database, but built on SQLite or MySQL or PostgreSQL.
    • [] [] Overfull \hbox (239.9925pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Check out the SQLite website.   Some key points:

    [] [] Overfull \hbox (82.49387pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10
  • SQLite is surely the most widely deployed database i n the world, in some sense.
    • [] [] Overfull \hbox (108.74364pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10
  • SQLite is vastly simpler to use and administer than pretty much all other databases.
    • [] [] Overfull \hbox (239.9925pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10
  • SQLite is public domain.  You can do absolutely anything you want with the source code.
    • [] [] Overfull \hbox (213.74272pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10
  • Learning about SQLite may server you well in non-Sag e related projects, since it can be used on its own.
    • [] [] [126] Overfull \hbox (113.99359pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Here's a complete example of using SQLite to make a d atabase of integer factorizations.

    [] [] Overfull \hbox (14.24446pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10 sage: # Make sure the database file isn't left over from a previous demo...[] [] Overfull \hbox (45.74419pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10 sage: cursor.execute("CREATE INDEX factorizations_idx ON factorizations(number)")[] [] Overfull \hbox (145.49332pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    We can look at our new database on the command line, completely independently of Sage/Python:

    [] [] Overfull \hbox (171.74309pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    By the way, the UNIQUE above makes it so you can't en ter another factorization of the same number.

    [] [] Overfull \hbox (24.74437pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10 ... z = cursor.execute('INSERT INTO factorizat ions VALUES(?,?)', t)[] [] [127] Overfull \hbox (3.74455pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10 sage: a = cursor.execute('SELECT * FROM factorizations O RDER BY number;')[] [] Overfull \hbox (187.49295pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    We use the command line again (we do not< /em> have to exit or reload!) and find:

    [] [] Overfull \hbox (890.98682pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Obviously, to use SQLite effectively, it helps enormo usly to know the SQL language.   Fortunately, you don't need to know very much, there are tons of examples on the web, many tutorials, books, and SQL isn 't hard to learn.  

    [] [] Overfull \hbox (507.74016pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Python documentation for the sqlite3 module: http://docs.pytho n.org/library/sqlite3.html

    [] [] Overfull \hbox (827.98737pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Next we'll spend a few moments on SQLAlchemy, which is a Python package inc luded standard with Sage, which can also be installed easily into any Python in stall. 

    [] [] Overfull \hbox (197.99286pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10
  • SQLAlchemy is the canonical "object relational database mapper" for Python.
    • [] [] Overfull \hbox (350.24153pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10
  • SQLAlchemy abstracts away the database backend, so t he same code/application can work with SQLite, PostgreSQL, Oracle, MySQL, etc.< /li>[] [] Overfull \hbox (192.7429pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10
  • SQLAlchemy has a large test suite, good documentatio n, and is a high quality polished product. 
    • [] [] Overfull \hbox (1405.48233pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    WARNING: As of this writing (May 27, 2011) the version of SQLAlchemy in the newest Sage (which is Sage-4.7) is the "ancient" 0.5.8 version.  So make sure to look at the right version of the SQLAlchemy docs here: http://www.sqlalchemy.org/docs/05/

    [] [] Overfull \hbox (66.744pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    We will use the file /tmp/sqlite1 for our demo.   ;Make sure it is deleted.

    [] [] [128] Overfull \hbox (297.74199pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Create a SQLite engine, which SQLalchemy will use. &n bsp;This is the only place below that SQLite is explicitly mentioned.

    [] [] Overfull \hbox (229.49258pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Use SQLalchemy to declare a new Python class, which w ill get mapped to a table in the above SQLlite database.

    [] [] Overfull \hbox (19.49442pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10 ... self.factorization = str(list(factor(numbe r))).replace(' ','')[] [] Overfull \hbox (208.49277pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Create the tables.  In this case, there is exact ly one, which corresponds to the IntFac class above.

    [] [] Overfull \hbox (3.74455pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    And add it to our session, so it will get tracked by the database.

    [] [] Overfull \hbox (575.98956pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Commit everything we have done so far.  After th is commit, the database exists separately on a disk on file, and we can inspect it using the sqlite3 command line program.

    [] [] Overfull \hbox (350.24153pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    We try adding the factorization of 6 again.  Thi s should give an error because number is the primary key, hence must be unique.

    [] [] [129] Overfull \hbox (533.98993pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10 sqlalchemy.orm.exc.FlushError: New instance with identity key (, (6,)) conflicts with pe rsistent instance [] [] Overfull \hbox (45.74419pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Once an error occurs the only option is to rollback t he whole transaction.

    [] [] Overfull \hbox (266.24226pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Let's make a few thousand factorization (like we did above) and include them all in one transaction in the database.

    [] [] Overfull \hbox (40.49423pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Using add_all should be more efficient than calling a dd many times. 

    [] [] Overfull \hbox (134.99341pt too wide) in paragraph at lines 6613--6613 [][][]\OT1/cmtt/m/n/10

    Now we have factorizations of all integers up to 1000 0.  We can do a query like above.

    [] [] (./sagebook.bbl [130]) Package atveryend Info: Empty hook `BeforeClearDocument' on input line 6621. [131 ] Package atveryend Info: Empty hook `AfterLastShipout' on input line 6621. (./sagebook.aux) Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 6621. Package rerunfilecheck Info: File `sagebook.out' has not changed. (rerunfilecheck) Checksum: E46E2EF70CCF0A89E41E762E395B4EAB;5597. LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: 10513 strings out of 493747 148630 string characters out of 3143546 290592 words of memory out of 3000000 11555 multiletter control sequences out of 15000+200000 14812 words of font info for 58 fonts, out of 3000000 for 9000 714 hyphenation exceptions out of 8191 36i,16n,51p,3526b,1732s stack positions out of 5000i,500n,10000p,200000b,50000s < /usr/local/texlive/2010/texmf-dist/fonts/type1/public/amsfonts/cmextra/cmex8.pf b>< /usr/local/texlive/2010/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb> Output written on sagebook.pdf (132 pages, 2623637 bytes). PDF statistics: 3581 PDF objects out of 4296 (max. 8388607) 3325 compressed objects within 34 object streams 2087 named destinations out of 2487 (max. 500000) 709 words of extra memory for PDF output out of 10000 (max. 10000000)