(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 7559169, 142602]*) (*NotebookOutlinePosition[ 7589957, 143716]*) (* CellTagsIndexPosition[ 7589913, 143712]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Technology News" }], "Title"], Cell[TextData[StyleBox["Roger Germundsson\nDirector of Research and \ Development\nWolfram Research, Inc", "Subtitle"]], "Subtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ I'm glad to see that so many people showed up at IMS and judging from how the \ last IMS conference went in Finland '97 I would expect this to be a great \ conference. \ \>", "Text"], Cell[TextData[{ "My name is Roger Germundsson and I'm Director of Research and Development \ at Wolfram Research. Today I'm planning to give a presentation of ", StyleBox["Mathematica", FontSlant->"Italic"], " 4 and some additional areas as time permits. " }], "Text"], Cell[TextData[{ "Note: To run this on your machine, change the path below to point to where \ you have this notebook and the corresponding datafiles. This presentation \ also assumes that you have a ", StyleBox["Mathematica", FontSlant->"Italic"], " version 3.0 installed in the standard place. If not then modify the style \ sheet appropriately. " }], "Text"], Cell[CellGroupData[{ Cell["Initialization", "Subsubsection"], Cell["Set directory to talk directory ... ", "Text"], Cell[BoxData[ \(\(SetDirectory[\ "\"\ ];\)\)], "Input", InitializationCell->True], Cell["Make Developer` and Experimental` contexts visible ... ", "Text"], Cell[BoxData[{ \(\(<< Developer`;\)\), "\[IndentingNewLine]", \(\(<< Experimental`;\)\)}], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["IMS Requests", "Section"], Cell["\<\ In the IMS97 conference there where of several requests made in the Q & A \ session and these are the ones I remember now off the top of my mind: \ \>", "Text"], Cell[TextData[{ "\[Bullet] ", StyleBox["Request1", FontSlant->"Italic"], ": Can you please update the report on warning messages, this was a great \ source of additional information on what might have generated a certain \ message etc. " }], "Text"], Cell["\<\ Done, this is now available in the help browser. You will, eventually see \ these linked to from the actual messages themselves. \ \>", "Text"], Cell[TextData[{ "\[Bullet] ", StyleBox["Request2", FontSlant->"Italic"], ": There should be a Java interface to ", StyleBox["MathLink", FontSlant->"Italic"], ". " }], "Text"], Cell[TextData[{ "Done, there is now an alpha version shipping with ", StyleBox["Mathematica", FontSlant->"Italic"], " 4. You will see something more comprehensive come out later, as now the \ JNI has more or less been settled. " }], "Text"], Cell[TextData[{ "\[Bullet] ", StyleBox["Request3", FontSlant->"Italic"], ": There should be a Fortran interface to ", StyleBox["MathLink", FontSlant->"Italic"], ". " }], "Text"], Cell[TextData[{ "Done, you will see a demo at the Developer Conference this fall. And a \ little bit later a release of the Fortran ", StyleBox["MathLink", FontSlant->"Italic"], " SDK." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Numerics", "Section"], Cell[TextData[{ "In numerics the basic theme is to make ", StyleBox["Mathematica", FontSlant->"Italic"], " faster and scalable to very large computations as well as round out the \ feature set. We do in fact have a multi-year program called ", StyleBox["gigaNumerics", FontWeight->"Bold", FontSlant->"Italic"], " that will produce deliverables along these lines for each subsequent ", StyleBox["Mathematica", FontSlant->"Italic"], " release and this upcoming version is the first such example. " }], "Text"], Cell[CellGroupData[{ Cell["Packed Array Technology", "Subsection"], Cell[TextData[{ "Packed Array technology, uses a special efficient representation of \ tensors (or arrays) of machine numbers and provide quite impressive memory \ and speed improvements. In essence this recognizes tensors (arrays) of \ machine numbers as a new (internal) fundamental type, i.e. much like \ individual machine numbers are made use of whenever possible and thus \ speeding up operations. This is an example of where we have take a general \ system like ", StyleBox["Mathematica", FontSlant->"Italic"], " and optimize its operation on a special class of expression, something \ that is ", StyleBox["much", FontSlant->"Italic"], " easier then taking a specialized system and try to generalize it. Best of \ all is that Packed Array technology is completely transparent to the user ... \ except for the speed improvement." }], "Text"], Cell["\<\ In fact this is very much in line with what microprocessor manufacturers such \ as Intel are doing with Pentium III (or MMX2), i.e. recognizing arrays of \ numbers as a new fundamental data type, although in their case it is much \ less transparent. \ \>", "Text"], Cell[CellGroupData[{ Cell["Memory and Speed", "Subsubsection"], Cell[TextData[{ "Let us connect two different kernels a 4.0 kernel ", StyleBox["(red border)", FontColor->RGBColor[1, 0, 0]], " and a 3.0 kernel ", StyleBox["(blue border).", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \($Version\)], "Input3.0"], Cell[BoxData[ \("Microsoft Windows 3.0 (September 17, 1997)"\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \($Version\)], "Input4.0"], Cell[BoxData[ \("4.0 for Microsoft Windows (April 21, 1999)"\)], "Output"] }, Open ]], Cell["\<\ This a 100,000 element list of integers in the new version as well as in 3.0.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[\ Random[Integer], \ {10^5}] // ByteCount\) // Timing\)], "Input3.0"], Cell[BoxData[ \({0.329999999999998294`\ Second, 2000024}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[\ Random[Integer], \ {10^5}] // ByteCount\) // Timing\)], "Input4.0"], Cell[BoxData[ \({0.3899999999999988`\ Second, 400056}\)], "Output"] }, Open ]], Cell["\<\ As can be seen there is some small performance improvement, but a rather \ large memory savings, i.e. using a fifth of the memory. We can do the same \ for real and complex numbers ... \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[\ Random[Real], \ {10^5}\ ] // ByteCount\) // Timing\)], "Input3.0"], Cell[BoxData[ \({0.490000000000001989`\ Second, 2000024}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[\ Random[Real], \ {10^5}\ ] // ByteCount\) // Timing\)], "Input4.0"], Cell[BoxData[ \({0.1700000000000017`\ Second, 800056}\)], "Output"] }, Open ]], Cell["Complex tensors ... ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[\ Random[Complex], \ {10^5}] // ByteCount\) // Timing\)], "Input3.0"], Cell[BoxData[ \({1.03999999999999914`\ Second, 6000024}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[\ Random[Complex], \ {10^5}] // ByteCount\) // Timing\)], "Input4.0"], Cell[BoxData[ \({0.3299999999999983`\ Second, 1600056}\)], "Output"] }, Open ]], Cell["\<\ For other types of operations the speed improvements can be vastly larger \ then this .... \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["System Integration", "Subsubsection"], Cell[TextData[{ "One of the really good features of Packed Arrays is that you typically \ will not have to know about them at all, i.e. this is all happening under the \ hood and automatically. In particular most things that you do in ", StyleBox["Mathematica", FontSlant->"Italic"], " when dealing tensors of numbers will see performance improvements \ automatically. These are some more examples: " }], "Text"], Cell[BoxData[ \(\(m\ = \ Table[Random[], {500}, \ {500}];\)\)], "Input3.0"], Cell[BoxData[ \(\(m\ = \ Table[Random[], {500}, \ {500}];\)\)], "Input4.0"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Sin[m];\) // Timing\)], "Input3.0"], Cell[BoxData[ \({2.0899999999999963`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Sin[m];\) // Timing\)], "Input4.0"], Cell[BoxData[ \({0.1600000000000037`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\((m + 1)\)\^100;\) // Timing\)], "Input3.0"], Cell[BoxData[ \({3.78999999999999914`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(\((m + 1)\)\^100;\) // Timing\), "Input3.0"]], "Input4.0"], Cell[BoxData[ \({0.5499999999999972`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Min[m];\) // Timing\)], "Input3.0"], Cell[BoxData[ \({3.63000000000000255`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Min[m];\) // Timing\)], "Input4.0"], Cell[BoxData[ \({0.060000000000002274`\ Second, Null}\)], "Output"] }, Open ]], Cell[TextData[{ "However in some case the result of a computation cannot be represented as \ a tensor of machine numbers and in those cases ", StyleBox["Mathematica", FontSlant->"Italic"], " automatically converts the PackedArray expression to a general expression \ and the operation just works they way it used to. " }], "Text"], Cell[TextData[{ "In this case ", StyleBox["Range", "InputWord"], " produces a Packed Array ... we need access to a Deveveloper` function to \ test whether something is represented as a packed array." }], "Text"], Cell[BoxData[ \(<< Developer`\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(lst\ = \ Range[10]\)], "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PackedArrayQ[\ lst\ ]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(lst\)\(\ \)\(+\)\(\ \)\(1\)\(\ \)\)\)], "Input"], Cell[BoxData[ \({2, 3, 4, 5, 6, 7, 8, 9, 10, 11}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PackedArrayQ[\ %\ ]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ ... but adding a symbol to it of course means that it no longer is a tensor \ or numbers. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lst\ + \ x\)], "Input"], Cell[BoxData[ \({1 + x, 2 + x, 3 + x, 4 + x, 5 + x, 6 + x, 7 + x, 8 + x, 9 + x, 10 + x}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PackedArrayQ[\ %\ ]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[TextData[{ "Some other cases where this happens is when the numbers in the tensor \ become too big to be able to be represented as machine numbers, i.e. where a \ C program would normally crash or generate an exception ... in ", StyleBox["Mathematica", FontSlant->"Italic"], " they are then represented as arbitrary precision numbers. " }], "Text"], Cell[TextData[{ "Packed Array technology is leading to performance improvements across the \ system, e.g. graphics, ", StyleBox["MathLink,", FontSlant->"Italic"], " importing and exporting of files and many other areas. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Completing the Features", "Subsubsection"], Cell[TextData[{ "One frequently needs to get parts of matrix (or general) tensor in order \ to do some further manipulation. We generalized the basic primitives for this \ somewhat in the new ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell["Suppose you have a general 3 by 5 matrix. ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((m\ = \ Table[\ a\_\(i, j\), \ {i, 3}, \ {j, 5}])\) // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(a\_\(1, 1\)\), \(a\_\(1, 2\)\), \(a\_\(1, 3\)\), \(a\_\(1, 4\)\), \(a\_\(1, 5\)\)}, {\(a\_\(2, 1\)\), \(a\_\(2, 2\)\), \(a\_\(2, 3\)\), \(a\_\(2, 4\)\), \(a\_\(2, 5\)\)}, {\(a\_\(3, 1\)\), \(a\_\(3, 2\)\), \(a\_\(3, 3\)\), \(a\_\(3, 4\)\), \(a\_\(3, 5\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "Interactively you can already easily pick out submatrices from such a \ matrix, e.g. row, column or general submatrix. ", StyleBox["Take", "InputWord"], " now takes several sequence specifications allowing it to deal with this \ nicely programmatically. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Take[m, \ All, \ {3}]\)], "Input"], Cell[BoxData[ \({{a\_\(1, 3\)}, {a\_\(2, 3\)}, {a\_\(3, 3\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The new keyword ", StyleBox["All", "InputWord"], " can also be used in ", StyleBox["Part", "InputWord"], ". The main difference between ", StyleBox["Take", "InputWord"], " and ", StyleBox["Part", "InputWord"], " is that ", StyleBox["Part", "InputWord"], " lowers the rank, i.e.if you use part to pick out a column you end up with \ a vector and not a matrix." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Part[m, \ All, \ 3\ ]\)], "Input"], Cell[BoxData[ \({a\_\(1, 3\), a\_\(2, 3\), a\_\(3, 3\)}\)], "Output"] }, Open ]], Cell["This picks out a submatrix ... ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Take[\ m, \ {2, 3}, \ {3, 5}\ ]\)], "Input"], Cell[BoxData[ \({{a\_\(2, 3\), a\_\(2, 4\), a\_\(2, 5\)}, {a\_\(3, 3\), a\_\(3, 4\), a\_\(3, 5\)}}\)], "Output"] }, Open ]], Cell["\<\ Of course all of this also works smoothly with Packed Arrays ... \ \>", "Text"], Cell[BoxData[ \(\(m\ = \ Table[\ Random[], \ {100}, \ {100}\ ];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(msub\ = \ Take[\ m, \ All, \ \(-20\)]; \ Dimensions[\ msub\ ]\)], "Input"], Cell[BoxData[ \({100, 20}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PackedArrayQ[\ msub\ ]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ ... but for larger matrices it is usually more informative to plot them \ instead of typesetting them or course ... something that also benefits from \ packed arrays\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ListDensityPlot[\ msub\ ];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: 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