(*********************************************************************** 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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For instance the following computes the discrete Fourier \ transform of a 500 by 500 matrix.\ \>", "Text"], Cell[BoxData[ \(\(m = \ Table[Random[], \ {500}, \ {500}\ ];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Fourier[\ m\ ];\) // Timing\)], "Input"], Cell[BoxData[ \({2.1499999999999773`\ Second, Null}\)], "Output"] }, Open ]], Cell["\<\ One thing worth noting about this implementation is that there are no \ restrictions such as having to have a size that is a power of two (as in some \ earlier algorithms in this field), or a tensor of a particular rank. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Convolution (Correlation)", "Subsubsection"], Cell["\<\ An operation closely related to the discrete Fourier transform is the \ convolution of data. 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", "Text"], Cell[BoxData[ \(\(Clear[\ a, \ b, \ c, \ d, \ e\ ];\)\)], "Input"], Cell[BoxData[ \(\(polys\ = \[IndentingNewLine]{\[IndentingNewLine]\(-120\) - 144\ b\ c + 6\ b\^2\ c\^3 - 648\ b\ d + 21\ b\^2\ c\^2\ d + 15\ b\^2\ c\ d\^2 + 9\ b\^2\ d\^3 - 8\ b\ c\^2\ e - 28\ b\ c\ d\ e + b\ d\^2\ e, \[IndentingNewLine]5184 - 240\ c - 432\ b\ c\^2 + 9\ b\^2\ c\^4 - 720\ b\ c\ d + 30\ b\^2\ c\^3\ d - 432\ b\ d\^2 + 39\ b\^2\ c\^2\ d\^2 + 18\ b\^2\ c\ d\^3 + 576\ c\ e - 24\ b\ c\^3\ e - 576\ d\ e - 16\ b\ c\^2\ d\ e + 16\ b\ c\ d\^2\ e + 24\ b\ d\^3\ e + 16\ c\^2\ e\^2 - 32\ c\ d\ e\^2 + 16\ d\^2\ e\^2, \[IndentingNewLine]5184 - 81\ b\ c\^2 + 216\ b\ c\ d - 162\ b\ d\^2 + 1008\ c\ e - 15\ b\ c\^3\ e - 1008\ d\ e + 15\ b\ c\^2\ d\ e + 40\ c\^2\ e\^2 - 80\ c\ d\ e\^2 + 40\ d\^2\ e\^2, \[IndentingNewLine]261 - 4\ b\ c\^2 + 4\ b\ c\ d - 3\ b\ d\^2 + 22\ c\ e - 22\ d\ e\[IndentingNewLine]};\)\)], "Input"], Cell["\<\ In this case using machine precision will not safely resolve the roots (you \ get a warning), so we use 200 digit precision ... (it doesn't complete in a \ reasonable time in version 3.0)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(solution\ = \ NSolve[\ polys \[Equal] 0, \ {b, \ c, \ d, e}, \ WorkingPrecision\ \[Rule] \ 200\ ];\) // Timing\)], "Input"], Cell[BoxData[ \({5.21999999999997`\ Second, Null}\)], "Output"] }, Open ]], Cell["This verifies the solution ... ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(Abs[\ polys /. solution\ ]\)\)\)], "Input"], Cell[BoxData[ \({{0``52.3536, 0``49.6568, 0``51.9243, 0``53.5829}, {0``52.3536, 0``49.6568, 0``51.9243, 0``53.5829}, {5.0978941156`0*^-57, 4.17619486`0*^-53, 7.9654595556623`0*^-59, 4.978412222289`0*^-60}, {1.244603055572`0*^-60, 5.0978941156`0*^-57, 1.9913648889156`0*^-59, 1.244603055572`0*^-60}, {0``71.4879, 0``70.6936, 0``70.867, 0``72.4602}, {0``71.4879, 0``70.6936, 0``70.867, 0``72.4602}, {7.074749280333`0*^-74, 2.8298997121333`0*^-73, 1.131959885`0*^-72, 1.768687320083`0*^-74}, {1.4149498560667`0*^-73, 2.26391977`0*^-72, 2.26391977`0*^-72, 3.537374640167`0*^-74}, {1.727233711`0*^-77, 2.635549486`0*^-82, 2.635549486`0*^-82, 8.236092143149`0*^-84}, {4.2168791773`0*^-81, 1.31777474290382`0*^-82, 6.5888737145191`0*^-83, 1.28688939737`0*^-85}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Systems of Sparse Linear Equations", "Subsubsection"], Cell["\<\ Another interesting class of problems to solve are of course large linear \ systems. These systems are often very sparse and we now also provide a sparse \ linear solver (as a developer function). \ \>", "Text"], Cell["Let us first define a sparse matrix structure ... ", "Text"], Cell[BoxData[ \(RandomSparseMatrix[ n_] := \[IndentingNewLine]Join[\[IndentingNewLine]Table[{j + 1, j} \[Rule] Random[], {j, n - 1}], \[IndentingNewLine]Table[{j, j} \[Rule] 2 + Random[], {j, n}], \[IndentingNewLine]Table[{j, j + 1} \[Rule] Random[], {j, n - 1}]\[IndentingNewLine]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(RandomSparseMatrix[\ 5\ ]\)], "Input"], Cell[BoxData[ \({{2, 1} \[Rule] 0.4584027098433146`, {3, 2} \[Rule] 0.34623945624717023`, {4, 3} \[Rule] 0.18384049881498712`, {5, 4} \[Rule] 0.4050506704886163`, {1, 1} \[Rule] 2.9872541236680648`, {2, 2} \[Rule] 2.891647770318401`, {3, 3} \[Rule] 2.2633614418269565`, {4, 4} \[Rule] 2.3362711811950145`, {5, 5} \[Rule] 2.1141347869107694`, {1, 2} \[Rule] 0.22468317866854765`, {2, 3} \[Rule] 0.7053799839518786`, {3, 4} \[Rule] 0.12314828360746874`, {4, 5} \[Rule] 0.723005868485362`}\)], "Output"] }, Open ]], Cell["\<\ Let us define a 100,000 \[Times] 100,000 matrix ... \ \>", "Text"], Cell[BoxData[{ \(\(A = RandomSparseMatrix[10^5];\)\), "\[IndentingNewLine]", \(\(b = Table[Random[], {10^5}];\)\)}], "Input"], Cell["Due to sparsity this still works remarkably fast ... ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(solution\ = SparseLinearSolve[A, \ b];\) // Timing\)], "Input"], Cell[BoxData[ \({5.659999999999968`\ Second, Null}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Minimization", "Subsubsection"], Cell["\<\ Similarly there are several robustness and performance improvements in the \ various optimization functions. 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s .02383 .33924 m .02752 .33785 L s .02199 .35084 m .02568 .34946 L s .02014 .36253 m .02383 .36116 L s .0164 .38616 m .02009 .38481 L s .0145 .39811 m .0182 .39677 L s .01259 .41015 m .0163 .40881 L s .01067 .42228 m .01438 .42095 L s .00678 .44682 m .01049 .4455 L s .00482 .45923 m .00853 .45792 L s .00284 .47173 m .00655 .47043 L s .00084 .48433 m .00456 .48303 L s .25 Mabswid .03716 .25514 m 0 .48963 L s 0 .48963 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s .40529 .59895 m .03716 .25514 L s .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid .811 .817 .855 r .38194 .61173 .40436 .62545 .44177 .59774 .41995 .58032 Metetra .731 .786 .897 r .41995 .58032 .44177 .59774 .47916 .58181 .45779 .56132 Metetra .661 .746 .913 r .45779 .56132 .47916 .58181 .51694 .57297 .49593 .54995 Metetra .621 .713 .911 r .49593 .54995 .51694 .57297 .55534 .56746 .53462 .54239 Metetra .614 .695 .901 r .53462 .54239 .55534 .56746 .59443 .56228 .57398 .53565 Metetra .627 .69 .889 r .57398 .53565 .59443 .56228 .63417 .55528 .61399 .52757 Metetra .65 .692 .876 r .61399 .52757 .63417 .55528 .67446 .54508 .65458 .51678 Metetra .673 .699 .866 r .65458 .51678 .67446 .54508 .71517 .53121 .69565 .50281 Metetra .691 .708 .861 r .69565 .50281 .71517 .53121 .75622 .51407 .73712 .48608 Metetra .701 .717 .862 r .73712 .48608 .75622 .51407 .7976 .49498 .779 .46789 Metetra .696 .724 .872 r .779 .46789 .7976 .49498 .83946 .47614 .82146 .45047 Metetra .666 .727 .895 r .82146 .45047 .83946 .47614 .88218 .46062 .86486 .4369 Metetra .586 .713 .931 r .86486 .4369 .88218 .46062 .92638 .45243 .90987 .43125 Metetra .389 .639 .955 r .90987 .43125 .92638 .45243 .9731 .45666 .95754 .43872 Metetra .827 .822 .844 r .3588 .5985 .38194 .61173 .41995 .58032 .39749 .56331 Metetra .752 .797 .889 r .39749 .56331 .41995 .58032 .45779 .56132 .43584 .54118 Metetra .682 .76 .91 r .43584 .54118 .45779 .56132 .49593 .54995 .47437 .52723 Metetra .639 .727 .912 r .47437 .52723 .49593 .54995 .53462 .54239 .51339 .5176 Metetra .626 .707 .904 r .51339 .5176 .53462 .54239 .57398 .53565 .55304 .50928 Metetra .634 .699 .893 r .55304 .50928 .57398 .53565 .61399 .52757 .59336 .5001 Metetra .652 .7 .881 r .59336 .5001 .61399 .52757 .65458 .51678 .63427 .48871 Metetra .672 .705 .873 r .63427 .48871 .65458 .51678 .69565 .50281 .67571 .47464 Metetra .687 .713 .868 r .67571 .47464 .69565 .50281 .73712 .48608 .71761 .4583 Metetra .693 .721 .871 r .71761 .4583 .73712 .48608 .779 .46789 .76 .44102 Metetra .684 .727 .883 r .76 .44102 .779 .46789 .82146 .45047 .80305 .42502 Metetra .646 .727 .908 r .80305 .42502 .82146 .45047 .86486 .4369 .84714 .41342 Metetra .549 .705 .943 r .84714 .41342 .86486 .4369 .90987 .43125 .89296 .41035 Metetra .318 .608 .951 r .89296 .41035 .90987 .43125 .95754 .43872 .94159 .42112 Metetra .841 .825 .833 r .3349 .58575 .3588 .5985 .39749 .56331 .37434 .54669 Metetra .771 .805 .881 r .37434 .54669 .39749 .56331 .43584 .54118 .41326 .52137 Metetra .703 .772 .906 r .41326 .52137 .43584 .54118 .47437 .52723 .45223 .5048 Metetra .657 .74 .911 r .45223 .5048 .47437 .52723 .51339 .5176 .49161 .49308 Metetra .638 .718 .906 r .49161 .49308 .51339 .5176 .55304 .50928 .5316 .48316 Metetra .641 .709 .896 r .5316 .48316 .55304 .50928 .59336 .5001 .57223 .47287 Metetra .654 .708 .887 r .57223 .47287 .59336 .5001 .63427 .48871 .61349 .46087 Metetra .67 .712 .879 r .61349 .46087 .63427 .48871 .67571 .47464 .65531 .44668 Metetra .682 .718 .876 r .65531 .44668 .67571 .47464 .71761 .4583 .69765 .43073 Metetra .685 .724 .88 r .69765 .43073 .71761 .4583 .76 .44102 .74056 .41435 Metetra .67 .729 .894 r .74056 .41435 .76 .44102 .80305 .42502 .7842 .39978 Metetra .623 .725 .92 r .7842 .39978 .80305 .42502 .84714 .41342 .82899 .39017 Metetra .506 .693 .952 r .82899 .39017 .84714 .41342 .89296 .41035 .87562 .38971 Metetra .24 .569 .939 r .87562 .38971 .89296 .41035 .94159 .42112 .92521 .40386 Metetra .853 .828 .822 r .31018 .57352 .3349 .58575 .37434 .54669 .35046 .53048 Metetra .789 .813 .872 r .35046 .53048 .37434 .54669 .41326 .52137 .39001 .5019 Metetra .722 .784 .901 r .39001 .5019 .41326 .52137 .45223 .5048 .42947 .48267 Metetra .674 .753 .91 r .42947 .48267 .45223 .5048 .49161 .49308 .46926 .46882 Metetra .65 .73 .908 r .46926 .46882 .49161 .49308 .5316 .48316 .5096 .45728 Metetra .648 .719 .9 r .5096 .45728 .5316 .48316 .57223 .47287 .55058 .44586 Metetra .657 .716 .892 r .55058 .44586 .57223 .47287 .61349 .46087 .5922 .43323 Metetra .669 .718 .885 r .5922 .43323 .61349 .46087 .65531 .44668 .63442 .41891 Metetra .677 .723 .884 r .63442 .41891 .65531 .44668 .69765 .43073 .67721 .40335 Metetra .675 .728 .89 r .67721 .40335 .69765 .43073 .74056 .41435 .72064 .38787 Metetra .654 .731 .905 r .72064 .38787 .74056 .41435 .7842 .39978 .76489 .37473 Metetra .596 .722 .932 r .76489 .37473 .7842 .39978 .82899 .39017 .81037 .36714 Metetra .457 .677 .959 r .81037 .36714 .82899 .39017 .87562 .38971 .85781 .36934 Metetra .155 .523 .918 r .85781 .36934 .87562 .38971 .92521 .40386 .90836 .38696 Metetra .864 .83 .812 r .28457 .56181 .31018 .57352 .35046 .53048 .32578 .51468 Metetra .805 .82 .863 r .32578 .51468 .35046 .53048 .39001 .5019 .36603 .48276 Metetra .741 .794 .895 r .36603 .48276 .39001 .5019 .42947 .48267 .40604 .46082 Metetra .69 .765 .909 r .40604 .46082 .42947 .48267 .46926 .46882 .44627 .4448 Metetra .662 .742 .909 r .44627 .4448 .46926 .46882 .5096 .45728 .487 .43161 Metetra .655 .728 .903 r .487 .43161 .5096 .45728 .55058 .44586 .52836 .41905 Metetra .659 .724 .897 r .52836 .41905 .55058 .44586 .5922 .43323 .57036 .40578 Metetra .667 .724 .892 r .57036 .40578 .5922 .43323 .63442 .41891 .61299 .39133 Metetra .672 .728 .892 r .61299 .39133 .63442 .41891 .67721 .40335 .65625 .37614 Metetra .665 .732 .899 r .65625 .37614 .67721 .40335 .72064 .38787 .70021 .36157 Metetra .637 .732 .917 r .70021 .36157 .72064 .38787 .76489 .37473 .74506 .34988 Metetra .566 .717 .944 r .74506 .34988 .76489 .37473 .81037 .36714 .79124 .34431 Metetra .402 .657 .962 r .79124 .34431 .81037 .36714 .85781 .36934 .8395 .34924 Metetra .066 .47 .888 r .8395 .34924 .85781 .36934 .90836 .38696 .89101 .37041 Metetra .874 .832 .802 r .25802 .55064 .28457 .56181 .32578 .51468 .30025 .4993 Metetra .819 .825 .854 r .30025 .4993 .32578 .51468 .36603 .48276 .34127 .46396 Metetra .758 .803 .889 r .34127 .46396 .36603 .48276 .40604 .46082 .38189 .43925 Metetra .706 .776 .906 r .38189 .43925 .40604 .46082 .44627 .4448 .42262 .42103 Metetra .674 .753 .91 r .42262 .42103 .44627 .4448 .487 .43161 .46378 .40616 Metetra .662 .738 .906 r .46378 .40616 .487 .43161 .52836 .41905 .50553 .39243 Metetra .662 .732 .902 r .50553 .39243 .52836 .41905 .57036 .40578 .54794 .3785 Metetra .665 .731 .898 r .54794 .3785 .57036 .40578 .61299 .39133 .591 .36391 Metetra .665 .733 .9 r .591 .36391 .61299 .39133 .65625 .37614 .63474 .34909 Metetra .654 .735 .909 r .63474 .34909 .65625 .37614 .70021 .36157 .67922 .33542 Metetra .617 .732 .928 r .67922 .33542 .70021 .36157 .74506 .34988 .72468 .32519 Metetra .532 .71 .954 r .72468 .32519 .74506 .34988 .79124 .34431 .77156 .32169 Metetra .34 .631 .96 r .77156 .32169 .79124 .34431 .8395 .34924 .82063 .32939 Metetra 0 .412 .848 r .82063 .32939 .8395 .34924 .89101 .37041 .8731 .35422 Metetra .883 .833 .792 r .23045 .54003 .25802 .55064 .30025 .4993 .27381 .48434 Metetra .833 .83 .844 r .27381 .48434 .30025 .4993 .34127 .46396 .31569 .4455 Metetra .774 .812 .883 r .31569 .4455 .34127 .46396 .38189 .43925 .35697 .41795 Metetra .721 .787 .904 r .35697 .41795 .38189 .43925 .42262 .42103 .39824 .39748 Metetra .686 .764 .91 r .39824 .39748 .42262 .42103 .46378 .40616 .43987 .38091 Metetra .669 .748 .909 r .43987 .38091 .46378 .40616 .50553 .39243 .48205 .36599 Metetra .664 .74 .906 r .48205 .36599 .50553 .39243 .54794 .3785 .52489 .35138 Metetra .663 .738 .905 r .52489 .35138 .54794 .3785 .591 .36391 .5684 .33663 Metetra .659 .738 .908 r .5684 .33663 .591 .36391 .63474 .34909 .61261 .32218 Metetra .641 .738 .919 r .61261 .32218 .63474 .34909 .67922 .33542 .65764 .30941 Metetra .595 .731 .939 r .65764 .30941 .67922 .33542 .72468 .32519 .70371 .30065 Metetra .493 .7 .962 r .70371 .30065 .72468 .32519 .77156 .32169 .75129 .29926 Metetra .272 .6 .951 r .75129 .29926 .77156 .32169 .82063 .32939 .80116 .30979 Metetra 0 .35 .801 r .80116 .30979 .82063 .32939 .8731 .35422 .85459 .33839 Metetra .891 .833 .783 r .20177 .53001 .23045 .54003 .27381 .48434 .24638 .46982 Metetra .845 .833 .835 r .24638 .46982 .27381 .48434 .31569 .4455 .28921 .42737 Metetra .789 .819 .876 r .28921 .42737 .31569 .4455 .35697 .41795 .33122 .39692 Metetra .736 .797 .9 r .33122 .39692 .35697 .41795 .39824 .39748 .37309 .37415 Metetra .698 .774 .91 r .37309 .37415 .39824 .39748 .43987 .38091 .41523 .35584 Metetra .676 .758 .912 r .41523 .35584 .43987 .38091 .48205 .36599 .45788 .33971 Metetra .667 .748 .911 r .45788 .33971 .48205 .36599 .52489 .35138 .50116 .32441 Metetra .661 .744 .912 r .50116 .32441 .52489 .35138 .5684 .33663 .54513 .30948 Metetra .651 .743 .917 r .54513 .30948 .5684 .33663 .61261 .32218 .58984 .2954 Metetra .627 .74 .929 r .58984 .2954 .61261 .32218 .65764 .30941 .63541 .28353 Metetra .571 .729 .949 r .63541 .28353 .65764 .30941 .70371 .30065 .6821 .27626 Metetra .449 .687 .967 r .6821 .27626 .70371 .30065 .75129 .29926 .73036 .277 Metetra .198 .562 .936 r .73036 .277 .75129 .29926 .80116 .30979 .78103 .29043 Metetra 0 .287 .747 r .78103 .29043 .80116 .30979 .85459 .33839 .83541 .32293 Metetra .898 .834 .774 r .1719 .52058 .20177 .53001 .24638 .46982 .2179 .45574 Metetra .856 .836 .826 r .2179 .45574 .24638 .46982 .28921 .42737 .26176 .40957 Metetra .803 .826 .869 r .26176 .40957 .28921 .42737 .33122 .39692 .30459 .37615 Metetra .75 .806 .897 r .30459 .37615 .33122 .39692 .37309 .37415 .34711 .35103 Metetra .71 .784 .91 r .34711 .35103 .37309 .37415 .41523 .35584 .3898 .33094 Metetra .684 .767 .915 r .3898 .33094 .41523 .35584 .45788 .33971 .43294 .31357 Metetra .669 .757 .916 r .43294 .31357 .45788 .33971 .50116 .32441 .4767 .29755 Metetra .659 .751 .918 r .4767 .29755 .50116 .32441 .54513 .30948 .52115 .28245 Metetra .643 .748 .925 r .52115 .28245 .54513 .30948 .58984 .2954 .56637 .26871 Metetra .611 .742 .939 r .56637 .26871 .58984 .2954 .63541 .28353 .61249 .25776 Metetra .543 .725 .958 r .61249 .25776 .63541 .28353 .6821 .27626 .65978 .25199 Metetra .4 .671 .969 r .65978 .25199 .6821 .27626 .73036 .277 .70874 .25491 Metetra .12 .518 .912 r .70874 .25491 .73036 .277 .78103 .29043 .7602 .27132 Metetra 0 .223 .689 r .7602 .27132 .78103 .29043 .83541 .32293 .81552 .30785 Metetra .905 .834 .765 r .14075 .51179 .1719 .52058 .2179 .45574 .18826 .44212 Metetra .866 .839 .817 r .18826 .44212 .2179 .45574 .26176 .40957 .23328 .39212 Metetra .816 .832 .862 r .23328 .39212 .26176 .40957 .30459 .37615 .27699 .35564 Metetra .764 .814 .893 r .27699 .35564 .30459 .37615 .34711 .35103 .32023 .32811 Metetra .721 .794 .909 r .32023 .32811 .34711 .35103 .3898 .33094 .36352 .3062 Metetra .691 .777 .917 r .36352 .3062 .3898 .33094 .43294 .31357 .40719 .28755 Metetra .672 .765 .92 r .40719 .28755 .43294 .31357 .4767 .29755 .45145 .27081 Metetra .656 .758 .925 r .45145 .27081 .4767 .29755 .52115 .28245 .4964 .2555 Metetra .634 .753 .933 r .4964 .2555 .52115 .28245 .56637 .26871 .54214 .24211 Metetra .594 .744 .948 r .54214 .24211 .56637 .26871 .61249 .25776 .58881 .23207 Metetra .512 .72 .966 r .58881 .23207 .61249 .25776 .65978 .25199 .63671 .22783 Metetra .346 .649 .967 r .63671 .22783 .65978 .25199 .70874 .25491 .68635 .23297 Metetra .038 .47 .881 r .68635 .23297 .70874 .25491 .7602 .27132 .73859 .25245 Metetra 0 .162 .63 r .73859 .25245 .7602 .27132 .81552 .30785 .79484 .29317 Metetra .91 .833 .757 r .1082 .50367 .14075 .51179 .18826 .44212 .15739 .42897 Metetra .875 .841 .809 r .15739 .42897 .18826 .44212 .23328 .39212 .20367 .375 Metetra .828 .837 .855 r .20367 .375 .23328 .39212 .27699 .35564 .24837 .33537 Metetra .777 .822 .888 r .24837 .33537 .27699 .35564 .32023 .32811 .29238 .30538 Metetra .732 .804 .908 r .29238 .30538 .32023 .32811 .36352 .3062 .33632 .2816 Metetra .699 .786 .919 r .33632 .2816 .36352 .3062 .40719 .28755 .38057 .26165 Metetra .674 .773 .925 r .38057 .26165 .40719 .28755 .45145 .27081 .42536 .24414 Metetra .653 .765 .931 r .42536 .24414 .45145 .27081 .4964 .2555 .47082 .22862 Metetra .625 .757 .941 r .47082 .22862 .4964 .2555 .54214 .24211 .51709 .21557 Metetra .575 .744 .957 r .51709 .21557 .54214 .24211 .58881 .23207 .56432 .20646 Metetra .478 .712 .973 r .56432 .20646 .58881 .23207 .63671 .22783 .61282 .20377 Metetra .287 .624 .961 r .61282 .20377 .63671 .22783 .68635 .23297 .66313 .21117 Metetra 0 .417 .842 r .66313 .21117 .68635 .23297 .73859 .25245 .71614 .23381 Metetra 0 .105 .571 r .71614 .23381 .73859 .25245 .79484 .29317 .7733 .27888 Metetra .915 .833 .75 r .07415 .49624 .1082 .50367 .15739 .42897 .12519 .41631 Metetra .883 .842 .8 r .12519 .41631 .15739 .42897 .20367 .375 .17286 .35823 Metetra .839 .841 .847 r .17286 .35823 .20367 .375 .24837 .33537 .21863 .31535 Metetra .789 .83 .883 r .21863 .31535 .24837 .33537 .29238 .30538 .2635 .28282 Metetra .743 .812 .907 r .2635 .28282 .29238 .30538 .33632 .2816 .30814 .25712 Metetra .706 .795 .92 r .30814 .25712 .33632 .2816 .38057 .26165 .353 .23583 Metetra .677 .782 .929 r .353 .23583 .38057 .26165 .42536 .24414 .39835 .21754 Metetra .65 .771 .938 r .39835 .21754 .42536 .24414 .47082 .22862 .44435 .20179 Metetra .614 .761 .949 r .44435 .20179 .47082 .22862 .51709 .21557 .49115 .18908 Metetra .554 .744 .965 r .49115 .18908 .51709 .21557 .56432 .20646 .53894 .18089 Metetra .44 .702 .977 r .53894 .18089 .56432 .20646 .61282 .20377 .58805 .17977 Metetra .223 .593 .948 r .58805 .17977 .61282 .20377 .66313 .21117 .63901 .1895 Metetra 0 .361 .797 r .63901 .1895 .66313 .21117 .71614 .23381 .69276 .21541 Metetra .481 0 0 r .69276 .21541 .71614 .23381 .7733 .27888 .75081 .26501 Metetra .92 .832 .742 r .03844 .48954 .07415 .49624 .12519 .41631 .09153 .40414 Metetra .89 .843 .792 r .09153 .40414 .12519 .41631 .17286 .35823 .14074 .3418 Metetra .849 .845 .84 r .14074 .3418 .17286 .35823 .21863 .31535 .18768 .29557 Metetra .801 .836 .878 r .18768 .29557 .21863 .31535 .2635 .28282 .23349 .26043 Metetra .754 .821 .905 r .23349 .26043 .2635 .28282 .30814 .25712 .2789 .23275 Metetra .713 .804 .922 r .2789 .23275 .30814 .25712 .353 .23583 .32441 .21009 Metetra .68 .79 .933 r .32441 .21009 .353 .23583 .39835 .21754 .37035 .19098 Metetra .646 .778 .944 r .37035 .19098 .39835 .21754 .44435 .20179 .4169 .17498 Metetra .603 .765 .957 r .4169 .17498 .44435 .20179 .49115 .18908 .46425 .1626 Metetra .531 .742 .972 r .46425 .1626 .49115 .18908 .53894 .18089 .51261 .15534 Metetra .398 .689 .978 r .51261 .15534 .53894 .18089 .58805 .17977 .5623 .15583 Metetra .156 .558 .929 r .5623 .15583 .58805 .17977 .63901 .1895 .6139 .16795 Metetra 0 .304 .748 r .6139 .16795 .63901 .1895 .69276 .21541 .66837 .19723 Metetra .532 0 0 r .66837 .19723 .69276 .21541 .75081 .26501 .72727 .25156 Metetra .924 .831 .735 r .00095 .48363 .03844 .48954 .09153 .40414 .0563 .39251 Metetra .897 .844 .784 r .0563 .39251 .09153 .40414 .14074 .3418 .10719 .32573 Metetra .858 .848 .832 r .10719 .32573 .14074 .3418 .18768 .29557 .15544 .27603 Metetra .812 .842 .873 r .15544 .27603 .18768 .29557 .23349 .26043 .20226 .23819 Metetra .764 .829 .903 r .20226 .23819 .23349 .26043 .2789 .23275 .2485 .20848 Metetra .721 .813 .922 r .2485 .20848 .2789 .23275 .32441 .21009 .29472 .18439 Metetra .682 .798 .937 r .29472 .18439 .32441 .21009 .37035 .19098 .34128 .16444 Metetra .643 .785 .95 r .34128 .16444 .37035 .19098 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.56868 L .03213 .54906 L .03718 .52943 L .03885 .52297 L .04229 .50981 L .04745 .49019 L .05269 .47057 L .05799 .45094 L .05848 .44914 L .06334 .43132 L .06876 .4117 L .07426 .39207 L .0781 .37851 L .07982 .37245 L .08546 .35283 L .09118 .3332 L .09698 .31358 L .09772 .31109 L .10286 .29396 L .10883 .27433 L .1149 .25471 L .11735 .24687 L .12105 .23509 L .12731 .21546 L .13369 .19584 L .13697 .18587 L .14017 .17622 L .14677 .15659 L .1535 .13697 L .15659 .12807 L .16035 .11735 L .16735 .09772 L .17451 .0781 L .17622 .07348 L .18182 .05848 L .1893 .03885 L .19584 .0221 L .19697 .01923 L s .98077 .64918 m .98026 .64717 L .97531 .62755 L .97033 .60793 L .9653 .5883 L .96115 .57223 L .96022 .56868 L .95508 .54906 L .94987 .52943 L .94459 .50981 L .94152 .4985 L .93926 .49019 L .93387 .47057 L .9284 .45094 L .92285 .43132 L .9219 .42797 L .91725 .4117 L .91156 .39207 L .90579 .37245 L .90228 .36065 L .89994 .35283 L .894 .3332 L .88797 .31358 L .88265 .29654 L .88184 .29396 L .87562 .27433 L .86929 .25471 L .86303 .23564 L .86285 .23509 L .8563 .21546 L .84962 .19584 L .84341 .17794 L .8428 .17622 L .83586 .15659 L .82876 .13697 L .82378 .12346 L .82151 .11735 L .8141 .09772 L .80649 .0781 L .80416 .07218 L .7987 .05848 L .7907 .03885 L .78454 .02411 L .78247 .01923 L s .01923 .65189 m .02042 .64717 L .02537 .62755 L .03035 .60793 L .03537 .5883 L .03885 .57484 L .04045 .56868 L .0456 .54906 L .05081 .52943 L .05608 .50981 L .05848 .50099 L .06142 .49019 L .06681 .47057 L .07228 .45094 L .07783 .43132 L .0781 .43035 L .08343 .4117 L .08912 .39207 L .09489 .37245 L .09772 .36292 L .10074 .35283 L .10667 .3332 L .11271 .31358 L .11735 .2987 L .11883 .29396 L .12505 .27433 L .13138 .25471 L .13697 .23769 L .13783 .23509 L .14438 .21546 L .15106 .19584 L .15659 .17988 L .15787 .17622 L .16482 .15659 L .17191 .13697 L .17622 .12528 L .17917 .11735 L .18658 .09772 L .19419 .0781 L .19584 .0739 L .20197 .05848 L .20997 .03885 L .21546 .02572 L .2182 .01923 L s .98077 .70441 m .97622 .68642 L .97125 .6668 L .96624 .64717 L .96117 .62755 L .96115 .62746 L .95604 .60793 L .95084 .5883 L .94558 .56868 L .94152 .55372 L .94025 .54906 L .93487 .52943 L .92942 .50981 L .92389 .49019 L .9219 .48319 L .9183 .47057 L .91263 .45094 L .90687 .43132 L .90228 .41587 L .90103 .4117 L .89511 .39207 L .8891 .37245 L .88299 .35283 L .88265 .35176 L .87679 .3332 L .87048 .31358 L .86406 .29396 L .86303 .29085 L .85753 .27433 L .85087 .25471 L .84408 .23509 L .84341 .23316 L .83717 .21546 L .8301 .19584 L .82378 .17867 L .82287 .17622 L .81549 .15659 L .80793 .13697 L .80416 .12739 L .80017 .11735 L .79221 .09772 L .78454 .07932 L .78402 .0781 L .7756 .05848 L .7669 .03885 L .76491 .03446 L .75792 .01923 L s .01923 .70732 m .01955 .70604 L .0245 .68642 L .02948 .6668 L .03449 .64717 L .03885 .63026 L .03956 .62755 L .04469 .60793 L .04988 .5883 L .05515 .56868 L .05848 .5564 L .06047 .54906 L .06585 .52943 L .07131 .50981 L .07684 .49019 L .0781 .48575 L .08243 .47057 L .0881 .45094 L .09386 .43132 L .09772 .41831 L .0997 .4117 L .10561 .39207 L .11163 .37245 L .11735 .35407 L .11774 .35283 L .12394 .3332 L .13025 .31358 L .13667 .29396 L .13697 .29305 L .1432 .27433 L .14985 .25471 L .15659 .23524 L .15665 .23509 L .16356 .21546 L .17063 .19584 L .17622 .18063 L .17785 .17622 L .18523 .15659 L .1928 .13697 L .19584 .12923 L .20055 .11735 L .20851 .09772 L .21546 .08104 L .2167 .0781 L .22512 .05848 L .23382 .03885 L .23509 .03606 L .2428 .01923 L s .98077 .764 m .97603 .74529 L .97106 .72567 L .96605 .70604 L .96115 .68705 L .96098 .68642 L .95585 .6668 L .95065 .64717 L .94539 .62755 L .94152 .61331 L .94006 .60793 L .93468 .5883 L .92922 .56868 L .92369 .54906 L .9219 .54277 L .91809 .52943 L .91242 .50981 L .90666 .49019 L .90228 .47545 L .90082 .47057 L .8949 .45094 L .88889 .43132 L .88277 .4117 L .88265 .41133 L .87657 .39207 L .87026 .37245 L .86383 .35283 L .86303 .35042 L .85729 .3332 L .85063 .31358 L .84384 .29396 L .84341 .29272 L .83692 .27433 L .82985 .25471 L .82378 .23824 L .82261 .23509 L .81523 .21546 L .80765 .19584 L .80416 .18695 L .79989 .17622 L .79193 .15659 L .78454 .13888 L .78373 .13697 L .7753 .11735 L .76659 .09772 L .76491 .09402 L .7576 .0781 L .74827 .05848 L .74529 .05237 L .73859 .03885 L Mistroke .72848 .01923 L Mfstroke .01923 .76717 m .0198 .76491 L .02476 .74529 L .02973 .72567 L .03474 .70604 L .03885 .69008 L .0398 .68642 L .04494 .6668 L .05014 .64717 L .0554 .62755 L .05848 .61621 L .06073 .60793 L .06611 .5883 L .07157 .56868 L .0771 .54906 L .0781 .54555 L .08269 .52943 L .08837 .50981 L .09413 .49019 L .09772 .47809 L .09997 .47057 L .10589 .45094 L .1119 .43132 L .11735 .41385 L .11802 .4117 L .12422 .39207 L .13053 .37245 L .13696 .35283 L .13697 .35281 L .14349 .3332 L .15015 .31358 L .15659 .29498 L .15695 .29396 L .16387 .27433 L .17094 .25471 L .17622 .24036 L .17817 .23509 L .18556 .21546 L .19313 .19584 L .19584 .18895 L .20089 .17622 L .20886 .15659 L .21546 .14075 L .21706 .13697 L .22549 .11735 L .2342 .09772 L .23509 .09576 L .24319 .0781 L .25252 .05848 L .25471 .05398 L Mistroke .2622 .03885 L .27231 .01923 L Mfstroke .98077 .82919 m .97939 .82378 L .97442 .80416 L .96944 .78454 L .96442 .76491 L .96115 .75223 L .95934 .74529 L .95419 .72567 L .94897 .70604 L .94368 .68642 L .94152 .67848 L .93834 .6668 L .93294 .64717 L .92746 .62755 L .9219 .60794 L .9219 .60793 L .91629 .5883 L .91059 .56868 L .9048 .54906 L .90228 .54061 L .89893 .52943 L .89299 .50981 L .88694 .49019 L .88265 .47649 L .88079 .47057 L .87456 .45094 L .86821 .43132 L .86303 .41558 L .86175 .4117 L .85518 .39207 L .84848 .37245 L .84341 .35788 L .84164 .35283 L .83468 .3332 L .82755 .31358 L .82378 .30338 L .82027 .29396 L .81283 .27433 L .80519 .25471 L .80416 .2521 L .79737 .23509 L .78933 .21546 L .78454 .20403 L .78106 .19584 L .77255 .17622 L .76491 .15917 L .76375 .15659 L .75466 .13697 L .74529 .11751 L .74521 .11735 L Mistroke .73541 .09772 L .72567 .07907 L .72515 .0781 L .71443 .05848 L .70604 .04383 L .70312 .03885 L .69116 .01923 L Mfstroke .01923 .83268 m .02149 .82378 L .02645 .80416 L .03142 .78454 L .03644 .76491 L .03885 .75558 L .04153 .74529 L .04668 .72567 L .0519 .70604 L .05719 .68642 L .05848 .68168 L .06253 .6668 L .06793 .64717 L .07341 .62755 L .0781 .611 L .07897 .60793 L .08458 .5883 L .09029 .56868 L .09608 .54906 L .09772 .54353 L .10193 .52943 L .10788 .50981 L .11393 .49019 L .11735 .47926 L .12007 .47057 L .12631 .45094 L .13266 .43132 L .13697 .41821 L .13912 .4117 L .14569 .39207 L .1524 .37245 L .15659 .36037 L .15923 .35283 L .16619 .3332 L .17332 .31358 L .17622 .30573 L .1806 .29396 L .18804 .27433 L .19568 .25471 L .19584 .25431 L .2035 .23509 L .21154 .21546 L .21546 .20609 L .2198 .19584 L .22832 .17622 L .23509 .16109 L .23712 .15659 L .24621 .13697 L .25471 .11929 L .25566 .11735 L Mistroke .26546 .09772 L .27433 .0807 L .27571 .0781 L .28644 .05848 L .29396 .04533 L .29774 .03885 L .30971 .01923 L Mfstroke .98077 .90192 m .97586 .88265 L .97089 .86303 L .9659 .84341 L .96115 .82495 L .96084 .82378 L .95572 .80416 L .95052 .78454 L .94525 .76491 L .94152 .75119 L .93992 .74529 L .93455 .72567 L .92909 .70604 L .92355 .68642 L .9219 .68064 L .91795 .6668 L .91228 .64717 L .90652 .62755 L .90228 .61331 L .90067 .60793 L .89476 .5883 L .88874 .56868 L .88265 .54918 L .88262 .54906 L .87642 .52943 L .87011 .50981 L .86367 .49019 L .86303 .48826 L .85714 .47057 L .85048 .45094 L .84367 .43132 L .84341 .43056 L .83676 .4117 L .82968 .39207 L .82378 .37606 L .82244 .37245 L .81506 .35283 L .80748 .3332 L .80416 .32478 L .79971 .31358 L .79174 .29396 L .78454 .2767 L .78354 .27433 L .77511 .25471 L .76639 .23509 L .76491 .23184 L .75739 .21546 L .74805 .19584 L .74529 .19018 L .73836 .17622 L Mistroke .72824 .15659 L .72567 .15174 L .71767 .13697 L .70654 .11735 L .70604 .1165 L .69479 .09772 L .68642 .08448 L .68227 .0781 L .66882 .05848 L .6668 .05567 L .65421 .03885 L .64717 .03007 L .63806 .01923 L Mfstroke .01923 .90586 m .02015 .90228 L .02513 .88265 L .03009 .86303 L .03508 .84341 L .03885 .82873 L .04014 .82378 L .04526 .80416 L .05046 .78454 L .05574 .76491 L .05848 .75481 L .06106 .74529 L .06644 .72567 L .0719 .70604 L .07744 .68642 L .0781 .6841 L .08303 .6668 L .0887 .64717 L .09447 .62755 L .09772 .6166 L .10031 .60793 L .10622 .5883 L .11225 .56868 L .11735 .55232 L .11836 .54906 L .12456 .52943 L .13088 .50981 L .13697 .49124 L .13732 .49019 L .14384 .47057 L .15051 .45094 L .15659 .43337 L .15731 .43132 L .16422 .4117 L .1713 .39207 L .17622 .37871 L .17854 .37245 L .18592 .35283 L .19351 .3332 L .19584 .32727 L .20127 .31358 L .20924 .29396 L .21546 .27903 L .21744 .27433 L .22588 .25471 L .2346 .23509 L .23509 .23401 L .24359 .21546 L .25293 .19584 L .25471 .19219 L Mistroke .26262 .17622 L .27274 .15659 L .27433 .15359 L .28331 .13697 L .29396 .11819 L .29445 .11735 L .30619 .09772 L .31358 .08601 L .31871 .0781 L .33217 .05848 L .3332 .05703 L .34677 .03885 L .35283 .03127 L .36292 .01923 L Mfstroke .5343 .98077 m .52943 .97965 L .50981 .97636 L .49019 .97626 L .47057 .97936 L .46431 .98077 L s .9795 .98077 m .97448 .96115 L .96953 .94152 L .96455 .9219 L .96115 .90865 L .9595 .90228 L .95436 .88265 L .94914 .86303 L .94385 .84341 L .94152 .83488 L .93852 .82378 L .93312 .80416 L .92764 .78454 L .92207 .76491 L .9219 .76432 L .91648 .74529 L .91078 .72567 L .90499 .70604 L .90228 .69697 L .89913 .68642 L .89319 .6668 L .88714 .64717 L .88265 .63284 L .881 .62755 L .87478 .60793 L .86843 .5883 L .86303 .57191 L .86196 .56868 L .85541 .54906 L .84871 .52943 L .84341 .5142 L .84187 .50981 L .83492 .49019 L .8278 .47057 L .82378 .4597 L .82053 .45094 L .8131 .43132 L .80546 .4117 L .80416 .40841 L .79765 .39207 L .78962 .37245 L .78454 .36033 L .78136 .35283 L .77286 .3332 L .76491 .31546 L .76406 .31358 L .75499 .29396 L .74555 .27433 L .74529 .27381 L .73576 .25471 L Mistroke .72567 .23537 L .72552 .23509 L .71482 .21546 L .70604 .20013 L .70353 .19584 L .69159 .17622 L .68642 .16811 L .67885 .15659 L .6668 .1393 L .66511 .13697 L .65013 .11735 L .64717 .11371 L .63349 .09772 L .62755 .09132 L .61442 .0781 L .60793 .07215 L .59142 .05848 L .5883 .05618 L .56868 .04343 L .5604 .03885 L .54906 .03389 L .52943 .02756 L .50981 .02445 L .49019 .02454 L .47057 .02785 L .45094 .03436 L .44077 .03885 L .43132 .04409 L .4117 .05703 L .40975 .05848 L .39207 .07318 L .38674 .0781 L .37245 .09255 L .36767 .09772 L .35283 .11512 L .35102 .11735 L .33604 .13697 L .3332 .14091 L .3223 .15659 L .31358 .16991 L .30957 .17622 L .29762 .19584 L .29396 .20212 L .28634 .21546 L .27563 .23509 L .27433 .23754 L .26539 .25471 L .25561 .27433 L .25471 .27617 L .24617 .29396 L Mistroke .23709 .31358 L .23509 .31802 L .2283 .3332 L .21979 .35283 L .21546 .36307 L .21154 .37245 L .2035 .39207 L .19584 .41134 L .1957 .4117 L .18806 .43132 L .18062 .45094 L .17622 .46282 L .17336 .47057 L .16623 .49019 L .15928 .50981 L .15659 .51751 L .15245 .52943 L .14575 .54906 L .13919 .56868 L .13697 .57541 L .13273 .5883 L .12638 .60793 L .12015 .62755 L .11735 .63653 L .11402 .64717 L .10796 .6668 L .10202 .68642 L .09772 .70085 L .09618 .70604 L .09038 .72567 L .08468 .74529 L .07908 .76491 L .0781 .76839 L .07352 .78454 L .06804 .80416 L .06264 .82378 L .05848 .83914 L .05732 .84341 L .05202 .86303 L .04679 .88265 L .04165 .90228 L .03885 .9131 L .0366 .9219 L .03162 .94152 L .02668 .96115 L .0217 .98077 L Mfstroke .6598 .98077 m .64717 .96463 L .64431 .96115 L .62755 .94215 L .62696 .94152 L .60793 .92287 L .60685 .9219 L .5883 .90678 L .58207 .90228 L .56868 .89389 L .54906 .88418 L .54591 .88265 L .52943 .87767 L .50981 .87436 L .49019 .87424 L .47057 .87731 L .45277 .88265 L .45094 .88358 L .43132 .89304 L .41647 .90228 L .4117 .9057 L .39207 .92155 L .39168 .9219 L .37245 .94059 L .37156 .94152 L .35421 .96115 L .35283 .96283 L .33873 .98077 L s .95338 .98077 m .94814 .96115 L .94281 .94152 L .94152 .93684 L .93749 .9219 L .93209 .90228 L .92657 .88265 L .9219 .86626 L .92099 .86303 L .9154 .84341 L .90968 .82378 L .90386 .80416 L .90228 .79889 L .89801 .78454 L .89205 .76491 L .88597 .74529 L .88265 .73474 L .87982 .72567 L .87359 .70604 L .8672 .68642 L .86303 .6738 L .86072 .6668 L .85415 .64717 L .84742 .62755 L .84341 .61608 L .84056 .60793 L .83359 .5883 L .82642 .56868 L .82378 .56157 L .81913 .54906 L .81166 .52943 L .80416 .51027 L .80398 .50981 L .79615 .49019 L .78807 .47057 L .78454 .46219 L .77977 .45094 L .77121 .43132 L .76491 .41732 L .76236 .4117 L .75323 .39207 L .74529 .37566 L .74372 .37245 L .73386 .35283 L .72567 .33722 L .72353 .3332 L .71272 .31358 L .70604 .30199 L .70132 .29396 L .68922 .27433 L Mistroke .68642 .26997 L .67632 .25471 L .6668 .24117 L .66237 .23509 L .64717 .21558 L .64708 .21546 L .63003 .19584 L .62755 .19321 L .61035 .17622 L .60793 .17404 L .5883 .1581 L .58628 .15659 L .56868 .14536 L .55192 .13697 L .54906 .13584 L .52943 .12953 L .50981 .12644 L .49019 .12656 L .47057 .12989 L .45094 .13644 L .44963 .13697 L .43132 .1462 L .41516 .15659 L .4117 .15918 L .39207 .17537 L .39113 .17622 L .37245 .19477 L .37145 .19584 L .35438 .21546 L .35283 .21738 L .3391 .23509 L .3332 .24321 L .32515 .25471 L .31358 .27226 L .31225 .27433 L .30015 .29396 L .29396 .30451 L .28875 .31358 L .27794 .3332 L .27433 .33999 L .26762 .35283 L .25775 .37245 L .25471 .37867 L .24825 .39207 L .2391 .4117 L .23509 .42057 L .23026 .43132 L .2217 .45094 L .21546 .46568 L .21341 .47057 L Mistroke .20532 .49019 L .19748 .50981 L .19584 .51401 L .18981 .52943 L .18233 .54906 L .17622 .56555 L .17506 .56868 L .16789 .5883 L .1609 .60793 L .15659 .6203 L .15407 .62755 L .14732 .64717 L .14074 .6668 L .13697 .67827 L .13428 .68642 L .12789 .70604 L .12164 .72567 L .11735 .73945 L .11551 .74529 L .10942 .76491 L .10346 .78454 L .09772 .80385 L .09763 .80416 L .0918 .82378 L .08607 .84341 L .08047 .86303 L .0781 .87146 L .07491 .88265 L .06939 .90228 L .06397 .9219 L .05868 .94152 L .05848 .94228 L .05335 .96115 L .0481 .98077 L Mfstroke .74471 .98077 m .73485 .96115 L .72567 .9435 L .72461 .94152 L .71384 .9219 L .70604 .90824 L .70253 .90228 L .69054 .88265 L .68642 .87617 L .67776 .86303 L .6668 .84728 L .66399 .84341 L .64895 .82378 L .64717 .82158 L .63224 .80416 L .62755 .79908 L .61311 .78454 L .60793 .77975 L .59005 .76491 L .5883 .76362 L .56868 .75067 L .5589 .74529 L .54906 .74092 L .52943 .73435 L .50981 .73096 L .49019 .73077 L .47057 .73376 L .45094 .73994 L .43868 .74529 L .43132 .74931 L .4117 .76186 L .40754 .76491 L .39207 .77761 L .38448 .78454 L .37245 .79654 L .36535 .80416 L .35283 .81866 L .34864 .82378 L .33361 .84341 L .3332 .84396 L .31983 .86303 L .31358 .87246 L .30705 .88265 L .29505 .90228 L .29396 .90413 L .28375 .9219 L .27433 .939 L .27298 .94152 L .26274 .96115 L .25471 .97704 L Mistroke .25289 .98077 L Mfstroke .91388 .98077 m .90806 .96115 L .90228 .94208 L .90212 .94152 L .89641 .9219 L .89038 .90228 L .88418 .88265 L .88265 .87788 L .87812 .86303 L .87184 .84341 L .86534 .82378 L .86303 .8169 L .85892 .80416 L .8523 .78454 L .84545 .76491 L .84341 .75914 L .83864 .74529 L .83161 .72567 L .82432 .70604 L .82378 .7046 L .81708 .68642 L .80952 .6668 L .80416 .65328 L .80179 .64717 L .79392 .62755 L .78571 .60793 L .78454 .60518 L .7774 .5883 L .76872 .56868 L .76491 .5603 L .75983 .54906 L .75057 .52943 L .74529 .51864 L .74097 .50981 L .73097 .49019 L .72567 .4802 L .72052 .47057 L .70952 .45094 L .70604 .44498 L .69796 .43132 L .68642 .41297 L .6856 .4117 L .67243 .39207 L .6668 .38419 L .65814 .37245 L .64717 .35863 L .64238 .35283 L .62755 .33628 L .62463 .3332 L .60793 .31716 L Mistroke .6039 .31358 L .5883 .30126 L .57778 .29396 L .56868 .28857 L .54906 .27911 L .53613 .27433 L .52943 .27286 L .50981 .26984 L .49019 .27003 L .47057 .27345 L .46677 .27433 L .45094 .28008 L .43132 .28994 L .42463 .29396 L .4117 .30301 L .39847 .31358 L .39207 .31931 L .37773 .3332 L .37245 .33882 L .35999 .35283 L .35283 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}, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Very High Precision Numbers", "Subsection"], Cell[TextData[{ "Packed Arrays deal with tensors containing many numbers, but small \ precision, i.e. machine precision. But in ", StyleBox["Mathematica", FontSlant->"Italic"], " you can just as easily have numbers with very high precision and this is \ another area that we have put a lot effort into in this release. In \ particular it is now feasible to work with ", StyleBox["very", FontSlant->"Italic"], " high precision numbers in ", StyleBox["Mathematica.", FontSlant->"Italic"], " For instance this computes a hundred thousand digits of \[Pi]. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(pi\ = \ N[\ Pi, \ 10\^5];\) // Timing\)], "Input"], Cell[BoxData[ \({6.590000000000032`\ Second, Null}\)], "Output"] }, Open ]], Cell["\<\ And this computes the square of this hundred thousand digit representation of \ \[Pi].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(pi\^2;\) // Timing\)], "Input"], Cell[BoxData[ \({0.160000000000025`\ Second, Null}\)], "Output"] }, Open ]], Cell[TextData[{ "In fact this is a good example of algorithm development in ", StyleBox["Mathematica.", FontSlant->"Italic"], " The algorithm used to multiply high precision numbers internally uses \ four different algorithms depending on the precision of the numbers involved \ with break points that have been optimized for different platforms. These \ include algorithms such as schoolbook multiplication, Karatsuba \ multiplication, FFT multiplication and interleaved Karatsuba and FFT \ multiplication. Well fortunately you don't have to think about all these \ things ... it just works great. In fact this will make it even easier to do \ record breaking computations in ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell["These are some more examples ... 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072452256326513410559240190274216248439140359989535394590944070469120914093870\ 012645600162374288021092764579310657922955249887275846101264836999892256959688\ 1592056001016552563756785667227966199`10000\)], "Output"] }, Closed]], Cell[TextData[{ "Being able to compute with very high precision numbers is great, but one \ reason for doing that is typically that one would like to examine the \ structure of these numbers (just as you want to examine the structure of \ matrices and pictures). There are several new or vastly improved functions in \ the new ", StyleBox["Mathematica", FontSlant->"Italic"], " that will allow you to do these things. " }], "Text"], Cell[CellGroupData[{ Cell["Digit Expansion", "Subsubsection"], Cell["\<\ For instance computing the digit expansion in some base for a number ... \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RealDigits[\ N[1/77, \ 20], \ 10, \ 19\ ]\)], "Input"], Cell[BoxData[ \({{1, 2, 9, 8, 7, 0, 1, 2, 9, 8, 7, 0, 1, 2, 9, 8, 7, 0, 1}, \(-1\)}\)], "Output"] }, Open ]], Cell["\<\ As can be seen from this expansion the sequence generated appears to cyclic. \ This is true in general for rational numbers and RealDigits will compute the \ period and represent it exactly, meaning that this is really an infinite \ digit expansion ... \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RealDigits[1/77]\)], "Input"], Cell[BoxData[ \({{{1, 2, 9, 8, 7, 0}}, \(-1\)}\)], "Output"] }, Open ]], Cell["\<\ FromDigits also knows how to use this information to get back to the exact \ number. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FromDigits[\ %\ ]\)], "Input"], Cell[BoxData[ \(1\/77\)], "Output"] }, Open ]], Cell["Here are two examples of plots of digit sequences ... ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ListPlot[\ First[\ RealDigits[\ N[\(3!\)/\((\(5!\) + 1)\), \ 100], \ 10, \ 100\ ]\ ], \ PlotJoined \[Rule] True\ ];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00952381 0.0147151 0.0654004 [ [.21429 .00222 -6 -9 ] [.21429 .00222 6 0 ] [.40476 .00222 -6 -9 ] [.40476 .00222 6 0 ] [.59524 .00222 -6 -9 ] [.59524 .00222 6 0 ] [.78571 .00222 -6 -9 ] [.78571 .00222 6 0 ] [.97619 .00222 -9 -9 ] [.97619 .00222 9 0 ] [.01131 .14552 -6 -4.5 ] [.01131 .14552 0 4.5 ] [.01131 .27632 -6 -4.5 ] [.01131 .27632 0 4.5 ] [.01131 .40712 -6 -4.5 ] [.01131 .40712 0 4.5 ] [.01131 .53792 -6 -4.5 ] [.01131 .53792 0 4.5 ] [ 0 0 0 0 ] 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