(*********************************************************************** 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[ 467957, 12430]*) (*NotebookOutlinePosition[ 468635, 12454]*) (* CellTagsIndexPosition[ 468591, 12450]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "How and why? 9.3\n", Cell[BoxData[ FormBox[ StyleBox[\(Mathematica\ Techniques\), "Subtitle", FontSize->18, FontWeight->"Plain"], TraditionalForm]], FormatType->StandardForm], "\n", StyleBox["By Allan Hayes", FontSize->14] }], "Title"], Cell["\<\ In this issue: labels along axes; faster sorting; removing repeats; axes on \ contour plots, messages, Graphics3D displaying; .\ \>", 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the following \ account.\ \>", "Text"], Cell[TextData[{ "When ", StyleBox["Mathematica", FontSlant->"Italic"], " is given a ", StyleBox["Graphics3D", FontFamily->"Courier"], " object, ", StyleBox["gr", FontFamily->"Courier"], ", to display on screen, it first has to find\n\[Bullet] the ", StyleBox["bounding box, ", FontSlant->"Italic"], "the part of user space that will be shown, given by the option ", StyleBox["PlotRange", "Input", FontWeight->"Plain"], StyleBox[" (", "Input", FontFamily->"Times", FontWeight->"Plain"], "its edges are drawn when the option ", StyleBox["Boxed", FontFamily->"Courier"], " has its default value ", StyleBox["True", FontFamily->"Courier"], ").\n\[Bullet] the relative edge lengths that the bounding box will appear \ to have (", StyleBox["option", FontFamily->"Times"], " ", StyleBox["BoxRatios", "Input", FontWeight->"Plain"], "), \n\[Bullet] the ", StyleBox["view point", FontSlant->"Italic"], ", from which the box will be viewed (", StyleBox["option", FontFamily->"Times"], " ", StyleBox["ViewPoint", "Input", FontWeight->"Plain"], "); \n\[Bullet] the ", StyleBox["view center", FontSlant->"Italic"], ", the point that we shall be looking towards, and which will usually be \ the center of the bounding box (", StyleBox["option", FontFamily->"Times"], " ", StyleBox["ViewCenter", "Input", FontWeight->"Plain"], ") \n\[Bullet] the ", StyleBox["view vertical", FontSlant->"Italic"], ", a vector from the view center that will appear to be vertical (", StyleBox["option", FontFamily->"Times"], " ", StyleBox["ViewVertical", "Input", FontWeight->"Plain"], "). " }], "Text"], Cell[TextData[{ "The bounding box and its contents are then projected onto the ", StyleBox["projection plane", FontSlant->"Italic"], ", the plane through the view center that is normal to the ", StyleBox["line of sight ", FontSlant->"Italic"], "(view center to view point). I'll try to explain how the options are \ implemented in terms of the several coordinate systems that are used; then I \ shall give some examples. While showing the examples I shall comment on how \ the display of the projection the notebook is affected by the options ", StyleBox["ViewCenter", FontFamily->"Courier"], StyleBox[", ", FontFamily->"Times"], StyleBox["SphericalRegion", FontFamily->"Courier"], " and ", StyleBox["AspectRatio", FontFamily->"Courier"], " " }], "Text"], Cell[TextData[{ StyleBox["In programming, we need to get the actual", FontFamily->"Times"], ", ", StyleBox["absolute", FontSlant->"Italic"], ", values, used (for example, although the default value of ", StyleBox["PlotRange", FontFamily->"Courier"], " may be ", StyleBox["Automatic", FontFamily->"Courier"], ", we need to know the numerical specification of ", StyleBox["PlotRange", FontFamily->"Courier"], " that will be used); we can find these by using the function ", StyleBox["AbsoluteOptions", FontFamily->"Courier"], ":" }], "Text"], Cell[BoxData[ \(AbsoluteOptions[ gr, \ {PlotRange, BoxRatios, ViewPoint, ViewCenter, ViewVertical}]\)], "Input", Evaluatable->False], Cell["The output will be of the form", "Text"], Cell[BoxData[ \(\({\ PlotRange \[Rule] {{xmin, xmax}, {ymin, ymax}, {zm\ in, zmax}}, BoxRatios \[Rule] {brx, bry, brz}, \[IndentingNewLine]ViewPoint \[Rule] {vpx, vpy, vpz}, \[IndentingNewLine]ViewCenter \[Rule] {vcx, vcy, vcz}, \[IndentingNewLine]ViewVertical \[Rule] {vvx, vvy, vvz}\[IndentingNewLine]};\)\)], "Input", PageWidth->WindowWidth, FontWeight->"Plain"], Cell[TextData[{ "where all symbols on the rights sides of the rules are real numbers.\nCall \ the right sides of the rules ", StyleBox["{pr,br,vp,vc,vv} ", FontFamily->"Courier"] }], "Text"], Cell[TextData[{ "The basic coordinate system is the ", StyleBox["user coordinate system", FontSlant->"Italic"], ", the one we use in specifications like Line[{{0,0,0},{1,2,3}}] and for \ plotting functions. Other coordinate systems will be specified by their \ origin and unit vectors ", StyleBox["in user coordinates", FontSlant->"Italic"], ". " }], "Text"], Cell[TextData[{ "The box ratios are implemented in the ", StyleBox["scaled coordinate system ", FontSlant->"Italic"], "with origin at the center of the bounding box and unit vectors ", StyleBox["pr.{-1,1} Max[br]/br", "Input", FontWeight->"Plain"], ". In this system the box edges have lengths ", StyleBox["br/Max[br]", "Input", FontWeight->"Plain"], StyleBox[", which are in the required proportions, and", "Input", FontFamily->"Times", FontWeight->"Plain"], " the longest edge has length 1. \nThe view point is also implemented in \ scaled coordinates. In user coordinates it is" }], "Text"], Cell["vpu = pr.{1,1}/2 + vp pr.{-1,1} Max[br]/br;", "Input", Evaluatable->False, FontWeight->"Plain"], Cell[TextData[{ StyleBox["The view center and view vertical are given ", FontFamily->"Times"], "in the ", StyleBox["CV-coordinate system", FontSlant->"Italic"], " with origin at ", StyleBox["Min/@pr", "Input", FontWeight->"Plain"], " (the \"lower left front\" of the bounding box) and unit vectors ", StyleBox["pr.{-1,1}", "Input", FontWeight->"Plain"], StyleBox["(", FontFamily->"Times"], "so in this system the box edges are all of unit length). \nIn user \ coordinates view center and view vertical are " }], "Text"], Cell["\<\ vcu = Min/@pr + vc pr.{-1,1} vvu = vc pr.{-1,1}\ \>", "Input", Evaluatable->False, FontWeight->"Plain"], Cell[TextData[{ "Conversely, ", StyleBox["vp", FontFamily->"Courier"], ", ", StyleBox["vc", FontFamily->"Courier"], " and ", StyleBox["vv", FontFamily->"Courier"], " are given in terms of ", StyleBox["vpu", FontFamily->"Courier"], ", ", StyleBox["vcu", FontFamily->"Courier"], " and ", StyleBox["vvu", FontFamily->"Courier"], StyleBox[" respectively by:", FontFamily->"Times"] }], "Text"], Cell[BoxData[{ \(vp\ = \ \((\ vpu - pr . {1, 1}/2\ )\)\ \(br/Max[br]\)/ pr . {\(-1\), 1}\), "\[IndentingNewLine]", \(vc = \((vcu - Min /@ pr)\)/pr . {\(-1\), 1}\), "\[IndentingNewLine]", \(vv = vvu/pr . {\(-1\), 1}\)}], "Input", Evaluatable->False, FontWeight->"Plain"], Cell["\<\ To use these conversions in a program we need first of all to find the \ absolute values of the plot range and the box ratios. Here is a function that \ lets us use user coordinates in specifying the \"view\" options\ \>", "Text"], Cell["\<\ FromUser[gr_, opts___?OptionQ]:= Block[{br,pr,$DisplayFunction = Identity}, {br,pr} = {BoxRatios,PlotRange}/.Flatten[{opts}]/.AbsoluteOptions[gr]; Show[gr,{opts}/.{ \t(ViewCenter->vcu_):>(ViewCenter-> (vcu - Min/@pr)/(pr.{-1,1}) ), \t(ViewVertical->vvu_):>(ViewVertical->vvu/( pr.{-1,1})), \t(ViewPoint->vpu_):> \t\t(ViewPoint->(( vpu - pr.{1,1}/2 ) br/Max[br]/ pr.{-1,1})) \t}] ] \ \>", "Input"], Cell[CellGroupData[{ Cell["Projection and lighting", "Subsubsection"], Cell[TextData[{ "To produce the display that we see on screen the bounding box and its \ contents are projected onto the ", StyleBox["projection plane ", FontSlant->"Italic"], "\[Dash] the plane through the view center that is normal (in scaled \ coordinates) to the ", StyleBox["line of sight ", FontSlant->"Italic"], "(view center to view point). \nTo describe this, a third coordinate \ system, the ", StyleBox["projection system", FontSlant->"Italic"], ", is needed. 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options ", StyleBox["ImageSize", FontFamily->"Courier"], ", ", StyleBox["ImageRegion", FontFamily->"Courier"], " and ", StyleBox["ImageMargins", FontFamily->"Courier"], "." }], "Text"], Cell[CellGroupData[{ Cell["Problem", "Subsubsection"], Cell["\<\ How can we show realistically the spiral approach of the view point to a \ graphics object? We will certainly have to alter the plot range.\ \>", "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Messages", "Subtitle"], Cell["\<\ Messages are generated during evaluation to indicate and warn of what has \ happened. Many are generated automatically but we can also program our own. \ Messages can be turned on and off. We can use the input that is used to \ generate the message to, for example, modify an evaluation or build up a data \ list.\ \>", "Text"], Cell["\<\ The Help Browser can help in understanding the built-in messages: to find out \ about the warning message that is generated below\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[a, {a, b, c}]\)], "Input"], Cell[BoxData[ \(Plot::"plln" \(\(:\)\(\ \)\) "Limiting value \!\(b\) in \!\({a, b, c}\) is not a machine-size real \ number."\)], "Message"], Cell[BoxData[ \(Plot[a, {a, b, c}]\)], "Output"] }, Open ]], Cell[TextData[{ " - select ", StyleBox["plln", FontFamily->"Courier", FontColor->RGBColor[0, 0, 1]], " (not ", StyleBox["Plot::plln", FontFamily->"Courier", FontColor->RGBColor[0, 0, 1]], ")\n- ", StyleBox[" Help > Find Selected Function", FontFamily->"Arial"], "\n- click button ", StyleBox["Other Information", FontFamily->"Arial"], " \n- click button ", StyleBox["Go To", FontFamily->"Arial"] }], "Text"], Cell[TextData[{ "We can set up our own messages:\n- assign a ", StyleBox["control string", FontSlant->"Italic"], " to message ", StyleBox["\"n\"", FontFamily->"Courier"], " for symbol ", StyleBox["s", FontFamily->"Courier"], StyleBox[" ( ", FontFamily->"Times"], StyleBox["s", FontFamily->"Courier"], StyleBox[" can be protected or even locked).", FontFamily->"Times"] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(MessageName[s, "\"]\), "=", " ", StyleBox["\"\<`1` before `2`\>\"", FontFamily->"Courier"]}], StyleBox[";", FontFamily->"Courier"]}]], "Input"], Cell[TextData[{ "Note:\n", StyleBox["Special form: ", FontFamily->"Times"], StyleBox["s::n", FontFamily->"Courier"], StyleBox[" or ", FontFamily->"Times"], StyleBox["s::\"n\"", FontFamily->"Courier"], StyleBox[" for ", FontFamily->"Times"], StyleBox["MessageName[s, \"n\"]", FontFamily->"Courier"], StyleBox[".", FontFamily->"Times"], StyleBox["\n", FontFamily->"Times", FontWeight->"Bold"], StyleBox["Attributes: ", FontFamily->"Times"], StyleBox["MessageName", FontFamily->"Courier"], StyleBox[" and ", FontFamily->"Times"], StyleBox["Messages", FontFamily->"Courier"], StyleBox[" , below, are ", FontFamily->"Times"], StyleBox["HoldFirst", FontFamily->"Courier"], StyleBox[".", FontFamily->"Times"] }], "Text"], Cell[TextData[{ "The corresponding rule is added to the list of messages for ", StyleBox["s", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", FontFamily->"Times"], StyleBox[" For user-defined messages these lists can be shown", FontFamily->"Times"], StyleBox[":", FontFamily->"Times", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Messages[s]\)], "Input"], Cell[BoxData[ \({HoldPattern[s::"n"] \[RuleDelayed] "`1` before `2`"}\)], "Output"] }, Open ]], Cell["\<\ - display a message with inserts for the items `1` and `2` (usually, we also \ get a beep) \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Message[MessageName[s, "\"], "\", \ "\"]\)], "Input"], Cell[BoxData[ \(s::"n" \(\(:\)\(\ \)\) "\!\(\"work\"\) before \!\(\"play\"\)"\)], "Message"] }, Open ]], Cell["- turn the message off :", "Text"], Cell[BoxData[ \(Off[s::n]\)], "Input"], Cell[TextData[{ "This just wraps the message name in ", StyleBox["$Off", FontFamily->"Courier"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s::n\)], "Input"], Cell[BoxData[ \($Off["`1` before `2`"]\)], "Output"] }, Open ]], Cell["The message is not now shown:", "Text"], Cell[BoxData[ \(Message[s::n, \ "\", \ "\"]\)], "Input"], Cell["- turn the message back on", "Text"], Cell[BoxData[ \(On[s::n]\)], "Input"], Cell[TextData[{ StyleBox["$Off", FontFamily->"Courier"], " has been removed:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s::n\)], "Input"], Cell[OutputFormData["\<\ \"`1` before `2`\"\ \>", "\<\ `1` before `2`\ \>"], "Output"] }, Open ]], Cell["and the message is shown:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Message[s::n, \ "\", \ "\"]\)], "Input"], Cell["s::n: work before play", "Message"] }, Open ]], Cell[TextData[{ "The most common messages to be turned off are those arising from \ spell-checking , these are turned off by evaluating ", StyleBox["Off[General::spell, General::spell1]", FontFamily->"Courier"], " (note the lower case ", StyleBox["s", FontFamily->"Courier"], ", and that we can turn on and off a sequence of message names)." }], "Text"], Cell[TextData[{ "\[Bullet] Turning on and turning off does not affect the display of usage \ messages with ", StyleBox["?", FontFamily->"Courier"], " and ", StyleBox["??", FontFamily->"Courier"], ", which use ", StyleBox["Information", FontFamily->"Courier"], " rather than ", StyleBox["Message", FontFamily->"Courier"], ". For example after" }], "Text"], Cell[BoxData[{ \(\(t::usage\ = \ "\";\)\), "\[IndentingNewLine]", \(Off[t::usage]\)}], "Input"], Cell["we still get", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?t\)\)], "Input"], Cell[BoxData[ \("about t"\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(?? t\)], "Input"], Cell[BoxData[ \("about t"\)], "Print"] }, Open ]], Cell[TextData[{ StyleBox["See also:", FontWeight->"Bold"], "\n", StyleBox["$MessageList", FontFamily->"Courier", FontWeight->"Bold"], ": the list of message names for messages so far generated (even if not \ displayed) since the line was input. It is reset to {} when the line has been \ processed.\n", StyleBox["MessageList[n]", FontFamily->"Courier", FontWeight->"Bold"], ": the list of message names of messages displayed during the evaluation \ of a ", StyleBox["previous", FontSlant->"Italic"], " ", StyleBox["Input[n]", FontFamily->"Courier", FontWeight->"Bold"], " (", StyleBox["$Line", FontFamily->"Courier", FontWeight->"Bold"], ", it is assigned a value only after the line has been processed);\n", StyleBox["Check[expr,failexpr]", FontFamily->"Courier", FontWeight->"Bold"], " gives ", StyleBox["expr*", FontFamily->"Courier", FontWeight->"Bold"], " (the value of ", StyleBox["expr", FontFamily->"Courier"], ") if no message was ", StyleBox["displayed", FontSlant->"Italic"], " in evaluating ", StyleBox["expr", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Times"], "(turned-off messages are not counted) , otherwise it gives ", StyleBox["failedexpr*", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", FontFamily->"Times"], StyleBox[" \nCheck[expr,failexpr,s::n..]", FontFamily->"Courier", FontWeight->"Bold"], " gives ", StyleBox["expr*", FontFamily->"Courier", FontWeight->"Bold"], " if no message with name in the, sequence ", StyleBox["s::n..", FontFamily->"Courier", FontWeight->"Bold"], " was displayed in evaluating ", StyleBox["expr", FontFamily->"Courier", FontWeight->"Bold"], " , otherwise it gives ", StyleBox["failedexpr*", FontFamily->"Courier", FontWeight->"Bold"], ". " }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Programming with Messages", FontWeight->"Bold"]], "Subsubsection"], Cell[TextData[StyleBox["Input checking and returning unchanged.", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "Warning messages can be very helpful to users. Here is some code with no \ warning messages: it only works as intended when ", StyleBox["x", FontFamily->"Courier"], " is an expression with an odd number of elements. You will see several way \ in which the user could make mistakes. Of course we must expect that adding \ tests and messages will slow down the computation; and, certainly, doing the \ extra programming can be quite time-consuming." }], "Text"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(middle[x_] := x[\([\)\(Ceiling[Length[x]/2]\)\(]\)]\)}], "Input"], Cell["We could protect the user much more:", "Text"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(middle[x_ /; OddQ[Length[x]]] := x[\([\)\(Ceiling[Length[x]/2]\)\(]\)]\)}], "Input"], Cell["\<\ But all that happens now is that with inappropriate entries the expression is \ returned unchanged:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(middle[{1, 2}]\)], "Input"], Cell[BoxData[ \(middle[{1, 2}]\)], "Output"] }, Open ]], Cell[TextData[{ "We can give more comprehensive protection and add helpful messages (one of \ the message names, ", StyleBox["middle::argx", FontFamily->"Courier"], ", uses a built-in definition (look it up as shown earlier); the other one, \ ", StyleBox["middle::nodd", FontFamily->"Courier"], " is specially defined):" }], "Text"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(middle[ x___] := \[IndentingNewLine]x[\([Ceiling[ Length[x]/2]]\)] /; \[IndentingNewLine]If[\[IndentingNewLine]\t Length[{x}] === 1, \[IndentingNewLine]\t\tIf[OddQ[Length[x]], True, Message[middle::nodd]], \[IndentingNewLine]Message[middle::argx, middle, \ Length[{x}]]\[IndentingNewLine]]\)}], "Input"], Cell[TextData[{ "Define ", StyleBox["middle::nodd", FontFamily->"Courier"], StyleBox[":", FontFamily->"Times"] }], "Text"], Cell[BoxData[ \(\(middle::nodd = \ "\";\)\)], "Input"], Cell["Tests", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(middle[{1, 2}]\)], "Input"], Cell[BoxData[ \(middle::"nodd" \(\(:\)\(\ \)\) "The argument of middle is expected to be normal with an odd number of \ elements"\)], "Message"], Cell[BoxData[ \(middle[{1, 2}]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(middle[1, 2]\)], "Input"], Cell[BoxData[ \(middle::"argx" \(\(:\)\(\ \)\) "\!\(middle\) called with \!\(2\) arguments; 1 argument is \ expected."\)], "Message"], Cell[BoxData[ \(middle[1, 2]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(middle[p[1, 2, 3]]\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[TextData[{ "Here is an alternative style, due to David Withoff. It is more \ complicated for simple examples, but using ", StyleBox["Throw", FontFamily->"Courier"], " and ", StyleBox["Catch", FontFamily->"Courier"], " lets us escape from deep within complex code." }], "Text"], Cell[BoxData[ \(Clear["\<`*\>"]\)], "Input", GeneratedCell->True], Cell[BoxData[{ \(middle[x___] := With[{result = middle1[x]}, result /; result =!= $Failed\[IndentingNewLine]]\), "\[IndentingNewLine]", \(middle1[ x___] := \ \[IndentingNewLine]Module[{tag}, \[IndentingNewLine]Catch[ If[\[IndentingNewLine]\t Length[{x}] === 1, \[IndentingNewLine]\t\tIf[OddQ[Length[x]], Throw[x[\([Ceiling[Length[x]/2]]\)], tag], Message[middle::nodd]; Throw[$Failed, \ tag]], \[IndentingNewLine]Message[ middle::argx, middle, \ Length[{x}]]; Throw[$Failed, tag]\[IndentingNewLine]], \[IndentingNewLine]tag]\ \[IndentingNewLine]]\[IndentingNewLine]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(middle[{1, 2}]\)], "Input"], Cell[BoxData[ \(middle::"nodd" \(\(:\)\(\ \)\) "The argument of middle is expected to be normal with an odd number of \ elements"\)], "Message"], Cell[BoxData[ \(middle[{1, 2}]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(middle[1, 2]\)], "Input"], Cell[BoxData[ \(middle::"argx" \(\(:\)\(\ \)\) "\!\(middle\) called with \!\(2\) arguments; 1 argument is \ expected."\)], "Message"], Cell[BoxData[ \(middle[1, 2]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(middle[p[1, 2, 3]]\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell["Using data supplied to Message", "Text", FontWeight->"Bold"], Cell["\<\ There is a problem \[Dash] messages are not output, they are printed. \ Moreover, we may not want the message to be displayed. We have to do a little \ programming: here are some examples.\ \>", "Text"], Cell[TextData[{ StyleBox["SilentCheck", FontFamily->"Courier"], ", below, is a variant of ", StyleBox["Check", FontFamily->"Courier"], ", based on an idea of Carl Woll for getting the benefit of ", StyleBox["Check", FontFamily->"Courier"], " without displaying any messages(the reason for the condition ", StyleBox["f_/;Not[MatchQ[f,_$Off]]", FontFamily->"Courier"], " will be explained later; the attributes ", StyleBox["HoldAll", FontFamily->"Courier"], " guarantees that the evaluation of expr take place on the right, inside ", StyleBox["Catch[expr, SilentCheck]", FontFamily->"Courier"], ")." }], "Text"], Cell[BoxData[ \(\(SilentCheckPlus::usage = \ \n\*"\"\\"";\)\)], "Input"], Cell[BoxData[{ \(\(ClearAll[SilentCheck];\)\), "\n", \(SetAttributes[SilentCheck, \ HoldAll]\), "\[IndentingNewLine]", \(SilentCheck[expr_, \ failexpr_] := \[IndentingNewLine]Block[{Message}, \ \[IndentingNewLine]Message[f_ /; Not[MatchQ[f, _$Off]], ___] := \ Throw[failexpr, SilentCheck]; \[IndentingNewLine]Catch[expr, SilentCheck]\[IndentingNewLine]]\)}], "Input"], Cell["Tests", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheck[FindMinimum[\(-Sin[Pi\ x]\), {x, 0, 0, 0.4}], \ ff]\)], "Input"], Cell[BoxData[ \(ff\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheck[FindMinimum[\(-Sin[Pi\ x]\), {x, 0, 0, 1}], \ ff]\)], "Input"], Cell[BoxData[ \({\(-1.`\), {x \[Rule] 0.5000000000000011`}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["SilentCheckPlus", FontFamily->"Courier"], " lets us use the inputs to ", StyleBox["Message", FontFamily->"Courier"], "." }], "Text"], Cell[BoxData[ \(\(SilentCheckPlus::usage = \ \n\*"\"\\"";\)\)], "Input"], Cell[BoxData[{ \(\(SetAttributes[SilentCheckPlus, \ HoldAll];\)\[IndentingNewLine]\), "\n", \(SilentCheckPlus[expr_, \ f : \ _?\((Not[MatchQ[#, _MessageName]] &)\), mns___MessageName] := \[IndentingNewLine]Block[{Message, \ acc = f[]}, \[IndentingNewLine]Message[ n_ /; Not[ MatchQ[n, _$Off]] && \(({mns} \[Equal] {} || MemberQ[{mns}, n])\), \[IndentingNewLine]r___] := AppendTo[ acc, {n, r}]; \[IndentingNewLine]\(If[acc === f[], #, acc] &\)[ expr]\[IndentingNewLine]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheckPlus[FindMinimum[\(-Sin[Pi\ x]\), {x, 0, 0, 0.4}], \ f]\)], "Input"], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"{", RowBox[{"\<\"Reached the point `1` which is outside the region `2`.\"\ \>", ",", TagBox["0.4`", HoldForm], ",", TagBox[\({{0.`, 0.4`}}\), HoldForm]}], "}"}], "]"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheckPlus[FindMinimum[\(-Sin[Pi\ x]\), {x, 0, 0, 1}], \ f]\)], "Input"], Cell[BoxData[ \({\(-1.`\), {x \[Rule] 0.5000000000000011`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheckPlus[1/0 + \ 0^0 + 0\ Infinity, f]\)], "Input"], Cell[BoxData[ RowBox[{"f", "[", RowBox[{ RowBox[{"{", RowBox[{"\<\"Infinite expression `1` encountered.\"\>", ",", TagBox[\(1\/0\), HoldForm]}], "}"}], ",", RowBox[{"{", RowBox[{\(Power::"indet"\), ",", TagBox[\(0\^0\), HoldForm]}], "}"}], ",", RowBox[{"{", RowBox[{\(\[Infinity]::"indet"\), ",", TagBox[ RowBox[{"0", " ", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], HoldForm]}], "}"}]}], "]"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheckPlus[1/0 + \ 0^0, f, \ Power::"\"]\)], "Input"], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"{", RowBox[{\(Power::"indet"\), ",", TagBox[\(0\^0\), HoldForm]}], "}"}], "]"}]], "Output"] }, Open ]], Cell[TextData[{ "\[Bullet]Notice that some built-in message names, here ", StyleBox["Power::indet", FontFamily->"Courier"], " and ", StyleBox["\[Infinity]::indet", FontFamily->"Courier"], ", do not have a control string assigned." }], "Text"], Cell[TextData[{ "\[Bullet] Now for an explanation of why ", StyleBox["Not[MatchQ[n,_$Off]]", FontFamily->"Courier"], " is used in ", StyleBox["SilentCheck", FontFamily->"Courier"], " and ", StyleBox["SilentCheckPlus", FontFamily->"Courier"], "." }], "Text"], Cell[TextData[{ "SilentCheckPlus2 and SilentCheck below are SilentCheckPlus and Silent \ Check with ", StyleBox["Not[MatchQ[n,_$Off]]", FontFamily->"Courier"], " deleted. " }], "Text"], Cell[BoxData[{ \(\(SetAttributes[SilentCheckPlus2, \ HoldAll];\)\[IndentingNewLine]\), "\n", \(SilentCheckPlus2[expr_, \ f : \ _?\((Not[MatchQ[#, _MessageName]] &)\), mns___MessageName] := \[IndentingNewLine]Block[{Message, \ acc = f[]}, \[IndentingNewLine]Message[ n_ /; \(({mns} \[Equal] {} || MemberQ[{mns}, n])\), \[IndentingNewLine]r___] := AppendTo[ acc, {n, r}]; \[IndentingNewLine]\(If[acc === f[], #, acc] &\)[ expr]\[IndentingNewLine]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Short[SilentCheckPlus2[Integrate[x, \ {x, 1, \ 2}], f], 4]\)], "Input"], Cell[BoxData[ TagBox[\(f[{$Off[], Integrate`NLtheoremDump`f}, {$Off[], Integrate`NLtheoremDump`dpoly$}, {$Off[], Integrate`NLtheoremDump`ro}, \[LeftSkeleton]500\[RightSkeleton], \ {$Off[], Integrate`NLtheoremDump`an1}, {$Off[], Integrate`NLtheoremDump`wi}, {$Off[], Integrate`NLtheoremDump`var}]\), (Short[ #, 4]&)]], "Output"] }, Open ]], Cell["\<\ We see that an enormous number of expressions like Message[$Off[],...] have \ been generated, in fact 505 of them. Their contribution to the output is \ probably not wanted, and the output is not as expected \[Dash] it should be \ 3/2. A second evaluation does not show this feature (this is why we quitted the \ kernel), so it could be due to loading the code for integrating.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Short[SilentCheckPlus2[Integrate[x, \ {x, 1, \ 2}], f], 4]\)], "Input"], Cell[BoxData[ TagBox[\(3\/2\), (Short[ #, 4]&)]], "Output"] }, Open ]], Cell["But a different integral give similar problems", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Short[SilentCheckPlus2[Integrate[1/x^2, \ {x, 1, \ Infinity}], \ f], 4]\)], "Input"], Cell[BoxData[ TagBox[\(f[{$Off[], Integrate`ImproperDump`numericQ}, {$Off[], Integrate`ImproperDump`newAssumption}, {$Off[], Integrate`ImproperDump`ImExp}, \[LeftSkeleton]1520\[RightSkeleton], \ {$Off[], System`MeijerGDump`MMeijerG}, {$Off[], System`MeijerGDump`ExpandMeijerG}]\), (Short[ #, 4]&)]], "Output"] }, Open ]], Cell[TextData[{ "In this case we can see what is happenning, but with ", StyleBox["SilentCheck2", FontFamily->"Courier"], " below, the there is no evidence." }], "Text"], Cell["Please quit the kernel.", "Text"], Cell[BoxData[{ \(SetAttributes[SilentCheck2, \ HoldAll]\), "\[IndentingNewLine]", \(SilentCheck2[expr_, \ failexpr_] := \[IndentingNewLine]Block[{Message}, \ \[IndentingNewLine]Message[___] := \ Throw[failexpr, SilentCheck2]; \[IndentingNewLine]Catch[expr, SilentCheck2]\[IndentingNewLine]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheck2[Integrate[x, \ {x, 1, \ 2}], \ $Failed]\)], "Input"], Cell[BoxData[ \($Failed\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SilentCheck2[Integrate[x, \ {x, 1, \ 2}], \ $Failed]\)], "Input"], Cell[BoxData[ \(3\/2\)], "Output"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 723}}, WindowSize->{618, 633}, WindowMargins->{{169, Automatic}, {Automatic, 12}}, ShowCellLabel->False ] (*********************************************************************** Cached data follows. 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