(*********************************************************************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 152234, 5057]*) (*NotebookOutlinePosition[ 153246, 5089]*) (* CellTagsIndexPosition[ 153202, 5085]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Finding Roots of Polynomials of Very High Order", \ "Title"]], "Title"], Cell[TextData[StyleBox["Richard R. Silbar and T. Goldman", "Author"]], \ "Author"], Cell[TextData[{ StyleBox["T-16,", "Address"], " ", StyleBox["Theoretical Division\nLos Alamos National Laboratory\nP. O. Box \ 1663\nLos Alamos, NM 87544\ne-mail: silbar@lanl.gov", "Address"] }], "Input"], Cell["", "Input"], Cell[TextData[{ StyleBox["Given a polynomial of very high order (~ 500) with all positive \ coefficients, we show how to find its (complex) roots using standard ", "Abstract", FontSlant->"Italic"], StyleBox["built-in ", FontSlant->"Italic"], " ", StyleBox["Mathematica functions. In particular, we can efficiently locate \ the root with the largest real part using a search technique based on \ Cauchy's contour integral formula.", "Abstract", FontSlant->"Italic"] }], "Abstract"], Cell[CellGroupData[{ Cell["Motivation", "Section", FormatType->TextForm], Cell[TextData[{ StyleBox["One of the major", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", StyleBox["goals", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", StyleBox["of RHIC, the Relativistic Heavy Ion Collider facility at \ Brookhaven National Laboratory, is to discover the expected phase transition \ where the bound quarks in ordinary nuclear matter consisting of protons and \ neutrons turn into a plasma of free quarks", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", StyleBox["and gluons. In trying to refine predictions of the temperature ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["T", "Text", FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" at which this phase transition occurs, we turned to a classic \ paper by Yang and Lee [1] which considers the case of transitions from a \ liquid to a gas phase for a monatomic gas, hoping to apply their analysis to \ the quark-gluon-plasma", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", StyleBox[" case.", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", StyleBox["These authors show that the grand partition function for the gas \ can be written as a polynomial in ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["y", "Text", FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\[DoubleStruckCapitalQ]\_V = \[Sum]\+\(N\ = \ 0\)\%M\( \ Q\_N\/\(N!\)\) y\^N\), " ", StyleBox[",", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], RowBox[{ RowBox[{ StyleBox["where", "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", "y"}], " ", "=", " ", RowBox[{\(\((2 \[Pi]\ m\ k\ T/h\^2)\)\^\(3/2\)\), RowBox[{ SuperscriptBox["e", RowBox[{\(-\[Mu]\), "/", StyleBox[ RowBox[{ StyleBox["k", FontSlant->"Italic"], "T"}]]}]], "."}]}]}]}], TextForm]], "NumberedEquation"], Cell[TextData[{ StyleBox["Here \[Mu] is the chemical potential per atom and ", "Text", FontWeight->"Plain"], StyleBox["m", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is its mass.", "Text", FontWeight->"Plain"], " ", StyleBox["The ", "Text", FontWeight->"Plain"], StyleBox["N", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["th term in the polynomial gives the relative probability of \ having ", "Text", FontWeight->"Plain"], StyleBox["N", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" atoms in a box of volume ", "Text", FontWeight->"Plain"], StyleBox["V", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", and ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is the maximum number of atoms that can be placed in that box.", "Text", FontWeight->"Plain"], " ", StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" can be a very large integer for cases of physical interest. \ Finally,", "Text", FontWeight->"Plain"], " ", StyleBox[Cell[BoxData[ StyleBox[\(Q\_N\), "TI"]], "Text"], "Text"], StyleBox[" is the configurational part of the partition function,", "Text", FontWeight->"Plain"] }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(Q\_N\), "=", RowBox[{"\[Integral]", RowBox[{"\[CenterDot]", "\[CenterDot]", "\[CenterDot]", RowBox[{\(\[Integral]\_V\), RowBox[{\(d\[Tau]\_1\[CenterDot]\[CenterDot]\[CenterDot] d\[Tau]\_N\), " ", SuperscriptBox["e", RowBox[{\(-U\), "/", StyleBox[ RowBox[{ StyleBox["k", FontSlant->"Italic"], "T"}]]}]]}]}]}]}]}], ","}], TextForm]], "NumberedEquation"], Cell[TextData[{ StyleBox["where ", "Text", FontWeight->"Plain"], StyleBox["U", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is the energy of the gas, with interparticle interactions \ included, and d", "Text", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{\(\[Tau]\_i\), " ", "is", " ", "the", " ", "volume", " ", "element", " ", "for", " ", "the", " ", StyleBox[ RowBox[{ StyleBox["i", FontSlant->"Italic"], "th"}]], " ", "atom"}], TraditionalForm]]], StyleBox[". Note that the coefficients of the polynomial \ \[DoubleStruckCapitalQ] are all positive.\n\nYang and Lee (henceforth YL) go \ on to note that, by the fundamental theorem of algebra, \ \[DoubleStruckCapitalQ] can be factorized in terms of its ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" roots.", "Text", FontWeight->"Plain"], " ", StyleBox["Because of the positivity of its coefficients, these roots are \ complex.", "Text", FontWeight->"Plain"], " ", StyleBox["(If ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is odd, there will be one real root.) They then proceed to \ discuss when phase transitions occur in terms of these roots, as we shall see \ below.\n\nSo, the question arises, how does one find the roots of a very \ high-order polynomial?", "Text", FontWeight->"Plain"], " ", StyleBox["For the cases of interest to us, we might have ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" as large as 500 or 1000. 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We were unable to find any literature (bearing in mind that we \ are physicists and not all that familiar with the mathematical literature) \ describing the methods we independently developed.", "Text", FontWeight->"Plain"], " ", StyleBox["Also, the method can easily be generalized to find roots other \ than the one of particular interest to us.", "Text", FontWeight->"Plain"], " ", StyleBox["Hence we felt it would be of interest to the readers of this \ journal to describe how we found such roots using ", "Text", FontWeight->"Plain"], StyleBox["Mathematica", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".\n\nFor the record, the notebook presented here was developed in \ Version", "Text", FontWeight->"Plain"], " ", StyleBox[ ValueBox["$VersionNumber"], "Text", FontWeight->"Plain"], StyleBox[".", "Text", FontWeight->"Plain"], StyleBox[ ValueBox["$ReleaseNumber"], "Text", FontWeight->"Plain"], StyleBox[".", "Text", FontWeight->"Plain"], StyleBox[ ValueBox["$MinorReleaseNumber"], "Text", FontWeight->"Plain"], StyleBox[", Platform X, running on a Sun Ultra 5 workstation (Solaris 8).", "Text", FontWeight->"Plain"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Finding", "Section"], " ", StyleBox["All Roots Using ", "Section"], StyleBox["Solve", "MB", FontSize->14], StyleBox[" and ", "Section"], StyleBox["NSolve", "MB", FontSize->14] }], "Section", FormatType->TextForm], Cell[CellGroupData[{ Cell["A Simplified Set of\.13 Coefficients", "Subsection", FormatType->TextForm], Cell[TextData[{ "For ", StyleBox["purposes of this paper let us ignore the question of what ", "Text", FontWeight->"Plain"], StyleBox["U", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is and simply assume that the ", "Text", FontWeight->"Plain"], Cell[BoxData[ StyleBox[\(Q\_N\), "TI"]]], StyleBox["/", "Text", FontWeight->"Plain"], StyleBox["N", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["! are given as a Negative Binomial Distribution [2].", "Text", FontWeight->"Plain"], " ", StyleBox["That is, we define the ", "Text", FontWeight->"Plain"], Cell[BoxData[ StyleBox[\(Q\_N\), "TI"]]], StyleBox["/", "Text", FontWeight->"Plain"], StyleBox["N", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["! by the formula", "Text", FontWeight->"Plain"] }], "Text"], Cell["NBD[n_, p_, r_] := Binomial[n+r-1,r-1] p^r (1-p)^n", "Input"], Cell[TextData[{ "Here we use ", StyleBox["n", FontSlant->"Italic"], " in place of the YL's index/exponent ", StyleBox["N", FontSlant->"Italic"], " to avoid confusion with the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["N", "MB"], ". 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380.625}, {242.188, 100}} -> {-2.81045, 0.434064, \ 0.0116009, 0.0150165}}] }, Open ]], Cell[TextData[{ "The roots come, of course, in complex conjugate pairs. This plot \ foreshadows what YL led us to expect (see their Fig. 2a), a nice oval-like \ figure where the ", StyleBox["principal roots", FontSlant->"Italic"], " (i.e., the pair with the largest real part) begins to pinch the positive \ real axis near 1. \n\nHow good is our answer for the \"last root\"?", " ", StyleBox["Mathematica", FontSlant->"Italic"], " tends to order its solutions in increasing order of real part, so that \ root is" }], "Text"], Cell[CellGroupData[{ Cell["rootsList50[[50]]", "Input"], Cell[OutputFormData["\<\ {1.0775965213685874, 0.20181531410746278}\ \>", "\<\ {1.0776, 0.201815}\ \>"], "Output"] }, Open ]], Cell[TextData[StyleBox["Setting (by hand)", "Text", FontWeight->"Plain"]], "Text"], Cell[CellGroupData[{ Cell["z = 1.0776 + 0.201815 I", "Input"], Cell[OutputFormData["\<\ 1.0776 + 0.201815*I\ \>", "\<\ 1.0776 + 0.201815 I\ \ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[50,z]" }], "Input"], Cell[OutputFormData["\<\ 0.00008538504310362516 + 0.00005907004023916751*I\ \>", "\<\ 0.000085385 + 0.00005907 I\ \>"], \ "Output"] }, Open ]], Cell[TextData[{ "This is close to zero, but one might worry about a lack of precision.", " ", "To compare, the last term of the ", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], " for this case at this value of ", StyleBox["z", FontSlant->"Italic"], " is" }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "{z^50, coeff = Coefficient[", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[50,y], y, 50]}\ncoeff * z^50" }], "Input"], Cell[OutputFormData["\<\ {-97.95307125472469 + 16.604079090027604*I, 0.0014443792322110563}\ \>", "\<\ {-97.9531 + 16.6041 I, 0.00144438}\ \>"], "Output"], Cell[OutputFormData["\<\ -0.14148138185161413 + 0.023982587007625726*I\ \>", "\<\ -0.141481 + 0.0239826 I\ \>"], "Output"] }, Open ]], Cell[TextData[{ "So, a great deal of cancellation ", StyleBox["is", FontSlant->"Italic"], " taking place to bring things down to the ", Cell[BoxData[ \(TraditionalForm\`10\^\(-5\)\)]], " level.", " ", "We can improve the last root by using ", StyleBox["FindRoot", "MB"], " with increased precision," }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "t = Timing[FindRoot[", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[50,y]== 0, {y,z}, WorkingPrecision->16]]" }], "Input"], Cell[OutputFormData["\<\ {0.08999999999999986*Second, {y -> 1.0775965228354394 + 0.2018153164788166*I}}\ \>", "\<\ {0.09 Second, {y -> 1.0776 + 0.201815 I}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "z = y /. t[[2]]\n", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[50,z]" }], "Input"], Cell[OutputFormData["\<\ 1.0775965228354394 + 0.2018153164788166*I\ \>", "\<\ 1.0776 + 0.201815 I\ \>"], "Output"], Cell[OutputFormData["\<\ 8.605512830106576*^-9 + 6.6940407592031015*^-9*I\ \>", "\<\ -9 -9 8.60551 10 + 6.69404 10 I\ 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of ", "Text", FontWeight->"Plain"], StyleBox["p", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" and ", "Text", FontWeight->"Plain"], StyleBox["r", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", at around ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 80.) \n\nWe will loop on values of ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" to see when we no longer find all of the ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" roots. If one pushes beyond this point, one no longer gets a \ nice looking oval for the plot of the roots in the complex plane. We will \ need to set the numerical precision (the third argument of ", "Text", FontWeight->"Plain"], StyleBox["NSolve", "MB"], StyleBox[") to a value higher than its default. Let us try a precision of \ 16 digits.", "Text", FontWeight->"Plain"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "t = Timing[{\n{p=0.2, r=5, prec = 16},\nbigrootList = {};\nDo[{lastM = M,\n\ \trootsM = NSolve[", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[M,y] == 0, y, prec],\n\trootsList = {Re[y],Im[y]} /. rootsM;,\n\t\ numberFound = Length[rootsList],\n\tbiggestRoot = {0,0};,\n\tDo[{\n\t\t\ If[rootsList[[i]][[1]] > biggestRoot[[1]],\n\t\t\tbiggestRoot = \ rootsList[[i]]]\n\t\t},{i,1,numberFound}],\n\tIf[biggestRoot[[2]] < 0, \n\t\t\ biggestRoot[[2]] = -biggestRoot[[2]]],\n\tPrint[M,\" \",numberFound,\" \ \", biggestRoot],\n\tbigrootList = Append[bigrootList, {M, \ biggestRoot[[2]]}],\n\tIf[numberFound != M, Break[]]\n\t},{M,20,200,20}],\n\ strM = ToString[lastM],\nMstring = StringJoin[\"M = \",strM]\n}];\nt[[1]]\n\ ListPlot[rootsList,PlotRange->{{-1.5,1.5},{-1.5,1.5}},\n\t\ PlotLabel->Mstring];" }], "Input"], Cell[OutputFormData["\<\ 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Setting ", "Text", FontWeight->"Plain"], StyleBox["prec", "Text", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 20 does find them all, but of course takes longer to compute. \ The ", "Text", FontWeight->"Plain"], StyleBox["biggestRoot", "Text", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" with that precision is {1.177555, 0.094048}, to be compared with \ the ", "Text", FontWeight->"Plain"], "{1.17633, 0.0955108}", StyleBox[" found with ", "Text", FontWeight->"Plain"], StyleBox["prec", "Text", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" = 16.\n\nIn any case, the plot of the roots for ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 120 shows that the \[OpenCurlyDoubleQuote]pinch\ \[CloseCurlyDoubleQuote] is even more pronounced than in the ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 50 case.\n\nIf we were to redo the last set of cell commands \ with ", "Text", FontWeight->"Plain"], StyleBox["r", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 10, we find that the break out of the loop occurs at ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 160, again missing 2 roots, with a similar (but denser) ", "Text", FontWeight->"Plain"], StyleBox["ListPlot", "Text", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". \n\nOne can go to larger ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["'s by increasing the precision variable.", "Text", FontWeight->"Plain"], " ", StyleBox["For example, by letting the computer run overnight for ", "Text", FontWeight->"Plain"], StyleBox["M", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 400, ", "Text", FontWeight->"Plain"], StyleBox["p", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 1/5, and ", "Text", FontWeight->"Plain"], StyleBox["r", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" = 5 with precision ", "Text", FontWeight->"Plain"], StyleBox["prec", "MB"], StyleBox[" set to 24, ", "Text", FontWeight->"Plain"], StyleBox["Mathematica", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" found all roots. The principal root in this case, in case anyone \ really wants to know, is", "Text", FontWeight->"Plain"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["z ", "Text", FontFamily->"Courier"], StyleBox["= 1.228256632499110795361 + \n\t0.02987699922123763270863 ", "Text", FontFamily->"Courier"], StyleBox["I;\n", "Text", FontFamily->"Courier"], StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[400,z]" }], "Input"], Cell[OutputFormData["\<\ -2.1009327610954642*^-10 - 5.44645217814832*^-10*I\ \>", "\<\ -10 -2.10093 10 - -10 5.44645 10 I\ \>"], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The ", "Text", FontWeight->"Plain"], StyleBox["ListPlot", "Text", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" of these roots essentially looks like a continuous curve that \ pinches the real axis. That is, \[DoubleStruckCapitalQ](400,", "Text", FontWeight->"Plain"], StyleBox["y", "Text", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[") gives ", "Text", FontWeight->"Plain"], StyleBox["a pretty good approximation of where a phase transition is, \ according to YL. ", "Text", FontWeight->"Plain"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Finding", "Section"], " ", StyleBox["the \"Largest\" Root", "Section"] }], "Subsection"], Cell[TextData[{ "By now it should be clear that the authors are particularly interested in \ the principal root of the ", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], " polynomial. We therefore sought to find a way to find just that root, \ which would presumably be faster than finding all of them, as above. 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", " ", StyleBox["Mathematica", FontSlant->"Italic"], " computes a non-zero integral much more quickly. Incidentally, if we had \ not offset the center of the contour circle by ", StyleBox["i", FontSlant->"Italic"], " \[Rho] from the real axis, and thus included both principal roots, the \ integral would be the sum of the two residues of the last two poles of 1/\ \[DoubleStruckCapitalQ]. 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This starting value is quite delicate for large M, as it is \ easy for ", "TR"], StyleBox["FindRoot", "MB"], StyleBox[" to jump over the crossover and get lost if it is not close \ enough to the desired answer.", "TR"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "FindRoot[", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[M,y] == 0,{y,1.23+I*0.02}]" }], "Input"], Cell["\<\ FindRoot::cvnwt: Newton's method failed to converge to the prescribed accuracy after 15 iterations.\ \>", "Message"], Cell[OutputFormData["\<\ {y -> 0.9821530179783111 + 0.2888086666515947*I}\ \>", "\<\ {y -> 0.982153 + 0.288809 I}\ \>"], \ "Output"] }, Open ]], Cell["which, of course, is way off. However,", "Text", FormatType->TextForm], Cell[CellGroupData[{ Cell[TextData[{ "FindRoot[", StyleBox["\[DoubleStruckCapitalQ]", "Text", FontWeight->"Plain"], "[M,y] == 0,{y,1.23+I*0.03}]" }], "Input"], Cell[OutputFormData["\<\ {y -> 1.228256632499111 + 0.029876999221237634*I}\ \>", "\<\ {y -> 1.22826 + 0.029877 I}\ \>"], \ "Output"] }, Open ]], Cell[TextData[{ "does find the root properly (and quickly).", " ", "A skeptic might complain at this point that we cheated, since we already \ knew the answer from ", StyleBox["NSolve", "MB"], " and could therefore choose a better starting point.\n\nThus, let us apply \ the method to a case where we ", StyleBox["don't", FontSlant->"Italic"], " know the answer, ", StyleBox["M", FontSlant->"Italic"], " = 600. Let's see if we can locate the principal root in this case by \ \[OpenCurlyDoubleQuote]sneaking up on it\[CloseCurlyDoubleQuote] from the \ right. We will also need to set the AccuracyGoal appropriately." }], "Text"], Cell[CellGroupData[{ Cell["\<\ {M = 600, p = 1/5, r = 5} {rho = 0.012, xstart = 1.26, xend = 1.20, decrem = 0.01}; {prevRes = 10.0^(-14), contIntgl = 0, residuum = 0}; {xmax = 0, xmin = 0, hitRes = 0}; Do[{ thisTime = Timing[contIntgl = I*rho* \tNIntegrate[(1/\[DoubleStruckCapitalQ][M,z[phi]])*Exp[I*phi], \t{phi,0,2*Pi}, AccuracyGoal->8]/(2*Pi*I)], Print[x0, \" \", thisTime[[1]]], Print[\" \", thisTime[[2]]], Print[\" \"], If[ hitRes == 0 && xmin == 0 && Abs[contIntgl] > 1000*Abs[prevRes], \t{xmax = x0, residuum = contIntgl, hitRes = 1, Continue[]}], If[ hitRes == 1 && xmax > 0 && Abs[contIntgl] < Abs[residuum]/1000, \t{xmin = x0+decrem, hitRes = 0}] } ,{x0, xstart, xend, -decrem}] Print[] Print[\"xmax, xmin = \", xmax, \" \", xmin] Print[residuum]\ \>", "Input"], Cell[OutputFormData["\<\ {600, 1/5, 5}\ \>", "\<\ 1 {600, -, 5} 5\ \>"], "Output"], Cell["\<\ 1.26 23.21 Second -14 -2.47104 10 - -14 4.14936 10 I 1.25 69.32 Second -16 4.3014 10 + -16 6.91411 10 I 1.24 85.1 Second -9 -8 6.90723 10 - 2.25022 10 I 1.23 85.06 Second -9 -8 6.90723 10 - 2.25022 10 I 1.22 69.58 Second -17 -5.27566 10 - -17 6.36464 10 I 1.21 53.75 Second -17 1.59367 10 - -17 1.4868 10 I 1.2 69.26 Second -19 6.72141 10 - -20 1.35306 10 I xmax, xmin = 1.24 1.23 -9 -8 6.90723 10 - 2.25022 10 I\ \>", "Print"] }, Open ]], Cell[TextData[{ "Thus the root is somewhere near ", StyleBox["z", FontSlant->"Italic"], " = 1.235 + 0.02 ", StyleBox["i", FontSlant->"Italic"], ", but we're not quite sure what imaginary part greater than \[Rho] to use \ in ", StyleBox["FindRoot", "MB"], ".", " ", "Thus we will loop on that quantity and see what happens." }], "Text", FormatType->TextForm], Cell[CellGroupData[{ Cell["\<\ {x0 = 1.235, imdecr = 0.005, imval = 2*rho} Off[FindRoot::cvnwt] Do[{ t = Timing[trialRoot = FindRoot[\[DoubleStruckCapitalQ][M,y] == 0, \t{y, x0+I*imval}]], Print[x0,\" \", imval,\" \", thisTime[[1]]], Print[\" \", y /. trialRoot], Print[\" \"] } ,{imval, 2*rho, rho, -imdecr}] \ \>", "Input"], Cell[OutputFormData["\<\ {1.235, 0.005, 0.024}\ \>", "\<\ {1.235, 0.005, \ 0.024}\ \>"], "Output"], Cell["\<\ 1.235 0.024 69.26 Second 0.957269 + 0.238265 I 1.235 0.019 69.26 Second 1.23551 + 0.020083 I 1.235 0.014 69.26 Second 1.0448 + 0.205013 I \ \>", "Print"] }, Open ]], Cell["\<\ The middle value was the one that worked. \.0cLet us check that it \ is a good root.\ \>", "Text", FormatType->TextForm], Cell[CellGroupData[{ Cell["\<\ trialRoot = FindRoot[\[DoubleStruckCapitalQ][M,y] == 0,{y, \ 1.235+I*0.019}, \tAccuracyGoal->8] z = y /. trialRoot \[DoubleStruckCapitalQ][M,z]\ \>", "Input", FormatType->InputForm], Cell[OutputFormData["\<\ {y -> 1.2355071192832479 + 0.020082998059690205*I}\ \>", "\<\ {y -> 1.23551 + 0.020083 I}\ \>"], \ "Output"], Cell[OutputFormData["\<\ 1.2355071192832479 + 0.020082998059690205*I\ \>", "\<\ 1.23551 + 0.020083 I\ \>"], "Output"], Cell[OutputFormData["\<\ 1.8251284927828237*^-9 + 4.7629100663471036*^-9*I\ \>", "\<\ -9 -9 1.82513 10 + 4.76291 10 I\ \>"], "Output"] }, Open ]], Cell[OutputFormData["\<\ 1.8251284927828237*^-9 + 4.7629100663471036*^-9*I\ \>", "\<\ -9 -9 1.82513 10 + 4.76291 10 I\ \>"], "Text"], Cell["which is as good as one can expect for this accuracy.", "Text", FormatType->TextForm] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Summary", "Section"]], "Subsection"], Cell[TextData[{ "First, contrary to our initial expectation, ", StyleBox["Mathematica", FontSlant->"Italic"], " can indeed find roots of very high-order polynomials. To find all of \ roots, one can use the ", StyleBox["Solve", "MB"], " function, but only up to order 80 or so. ", StyleBox["NSolve", "MB"], " works better, particularly when option ", StyleBox["AccuracyGoal", "MB"], " is set high enough, but it begins to chew up a lot of CPU cycles by the \ time one gets to order 400 or so. \n\nTo find a particular root (which for \ our particular problem was the principal root, but need not be) we use a \ trick using small circular contour integration in the complex plane. Using a \ ", StyleBox["Do", "MB"], " loop, we look for a ", "non-vanishing ", " residue to get a rough idea of where the root is. This result can then be \ sharpened using that value (or perhaps searching around near it) as the \ starting value for ", StyleBox["FindRoot", "MB"], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["About the Authors", "SectionAboutAuthor"]], \ "Subsection"], Cell["\<\ Both of us are theoretical physicists at the Los Alamos National \ Laboratory working at the intersection of nuclear and particle physics. Dick \ Silbar got his Ph. D. from the University of Michigan and came to Los Alamos \ to participate in the physics program of the Los Alamos Meson Physics \ Facility (LAMPF). He retired from LANL in 1993 to form a software development \ company, WhistleSoft, Inc., which specializes in computer-based tutorials in \ areas such as accelerator physics. He continues working at LANL as a \ consultant with his old group. Terry Goldman got his Ph. D. from Harvard, and returned to Los Alamos as a \ permanent staff member after post-doctoral appointments at SLAC, the Los \ Alamos Particle Theory group, and Cal Tech. He recently stepped down as group \ leader of the Medium Energy Nuclear Theory group and is continuing his \ research in the areas of neutrino and hadronic physics.\ \>", \ "TextAboutAuthor"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["References", "Section"]], "Subsection"], Cell[TextData[{ "\[ThickSpace]\[MediumSpace][", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]", StyleBox["C. N. Yang and T. D. Lee, ", "Reference", FontWeight->"Plain"], StyleBox["\"Statistical Theory of Equations of State and Phase Transitions, \ I:", "Reference", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " ", " ", StyleBox["Theory of Condensation,\[CloseCurlyDoubleQuote]", "Reference", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" Phys. Rev. ", "Reference", FontWeight->"Plain"], StyleBox["87", "Reference", FontWeight->"Bold"], StyleBox[", p. 404, 1952.", "Reference", FontWeight->"Plain"] }], "Reference"], Cell[TextData[{ "\[ThickSpace]\[MediumSpace][", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]This probability distribution works well for \ describing the multiplicity distributions in high energy proton-proton \ collisions, so one might expect it to also work for nucleus-nucleus \ collisions at the same energies.", " ", "See A. Giovannini and L. Van Hove, Z. Phys. ", StyleBox["C30", FontWeight->"Bold"], ", p. 391, 1986; P. Carruthers and C. C. Shih, Phys. Rev. ", StyleBox["D34", FontWeight->"Bold"], ", p. 2710, 1986; and C. C. Shih and P. Carruthers, Int. J. Mod. Phys. ", StyleBox["A2", FontWeight->"Bold"], ", 1447, 1987." }], "Reference"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowToolbars->"RulerBar", WindowSize->{962, 729}, WindowMargins->{{109, Automatic}, {68, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", "silbar", \ "Goldman", "HIC", "RootsPaper"}, "NewPaper.nb.ps", CharacterEncoding -> \ "ISO8859-1"], "Magnification"->1}, StyleDefinitions -> "TMJv4Style.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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