(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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[ 524806, 9986]*) (*NotebookOutlinePosition[ 525770, 10016]*) (* CellTagsIndexPosition[ 525726, 10012]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["LaguerreNSolve", "Title"], Cell[TextData[StyleBox["T. M. Jonassen & R. Ghulghazaryan", FontSize->14, FontSlant->"Italic"]], "Text"], Cell[TextData[{ "The following example implements the ", StyleBox["laguer()", FontWeight->"Bold"], " and ", StyleBox["zroots()", FontWeight->"Bold"], " functions from NRIC to imitate the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["NSolve", FontWeight->"Bold"], ". The syntax is slightly different, as ", StyleBox["NSolve", FontWeight->"Bold"], " uses ", StyleBox["NSolve[", FontWeight->"Bold"], StyleBox["poly==0", FontSlant->"Italic"], ",", StyleBox["var", FontSlant->"Italic"], StyleBox["]", FontWeight->"Bold"], " while ", StyleBox["LagerreNSolve", FontWeight->"Bold"], " uses ", StyleBox["LaguerreNSolve[", FontWeight->"Bold"], StyleBox["poly,var", FontSlant->"Italic"], StyleBox["]", FontWeight->"Bold"], " to solve ", StyleBox["poly==0", FontSlant->"Italic"], ", that is only the left hand side of the equation appears as an argument, \ and the zero on the right hand side is left out." }], "Text"], Cell[CellGroupData[{ Cell["Utillity functions", "Section"], Cell[TextData[{ "We will need a few simple functions in order to define ", StyleBox["LaguerreNSolve", FontWeight->"Bold"], ". The function of these are easily understood and not commented here:" }], "Text"], Cell[BoxData[ \(\(rtc[l_] := {Re[l], Im[l]};\)\)], "Input"], Cell[BoxData[ \(makelist[poly_, var_] := Flatten[Map[rtc, CoefficientList[poly, var]]]\)], "Input"], Cell[BoxData[ \(tocomp[{\[Alpha]_, \[Beta]_}] := \[Alpha] + \[ImaginaryI]\ \[Beta]\)], \ "Input"], Cell[BoxData[ \(br[x_] := {x}\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["External function", "Section"], Cell[TextData[{ "We must install the external program ", StyleBox["findroot", FontWeight->"Bold"], ". Note that this program must be running under use of ", StyleBox["LaguerreNSolve", FontWeight->"Bold"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(link = Install["\"]\)], "Input"], Cell[OutputFormData["\<\ LinkObject[\"/home/torejo/mathbin/findroot\", 2, \ 2]\ \>", "\<\ LinkObject[/home/torejo/mathbin/findroot, 2, 2]\ \>"], "Output"] }, Open ]], Cell["\<\ The following terminates the external program (do not use this now \ -:)):\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Uninstall[link]\)], "Input"], Cell[OutputFormData["\<\ \"/home/torejo/mathbin/findroot\"\ \>", \ "\<\ /home/torejo/mathbin/findroot\ \>"], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["LaguerreNSolve", FontWeight->"Plain"], " definition" }], "Section"], Cell[TextData[{ "The construction of ", StyleBox["LaguerreNSolve", FontWeight->"Bold"], " is straight forward. The main function is the external function ", StyleBox["newroots", FontWeight->"Bold"], " defined by the external program ", StyleBox["findroot", FontWeight->"Bold"], ". The rest of the code is simply formatting stuff. The output is of the \ same form as of ", StyleBox["NSolve", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[ \(LaguerreNSolve[poly_, var_] := Map[br, Map[Rule[var, #] &, Chop[Map[tocomp, Evaluate[ newroots[Evaluate[N[makelist[poly, var]]]]]]]]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Limitations", "Section"], Cell[TextData[{ "The polynomial must have numerical coefficients, and be a polynomial in \ one variable only. The polynomial must have non-negative integer exponents. \ If the degree of the polynomial is too large, and the polynomial is strongly \ oscillating there is no guarantee that the returned numbers has anything to \ do with the true solutions. This is also the case for ", StyleBox["NSolve", FontWeight->"Bold"], ". See ", StyleBox["Example 2", FontWeight->"Bold", FontSlant->"Italic"], " below. Indeed, this is in an other context, our main problem, and this \ will be solved by another method. The code has not been extensively tested, \ and the error handeling could be better." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Examples", "Section"], Cell[TextData[{ "Here are some simple use of ", StyleBox["LaguerreNSolve", FontWeight->"Bold"], ". As in ", StyleBox["NSolve", FontWeight->"Bold"], ", the solutions are returned as a list of rules:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[\((x - 3)\) \((x + 2)\), x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -2.}, {x -> 3.}}\ \>", "\<\ {{x -> -2.}, {x \ -> 3.}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[x\^5 - 4 x\^4 + 2 x\^3 - x + 1, x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -0.6863181826239249}, {x -> 0.2392423627450048 + 0.7012742484709023*I}, {x -> 0.2392423627461626 - 0.7012742484711563*I}, {x -> 0.7725389689523162}, {x -> 3.4352944881804417}}\ \>", "\<\ {{x -> \ -0.686318}, {x -> 0.239242 + 0.701274 I}, {x -> 0.239242 - 0.701274 I}, {x -> 0.772539}, {x -> 3.43529}}\ \>"], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[z\^3 + 1, z]\)], "Input"], Cell[OutputFormData["\<\ {{z -> -1.}, {z -> 0.5 - 0.8660254037844387*I}, {z -> 0.5 + 0.8660254037844387*I}}\ \>", "\<\ {{z -> -1.}, {z -> 0.5 - \ 0.866025 I}, {z -> 0.5 + 0.866025 I}}\ \>"], "Output"] }, Open ]], Cell["\<\ The argument can be anything that evaluates to a polynomial of the \ desired form:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[Product[s - Random[], {i, 7}], s]\)], "Input"], Cell[OutputFormData["\<\ {{s -> 0.015315906215420826}, {s -> \ 0.21067828556478957}, {s -> 0.2874368023485597}, {s -> 0.5383410264051484}, {s -> 0.5734057276546796}, {s -> 0.8742031541254628}, {s -> 0.9886535168190954}}\ \>", "\<\ {{s -> 0.0153159}, {s -> 0.210678}, {s \ -> 0.287437}, {s -> 0.538341}, {s -> 0.573406}, {s -> 0.874203}, {s -> 0.988654}}\ \>"], "Output"] }, Open ]], Cell["Here is an example with complex coefficients:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[u\^7 + 3 \[ImaginaryI]\ u\^5 - 2, u]\)], "Input"], Cell[OutputFormData["\<\ {{u -> -1.228754409685915 + 1.1848091301946988*I}, {u -> -0.8374174374271284 - 0.3042632000906518*I}, {u -> -0.5527291383100232 + 0.8186939766838252*I}, {u -> -0.04752635909927073 - 0.9093426672480563*I}, {u -> 0.5267357344401329 + 0.7071734266396297*I}, {u -> 0.9125564806165627 - 0.23827712383441021*I}, {u -> 1.2271351294656418 - 1.2587935423450354*I}}\ \>", "\<\ {{u -> -1.22875 \ + 1.18481 I}, {u -> -0.837417 - 0.304263 I}, {u -> -0.552729 + 0.818694 I}, {u -> -0.0475264 - 0.909343 I}, {u -> 0.526736 + 0.707173 I}, {u -> 0.912556 - 0.238277 I}, {u -> 1.22714 - 1.25879 I}}\ \>"], "Output"] }, Open ]], Cell[TextData[{ StyleBox["LaguerreNSolve", FontWeight->"Bold"], " fails if the polynomial contains non-numerical coefficients:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[x\^2 + \[Alpha]\ x + 1, x]\)], "Input"], Cell[OutputFormData["\<\ $Failed\ \>", "\<\ $Failed\ \>"], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["NSolve", FontWeight->"Plain"], " comparison" }], "Section"], Cell[CellGroupData[{ Cell["Example 1", "Subsection"], Cell["Here is a simple 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We solve the result to compare with the output from ", StyleBox["LaguerreNSolve", FontWeight->"Bold"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(nsol = NSolve[poly \[Equal] 0, x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -4.995155577973647}, {x -> \ -1.0466512840819044}, {x -> -0.430783795455566 - 0.6363345341585076*I}, {x -> -0.430783795455566 + 0.6363345341585076*I}, {x -> 0.15340912267704188 - 0.9502936384767037*I}, {x -> 0.15340912267704188 + 0.9502936384767037*I}, {x -> 0.7982781038063 - 0.24880998570792962*I}, {x -> 0.7982781038063 + 0.24880998570792962*I}}\ \>", "\<\ {{x -> -4.99516}, \ {x -> -1.04665}, {x -> -0.430784 - 0.636335 I}, {x -> -0.430784 + 0.636335 I}, {x -> 0.153409 - 0.950294 I}, {x -> 0.153409 + 0.950294 I}, {x -> 0.798278 - 0.24881 I}, {x -> 0.798278 + 0.24881 I}}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "Here is the result from ", StyleBox["LaguerreNSolve", FontWeight->"Bold"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lagnsol = LaguerreNSolve[poly, x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -4.995155577970307}, {x -> \ -1.0466512842953706}, {x -> -0.4307837954555671 - 0.6363345341585015*I}, {x -> -0.4307837954555598 + 0.6363345341585078*I}, {x -> 0.15340912238828344 - 0.9502936385702011*I}, {x -> 0.15340912242440954 + 0.9502936387073064*I}, {x -> 0.7982781040174095 - 0.24880998661688272*I}, {x -> 0.7982781043467029 + 0.24880998646938998*I}}\ \>", "\<\ {{x -> \ -4.99516}, {x -> -1.04665}, {x -> -0.430784 - 0.636335 I}, {x -> -0.430784 + 0.636335 I}, {x -> 0.153409 - 0.950294 I}, {x -> 0.153409 + 0.950294 I}, {x -> 0.798278 - 0.24881 I}, {x -> 0.798278 + 0.24881 I}}\ \>"], "Output"] }, Open ]], Cell["We store the result in two vectors:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(vec1 = x /. nsol\)], "Input"], Cell[OutputFormData["\<\ {-4.995155577973647, -1.0466512840819044, -0.430783795455566 - 0.6363345341585076*I, -0.430783795455566 + 0.6363345341585076*I, 0.15340912267704188 - 0.9502936384767037*I, 0.15340912267704188 + 0.9502936384767037*I, 0.7982781038063 - 0.24880998570792962*I, 0.7982781038063 + 0.24880998570792962*I}\ \>", "\<\ {-4.99516, -1.04665, \ -0.430784 - 0.636335 I, -0.430784 + 0.636335 I, 0.153409 - 0.950294 I, 0.153409 + 0.950294 I, 0.798278 - 0.24881 I, 0.798278 + 0.24881 I}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(vec2 = x /. lagnsol\)], "Input"], Cell[OutputFormData["\<\ {-4.995155577970307, -1.0466512842953706, -0.4307837954555671 - 0.6363345341585015*I, -0.4307837954555598 + 0.6363345341585078*I, 0.15340912238828344 - 0.9502936385702011*I, 0.15340912242440954 + 0.9502936387073064*I, 0.7982781040174095 - 0.24880998661688272*I, 0.7982781043467029 + 0.24880998646938998*I}\ \>", "\<\ {-4.99516, -1.04665, \ -0.430784 - 0.636335 I, -0.430784 + 0.636335 I, 0.153409 - 0.950294 I, 0.153409 + 0.950294 I, 0.798278 - 0.24881 I, 0.798278 + 0.24881 I}\ \>"], "Output"] }, Open ]], Cell["And compute the difference:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(diff = vec1 - vec2\)], "Input"], Cell[OutputFormData["\<\ {-3.340439036492171*^-12, 2.1346613365835765*^-10, 1.1102230246251565*^-15 - 6.106226635438361*^-15*I, -6.217248937900877*^-15 - 2.220446049250313*^-16*I, 2.8875843383069366*^-10 + 9.349743201880756*^-11*I, 2.526323317209034*^-10 - 2.3060275911035433*^-10*I, -2.1110946324398583*^-10 + 9.089531072259405*^-10*I, -5.404028335931343*^-10 - 7.61460366982547*^-10*I}\ \>", "\<\ \ -12 -10 -15 -15 {-3.34044 10 , 2.13466 10 , 1.11022 10 - 6.10623 10 I, -15 -16 -10 -11 -6.21725 10 - 2.22045 10 I, 2.88758 10 + 9.34974 10 I, -10 -10 -10 -10 2.52632 10 - 2.30603 10 I, -2.11109 10 + 9.08953 10 I, -10 -10 -5.40403 10 - 7.6146 10 I}\ \>"], "Output"] }, Open ]], Cell["The over all difference is quite acceptable:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\@Dot[Abs[diff], Abs[diff]]\)], "Input"], Cell[OutputFormData["\<\ 1.4132667649524917*^-9\ \>", "\<\ -9 1.41327 10\ \>"], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Example 2", "Subsection"], Cell["We will consider the following quadratic polynomial:", "Text"], Cell[BoxData[ \(f[x_] := x\^2 - 3\)], "Input"], Cell[TextData[{ "Iterates of this polynomial is know to have only real roots. That is ", Cell[BoxData[ \(TraditionalForm\`\(f\^n\)(x) = f\[SmallCircle]f\[SmallCircle]\[CenterEllipsis]\[SmallCircle]\(f( x)\)\)]], ", a ", Cell[BoxData[ \(TraditionalForm\`2\^n\)]], "degree polynomial, has only real roots. In fact, the closure of the union \ of roots of all these polynomials as ", Cell[BoxData[ \(TraditionalForm\`n \[RightArrow] \[Infinity]\)]], " is a Cantor set contained in \[DoubleStruckCapitalR]. A Cantor set is an \ uncountable closed, totally disconnected perfect set. In ", StyleBox["Mathematica", FontSlant->"Italic"], " language we can form these compositions by ", StyleBox["Nest", FontWeight->"Bold"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[Nest[f, x, i] // Expand], {i, 1, 4}]\)], "Input"], Cell["\<\ 2 -3 + x 2 4 6 - 6 x + x 2 4 6 8 33 - 72 x + 48 x - 12 x + x 2 4 6 8 10 12 1086 - 4752 x + 8352 x - 7704 x + 4098 x - 1296 x + 240 x - 14 16 24 x + x\ \>", "Print"] }, Open ]], Cell[TextData[{ "Here are the ", StyleBox["NSolve", FontWeight->"Bold"], " solution to ", Cell[BoxData[ \(TraditionalForm\`f(f(f(x))) = 0\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NSolve[Nest[f, x, 3] \[Equal] 0, x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -2.274934668767676}, {x -> \ -2.0312637693343696}, {x -> -1.3689293259293962}, {x -> -0.9081146694327342}, {x -> 0.9081146694327349}, {x -> 1.3689293259293938}, {x -> 2.0312637693343767}, {x -> 2.2749346687676706}}\ \>", "\<\ {{x -> \ -2.27493}, {x -> -2.03126}, {x -> -1.36893}, {x -> -0.908115}, {x -> 0.908115}, {x -> 1.36893}, {x -> 2.03126}, {x -> 2.27493}}\ \>"], \ "Output"] }, Open ]], Cell[TextData[{ "We get the same solutions with ", StyleBox["LaguerreNSolve", FontWeight->"Bold"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[Nest[f, x, 3], x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -2.2749345857241856}, {x -> \ -2.03126378584866}, {x -> -1.3689293204664816}, {x -> -0.9081146667416294}, {x -> 0.9081146667217679}, {x -> 1.3689293203988593}, {x -> 2.0312624295049324}, {x -> 2.274935942155397}}\ \>", "\<\ {{x -> \ -2.27493}, {x -> -2.03126}, {x -> -1.36893}, {x -> -0.908115}, {x -> 0.908115}, {x -> 1.36893}, {x -> 2.03126}, {x -> 2.27494}}\ \>"], \ "Output"] }, Open ]], Cell["The fifth iterate is strongly oscillating:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[Nest[f, x, 5], {x, \(-2.4\), 2.4}, PlotStyle \[Rule] {RGBColor[1, 0, 0]}, AxesLabel \[Rule] {"\", \*"\"\<\!\(f\^5\)(x)\>\""}, PlotPoints \[Rule] 1000, PlotRange 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StyleBox["NSolve", FontWeight->"Bold"], " finds in this case only real roots (but we do not know anything about the \ correctness of these values):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NSolve[Nest[f, x, 5] \[Equal] 0, x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -2.301454576824469}, {x -> \ -2.2897832132896854}, {x -> -2.256130919380754}, {x -> -2.2309063334578227}, {x -> -2.108628282196158}, {x -> -2.0681284466352774}, {x -> -1.9960548632301973}, {x -> -1.9625266886132906}, {x -> -1.4657736340761351}, {x -> -1.4197732107315386}, {x -> -1.312579997610831}, {x -> -1.2464609310315484}, {x -> -1.01148642674667}, {x -> -0.953835208692754}, {x -> -0.8700299565317963}, {x -> -0.8386163646276246}, {x -> 0.8386163646697032}, {x -> 0.8700299564122256}, {x -> 0.9538352090592697}, {x -> 1.0114864262735856}, {x -> 1.2464609351012046}, {x -> 1.3125799877245488}, {x -> 1.4197732302398534}, {x -> 1.4657736194800355}, {x -> 1.9625268362170654}, {x -> 1.996054621936738}, {x -> 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The \ reason for this is the use of ", StyleBox["zroots", FontWeight->"Bold"], " inside of the program ", StyleBox["findroot", FontWeight->"Bold"], ". The main objective is to develop a method to find all roots with a great \ accuracy. On the other hand ", StyleBox["NSolve", FontWeight->"Bold"], " fails for ", Cell[BoxData[ \(TraditionalForm\`\(f\^6\)(x) = 0\)]], " as shown below." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LaguerreNSolve[Nest[f, x, 5], x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -2.3960729213947998 - 0.03894775049496833*I}, \ {x -> -2.3663806444855062 + 0.0952581861366318*I}, {x -> -2.3009962755954847 - 0.15763449022260625*I}, {x -> -2.2255229860235306 + 0.17502848273580235*I}, {x -> -2.1070233966027225 - 0.20576957719809527*I}, {x -> -2.0443740010303166 + 0.17358711851061195*I}, {x -> -1.8963154233651793 - 0.12474502895248098*I}, {x -> -1.8774179561968478 + 0.08344327396803186*I}, {x -> -1.4838324800565539 - 0.0002202025865549272*I}, {x -> -1.3647796531373588}, {x -> -1.358095655015654}, {x -> -1.2365749552222838}, {x -> -1.0157746661299014}, {x -> -0.9478989314619755}, {x -> -0.8744105609158098}, {x -> -0.8366972417182267}, {x -> 0.8357675659759694}, {x -> 0.8770116504699903}, {x -> 0.9441551302179961}, {x -> 1.0180592627731773}, {x -> 1.2602247400278939}, {x -> 1.2829814095398642}, {x -> 1.4495469548950417 - 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0.0350914 I}, {x -> 1.44985 + 0.0353637 I}, {x -> 1.89842 - 0.0450602 I}, {x -> 1.95224 + 0.0999154 I}, {x -> 2.04431 - 0.150593 I}, {x -> 2.11356 + 0.127718 I}, {x -> 2.23926 - 0.146672 I}, {x -> 2.24684 + 0.118476 I}, {x -> 2.35839 + 0.0572867 I}, {x -> 2.36156 - 0.061343 I}}\ \>"], "Output"] }, Open ]], Cell[TextData[{ StyleBox["NSolve", FontWeight->"Bold"], " fails for ", Cell[BoxData[ \(TraditionalForm\`\(f\^6\)(x) = 0\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NSolve[Nest[f, x, 6] \[Equal] 0, x]\)], "Input"], Cell[OutputFormData["\<\ {{x -> -2.553463864768054}, {x -> \ -2.510152918406173}, {x -> -2.4656450299249464 - 0.26084574089016205*I}, {x -> -2.4656450299249464 + 0.26084574089016205*I}, {x -> -2.440046364642811}, {x -> -2.3336590300590205 - 0.3135107708143643*I}, {x -> -2.3336590300590205 + 0.3135107708143643*I}, {x -> -2.203586260156875 - 0.3628414866584539*I}, {x -> -2.203586260156875 + 0.3628414866584539*I}, {x -> -2.0600543608489854 - 0.3842878243662537*I}, {x -> -2.0600543608489854 + 0.3842878243662537*I}, {x -> -2.053279742183815}, {x -> -1.8996185578925264 - 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0.320141 I}, {x -> 1.5091 + 0.320141 I}, {x -> 1.71303 - 0.357537 I}, {x -> 1.71303 + 0.357537 I}, {x -> 1.85668}, {x -> 1.89912 - 0.380221 I}, {x -> 1.89912 + 0.380221 I}, {x -> 2.05946 - 0.372413 I}, {x -> 2.05946 + 0.372413 I}, {x -> 2.17168 - 0.53068 I}, {x -> 2.17168 + 0.53068 I}, {x -> 2.19254 - 0.329469 I}, {x -> 2.19254 + 0.329469 I}, {x -> 2.3703 - 0.243527 I}, {x -> 2.3703 + 0.243527 I}, {x -> 2.6827 - 0.44807 I}, {x -> 2.6827 + 0.44807 I}, {x -> 2.90228}}\ \>"], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "NRIC W.H. Press, S. A. Teukolsky, W. T. Vetterling & B. P. Flannery: ", StyleBox["Numerical Recipes in C", FontSlant->"Italic"], ". Cambridge Univerity Press", ". ", StyleBox["1992", FontWeight->"Bold"] }], "Text", FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell["Compilation", "Section"], Cell[TextData[{ "On a Linux platform the ", StyleBox["MathLink", FontSlant->"Italic"], " program is compiled by:" }], "Text"], Cell["\<\ mcc -o findroot -O2 -march=i686 findroot.tm findroot.c -lm \ -lML\ \>", "Program"], Cell[TextData[{ "My suggestion is to keep all working ", StyleBox["MathLink", FontSlant->"Italic"], " programs in one directory. On my machine I have used the directory ", StyleBox["/home/torejo/mathbin", FontFamily->"Lucidatypewriter"], " for private programs." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["MathLink template file", "Section"], Cell[TextData[{ "This is the ", StyleBox["MathLink", FontSlant->"Italic"], " template file, here called ", StyleBox["findroot.tm", FontWeight->"Bold", FontSlant->"Italic"], ":" }], "Text"], Cell["\<\ :Begin: :Function: newroots :Pattern: newroots[c_List] :Arguments: {c} :ArgumentTypes: {RealList} :ReturnType: Manual :End:\ \>", "Program"] }, Open ]], Cell[CellGroupData[{ Cell["C code for external functions", "Section"], Cell[TextData[{ "This is the C code, here called ", StyleBox["findroot.c", FontWeight->"Bold", FontSlant->"Italic"], ":" }], "Text"], Cell["\<\ /* MathLink program for LaguerreNSolve */ /* Compile with : mcc -o findroot -O2 -march=i686 findroot.tm findroot.c -lm \ -lML */ #include #include #include \"/usr/local/Wolfram/Mathematica/4.2/AddOns/MathLink/DeveloperKit/\ Linux/CompilerAdditions/mathlink.h\" #include #include /* #define's for laguerre() */ #define EPSS 1.0e-7 #define MR 8 #define MT 10 #define MAXIT (MT*MR) /* #define's for zroots */ #define EPS 2.0e-6 #define MAXM 100 /* Macro used in laguerre() */ static double maxarg1, maxarg2; #define FMAX(a,b) (maxarg1=(a),maxarg2=(b),(maxarg1)>(maxarg2) ? (maxarg1) : \ (maxarg2)) /* Find one complex root, a root finder from NRIC */ void laguerre(complex *a, int m, complex *x, int *its) { \tint iter,j; \tdouble abx,abp,abm,err; \tcomplex dx,x1,b,d,f,g,h,sq,gp,gm,g2; \tstatic double frac[MR+1] = {0.0,0.5,0.25,0.75,0.13,0.38,0.62,0.88,1.0}; \tfor (iter=1;iter<=MAXIT;iter++) { \t\t*its=iter; \t\tb=a[m]; \t\terr=cabs(b); \t\td=0.0; \t\tf=0.0; \t\tabx=cabs(*x); \t\tfor (j=m-1;j>=0;j--) { \t\t\tf=(*x)*f+d; \t\t\td=(*x)*d+b; \t\t\tb=(*x)*b+a[j]; \t\t\terr=cabs(b)+abx*err; \t\t} \t\terr *= EPSS; \t\tif (cabs(b) <= err) return; \t\tg=d/b; \t\tg2=g*g; \t\th=g2-(2.0*(f/b)); \t\tsq=csqrt((double) (m-1)*((double) m*h-g2)); \t\tgp=g+sq; \t\tgm=g-sq; \t\tabp=cabs(gp); \t\tabm=cabs(gm); \t\tif (abp < abm) gp=gm; \t\tdx = FMAX(abp,abm) > 0.0 ? ((double)m+0.0*I)/gp \t\t\t: exp(log(1+abx))*(cos((double)iter)+I*sin((double)iter)); \t\tx1=(*x)-dx; \t\tif (creal(*x) == creal(x1) && cimag(*x) == cimag(x1)) return; \t\tif (iter % MT) *x=x1; \t\telse *x = *x-(frac[iter/MT]-dx); \t} \tfprintf(stderr,\"too many iterations in laguerre\\n\"); \treturn; } /* Find all roots, a frontend for laguerre() */ void zroots(complex *a, int m, complex *roots, int polish) { \tint i,its,j,jj; \tcomplex x,b,c,ad[MAXM]; \tfor (j=0;j<=m;j++) ad[j]=a[j]; \tfor (j=m;j>=1;j--) { \t\tx=0.0+0.0*I; \t\tlaguerre(ad,j,&x,&its); \t\tif (fabs(cimag(x)) <= 2.0*EPS*fabs(creal(x))) x=creal(x)+0.0*I; \t\troots[j]=x; \t\tb=ad[j]; \t\tfor (jj=j-1;jj>=0;jj--) { \t\t\tc=ad[jj]; \t\t\tad[jj]=b; \t\t\tb=x*b+c; \t\t} \t} \tif (polish) \t\tfor (j=1;j<=m;j++) \t\t\tlaguerre(a,m,&roots[j],&its); \tfor (j=2;j<=m;j++) { \t\tx=roots[j]; \t\tfor (i=j-1;i>=1;i--) { \t\t\tif (creal(roots[i]) <= creal(x)) break; \t\t\troots[i+1]=roots[i]; \t\t} \t\troots[i+1]=x; \t} } void newroots(double *c, long length) { complex *a; complex *x, *x1; double *retarr; long dp[2]; int i,val; int deg; deg = (int)(length/2) - 1; dp[0] = deg; dp[1] = 2; /* Allocate space for \"a\" and \"x1\" and \"retarr\" */ a = (complex *)mall