# Message Boards

Posted 9 years ago
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 I want to find the roots of an equation using FindRoot as a function of a real parameter K and then plot the real and imaginary solutions vs. the parameter K. The function I'm trying to find the roots is: f[\[CapitalOmega]_,K_]:=1.\[VeryThinSpace]+(0.00021600988594924262 K^10+(3.81005811091083*^-6-5.617765774643865*^-9 I) K^2 \[CapitalOmega]^8-(1.938898556339015*^-8-4.736384338194926*^-11 I) \[CapitalOmega]^10+K^4 \[CapitalOmega]^6 ((-0.00024705957314623457+1.7313407860214411*^-7 I)+1.3234889800848443*^-23 \[CapitalOmega]^2)+K^6 \[CapitalOmega]^4 ((0.005208736011404687\[VeryThinSpace]+8.502422744684232*^-8 I)+4.235164736271502*^-22 \[CapitalOmega]^2)+K^8 \[CapitalOmega]^2 ((0.0021316316776359315\[VeryThinSpace]+1.0578397620930897*^-8 I)+6.776263578034403*^-21 \[CapitalOmega]^2))/(2.138711742071709*^-6 K^12+0.000021082195553663905 K^10 \[CapitalOmega]^2+0.000051246570978881814 K^8 \[CapitalOmega]^4-3.481280809505708*^-6 K^6 \[CapitalOmega]^6+8.665879028798112*^-8 K^4 \[CapitalOmega]^8-9.45611694948034*^-10 K^2 \[CapitalOmega]^10+3.831913991301718*^-12 \[CapitalOmega]^12) The real solutions Wr : Subscript[W, r][K_,\[Mu]_]:=Re[\[CapitalOmega]/.FindRoot[1.+(0.00021600988594924262 K^10+(3.81005811091083*^-6-5.617765774643865*^-9 I) K^2 \[CapitalOmega]^8-(1.938898556339015*^-8-4.736384338194926*^-11 I) \[CapitalOmega]^10+K^4 \[CapitalOmega]^6 ((-0.00024705957314623457+1.7313407860214411*^-7 I)+1.3234889800848443*^-23 \[CapitalOmega]^2)+K^6 \[CapitalOmega]^4 ((0.005208736011404687+8.502422744684232*^-8 I)+4.235164736271502*^-22 \[CapitalOmega]^2)+K^8 \[CapitalOmega]^2 ((0.0021316316776359315+1.0578397620930897*^-8 I)+6.776263578034403*^-21 \[CapitalOmega]^2))/(2.138711742071709*^-6 K^12+0.000021082195553663905 K^10 \[CapitalOmega]^2+0.000051246570978881814 K^8 \[CapitalOmega]^4-3.481280809505708*^-6 K^6 \[CapitalOmega]^6+8.665879028798112*^-8 K^4 \[CapitalOmega]^8-9.45611694948034*^-10 K^2 \[CapitalOmega]^10+3.831913991301718*^-12 \[CapitalOmega]^12),{\[CapitalOmega],K/Sqrt[\[Mu] (1+K^2)]}]]; And the imaginary solutions Wi : Subscript[W, i][K_,\[Mu]_]:=Im[\[CapitalOmega]/.FindRoot[1.\[VeryThinSpace]+(0.00021600988594924262 K^10+(3.81005811091083*^-6-5.617765774643865*^-9 I) K^2 \[CapitalOmega]^8-(1.938898556339015*^-8-4.736384338194926*^-11 I) \[CapitalOmega]^10+K^4 \[CapitalOmega]^6 ((-0.00024705957314623457+1.7313407860214411*^-7 I)+1.3234889800848443*^-23 \[CapitalOmega]^2)+K^6 \[CapitalOmega]^4 ((0.005208736011404687\[VeryThinSpace]+8.502422744684232*^-8 I)+4.235164736271502*^-22 \[CapitalOmega]^2)+K^8 \[CapitalOmega]^2 ((0.0021316316776359315\[VeryThinSpace]+1.0578397620930897*^-8 I)+6.776263578034403*^-21 \[CapitalOmega]^2))/(2.138711742071709*^-6 K^12+0.000021082195553663905 K^10 \[CapitalOmega]^2+0.000051246570978881814 K^8 \[CapitalOmega]^4-3.481280809505708*^-6 K^6 \[CapitalOmega]^6+8.665879028798112*^-8 K^4 \[CapitalOmega]^8-9.45611694948034*^-10 K^2 \[CapitalOmega]^10+3.831913991301718*^-12 \[CapitalOmega]^12),{\[CapitalOmega],K/Sqrt[\[Mu](1+K^2)]}]]; When I plotted the real part Wr I got the following numerical roundoff error : Block[{\[Mu] = 100}, Plot[{Subscript[W, r][K, \[Mu]]}, {K, 0, 5}, Frame -> True, PlotRange -> {0, 0.2}, PlotRangePadding -> 0, WorkingPrecision -> 100]] And When I plotted the imaginary part Wi I got also the numerical roundoff error : Block[{\[Mu] = 100}, Plot[{-Subscript[W, i][K, \[Mu]] }, {K, 0, 5}, Frame -> True, PlotRange -> {0, 0.02}, PlotRangePadding -> 0, WorkingPrecision -> 100]] Please, Can you give me now some hint here? I confess that I' m not very savvy with Mathematica and don' t really know which methods are implemented. Attachments:
 One could do In[74]:= Clear[f] f[\[CapitalOmega]_, K_] := 1 + (0.00021600988594924262 K^10 + (3.81005811091083 10^(-6) - 5.617765774643865 10^(-9) I) K^2 \[CapitalOmega]^8 - \ (1.938898556339015 10^(-8) - 4.736384338194926 10^(-11) I) \[CapitalOmega]^10 + K^4 \[CapitalOmega]^6 ((-0.00024705957314623457 + 1.7313407860214411 10^(-7) I) + 1.3234889800848443 10^(-23) \[CapitalOmega]^2) + K^6 \[CapitalOmega]^4 ((0.005208736011404687 + 8.502422744684232 10^(-8) I) + 4.235164736271502 10^(-22) \[CapitalOmega]^2) + K^8 \[CapitalOmega]^2 ((0.0021316316776359315 + 1.0578397620930897 10^(-8) I) + 6.776263578034403 10^(-21) \ \[CapitalOmega]^2))/(2.138711742071709 10^(-6) K^12 + 0.000021082195553663905 K^10 \[CapitalOmega]^2 + 0.000051246570978881814 K^8 \[CapitalOmega]^4 - 3.481280809505708 10^(-6) K^6 \[CapitalOmega]^6 + 8.665879028798112 10^(-8) K^4 \[CapitalOmega]^8 - 9.45611694948034 10^(-10) K^2 \[CapitalOmega]^10 + 3.831913991301718 10^(-12) \[CapitalOmega]^12) In[76]:= (* The solutions W *) Clear[W] W[K_, \[Mu]_] := With[{preC = 100}, FindRoot[SetPrecision[f[\[CapitalOmega], K], preC], {\[CapitalOmega], K/Sqrt[\[Mu]/(1 + K^2)]}, WorkingPrecision -> preC] ][[-1, -1]] In[78]:= {Re[#], Im[#]} &[W[3.1, 100]] During evaluation of In[78]:= FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than 100. digits of working precision to meet these tolerances. >> Out[78]= {-1624.8869305883541028830136870262106382101189280581163075734345152591252489\80638534881282900723186087, 34717.52852573805626771928431449226093876668088758334764286400786043682850613291079441186321008020593} In[90]:= Off[FindRoot::lstol] ListPlot[Table[{Re[#], Im[#]} &[W[x, 100]], {x, 0.1, 5, 0.001}], Frame -> True, PlotLabel -> "K\[Element]{0.1,5} \[Mu]=100", PlotRange -> All, PerformanceGoal -> "Quality", PlotMarkers -> {Automatic, 6}] On[FindRoot::lstol] to plot an Argand diagram of the roots foundOne could also do a Off[FindRoot::lstol] ParametricPlot[{Re[#], Im[#]} &[W[x, 100]], {x, 0.1, 5}, Frame -> True, PlotLabel -> "K\[Element]{0.1,5} \[Mu]=100", PlotRange -> All, PerformanceGoal -> "Quality"] On[FindRoot::lstol] `to see some structure I'm unable to decipher. If one enters W[0,100] a divergency is produced, because K = 0 and the search of FindRoot starts at [CapitalOmega] = 0, so the denominator is really 0.