Finding a sine wave

*To*: mathgroup at smc.vnet.net*Subject*: [mg95341] Finding a sine wave*From*: Hugh Goyder <h.g.d.goyder at cranfield.ac.uk>*Date*: Fri, 16 Jan 2009 06:07:49 -0500 (EST)

An experiment will give me three coordinates which lie on a sine wave. I have to find the sine wave efficiently. There are three unknowns the sine wave amplitude, A, the wavelength of the sine wave, L, and the phase, ph. I also know that the wavelength is larger than the interval containing my measurement points. I think this condition removes possible multiple solutions. Below I give two methods which need improving. The first method uses FindRoot. This methods works about 60% of the time. I give six examples where it fails. The failure may be due to poor initial guesses. In the second method I use FindFit. This is not quite the correct method because I have an equal number of equations and unknowns. Some of the failures here, I think, are due to there being no error for the algorithm to work with. I give examples of failures. I have also tried FindInstance, Reduce and NSolve but I don't think these are appropriate. Here are some questions 1. I would really like a symbolic solution rather than an iterative one. Is such a solution possible? 2. Can anyone improve on the methods below to make them more robust? 3. I have some control over my x-locations. How can I work out best x locations given an estimate of the wavelength L? Many thanks for all answers. xx = Sort[Join[{0}, RandomReal[{0, 10}, 2]]] yy = RandomReal[{-25, 25}, 3] sol = FindRoot[{yy[[1]] - A*Sin[ph], yy[[2]] - A*Sin[2*Pi*(xx[[2]]/L) + ph], yy[[3]] - A*Sin[2*Pi*(xx[[3]]/L) + ph]}, {{A, Max[Abs[yy]]}, {L, Max[xx]}, {ph, 0}}] Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, Epilog -> {PointSize[0.02], (Point[#1] & ) /@ Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}] FindRootFailures = {{{0, 3.781264462608982, 3.797055100562352}, {-22.948348087068737, 1.4581744078038472, -14.242676740704574}}, {{0, 4.424069570670131, 8.861716396743098}, {-6.444538773775843, -10.787309362608688, 15.579334094330942}}, {{0, 4.424069570670131, 8.861716396743098}, {-5.944536471563289, 10.627536107497393, -21.294232497202316}}, {{0, 1.3680556047878967, 8.546115267250002}, {4.128528863849845, 2.9017293848933923, -22.51610539815371}}, {{0, 1.3738869718371616, 5.689309423462079}, {20.553993773523437, -16.972841620064592, 16.61185061676568}}, {{0, 1.0408828831133632, 8.484645515699821}, {-3.7267589861478045, 4.1016610850387, -24.600635061804443}}}; (({xx, yy} = #1; sol = FindRoot[{yy[[1]] - A*Sin[ph], yy[[2]] - A*Sin[2*Pi*(xx[[2]]/L) + ph], yy[[3]] - A*Sin[2*Pi*(xx[[3]]/L) + ph]}, {{A, Max[Abs[yy]]}, {L, Max[xx]}, {ph, 0}}]; Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, Epilog -> {PointSize[0.02], (Point[#1] & ) /@ Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}]) & ) /@ FindRootFailures xx = Sort[Join[{0}, RandomReal[{0, 10}, 2]]] yy = RandomReal[{-25, 25}, 3] sol = FindFit[ Transpose[{xx, yy}], {A*Sin[2*Pi*(x/L) + ph], Max[Abs[xx]] < L}, {A, L, ph}, x] Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, Epilog -> {PointSize[0.02], (Point[#1] & ) /@ Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}] FindFitFailures = {{{0, 2.463263668380835, 4.3190892163093615}, {-1.8407827676541144, -8.736574079785198, 12.661520984622932}}, {{0, 3.894521446091823, 9.937403619870642}, {12.381369822165155, 15.840399165432128, -6.914634137727327}}, {{0, 8.087369725271945, 8.343899312282815}, {24.73795103895976, -13.396248713970305, 7.150470311065216}}, {{0, 1.5480031866178834, 9.575255260205617}, {-8.163720246784278, 13.373468882892958, -18.018462091502098}}, {{0, 0.6152784485896601, 0.818296772602134}, {-1.858698140836046, 3.695113783904491, -1.3186989232026658}}, {{0, 4.743093154057316, 6.028314583406327}, {-16.60893277597446, -17.413392198343093, -18.54081837965986}}}; (({xx, yy} = #1; sol = FindFit[ Transpose[{xx, yy}], {A*Sin[2*Pi*(x/L) + ph], Max[Abs[xx]] < L}, {A, L, ph}, x]; Plot[Evaluate[A*Sin[2*Pi*(x/L) + ph] /. sol], {x, 0, 10}, Epilog -> {PointSize[0.02], (Point[#1] & ) /@ Transpose[{xx, yy}]}, PlotRange -> {{0, 10}, {-25, 25}}]) & ) /@ FindFitFailures

**Follow-Ups**:**Re: Finding a sine wave***From:*DrMajorBob <btreat1@austin.rr.com>