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Re: Inverse function solution

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Solve[ {x == Cos[u], y == Cos[u + v] }, {u, v} ]

Its closed/analytic solution is not possible, even numerically.

The known solutions are ellipses from sine waves with a phase difference, having x^2, x y and y^2 terms, as also sketched in Lissajous curves:

ParametricPlot[{Cos[u], Cos[u + v]}, {u, -Pi, Pi}, {v, -Pi, Pi}]

Can there be a work around?

Narasimham



Hi, Narasimham,

It can be solved numerically, though, since it is double-periodic it should be done carefully. This is a fast shot of how it can be done:

Here I put y=1 and vary only x:

lst = Table[{x,
   FindRoot[{x == Cos[u],
      1 == Cos[u + v]}, {{u, 0.1}, {v, 0.1}}] /. {x_ -> a_,
      y_ -> b_} -> {a, b}}, {x, 0., 0.4, 0.05}]

This is the outcome:

{{0., {7.85398, -7.85398}}, {0.05, {7.80396, -7.80396}}, {0.1, \
{7.75381, -7.75381}}, {0.15, {7.70341, -7.70341}}, {0.2, {7.65262, \
-7.65262}}, {0.25, {7.6013, -7.6013}}, {0.3, {7.54929, -7.54929}}, \
{0.35, {7.49641, -7.49641}}, {0.4, {5.12391, -5.12391}}}

Let us check, if it is right:

lst /. {x_, {y_, z_}} -> {x == Cos[y], 1 == Cos[y + z]}

{{False, True}, {True, True}, {True, True}, {True, True}, {True,
  True}, {True, True}, {True, True}, {True, True}, {True, True}}

So, the result is right except the one in the first parentheses. Let us plot it. Try this:

ListPlot[{lst /. {x_, {y_, z_}} -> {x, y},
  lst /. {x_, {y_, z_}} -> {x, z}}]

Have fun, Alexei


Alexei BOULBITCH, Dr., habil.
IEE S.A.
ZAE Weiergewan,
11, rue Edmond Reuter,
L-5326 Contern, LUXEMBOURG

Office phone :  +352-2454-2566
Office fax:       +352-2454-3566
mobile phone:  +49 151 52 40 66 44

e-mail: alexei.boulbitch@iee.lu

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Re: Inverse function solution Alexei Boulbitch <Alexei.Boulbitch@iee.lu> - 2014-04-30 05:40 +0000

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