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How to add four arctangents efficiently

  1. May 3, 2013 #1
    I need to compute the addition of four arctangents:

    [itex]\alpha = \arctan(x_1) + \arctan(x_2) + \arctan(x_3) + \arctan(x_4) [/itex]

    Rather than call four arctangent functions, I was thinking of using the arctangent addition formula:

    [itex]\arctan(u) + \arctan(v) = \arctan(\frac{u + v}{1 - uv}) + \pi n[/itex], for some [itex]n \in N[/itex]

    I could invoke the above property three times and I would only have to call the arctan function once. My problem is how to keep track of the integer [itex]n[/itex] throughout so that the result is correct.

    My initial variables [itex]x_1[/itex] to [itex]x_4[/itex] are all positive so the result should be an angle [itex]0 \leq \alpha < 2\pi[/itex]. I suspect the answer lies in keeping track of the sign of the [itex]1 - uv[/itex] denominators and introducing some factor of [itex]\pi[/itex] correction if they go negative.

    I'll work this through the weekend but I thought I would post this here in case someone figured this out already.

    Thank you,
    manuel
     
  2. jcsd
  3. May 3, 2013 #2
  4. May 3, 2013 #3

    Ray Vickson

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    If ##\arctan(x_1) + \arctan(x_2) = \arctan(y_1) + \pi n_1## and ##\arctan(x_3) + \arctan(x_4) = \arctan(y_2) + \pi n_2,## we then have ##\sum_{i=1}^4 \arctan(x_i) = \arctan(y_1) + \arctan(y_2) + \pi (n_1 + n_2),## which has the form ##\arctan(z) + \pi (n_1 + n_2 + n_3).## Here,
    [tex] z = \frac{y_1 + y_2}{1-y_1 y_2}.[/tex]
    We might as well just say that ##n_1 + n_2 + n_3 = n,## an integer.
     
    Last edited: May 3, 2013
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