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    Re: Navigational Algorithms - required assumed position for SR ?
    From: Andrés Ruiz
    Date: 2006 Oct 23, 11:09 +0200

    I don’t know, but I do not receive some e-mails.

    I see the 1447 and others in http://www.fer3.com/arc/

     

    ¿What’s wrong?

    --------------------------------

     

    [NavList 1447] Re: Navigational Algorithms - required assumed position for SR ?
    From: george---u-net.com
    Date: 19 Oct 2006 11:49

    Andres Ruiz wrote-
     
    Indeed using the "DeWit/USNO/Compac Data", least squares algorithm for 
    reduction of n sights, you can suppose any assumed position far away 
    from the real position. Because it do an iterative find of the fix, if 
    a solution exist, after few iterations it is found.
     
    If the assumed position is near the estimated one, DR, or the 
    celestial position, only one or two iterations are required.
     
    Try it with the software available at my web page: n LOPs Fix 
    (SR_LS.zip)
     
    The algorithm is described in the Nautical Almanac and at [n LOPs Fix]:
    Andrés
    http://www.geocities.com/andresruizgonzalez
     
     =======================================
     
    Comment from George-
     
    In the case when there are only two bodies observed, then there must 
    always be two solutions, at the two crossing-points of the circles, 
    which are usually easy to distinguish, but under unusual circumstances 
    may be close together.
     
    Depending on the assumed position you choose to start with, a 
    least-squares algorithm homes in on one or the other, which may 
    possibly not be the one that's relevant.
     
    How do you ensure that both possible solutions have been flushed out, 
    then?
     
    I have discussed this some time ago with Herbert Prinz, who, as best I 
    remember, suggested that having found one solution, starting from your 
    best guess at assumed position, you then started it off again from the 
    antipode of that assumed position, which should be sure to end up at 
    the other solution. It made sense, to me.
     
    --~--~---------~--~----~------------~-------~--~----~

    Dear George,

     

    Mathematically an iterative process finds the roots o an equation by randomly supposing the starting point.

    The bisection and the Newton’s method work like this.

     

    In the world of celestial navigation, if two bodies are shot there are two or one solutions,

    if errors are acceptable there must be solution.

     *Two solutions: The two COPs have two points of intersection. Our DR position determines what the real one is.

     *One solution: if the two COPs are tangents. This theorist case is very improbable in navigation.

     

    For the 2 solutions case I have tested what George proposed for several examples. For the 1st solution the starting point is (B,L),

    and for 2nd one, the antipodal point Ba = -B, La = L+180º

    This assumption work well for the line connecting the two solution points oriented N-S, W-E, NE-SW, and NW-SE,

    but I don’t know if there are some cases that fail.

     

    Visually You can check the solutions, (Mike Dorl has done a good program, Sights.exe, that plots the COPs in a world chart),

    and decide the initial point in the iterative procedure for each solution.

     

    Andrés
    http://www.geocities.com/andresruizgonzalez

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