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    Re: The Maths behind DSD
    From: Paul Hirose
    Date: 2017 Oct 9, 17:02 -0700

    Let's look at linear interpolation in a table of sines:
    
    sin 45 = .707107
                     .012233
    sin 46 = .719340
    
    (The .012233 is the difference between the adjacent sines.) Suppose we
    want the sine of 45.5°. That's midway between table entries, so the
    interpolating factor (often called "p") is 0.5. In a linear
    interpolation you multiply p and the difference between entries
    (.012233) to obtain the increment to be added to sin 45. Thus the
    linearly interpolated sine is .713224. But a calculator says sin 45.5 is
    actually .713250, so the linear interpolation was good to only 4 digits.
    
    The full accuracy of the table can be realized if second differences are
    applied. Begin by expanding the table to show the first and second
    differences:
    
    sin 44 = .694658
                     12449
    sin 45 = .707107     -216
                     12233
    sin 46 = .719340     -219
                     12014
    sin 47 = .731354
    
    It's common practice to write the differences as integers, so the first
    difference between sin 45 and sin 46 is simply 12233. The two applicable
    second differences are -216 and -219.
    
    Bessel's interpolation formula is a linear interpolation plus a
    correction which includes second differences. The interpolating factor p
    is 0.5 as before. The coefficient B2 is defined as p * (p - 1) / 4.
    Since p is always between 0 and 1, B2 is always negative. In this case
    it's -.0625. The Bessel formula is
    
    increment = p * 12233 + B2 * (-216 + -219) = 6144
    
    Mentally convert the increment to .006144 and apply it: sin 45.5 =
    .707107 + .006144 = .713251. That's off by only one unit in the last place.
    
    How do you know when to apply second differences? If they are ignored,
    the maximum error is about 1/8 of the nearest second difference. So in
    the above example, the max error of a linear interpolation between 45°
    and 46° should be about 28 units in the last place. Actual error of the
    linear interpolation was 26.
    
    Bessel's formula also provides for higher order differences. Maximum
    error is 1/120 of the third difference if it's ignored. In my example
    the third difference is only 3, so its effect would be insignificant.
    
    In the old days when tables were computed by hand, interpolation with
    higher order differences was a big labor saver. A table could be
    computed with high accuracy but a relatively coarse interval, then the
    gaps filled by interpolation.
    

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