arcsinc( 1-y ), for small y

[Albert Chan]

The book Numerical Method that work showed Acton's favorite way of doing interpolation (no difference table)
A variation of Aitken scheme of intepolation (p 94), using post #8 as an example

Solve x, such that sinc(x) = k = 1/6.5 ~ 0.153846154

Using my asympotic formula, x ~ Pi/(1 + k + 1.244 k^3) ~ 2.712066499

sinc(2.712) = 0.1535768820, error ~ -0.00027, thus x too high (sinc(x) is decreasing for x = [0, 4.4934])
sinc(2.711) = 0.1539688052, error ~ +0.00013, thus x too low
sinc(2.7115) = 0.1537728267, error ~ -0.00007, used for 3 point quadratic fit.

Acton variation is to sort the values, interpolated estimate up on top.
This ensures top numbers likely the best estimate, with other entries very "tight"
Mistakes during manual calculations are easier to spot.

All entries below are linear interpolation (against top entry), for k = 1/6.5

PHP Code:

sinc(x)       x         
0.1537728267  2.71 15
0.1539688052  2.71 10  1312 920
0.1535768820  2.71 20  1312 888  910 --> x = 2.711312910 


Quadratic fit might not be needed, if interpolated estimate used as 3rd point

PHP Code:

sinc(x)       x         
0.1539688052  2.711
0.1535768820  2.712        2.711312947
0.1538461395  2.711312947  2.711312910 --> x = 2.711312910 


For Casio FX-115MS, Interpolation (A,B), (C,D) Formula:
With CALC: 0A + B + (C-A)^-1(D-B)(X-A)
Or, SOLVE: (X-A)(B-0C-D) + (Y-B)(C-A)

update: instead of manually getting the points to fit, we can use iterations:

sinc(pi/2) = 1/(pi/2) = 2/pi ≈ 0.6366197724
If k ≥ 2/pi: x = sin(x)/k
If k < 2/pi: x = pi - asin(kx)

For above example, k < 2/pi, we get 2.712 → 2.7111966 → 2.711332599 ...
Using Aitken Δ² method, above 3 numbers extrapolated to 2.711312910