Bürgi's Kunstweg to Calculate Sines

[Albert Chan]

(05-08-2022 01:19 PM)Thomas Klemm Wrote:  We can therefore describe the double difference operation with the matrix \( \Delta \):

\(
\Delta = \left[\begin{matrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 2
\end{matrix}\right]
\)

Asymmetry of matrix, [2,-1] top, [-2,2] bottom , we should match edge cases coefficients.

Lower bound, α = β, sin(α-β) = sin(0°) = 0

Δ² = sin(α+β) - 2 sin(α) + sin(α-β) = -dot([sin(α), sin(α+β)], [2, -1])

Upper bound, α = 90°, sin(α+β) = sin(α-β) = cos(β)

Δ² = sin(α+β) - 2 sin(α) + sin(α-β) = -dot([sin(α-β), sin(α)], [-2,2])

Quote:Note: What appeared to be simple turned out to be a problem.
The separator in the matrix has to be a tabulator.
But I couldn't figure out how to do that when you copy it from this page.
It is always replaced by a blank.
Thus my recommendation for now is to copy the matrix into a text-editor, replace the blanks by tabs and copy the result into Free42.

Another way is to click QUOTE, copy raw matrix data (with tabs), then paste to Free42