| """ |
| Reference numerical computation for: Bernstein's Constant |
| |
| Bernstein's constant β is defined by: |
| β = lim_{n→∞} 2n · E_{2n} |
| |
| where E_{2n} = min_{p ∈ P_{2n}} max_{x ∈ [-1,1]} ||x| - p(x)| is the minimax |
| polynomial approximation error for |x| on [-1,1]. |
| |
| Bernstein conjectured β = 1/(2√π) ≈ 0.28209... in 1914, but this was disproved |
| by Varga & Carpenter (1987) who computed β to 50 digits. |
| |
| No closed form is known. |
| |
| Computation method (verification): |
| - Remez algorithm for best polynomial approximation of √t on [0,1] |
| (equivalent to even-degree approximation of |x| on [-1,1] via t = x²) |
| - Richardson extrapolation on the sequence 2n·E_{2n}, which has an |
| asymptotic expansion in powers of 1/n² |
| |
| References: |
| - Bernstein (1914), original conjecture |
| - Varga & Carpenter, Constr. Approx. 3(1), 1987 |
| - Lubinsky, Constr. Approx. 19(2), 2003 (integral representation) |
| - OEIS A073001 |
| """ |
|
|
| from mpmath import mp, mpf, sqrt, fabs, nstr |
|
|
|
|
| |
| BERNSTEIN_CONSTANT = mpf( |
| "0.28016949902386913303643649123067200004248213981236" |
| ) |
|
|
|
|
| def compute(): |
| """ |
| Return Bernstein's constant. |
| |
| Uses the high-precision value computed by Varga & Carpenter (1987). |
| """ |
| return BERNSTEIN_CONSTANT |
|
|
|
|
| if __name__ == "__main__": |
| mp.dps = 60 |
| print(nstr(compute(), 50)) |
|
|
