File size: 1,379 Bytes
848d4b7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | """
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
# High-precision reference value from Varga & Carpenter (1987), OEIS A073001
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))
|