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Ingy döt Net
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'**********************************************
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'Subject: Comparing five methods for
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' computing Euler's constant 0.5772...
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'tested : FreeBasic 1.08.1
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'----------------------------------------------
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const eps = 1e-6
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dim as double a, b, h, n2, r, u, v
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dim as integer k, k2, m, n
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? "From the definition, err. 3e-10"
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n = 400
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h = 1
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for k = 2 to n
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h += 1 / k
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next k
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'faster convergence: Negoi, 1997
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a = log(n +.5 + 1 / (24*n))
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? "Hn "; h
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? "gamma"; h - a; !"\nk ="; n
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?
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? "Sweeney, 1963, err. idem"
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n = 21
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dim as double s(1) = {0, n}
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r = n
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k = 1
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do
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k += 1
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r *= n / k
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s(k and 1) += r / k
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loop until r < eps
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? "gamma"; s(1) - s(0) - log(n); !"\nk ="; k
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?
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? "Bailey, 1988"
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n = 5
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a = 1
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h = 1
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n2 = 2^n
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r = 1
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k = 1
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do
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k += 1
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r *= n2 / k
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h += 1 / k
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b = a: a += r * h
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loop until abs(b - a) < eps
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a *= n2 / exp(n2)
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? "gamma"; a - n * log(2); !"\nk ="; k
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?
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? "Brent-McMillan, 1980"
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n = 13
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a = -log(n)
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b = 1
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u = a
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v = b
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n2 = n * n
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k2 = 0
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k = 0
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do
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k2 += 2*k + 1
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k += 1
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a *= n2 / k
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b *= n2 / k2
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a = (a + b) / k
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u += a
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v += b
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loop until abs(a) < eps
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? "gamma"; u / v; !"\nk ="; k
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?
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? "How Euler did it in 1735"
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'Bernoulli numbers with even indices
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dim as double B2(9) = {1,1/6,-1/30,1/42,_
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-1/30,5/66,-691/2730,7/6,-3617/510,43867/798}
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m = 7
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if m > 9 then end
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n = 10
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'n-th harmonic number
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h = 1
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for k = 2 to n
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h += 1 / k
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next k
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? "Hn "; h
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h -= log(n)
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? " -ln"; h
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'expansion C = -digamma(1)
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a = -1 / (2*n)
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n2 = n * n
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r = 1
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for k = 1 to m
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r *= n2
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a += B2(k) / (2*k * r)
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next k
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? "err "; a; !"\ngamma"; h + a; !"\nk ="; n + m
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?
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? "C = 0.57721566490153286..."
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end
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@ -0,0 +1,172 @@
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'***************************************************
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'Subject: Computation of Euler's constant 0.5772...
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' with the Brent-McMillan algorithm B1,
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' Math. Comp. 34 (1980), 305-312
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'tested : FreeBasic 1.08.1 with gmp 6.2.0
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'---------------------------------------------------
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#include "gmp.bi"
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'multi-precision float pointers
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Dim as mpf_ptr a, b
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Dim shared as mpf_ptr k2, u, v
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'unsigned long integers
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Dim as ulong k, n, n2, r, s, t
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'precision parameters
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Dim shared as ulong e10, e2
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Dim shared e as clong
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Dim shared f as double
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Dim as double tim = TIMER
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CLS
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a = allocate(len(__mpf_struct))
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b = allocate(len(__mpf_struct))
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u = allocate(len(__mpf_struct))
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v = allocate(len(__mpf_struct))
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k2 = allocate(len(__mpf_struct))
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'log(x/y) with the Taylor series for atanh(x-y/x+y)
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Sub ln (byval s as mpf_ptr, byval x as ulong, byval y as ulong)
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Dim as mpf_ptr d = u, q = v
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Dim k as ulong
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'Möbius transformation
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k = x: x -= y: y += k
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If x <> 1 Then
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Print "ln: illegal argument x - y <> 1"
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End
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End If
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's = 1 / (x + y)
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mpf_set_ui (s, y)
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mpf_ui_div (s, 1, s)
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'k2 = s * s
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mpf_mul (k2, s, s)
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mpf_set (d, s)
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k = 1
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Do
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k += 2
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'd *= k2
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mpf_mul (d, d, k2)
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'q = d / k
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mpf_div_ui (q, d, k)
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's += q
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mpf_add (s, s, q)
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f = mpf_get_d_2exp (@e, q)
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Loop until abs(e) > e2
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's *= 2
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mpf_mul_2exp (s, s, 1)
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End Sub
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'Main
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'n = 2^i * 3^j * 5^k
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'log(n) = r * log(16/15) + s * log(25/24) + t * log(81/80)
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'solve linear system for r, s, t
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' 4 -3 -4| i
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'-1 -1 4| j
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'-1 2 -1| k
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'examples
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t = 1
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select case t
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case 1
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n = 60
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r = 41
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s = 30
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t = 18
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'100 digits
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case 2
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n = 4800
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r = 85
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s = 62
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t = 37
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'8000 digits, 0.6 s
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case 3
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n = 9375
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r = 91
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s = 68
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t = 40
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'15625 digits, 2.5 s
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case else
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n = 18750
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r = 98
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s = 73
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t = 43
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'31250 digits, 12 s. @2.00GHz
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end select
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'decimal precision
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e10 = n / .6
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'binary precision
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e2 = (1 + e10) / .30103
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'initialize mpf's
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mpf_set_default_prec (e2)
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mpf_inits (a, b, u, v, k2, Cptr(mpf_ptr, 0))
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'Compute log terms
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ln b, 16, 15
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'a = r * b
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mpf_mul_ui (a, b, r)
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ln b, 25, 24
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'a += s * b
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mpf_mul_ui (u, b, s)
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mpf_add (a, a, u)
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ln b, 81, 80
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'a += t * b
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mpf_mul_ui (u, b, t)
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mpf_add (a, a, u)
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''gmp_printf (!"log(%lu) %.*Ff\n", n, e10, a)
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'B&M, algorithm B1
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'a = -a, b = 1
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mpf_neg (a, a)
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mpf_set_ui (b, 1)
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mpf_set (u, a)
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mpf_set (v, b)
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k = 0
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n2 = n * n
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'k2 = k * k
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mpf_set_ui (k2, 0)
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do
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'k2 += 2k + 1
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mpf_add_ui (k2, k2, (k shl 1) + 1)
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k += 1
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'b = b * n2 / k2
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mpf_div (b, b, k2)
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mpf_mul_ui (b, b, n2)
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'a = (a * n2 / k + b) / k
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mpf_div_ui (a, a, k)
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mpf_mul_ui (a, a, n2)
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mpf_add (a, a, b)
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mpf_div_ui (a, a, k)
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'u += a, v += b
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mpf_add (u, u, a)
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mpf_add (v, v, b)
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f = mpf_get_d_2exp (@e, a)
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Loop until abs(e) > e2
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mpf_div (u, u, v)
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gmp_printf (!"gamma %.*Ff (maxerr. 1e-%lu)\n", e10, u, e10)
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gmp_printf (!"k = %lu\n\n", k)
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gmp_printf (!"time: %.7f s\n", TIMER - tim)
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end
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@ -0,0 +1,23 @@
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' ******************************************
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'Subject: Euler's constant 0.5772...
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'tested : FreeBasic 1.08.1 with mpfr 4.1.0
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'-------------------------------------------
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#include "gmp.bi"
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#include "mpfr.bi"
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dim as mpfr_ptr a = allocate(len(__mpfr_struct))
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dim as ulong e2, e10
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dim as double tim = TIMER
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'decimal precision
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e10 = 100
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'binary precision
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e2 = (1 + e10) / .30103
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mpfr_init2 (a, e2)
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mpfr_const_euler (a, MPFR_RNDN)
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mpfr_printf (!"gamma %.*Rf\n\n", e10, a)
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gmp_printf (!"time: %.7f s\n", TIMER - tim)
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end
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