96d1400654
* bernoulli -> bernoulli-number * farey -> farey-sequence * fibonacci/mod -> modular-fibonacci * order -> unit-group-order * orders -> unit-group-orders Documented `make-fibonacci' and `make-modular-fibonacci' Reworked text about loading external libraries in docs for `math/bigfloat' Removed type aliases like Z, Q, Prime (I like them, but TR was printing them in unexpected places like array return types)
70 lines
2.6 KiB
Racket
70 lines
2.6 KiB
Racket
#lang typed/racket/base
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(require racket/fixnum
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"../../flonum.rkt"
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"../../base.rkt"
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"../number-theory/bernoulli.rkt"
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"../number-theory/factorial.rkt"
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"bigfloat-struct.rkt")
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(provide bfhurwitz-zeta)
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(define 0.5b0 (parameterize ([bf-precision 2]) (bf 0.5)))
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(: bfhurwitz-zeta-series (Bigfloat Bigfloat -> Bigfloat))
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(define (bfhurwitz-zeta-series s q)
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(define eps (bf* 0.5b0 epsilon.bf))
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(let loop ([i 0] [y 0.bf])
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(define dy (bfexpt (bf+ q (bf i)) (bf- s)))
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(define new-y (bf+ y dy))
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(cond [(or ((bfabs dy) . bf<= . (bf* eps new-y))
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(not (bfrational? new-y)))
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new-y]
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[else
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(loop (+ i 1) new-y)])))
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(: bfhurwitz-zeta-euler-maclaurin (Bigfloat Bigfloat -> Bigfloat))
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(define (bfhurwitz-zeta-euler-maclaurin s q)
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(define n (exact-ceiling (+ (* 0.5 (bf-precision) (/ (log 2) (log 10))) 10)))
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(: f (Integer -> Bigfloat))
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(define (f k) (bfexpt (bf+ (bf k) q) (bf- s)))
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(define fn (f n))
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(define n+q (bf+ (bf n) q))
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(define sqr-n+q (bf* n+q n+q))
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(define eps epsilon.bf)
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(define y0
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(for/fold: ([y0 : Bigfloat (bf* fn (bf+ (bf/ n+q (bf- s 1.bf)) 0.5b0))]
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) ([k (in-range n)])
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(bf+ y0 (f k))))
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(define max-k 100)
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(let: loop : Bigfloat ([y : Bigfloat y0]
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[numer : Bigfloat s]
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[denom : Bigfloat (bf/ fn n+q)]
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[k : Nonnegative-Fixnum 0])
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(define ck (bf (/ (bernoulli-number (* 2 (fx+ k 1))) (factorial (* 2 (fx+ k 1))))))
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(define dy (bf* (bf* numer denom) ck))
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(define new-y (bf+ y dy))
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(cond [((bfabs dy) . bf<= . (bf* eps (bfabs new-y)))
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new-y]
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[else
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(define k.bf (bf k))
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(loop new-y
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(bf* (bf* numer (bf+ s (bf+ (bf* 2.bf k.bf) 1.bf)))
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(bf+ s (bf+ (bf* 2.bf k.bf) 2.bf)))
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(bf/ denom sqr-n+q)
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(fx+ k 1))])))
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(: bfhurwitz-zeta (Bigfloat Bigfloat -> Bigfloat))
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(define (bfhurwitz-zeta s q)
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(cond [(s . bf<= . 1.bf) (if (bf= s 1.bf) +inf.bf +nan.bf)]
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[(q . bf<= . 0.bf) (if (bf= q 0.bf) +inf.bf +nan.bf)]
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[(s . bf> . (bf/ (bflog (bf* 0.5b0 epsilon.bf)) (bf- (bflog q) (bflog1p q))))
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;; At this point, only the first term in the series is necessary; the condition can had by
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;; solving for s in (q+1)^-s < 0.5 * epsilon.0 * q^-s
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(bfexpt q (bf- s))]
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[(s . bf> . (bf+ (bf* 2.bf q) (bf 15)))
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;; Determined experimentally that the series computes fewer total iterations here
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(bfhurwitz-zeta-series s q)]
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[else
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(bfhurwitz-zeta-euler-maclaurin s q)]))
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