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import "lib/github.com/diku-dk/complex/complex"
import "lib/github.com/diku-dk/cpprandom/random"
module rand_engine = minstd_rand
module uniform_real = uniform_real_distribution f32 rand_engine
module cplx = mk_complex f32
type complex = cplx.complex
let N = 22i32
let PolyN = N + 1
type poly = [PolyN]f32
-- First element of pair steps fastest
let iota2 (n: i32) (m: i32): [](i32, i32) =
flatten (map (\y -> map (\x -> (x, y)) (iota n)) (iota m))
let evaln_c (p: poly) (nterms: i32) (pt: complex): complex =
foldr (\coef accum -> cplx.mk_re coef cplx.+ pt cplx.* accum)
(cplx.mk_re p[nterms-1]) (take (nterms - 1) p)
let eval_c (p: poly) (pt: complex): complex = evaln_c p (length p) pt
let evaln_d (p: poly) (nterms: i32) (pt: f32): f32 =
foldr (\coef accum -> coef + pt * accum)
p[nterms-1] (take (nterms - 1) p)
let eval_d (p: poly) (pt: f32): f32 = evaln_d p (length p) pt
let derivative (p: poly): *poly =
map (\(i, v) -> f32.i32 i * v) (zip (1..<PolyN) (tail p)) ++ [0]
-- Cauchy's bound: https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#Lagrange's_and_Cauchy's_bounds
let max_root_norm (p: poly): f32 =
1 + f32.maximum (map (\coef -> f32.abs (coef / p[PolyN-1])) (init p))
module aberth = {
type approx = [N]complex
-- bound is 's' in the stop condition formulated at p.189-190 of
-- https://link.springer.com/article/10.1007%2FBF02207694
type context = {p: poly, deriv: poly, bound: poly, radius: f32}
let gen_coord (r: f32) (rng: *rand_engine.rng): *(rand_engine.rng, f32) =
uniform_real.rand (-r, r) rng
let gen_coord_c (r: f32) (rng: rand_engine.rng): (rand_engine.rng, complex) =
let (rng, x) = gen_coord r rng
let (rng, y) = gen_coord r rng
in (rng, cplx.mk x y)
let generate (ctx: context) (rng: *rand_engine.rng): *(rand_engine.rng, approx) =
let rngs = rand_engine.split_rng N rng
let (rngs, approx) = unzip (map (gen_coord_c ctx.radius) rngs)
let rng = rand_engine.join_rng rngs
in (rng, approx)
let compute_bound_poly (p: poly): *poly =
map2 (\coef i -> f32.abs coef * f32.i32 (4 * i + 1)) p (0..<PolyN)
let initialise (p: *poly): *context =
let deriv = derivative p
let bound = compute_bound_poly p
let radius = max_root_norm p
in {p, deriv, bound, radius}
-- Jacobi-style step where the new elements are computed in parallel from
-- the previous values
let step (ctx: context) (approx: *approx): *(bool, approx) =
let pvals = map (eval_c ctx.p) approx
let derivvals = map (evaln_c ctx.deriv (PolyN - 1)) approx
let quos = map2 (cplx./) pvals derivvals
let sums = map (\i ->
reduce_comm (cplx.+) (cplx.mk_re 0.0)
(map (\j ->
if i == j then cplx.mk_re 0.0
else cplx.mk_re 1.0 cplx./
(approx[i] cplx.- approx[j]))
(0..<N)))
(0..<N)
let offsets = map2 (\quo sum -> quo cplx./ (cplx.mk_re 1.0 cplx.- quo cplx.* sum))
quos sums
let approx = map2 (cplx.-) approx offsets
let svals = map (eval_d ctx.bound <-< cplx.mag) approx
let conditions = map2 (\p s -> cplx.mag p <= 1e-5 * s) pvals svals
let all_converged = all id conditions
in (all_converged, approx)
let iterate (ctx: context) (rng: *rand_engine.rng): (rand_engine.rng, *approx) =
let (rng, approx) = generate ctx rng
let (init_conv, approx) = step ctx approx
let (rng, _, _, _, approx) =
loop (rng, conv, tries, step_idx, approx) =
(rng, init_conv, 1, 1: i32, approx)
while !conv
do if step_idx + 1 > tries * 100
then let (rng, approx) = generate ctx rng
let (conv, approx) = step ctx approx
in (rng, conv, tries + 1, 0, approx)
else let (conv, approx) = step ctx approx
in (rng, conv, tries, step_idx + 1, approx)
in (rng, approx)
let aberth (p: *poly) (rng: *rand_engine.rng): *(rand_engine.rng, approx) =
iterate (initialise p) rng
}
-- Set the constant coefficient to 1; nextDerbyshire will never change it
let init_derbyshire: poly =
[1] ++ replicate (PolyN - 1) (-1)
let next_derbyshire (p: *poly): *(bool, poly) =
let (_, p, looped) =
loop (i, p, cont) = (0, p, true)
while cont && i < length p
do if p[i] == -1
then (i, p with [i] = 1, false)
else (i + 1, p with [i] = -1, true)
in (looped, p)
let derbyshire_at_index (index: i32): *poly =
let bitfield = (index << 1) + 1
in tabulate PolyN (\i -> f32.i32 (i32.get_bit i bitfield * 2 - 1))
let calc_index (value: f32) (left: f32) (right: f32) (steps: i32): i32 =
i32.f32 ((value - left) / (right - left) * (f32.i32 steps - 1) + 0.5)
let point_index
(width: i32) (height: i32)
(bottom_left: complex) (top_right: complex)
(pt: complex)
: i32 =
-- Range for 'yi' is reversed because image coordinates go down in the y
-- direction, while complex coordinates go up in the y direction
let xi = calc_index (cplx.re pt) (cplx.re bottom_left) (cplx.re top_right) width
let yi = calc_index (cplx.im pt) (cplx.im top_right) (cplx.im bottom_left) height
in if 0 <= xi && xi < width && 0 <= yi && yi < height
then width * yi + xi
else -1
entry main_job
(start_index: i32) (num_polys: i32)
(width: i32) (height: i32)
(left: f32) (top: f32) (right: f32) (bottom: f32)
(seed: i32)
: []i32 =
-- Unnecessary to give each polynomial a different seed
let rng = rand_engine.rng_from_seed [seed]
let bottom_left = cplx.mk left bottom
let top_right = cplx.mk right top
let indices = flatten
(map (\idx ->
let p = derbyshire_at_index idx
let (_, pts) = aberth.aberth p rng
in map (point_index width height bottom_left top_right) pts)
(start_index ..< start_index + num_polys))
let filtered = filter (\i -> i != -1) indices
in reduce_by_index (replicate (width * height) 0)
(+) 0
filtered
(replicate (length filtered) 1)
entry main_all
(width: i32) (height: i32)
(left: f32) (top: f32) (right: f32) (bottom: f32)
(seed: i32)
: []i32 =
main_job 0 (1 << N) width height left top right bottom seed
entry get_N: i32 = N
|
