diff options
Diffstat (limited to 'aberth')
| -rw-r--r-- | aberth/lib.fut | 82 |
1 files changed, 77 insertions, 5 deletions
diff --git a/aberth/lib.fut b/aberth/lib.fut index 9e79197..a52d8b6 100644 --- a/aberth/lib.fut +++ b/aberth/lib.fut @@ -12,6 +12,10 @@ let PolyN = 19i32 type poly = [PolyN]f64 +-- 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 -> c64.mk_re coef c64.+ pt c64.* accum) (c64.mk_re p[nterms-1]) (take (nterms - 1) p) @@ -25,16 +29,19 @@ let evaln_d (p: poly) (nterms: i32) (pt: f64): f64 = let eval_d (p: poly) (pt: f64): f64 = evaln_d p (length p) pt let derivative (p: poly): *poly = - map (\(i, v) -> f64.i32 i * v) (zip (1...PolyN-1) (take (PolyN - 1) p)) ++ [0] + map (\(i, v) -> f64.i32 i * v) (zip (1..<PolyN) (take (PolyN - 1) 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): f64 = 1 + f64.maximum (map (\coef -> f64.abs (coef / p[PolyN-1])) p) module aberth = { type approx = [N]complex - type state = {p: poly, deriv: poly, bound: poly, approx: approx, radius: f64} + -- 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: f64} - let gen_coord (r: f64) (rng: rand_engine.rng): (rand_engine.rng, f64) = + let gen_coord (r: f64) (rng: *rand_engine.rng): *(rand_engine.rng, f64) = uniform_real.rand (-r, r) rng let gen_coord_c (r: f64) (rng: rand_engine.rng): (rand_engine.rng, complex) = @@ -42,13 +49,78 @@ module aberth = { let (rng, y) = gen_coord r rng in (rng, c64.mk x y) - let regenerate (rng: rand_engine.rng) (s: state): (rand_engine.rng, *approx) = + 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 s.radius) rngs) + 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 -> f64.abs coef * f64.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} + + -- Lagrange-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 (c64./) pvals derivvals + let sums = map (\i -> + reduce_comm (c64.+) (c64.mk_re 0.0) + (map (\j -> + if i == j then c64.mk_re 0.0 + else c64.mk_re 1.0 c64./ + (approx[i] c64.- approx[j])) + (0..<N))) + (0..<N) + let offsets = map2 (\quo sum -> quo c64./ (c64.mk_re 1.0 c64.- quo c64.* sum)) + quos sums + let approx = map2 (c64.-) approx offsets + let svals = map (eval_d ctx.bound <-< c64.mag) approx + let conditions = map2 (\p s -> c64.mag p > 1e-9 * 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 -> f64.i32 (i32.get_bit i bitfield * 2 - 1)) entry main: i32 = 42 |