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-rw-r--r--aberth/lib.fut82
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