301 lines
11 KiB
Go
301 lines
11 KiB
Go
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// Copyright (c) 2026 Petr Balvín <opensource@petrbalvin.org> (https://petrbalvin.org)
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// SPDX-License-Identifier: MIT
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package core
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import "math"
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// Modified Bessel functions of the first and second kind, the workhorses
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// of heat conduction, waveguides and filtered noise. I₀ and I₁ use the
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// polynomial fits below 3.75 (Abramowitz & Stegun 9.8.1 and 9.8.2) and
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// their integral representations beyond, K₀ and K₁ integrate their
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// kernels directly, and integer orders follow by the stable direction
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// of the recurrence: upward for I, downward for K.
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// All are element-wise over real arrays; ints and float32 promote.
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// BesselI0 returns the modified Bessel function of the first kind of
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// order zero, I₀(x), of each element. I₀ is even with I₀(0) = 1.
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func BesselI0(a *Array) (*Array, error) {
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return a.realFunc("BesselI0", besselI0)
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}
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// BesselI1 returns the modified Bessel function of the first kind of
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// order one, I₁(x), of each element. I₁ is odd with I₁(0) = 0.
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func BesselI1(a *Array) (*Array, error) {
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return a.realFunc("BesselI1", besselI1)
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}
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// BesselK0 returns the modified Bessel function of the second kind of
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// order zero, K₀(x), of each element, defined for x > 0. K₀ diverges
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// logarithmically at 0 and returns +Inf there.
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func BesselK0(a *Array) (*Array, error) {
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return a.realFunc("BesselK0", besselK0)
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}
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// BesselK1 returns the modified Bessel function of the second kind of
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// order one, K₁(x), of each element, defined for x > 0. K₁ diverges
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// like 1/x at 0 and returns +Inf there.
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func BesselK1(a *Array) (*Array, error) {
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return a.realFunc("BesselK1", besselK1)
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}
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// BesselIn returns the modified Bessel function of the first kind of
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// integer order n, Iₙ(x), of each element. The recurrence runs in its
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// stable direction: orders well below |x| climb upward from the
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// quadrature I₀ and I₁ at O(n) cost, and orders comparable to or above
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// it run Miller's downward algorithm, which is where the upward climb
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// would amplify the parasitic K component the seeds carry. The
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// downward walk starts above the turning point at order |x|, so it
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// costs O(n + |x|) steps: for an order far below a large argument that
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// is the difference between a microsecond and minutes, which is why
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// the climb exists.
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func BesselIn(n int, a *Array) (*Array, error) {
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if n < 0 {
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return nil, errf("BesselIn: order must be ≥ 0, got %d", n)
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}
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return a.realFunc("BesselIn", func(x float64) float64 {
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if n == 0 {
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return besselI0(x)
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}
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if n == 1 {
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return besselI1(x)
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}
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// The relative contamination the climb suffers grows roughly
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// like e^{n²/x}, and the seeds it climbs from are only as good
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// as besselI0 and besselI1 are there: below 3.75 those are
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// polynomial fits worth about 2.5e-8, so a short climb over a
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// small argument multiplies a weak seed by the condition number
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// and lands coarser than the downward walk, which renormalises
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// through the same seed once. The climb is therefore taken only
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// where its seeds are strong, from besselIUpwardMinX up, and the
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// split follows n² ≤ 4x inside that: above it the downward walk
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// is both accurate and affordable, since the order is then
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// comparable to the argument and O(n + |x|) steps is the
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// answer's own price.
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ax := math.Abs(x)
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if ax >= besselIUpwardMinX && float64(n)*float64(n) <= 4*ax {
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return besselIUpward(n, x)
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}
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return besselInMiller(n, x)
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})
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}
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// besselIUpwardMinX is the argument from which the upward climb beats
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// Miller's downward walk on accuracy as well as on cost: the asymptotic
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// I₀ and I₁ it climbs from are near full precision there, while the
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// polynomial pair below 3.75 is not. Miller's own cost is O(n + |x|), so
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// staying with it under this bound costs nothing that matters.
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const besselIUpwardMinX = 12
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// besselIUpward climbs I₀ and I₁ to order n by the upward recurrence
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// Iₖ₊₁ = Iₖ₋₁ − (2k/x)·Iₖ (the modified kind carries the minus sign
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// where J carries a plus), the stable direction while the order stays
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// well below |x|: the parasitic K component the seeds carry decays
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// relative to I there, whereas near and above the argument the same
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// recurrence amplifies it, which is what besselInMiller's downward walk
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// avoids. The quadrature seeds carry the absolute scale and the parity,
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// so a negative argument needs no extra handling. The caller guarantees
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// x ≠ 0, 2 ≤ n, n² ≤ 4|x| and |x| ≥ besselIUpwardMinX.
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func besselIUpward(n int, x float64) float64 {
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i0, i1 := besselI0(x), besselI1(x)
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// The quadrature overflows to Inf once |x| passes about 714, where
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// the true value overflows too: report that instead of letting the
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// recurrence fold Inf - Inf into NaN. The sign follows the parity
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// Iₙ(−x) = (−1)ⁿ Iₙ(x).
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if math.IsInf(i0, 0) || math.IsInf(i1, 0) {
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if x < 0 && n%2 == 1 {
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return math.Inf(-1)
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}
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return math.Inf(1)
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}
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for k := 1; k < n; k++ {
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i0, i1 = i1, i0-2*float64(k)/x*i1
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}
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return i1
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}
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// besselInMiller evaluates I_n by the downward Miller recurrence. The
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// upward recurrence for I is unstable once the order passes the
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// argument (the terms subtract nearly equal neighbours), while the
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// downward direction amplifies no rounding. The arbitrary seed scale
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// is removed against the exact I_0 at the end.
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func besselInMiller(n int, x float64) float64 {
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if x == 0 {
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if n == 0 {
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return 1
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}
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return 0
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}
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// |x| keeps the start above n for negative arguments too: a start
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// at or below n would never pass the order on the way down and
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// renormalise against a garbage seed.
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start := n + int(math.Abs(x)) + 40
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jp, j := 0.0, 1.0 // j_{k+1}, j_k, seeded at k = start
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jn := 0.0
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for k := start; k >= 1; k-- {
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if k == n {
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jn = j
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}
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// Downward recurrence: j_{k−1} = j_{k+1} + 2k/x·j_k.
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jp, j = j, jp+2*float64(k)/x*j
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if aj := math.Abs(j); aj > 1e200 {
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// I grows monotonically down the walk, and the unscaled
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// seed overflows for tiny x or high orders whose true
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// value is representable; a common rescale cancels in the
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// final ratio.
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jp /= aj
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j /= aj
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jn /= aj
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}
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}
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// j now holds the unscaled I_0. The quotient is formed before the
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// multiplication: above about x = 300 both walk values pass 1e100,
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// and the product I₀·jₙ overflows even though the answer, which is
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// the quotient times the seed scale, is an ordinary number. Taking
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// the product first turned every such evaluation into +Inf.
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return besselI0(x) * (jn / j)
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}
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// BesselKn returns the modified Bessel function of the second kind of
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// integer order n, Kₙ(x), of each element, by the upward recurrence
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// Kₙ₊₁ = Kₙ₋₁ + 2n/x·Kₙ from K₀ and K₁, which is the stable direction
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// for the second-kind functions. K is defined for x > 0 at every
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// order: outside that domain the answer is not order-dependent but
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// undefined, so any element that is NaN or non-positive is an error
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// naming the element, the same contract BesselY enforces at one
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// point, never a silent NaN or a spurious +Inf.
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func BesselKn(n int, a *Array) (*Array, error) {
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if n < 0 {
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return nil, errf("BesselKn: order must be ≥ 0, got %d", n)
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}
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if a.dt == Complex {
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return nil, errf("BesselKn: complex arrays are not supported")
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}
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for i := range a.Len() {
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if x := a.floatAt(i); math.IsNaN(x) || x <= 0 {
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return nil, errf("BesselKn: the argument must be positive, got %g at element %d", x, i)
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}
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}
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return a.realFunc("BesselKn", func(x float64) float64 {
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k0, k1 := besselK0(x), besselK1(x)
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if n == 0 {
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return k0
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}
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if n == 1 {
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return k1
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}
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km := k0
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k := k1
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for j := 1; j < n; j++ {
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km, k = k, km+2*float64(j)/x*k
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}
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return k
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})
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}
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// besselI0 evaluates I₀: the standard 3.75-piecewise polynomial fit
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// below, and the exact integral representation I₀ = (1/π)∫₀^π
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// e^(x·cos θ) dθ above, where the polynomial's accuracy degrades.
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func besselI0(x float64) float64 {
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ax := math.Abs(x)
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if ax < 3.75 {
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t := x / 3.75
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t2 := t * t
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return 1 + t2*(3.5156229+t2*(3.0899424+t2*(1.2067492+t2*(0.2659732+t2*(0.0360768+t2*0.0045813)))))
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}
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return quadratureBesselI(x, false)
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}
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// besselI1 evaluates I₁: the polynomial fit below 3.75, the integral
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// representation I₁ = (1/π)∫₀^π e^(x·cos θ)·cos θ dθ above. The odd
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// parity comes out of the quadrature by itself; the small branch
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// applies the sign of x explicitly.
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func besselI1(x float64) float64 {
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ax := math.Abs(x)
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var r float64
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if ax < 3.75 {
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t := x / 3.75
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t2 := t * t
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r = ax * (0.5 + t2*(0.87890594+t2*(0.51498869+t2*(0.15084934+t2*(0.02658733+t2*(0.00301532+t2*0.00032411))))))
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if x < 0 {
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r = -r
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}
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return r
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}
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return quadratureBesselI(x, true)
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}
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// quadratureBesselI evaluates I₀ (weight false) or I₁ (weight true)
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// through the exact integral representations over θ ∈ [0, π] by
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// composite Simpson with 2048 intervals, which resolves the peak at
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// θ = 0 for any practical |x|.
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func quadratureBesselI(x float64, weightCos bool) float64 {
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const intervals = 2048
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h := math.Pi / float64(intervals)
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kern := func(th float64) float64 {
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v := math.Exp(x * math.Cos(th))
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if weightCos {
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v *= math.Cos(th)
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}
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return v
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}
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sum := kern(0) + kern(math.Pi)
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for i := 1; i < intervals; i++ {
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w := 2.0
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if i%2 == 1 {
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w = 4.0
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}
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sum += w * kern(float64(i)*h)
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}
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return sum * h / (3 * math.Pi)
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}
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// besselK0 evaluates K₀(x) = ∫₀^∞ e^(−x·cosh t) dt by composite
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// Simpson quadrature over a truncated domain. The tail beyond the
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// cutoff falls below 1e-100 for every x > 0, and the integrand is
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// smooth, so the quadrature carries roughly ten significant digits.
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// That is the deliberate trade: a provably correct entry-level K
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// without hand-typed approximation coefficients.
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func besselK0(x float64) float64 {
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if x <= 0 {
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return math.Inf(1)
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}
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return integrateCoshKernel(x, func(_ float64, coshU float64) float64 {
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return math.Exp(-x * coshU)
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})
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}
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// besselK1 evaluates K₁(x) = ∫₀^∞ e^(−x·cosh t)·cosh t dt, the
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// derivative partner of K₀, by the same quadrature.
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func besselK1(x float64) float64 {
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if x <= 0 {
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return math.Inf(1)
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}
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return integrateCoshKernel(x, func(_ float64, coshU float64) float64 {
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return math.Exp(-x*coshU) * coshU
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})
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}
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// integrateCoshKernel integrates ∫₀^∞ kernel(u; x) du where the
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// kernel carries the factor e^(−x·cosh u). The cutoff follows from
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// requiring the tail e^(−x·e^U/2) below 1e-100; for large x the
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// result legitimately underflows to zero like the true value.
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func integrateCoshKernel(x float64, kernel func(float64, float64) float64) float64 {
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const intervals = 2001
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u := math.Log(200/x) + 2
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if u < 2 {
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u = 2
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}
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h := u / float64(intervals)
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sum := kernel(0, 1) + kernel(u, math.Cosh(u))
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for i := 1; i < intervals; i++ {
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tu := float64(i) * h
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w := 2.0
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if i%2 == 1 {
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w = 4.0
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}
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sum += w * kernel(tu, math.Cosh(tu))
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}
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return sum * h / 3
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}
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