From 788d6214087b1d5d9e46b374da358367e4277f6f Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 30 Apr 2026 15:44:17 -0700 Subject: [PATCH 01/15] Use backward stable algorithm with correction for infinite condition number Resolves https://github.com/PlasmaControl/DESC/pull/2157#discussion_r3106446266 --- desc/integrals/_bounce_utils.py | 8 +- desc/integrals/_interp_utils.py | 183 ++++++++--------------- desc/integrals/bounce_integral.py | 16 +- tests/inputs/master_compute_data_rpz.pkl | Bin 8802444 -> 8791428 bytes tests/test_compute_everything.py | 5 +- tests/test_integrals.py | 15 +- tests/test_interp_utils.py | 24 --- 7 files changed, 86 insertions(+), 165 deletions(-) diff --git a/desc/integrals/_bounce_utils.py b/desc/integrals/_bounce_utils.py index 9598dbd33c..f371a68a74 100644 --- a/desc/integrals/_bounce_utils.py +++ b/desc/integrals/_bounce_utils.py @@ -26,7 +26,7 @@ from desc.backend import dct, ifft, jax, jnp from desc.integrals._interp_utils import ( _JF_BUG, - _eps, + _root_eps, chebder, nufft1d2r, nufft2d2r, @@ -370,10 +370,11 @@ def bounce_points_jvp(num_well, primals, tangents): dB_dz += dB_dt * dt_dz dB_do += dB_dt * dt_do + regularization = _root_eps() dB_dz = jnp.where( - jnp.abs(dB_dz) > _eps, + jnp.abs(dB_dz) > regularization, dB_dz, - dB_dz + jnp.copysign(_eps, dB_dz.real), + dB_dz + jnp.copysign(regularization, dB_dz.real), ) dz = jnp.where(mask, (dp[..., None] - dB_do) / dB_dz, 0.0) @@ -702,6 +703,7 @@ def get_mins(knots, B, num_mins=-1, fill_value=0.0): a_min=jnp.array([0.0]), a_max=jnp.diff(knots), sentinel=0.0, + distinct=False, ) b = flatten_mat((poly_val(x=mins, c=b[..., None, :], der=True) > 0) & (mins > 0)) mins = flatten_mat( diff --git a/desc/integrals/_interp_utils.py b/desc/integrals/_interp_utils.py index 4699618eed..72912cbb61 100644 --- a/desc/integrals/_interp_utils.py +++ b/desc/integrals/_interp_utils.py @@ -257,18 +257,15 @@ def _subtract_last(c, k): ) -def _filter_distinct(r, sentinel, eps): - """Set all but one of matching adjacent elements in ``r`` to ``sentinel``.""" - # eps needs to be low enough that close distinct roots do not get removed. - # Otherwise, algorithms relying on continuity will fail. - mask = jnp.isclose(jnp.diff(r, axis=-1, prepend=sentinel), 0, atol=eps) - return jnp.where(mask, sentinel, r) - - _root_companion = jnp.vectorize( partial(jnp.roots, strip_zeros=False), signature="(m)->(n)" ) -_eps = max(jnp.finfo(jnp.array(1.0).dtype).eps, 2.5e-12) + + +def _root_eps(): + # Safer to make this a callable since output depends on whether + # double precision is enabled before it is called. + return max(jnp.finfo(jnp.array(1.0).dtype).eps, 1e-11) @partial(jax.custom_jvp, nondiff_argnums=(4, 5, 6, 7)) @@ -279,7 +276,7 @@ def polyroot_vec( a_max=None, sort=False, sentinel=jnp.nan, - eps=_eps, + eps=-1.0, distinct=False, ): """Roots of polynomial with given coefficients. @@ -320,44 +317,67 @@ def polyroot_vec( The roots of the polynomial, iterated over the last axis. """ - get_only_real_roots = not (a_min is None and a_max is None) - num_coef = c.shape[-1] - distinct = distinct and num_coef > 2 - func = {2: _root_linear, 3: _root_quadratic, 4: _root_cubic} - - if ( - num_coef in func - and get_only_real_roots - and jnp.isrealobj(c) - and jnp.isrealobj(k) - ): - # Compute from analytic formula to avoid the issue of complex roots with small - # imaginary parts. Also consumes less memory. - c = jnp.moveaxis(c, -1, 0) - r = func[num_coef](*c[:-1], c[-1] - k, sentinel, eps, distinct) - if num_coef == 2: - r = r[jnp.newaxis] - r = jnp.moveaxis(r, 0, -1) - - # We already filtered distinct roots for quadratics. - distinct = distinct and num_coef > 3 - else: - r = _root_companion(_subtract_last(c, k)) + if eps < 0: + eps = _root_eps() + + degree = c.shape[-1] - 1 + distinct = distinct and (degree > 1) + + # The analytical formulae in numerical.recipes/book.html, page 228 + # are not backwards stable. They can generate fake root with O(1) residual, + # so we use eigenvalue solve on local power basis companion matrix. + r = _root_companion(_subtract_last(c, k)) + + if get_only_real_roots := not (a_min is None and a_max is None): + # If the complex part is too big, then these would not be real roots of + # a nearby perturbed problem, so we set to nan so that they are not + # classified as candidates after the correction step. + r = jnp.where(jnp.abs(r.imag) <= eps**0.5, r.real, jnp.nan) + + # Schröder first kind correction maps the roots of the perturbed problem + # back to the roots of the original problem. + k = jnp.expand_dims(k, -1) + p0 = poly_val(x=r, c=c[..., None, :]) - k + p1 = poly_val(x=r, c=c[..., None, :], der=True) + p2 = poly_val(x=r, c=c[..., None, :-1] * jnp.arange(degree, 0, -1), der=True) + candidate = r - (p0 * p1) / (p1**2 - p0 * p2) + r = jnp.where( + jnp.abs(poly_val(x=candidate, c=c[..., None, :]) - k) < jnp.abs(p0), + candidate, + r, + ) - if get_only_real_roots: - a_min = -jnp.inf if a_min is None else a_min[..., jnp.newaxis] - a_max = +jnp.inf if a_max is None else a_max[..., jnp.newaxis] + if distinct: + # Then we need to ensure the return roots have a consistent multiplicity, + # so we discard roots within a few machine precision of extrema. This is + # mathematically justified as they could just as well have not been detected + # in the nearby perturbed problem. Application purpose is that, + # if we return only one root at a given extrema (e.g. if we only found one + # since the root of the perturbed problem was single multiplicity) and return + # multiple roots for another extrema (e.g. if we found 2 or 3 in the nearby + # perturbed problem which were merged by the correction step), then downstream + # algorithms which assume continuity (intermediate value theorem) of the + # underlying function will be misled. r = jnp.where( - (jnp.abs(r.imag) <= eps) & (a_min <= r.real) & (r.real <= a_max), - r.real, + jnp.abs(poly_val(x=r, c=c[..., None, :], der=True)) > eps, + r, sentinel, ) + if get_only_real_roots: + a_min = -jnp.inf if a_min is None else a_min[..., jnp.newaxis] + a_max = +jnp.inf if a_max is None else a_max[..., jnp.newaxis] + r = jnp.where((a_min <= r) & (r <= a_max), r, sentinel) + if sort or distinct: - r = jnp.sort(r, axis=-1) + r = jnp.sort(r, stable=False) if distinct: - r = _filter_distinct(r, sentinel, eps) - assert r.shape[-1] == num_coef - 1 + r = jnp.where( + jnp.isclose(jnp.diff(r, prepend=sentinel), 0.0, atol=eps), + sentinel, + r, + ) + assert r.shape[-1] == degree return r @@ -377,6 +397,8 @@ def _polyroot_vec_jvp(sort, sentinel, eps, distinct, primals, tangents): c, k, a_min, a_max = primals dc, dk, _, _ = tangents + if eps < 0: + eps = _root_eps() r = polyroot_vec(c, k, a_min, a_max, sort, sentinel, eps, distinct) dc_dr = poly_val(x=r, c=c[..., None, :], der=True) @@ -393,85 +415,6 @@ def _polyroot_vec_jvp(sort, sentinel, eps, distinct, primals, tangents): return r, dr -def _root_cubic(a, b, c, d, sentinel, eps, distinct): - """Return real cubic root assuming real coefficients.""" - # numerical.recipes/book.html, page 228 - - def irreducible(Q, R, b): - # Three irrational real roots. - theta = jnp.arccos(R / jnp.sqrt(Q**3)) - return ( - -2 - * jnp.sqrt(Q) - * jnp.stack( - [ - jnp.cos(theta / 3), - jnp.cos((theta + 2 * jnp.pi) / 3), - jnp.cos((theta - 2 * jnp.pi) / 3), - ] - ) - - b / 3 - ) - - def reducible(Q, R, b): - # One real and two complex roots. - A = -jnp.sign(R) * jnp.cbrt(jnp.abs(R) + jnp.sqrt(jnp.abs(R**2 - Q**3))) - B = Q / A - r1 = (A + B) - b / 3 - return _concat_sentinel(r1[jnp.newaxis], sentinel, num=2) - - def root(b, c, d): - b = b / a - c = c / a - Q = (b**2 - 3 * c) / 9 - R = (2 * b**3 - 9 * b * c) / 54 + d / (2 * a) - return jnp.where( - R**2 < Q**3, - irreducible(jnp.abs(Q), R, b), - reducible(Q, R, b), - ) - - return jnp.where( - # Tests catch failure here if eps < 1e-12 for double precision. - jnp.abs(a) <= eps, - _concat_sentinel( - _root_quadratic(b, c, d, sentinel, eps, distinct), - sentinel, - ), - root(b, c, d), - ) - - -def _root_quadratic(a, b, c, sentinel, eps, distinct): - """Return real quadratic root assuming real coefficients.""" - # numerical.recipes/book.html, page 227 - - discriminant = b**2 - 4 * a * c - q = -0.5 * (b + jnp.sign(b) * jnp.sqrt(jnp.abs(discriminant))) - r1 = jnp.where( - discriminant < 0, - sentinel, - jnp.where(a == 0, _root_linear(b, c, sentinel, eps), q / a), - ) - r2 = jnp.where( - # more robust to remove repeated roots with discriminant - (discriminant < 0) | (distinct & (discriminant <= eps)), - sentinel, - c / q, - ) - return jnp.stack([r1, r2]) - - -def _root_linear(a, b, sentinel, eps, distinct=False): - """Return real linear root assuming real coefficients.""" - return jnp.where((a == 0) & (jnp.abs(b) <= eps), 0.0, -b / a) - - -def _concat_sentinel(r, sentinel, num=1): - """Concatenate ``sentinel`` ``num`` times to ``r`` on first axis.""" - return jnp.concatenate((r, jnp.broadcast_to(sentinel, (num, *r.shape[1:])))) - - # TODO: replace the inner loop in orthax with this def chebder(c, m=1, scl=1.0, axis=0, keepdims=False): """Same as orthax.chebder but fast enough to use in optimization loop.""" diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 6a22a75139..d44d58efdb 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -49,7 +49,7 @@ ) from desc.integrals._interp_utils import ( _JF_BUG, - _eps, + _root_eps, interp1d_Hermite_vec, interp1d_vec, nufft2d2r, @@ -106,11 +106,6 @@ def pitch_quad(min_B, max_B, num_pitch, **kwargs): """ if isinstance(num_pitch, int): - errorif( - num_pitch > 1e5, - msg="Floating point error impedes detection of bounce points " - f"near global extrema. Choose {num_pitch} < 1e5.", - ) simp = kwargs.get("simp", True) num_pitch = simpson2(num_pitch) if simp else uniform(num_pitch) @@ -674,7 +669,7 @@ def points(self, pitch_inv, num_well=None): # size of 10⁻⁸ is reduced from 10³ to 10⁻⁶ for Γ_c # (which has the singular weight 1/v_∥). z1, z2 = self._B.intersect1d( - self._swap_axes(pitch_inv), num_intersect=num_well, eps=_eps + self._swap_axes(pitch_inv), num_intersect=num_well, eps=_root_eps() ) z1 = move(z1) z2 = move(z2) @@ -983,7 +978,7 @@ def interp_to_argmin(self, f, points, *, nufft_eps=-1.0, **kwargs): # such that all bounce points are at ζ >= 0; and therefore, # junk values in B_mins cannot be selected in argmin. mins, B_mins = ( - self._B.extrema1d(1, num_mins, fill_value=0.0, eps=_eps) + self._B.extrema1d(1, num_mins, fill_value=0.0, eps=_root_eps()) if isinstance(self._B, PiecewiseChebyshevSeries) else get_mins(self._c["knots"], self._B, num_mins, fill_value=0.0) ) @@ -1111,7 +1106,7 @@ def plot(self, l, m, pitch_inv=None, **kwargs): B = B[m] B = PiecewiseChebyshevSeries(B, domain) if pitch_inv is not None: - kwargs["z1"], kwargs["z2"] = B.intersect1d(pitch_inv, eps=_eps) + kwargs["z1"], kwargs["z2"] = B.intersect1d(pitch_inv, eps=_root_eps()) kwargs["k"] = pitch_inv return B.plot1d(B.cheb, **kwargs) @@ -1907,7 +1902,8 @@ def guess( """ errorif( - (surf_batch_size > 1) and (pitch_batch_size is not None), + (surf_batch_size is None or surf_batch_size > 1) + and (pitch_batch_size is not None), msg=f"Expected pitch_batch_size to be None, got {pitch_batch_size}.", ) diff --git a/tests/inputs/master_compute_data_rpz.pkl b/tests/inputs/master_compute_data_rpz.pkl index ce8720cf526707fd035dca15071ceb97d916c692..608a3407d583430dc3560f17064944906304565f 100644 GIT binary patch delta 5601068 zcmeF4c|c9wzyF=nF=USFC_^$;L^72jYnx@tOo&24iAu)Uii}Z8HW3*jLnWn%`kabJ zG|%&#sZ5#EZ+o6+k?QQTAD4T-_x`qj-tDzN=d}C0)@N929p!NDq+n0n34vgjz_s;J z!SGfsN7;ruP1~vp^ws*bA3JG7@@phf 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