# SPDX-FileCopyrightText: 2026 John C. Travers and contributors
#
# SPDX-License-Identifier: GPL-3.0-or-later
"""The reference's interaction-picture Adams march (SEP 13, SEP 14).
One accepted step of the PEC method costs one expensive source evaluation, against
seven for the reference backend's IP-DOPRI5. The march keeps a short history of
**transported** derivatives, re-anchored at every accepted step, and decides each step
on a local-defect estimate formed from that history without another evaluation (SEP 13
sections 4.1--4.3).
The interaction picture is anchored at the start of each step, ``z_n``. With
``U(z_2, z_1) = exp(Phi(z_2) - Phi(z_1))``, the variable ``v(z) = U(z_n, z) a(z)``
obeys ``v' = g(z) = U(z_n, z) N(U(z, z_n) v, z)``. A history derivative is stored in
the current anchor's frame; accepting a step re-anchors every one of them by the step's
own propagator, which is the multiplication the Runge-Kutta march already performs on
its state, so no table is queried backwards. Every derivative keeps its position and
the identity of the field it was evaluated at -- the accepted field, or the predicted
one PEC stores in its place.
The march is shared by its two copies: a backend supplies two hooks of the step driver,
``_adams_source`` (one source evaluation at a lab-frame field, returned in the step's
anchor frame) and ``_adams_propagator``, and the driver's own state, landing and
recording do the rest. ``numpy_ref`` evaluates through its reference passes (SEP 14
phase 2).
Lifecycle, owned here and not by the shared driver's Runge-Kutta march (SEP 13
section 6.1):
- a trial computes its predictor, its single source call and its estimate without
touching the history; only an accepted step commits, so a rejection leaves the
accepted history exactly as it was;
- a hard kink is landed on and the history restarted on its far side; a save plane is
a pure observation and keeps the history;
- a load or a resume starts afresh, at order one, and its startup calls are counted.
**Who marches.** The NumPy reference marches with
:class:`solprop.backends.host.adams.HostAdamsMarch`; the native backend's session
marches natively from one policy record (SEP 39, SEP 47). This module holds the
reference's kernels, its host engine :class:`HostHistory` and the tables; the call
ledger both backends keep is :mod:`solprop.backends.ledger`.
"""
import logging
import math
from collections.abc import Callable, Mapping
from dataclasses import dataclass, field
from typing import Any, Literal, Protocol, cast
import numpy as np
from jaxtyping import Complex
from scipy.linalg import expm
from solprop._helpers import PLAIN, Helpers
from solprop.backends.exponential import (
CarrierReference,
carrier_reference,
phi_functions,
)
from solprop.backends.host.driver import HostDriver
from solprop.backends.ledger import source_absent
from solprop.backends.numpy_ref import passes
from solprop.exceptions import BackendError
from solprop.stepper.multistep import (
Multistep,
integrated_weights,
lagrange_coefficients,
)
from solprop.translate.linop import phi_rate
__all__ = [
"AdamsEngine",
"AdamsKernels",
"CoupledBlock",
"CoupledExponentialKernels",
"ExponentialKernels",
"HostHistory",
"compensation_tables",
"coupled_moments",
"exponential_tables",
"factored_defect_rows",
"fused_tables",
"trial_tables",
]
logger = logging.getLogger(__name__)
_State = Complex[np.ndarray, "n_batch n_modes n_pol n_omega"]
_StoreRows = Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"]
_Shift = Complex[np.ndarray, "#n_batch 1 1 #n_omega"]
"""A carrier frame's shift: one per batch member (SEP 35); a bin axis broadcasts."""
_Factors = Complex[np.ndarray, "n_used #n_batch 1 1 #n_omega"]
"""A carrier frame's rotation of each history row a trial reads, in its slots' order."""
@dataclass
class _DenseStep:
"""The last accepted step, in its anchor's frame, for dense observation."""
z: float
dz: float
start: _State
nodes: np.ndarray
slopes: list[_State]
[docs]
class AdamsKernels:
"""The field arithmetic of a fused Adams step, in NumPy (SEP 14 phase 2).
Three passes serve a step of the one-order-higher PEC family: the predictor's
lab-frame field, the corrected field with the defects of the current and the
neighbouring orders, and the history's transport to the new anchor. The history is
a store of rows, newest first by ``slots``. This is the reference a compiled form
is checked against.
Parameters
----------
helpers : solprop._helpers.Helpers, optional
The owning backend's implementation of the Adams basis (SEP 29 section 6.6);
the plain Python forms by default.
"""
def __init__(self, *, helpers: Helpers = PLAIN) -> None:
"""Hold the backend's basis implementation."""
self._helpers = helpers
[docs]
def weights(
self, nodes: list[float] | np.ndarray, upper: float = 1.0
) -> np.ndarray:
"""Return the integrated weights of an interpolant on distinct nodes.
Parameters
----------
nodes : list of float or numpy.ndarray
The normalised nodes, accepted positions and so distinct.
upper : float, optional
The integral's upper limit.
Returns
-------
numpy.ndarray
One weight per node, as ``integrated_weights`` gives them.
"""
return integrated_weights(nodes, upper, check=False, helpers=self._helpers)
[docs]
def predict( # noqa: PLR0917 - called positionally
self,
state: _State,
propagator: _State,
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
weights: np.ndarray,
dz: float,
) -> _State:
"""Return the predictor's lab-frame field ``U (a_n + dz sum_t w_t g_t)``.
Parameters
----------
state : numpy.ndarray
The accepted state at the anchor.
propagator : numpy.ndarray
``U(z_n + dz, z_n)``.
store : numpy.ndarray
The history rows.
slots : numpy.ndarray
The rows the predictor combines, newest first, ``int64``.
weights : numpy.ndarray
One integrated weight per row.
dz : float
The step, m.
Returns
-------
numpy.ndarray
The predicted field at the end of the step.
"""
total = np.zeros_like(state)
for weight, slot in zip(weights, slots, strict=True):
total = passes.accumulate(total, store[slot], float(weight))
return propagator * (state + dz * total)
[docs]
def correct( # noqa: PLR0917 - buffers are a kernel's parameters
self,
state: _State,
propagator: _State,
source: _State,
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
corrector: np.ndarray,
rows: np.ndarray,
dz: float,
mask: np.ndarray,
divisor: np.ndarray,
) -> tuple[_State, np.ndarray]:
"""Return the corrected lab-frame field and the scaled norms of the defects.
With ``X = (source, store[slots[0]], ...)``, the field is
``U (a_n + dz sum_t c_t X_t)`` and each norm is the weak norm of
``U dz sum_t rows[r, t] X_t`` over the in-band bins.
Parameters
----------
state, propagator : numpy.ndarray
As for :meth:`predict`.
source : numpy.ndarray
The source at the predicted field, in the anchor's frame.
store : numpy.ndarray
The history rows.
slots : numpy.ndarray
The rows after the source, newest first, ``int64``.
corrector : numpy.ndarray
The corrector's weights over ``X``, zero past its own nodes.
rows : numpy.ndarray
One defect's weights over ``X`` per row, shape ``(n_rows, 1 + n_slots)``.
dz : float
The step, m.
mask : numpy.ndarray
The in-band bins, Boolean.
divisor : numpy.ndarray
The norm's scale per bin.
Returns
-------
tuple
The corrected field and one norm per row.
"""
values = [source, *(store[slot] for slot in slots)]
total = np.zeros_like(state)
for weight, value in zip(corrector, values, strict=True):
total = passes.accumulate(total, value, float(weight))
norms = np.empty(rows.shape[0])
for index, row in enumerate(rows):
defect = np.zeros_like(state)
for weight, value in zip(row, values, strict=True):
defect = passes.accumulate(defect, value, float(weight))
scaled = (propagator * (dz * defect))[..., mask] / divisor[mask]
norms[index] = float(np.linalg.norm(scaled))
return propagator * (state + dz * total), norms
[docs]
def transport(
self,
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
propagator: _State,
new_slot: int,
source: _State,
) -> None:
"""Re-anchor the history at the step's end and store the new derivative.
Parameters
----------
store : numpy.ndarray
The history rows, updated in place.
slots : numpy.ndarray
The rows of the kept history, ``int64``.
propagator : numpy.ndarray
``U(z_n + dz, z_n)``.
new_slot : int
The free row the new derivative goes to.
source : numpy.ndarray
The new derivative in the step's anchor frame.
"""
for slot in slots:
store[slot] *= propagator
np.multiply(propagator, source, out=store[new_slot])
[docs]
@dataclass(frozen=True)
class CoupledBlock:
"""A step's frozen coupled block on the bins that carry one (SEP 16 phase 5).
Beside a coupled flow the cumulative diagonal ``Phi`` is a multiple of the identity
in the channel axis (``docs/dev/kernel-contract.md``, ``coupled_flow``), so the
step's reference is ``B = (Delta Phi / h) I + G_s`` with ``G_s = Z T Z^H`` the
frozen block of the section the step is anchored in, and the two parts commute.
Attributes
----------
active : numpy.ndarray
Which bins carry a block, bool, shape ``(n_omega,)``.
vectors, triangular : numpy.ndarray
``Z`` and ``T`` of the section's block, 1/m for ``T``, shape
``(n_active, n_modes, n_modes)``.
section : int
The section the step is anchored in.
end_weights : numpy.ndarray or None
The flow's term weights at the step's end, read on the step's side of a
section edge (``StageScalars.flow_weights``); ``None`` without a residual.
residual : callable or None
``(state, weights) -> R_s(z) state`` for a frozen-residual flow: the
generator at the position the weights belong to less this section's frozen
block, on the active bins the error norm scores and zero elsewhere, as the
Runge-Kutta stage applies it; ``None`` for a constant-Schur flow.
basis : tuple or None
The section's eigenbasis ``(W, Y, lambda, route)``, as
:func:`solprop.translate.coupled.schur_eigenbasis` returns it: the bins on the
route take their matrix functions as ``W f(x + h lambda) Y``, the compiled
backends' evaluation. ``None``, the reference's, takes every bin from the
augmented exponential.
"""
active: np.ndarray
vectors: np.ndarray
triangular: np.ndarray
section: int
end_weights: np.ndarray | None = None
residual: Callable[[np.ndarray, np.ndarray], np.ndarray] | None = None
basis: tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray] | None = None
@dataclass(frozen=True)
class _MatrixMoments:
"""A coupled trial's flow and moments: scalar per element, matrices on a block.
Off the block's bins the step's flow and moments are the diagonal ones, per
element; on them each is a matrix in the channel axis, applied to the channel
vector of every batch row and polarisation. ``matrices`` and ``flow_matrix`` are
laid out by bin, ``(..., n_bins, n_modes, n_modes)``; the products read the same
matrices with the bin axis last, as the state holds it.
"""
scalar: np.ndarray
flow_scalar: np.ndarray
matrices: np.ndarray
flow_matrix: np.ndarray
active: np.ndarray
_columns: np.ndarray = field(init=False, repr=False, compare=False)
_flow_columns: np.ndarray = field(init=False, repr=False, compare=False)
def __post_init__(self) -> None:
"""Hold the matrices with the bin axis last, for the products."""
object.__setattr__(
self, "_columns", np.ascontiguousarray(np.moveaxis(self.matrices, -3, -1))
)
object.__setattr__(
self,
"_flow_columns",
np.ascontiguousarray(np.moveaxis(self.flow_matrix, -3, -1)),
)
def _product(
self, columns: np.ndarray, scalar: np.ndarray, values: _State
) -> _State:
"""Return the scalar product off the block and the matrix product on it.
- ``scalar``: the diagonal factor, in the state's layout.
- ``values``: the array, in the state's layout.
"""
out = scalar * values
selected = values[..., self.active]
block = columns[:, 0][np.newaxis, :, np.newaxis] * selected[:, np.newaxis, 0]
for column in range(1, columns.shape[1]):
block += (
columns[:, column][np.newaxis, :, np.newaxis]
* selected[:, np.newaxis, column]
)
out[..., self.active] = block
return out
def apply(self, k: int, values: _State) -> _State:
"""Return ``M_k`` applied to a state-shaped array.
- ``values``: the array, in the state's layout.
"""
columns = self._columns
return self._product(columns[k], self.scalar[k], values)
def flow(self, values: _State) -> _State:
"""Return the step's flow ``exp(h B)`` applied to a state-shaped array.
- ``values``: the array, in the state's layout.
"""
columns = self._flow_columns
return self._product(columns, self.flow_scalar, values)
def _through(
pair: tuple[np.ndarray, np.ndarray], values: np.ndarray
) -> Complex[np.ndarray, "n_bins n_modes_out n_modes_in"]:
"""Return ``W diag(values) Y`` per bin, for an eigenbasis pair ``(W, Y)``.
- ``pair``: ``W`` and ``Y``, shape ``(n_bins, n_modes, n_modes)``.
- ``values``: the function's values at the eigenvalues, shape ``(n_bins, n_modes)``.
"""
left, right = pair
return (left * values[:, np.newaxis, :]) @ right
def _eigen_functions( # noqa: PLR0917 - called through a reference
shift: np.ndarray,
dz: float,
count: int,
basis: tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray],
matrices: np.ndarray,
flow_matrix: np.ndarray,
) -> None:
"""Write the flow and moments of the bins on an eigenbasis route, in NumPy.
- ``dz``: the step, m.
- ``basis``: ``(W, Y, lambda, route)`` of :attr:`CoupledBlock.basis`.
"""
left, right, eigenvalues, route = basis
if not np.any(route):
return
arguments = shift[route, np.newaxis] + dz * eigenvalues[route]
pair = (left[route], right[route])
flow_matrix[route] = _through(pair, np.exp(arguments))
if count:
phi = phi_functions(arguments, count)
for k in range(count):
matrices[k, route] = math.factorial(k) * _through(pair, phi[k + 1])
[docs]
def coupled_moments(
exponent: _State,
count: int,
block: CoupledBlock,
dz: float,
*,
eigen: Callable[..., None] = _eigen_functions,
diagonal: Callable[[np.ndarray, int], np.ndarray] | None = None,
carrier: _Shift | None = None,
) -> _MatrixMoments:
"""Return a coupled trial's flow and moments ``k! phi_{k+1}(h B)``, ``k < count``.
The reference's evaluation (SEP 16 section 4.5): on each bin of the block the
functions of ``X = Delta Phi I + h T``, upper triangular, are read off the first
block row of the exponential of the augmented matrix
``[[X, I, 0, ...], [0, 0, I, ...], ..., [0, ..., 0]]`` -- ``exp(X)``,
``phi_1(X)``, ..., ``phi_count(X)`` -- and carried back by ``Z``; off the block the
diagonal functions of :class:`ExponentialKernels`. With the block's eigenbasis
(``CoupledBlock.basis``), the bins on its route instead take ``W f(Delta Phi + h
lambda) Y``, every bin in one pass: the route and its gate are the compiled flow
actions' (:func:`solprop.translate.coupled.eigenbasis_limit`), and the bins off it
keep the augmented exponential.
Parameters
----------
exponent : numpy.ndarray
``Delta Phi`` over the step, the same in every channel.
count : int
The moments formed.
block : CoupledBlock
The step's frozen block.
dz : float
The step, m.
eigen : callable, optional
Writes the eigenbasis route's bins, ``(shift, dz, count, basis, matrices,
flow_matrix)``: ``_eigen_functions``, or a compiled backend's kernel.
diagonal : callable or None, optional
``(exponent, count) -> moments`` off the block; ``None`` for the reference's
:meth:`ExponentialKernels.moments`.
carrier : numpy.ndarray or None, optional
A carrier frame's ``sigma`` (SEP 35 section 4.5), broadcasting against
`exponent`. The moments are then those of the shifted block
``h B - sigma I``, and the flow is ``exp(sigma)`` times its ``phi_0``:
``sigma I`` commutes with ``B``, so ``exp(h B) = exp(sigma) exp(h B - sigma
I)`` and one evaluation gives both, in the augmented and eigenbasis paths
alike. ``None`` for the unshifted step.
Returns
-------
_MatrixMoments
The flow ``exp(h B)`` and the moments.
"""
unshifted = exponent
if carrier is not None:
exponent = np.ascontiguousarray(exponent - carrier)
active = block.active
# The block's bins take matrix moments below: the scalar ones are formed off it.
scalar = np.zeros((count, *exponent.shape), dtype=np.complex128)
if count and not np.all(active):
moments = ExponentialKernels().moments if diagonal is None else diagonal
scalar[..., ~active] = moments(
np.ascontiguousarray(exponent[..., ~active]), count
)
flow_scalar = np.exp(unshifted)
n_modes = block.triangular.shape[-1]
shift = exponent[0, 0, 0, active]
matrices = np.empty((count, *block.triangular.shape), dtype=np.complex128)
flow_matrix = np.empty(block.triangular.shape, dtype=np.complex128)
size = n_modes * (count + 1)
identity = np.eye(n_modes)
route = np.zeros(block.triangular.shape[0], dtype=bool)
if block.basis is not None:
route = block.basis[3]
eigen(shift, dz, count, block.basis, matrices, flow_matrix)
for index in np.flatnonzero(~route):
augmented = np.zeros((size, size), dtype=np.complex128)
augmented[:n_modes, :n_modes] = (
shift[index] * identity + dz * block.triangular[index]
)
for row in range(count):
start = (row + 1) * n_modes
augmented[row * n_modes : start, start : start + n_modes] = identity
functions = expm(augmented)[:n_modes]
vectors = block.vectors[index]
adjoint = np.conj(vectors.T)
flow_matrix[index] = vectors @ functions[:, :n_modes] @ adjoint
for k in range(count):
column = (k + 1) * n_modes
matrices[k, index] = (
math.factorial(k)
* vectors
[docs]
@ functions[:, column : column + n_modes]
@ adjoint
)
if carrier is not None:
factor = np.exp(np.broadcast_to(carrier, unshifted.shape)[0, 0, 0, active])
flow_matrix = factor[:, np.newaxis, np.newaxis] * flow_matrix
return _MatrixMoments(scalar, flow_scalar, matrices, flow_matrix, active)
class ExponentialKernels:
"""The field arithmetic of an exponential Adams step, in NumPy (SEP 16 phase 2).
The history is a store of **untransported** source rows, newest first by
``slots``. A trial forms ``M_k = k! phi_{k+1}(x)``, ``x = h D`` per element, once,
and applies every formula it needs in the monomial basis of its interpolant,
``h sum_k M_k (sum_j c_jk F_j)`` (``solprop.backends.exponential``), so no
per-node weight volume is formed. Accepting a step re-anchors nothing. The
native backend's kernels are checked against these passes.
Under a carrier frame (SEP 35) the same passes take the moments at ``x - sigma``
and rows rotated by ``exp(sigma (1 - theta_j))``; the caller forms both
(:class:`HostHistory`), so the passes themselves do not change.
Parameters
----------
helpers : solprop._helpers.Helpers, optional
The owning backend's implementation of the Adams basis (SEP 29 section 6.6);
the plain Python forms by default.
"""
def __init__(self, *, helpers: Helpers = PLAIN) -> None:
"""Hold the backend's basis implementation."""
self._helpers = helpers
[docs]
def coupled(self) -> "CoupledExponentialKernels":
"""Return the passes this backend takes beside a coupled flow.
Returns
-------
CoupledExponentialKernels
The reference's; a compiled backend returns its own.
"""
return CoupledExponentialKernels(helpers=self._helpers)
def _times_moment(self, moments: Any, k: int, values: _State) -> _State:
"""Return ``M_k values``: elementwise here, through a block in a subclass.
- ``values``: the array, in the state's layout.
"""
return moments[k] * values
def _times_flow(self, flow: Any, values: _State) -> _State:
"""Return the step's flow applied to an array, elementwise here.
- ``values``: the array, in the state's layout.
"""
return flow * values
[docs]
def moments(
self, exponent: _State, count: int, *, shift: _Shift | None = None
) -> Complex[np.ndarray, "n_k n_batch n_modes n_pol n_omega"]:
"""Return ``k! phi_{k+1}(x)`` for ``k < count`` along a new leading axis.
Parameters
----------
exponent : numpy.ndarray
``x = Phi(z + h) - Phi(z)``, in the canonical layout.
count : int
The most interpolation nodes any formula of the trial has.
shift : numpy.ndarray or None, optional
A carrier frame's ``sigma`` (SEP 35): the moments are then those of
``x - sigma``, and the step's flow stays ``exp(x)`` (:meth:`flow`).
Returns
-------
numpy.ndarray
Shape ``(count, *x.shape)``.
"""
if shift is not None:
exponent = exponent - shift
phi = phi_functions(exponent, count)
factors = np.array([math.factorial(k) for k in range(count)], dtype=np.float64)
return phi[1:] * factors.reshape((count, *([1] * exponent.ndim)))
[docs]
def carry( # noqa: PLR0917 - called positionally
self,
exponent: _State,
dz: float,
rates: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
states: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
spans: np.ndarray,
) -> Complex[np.ndarray, "n_used n_batch n_modes n_pol n_omega"]:
"""Return each history sample's compensation carried to the step's end.
``exp(s_j x) (D(z_j) - x / dz) u_j`` with ``s_j = 1 - theta_j``, the
difference of rates formed before the state multiplies it (SEP 16 4.2).
Parameters
----------
exponent : numpy.ndarray
``x = Phi(z + h) - Phi(z)``.
dz : float
The step, m.
rates, states : numpy.ndarray
The history's rates and states, by slot.
slots : numpy.ndarray
The samples carried, newest first, ``int64``.
spans : numpy.ndarray
``1 - theta_j`` per sample: the flow's length in steps.
Returns
-------
numpy.ndarray
One carried row per sample.
"""
reference = exponent / dz
return np.stack(
[
np.exp(span * exponent) * ((rates[slot] - reference) * states[slot])
for slot, span in zip(slots, spans, strict=True)
]
)
[docs]
def flow(self, exponent: _State) -> _State:
"""Return the step's exact linear flow ``exp(x)``.
Parameters
----------
exponent : numpy.ndarray
``x = Phi(z + h) - Phi(z)``.
Returns
-------
numpy.ndarray
``exp(x)``, the propagator the interaction formulation would apply.
"""
return np.exp(exponent)
[docs]
def apply(
self,
moments: Complex[np.ndarray, "n_k n_batch n_modes n_pol n_omega"],
coefficients: np.ndarray,
values: list[_State],
dz: float,
) -> _State:
"""Return ``dz sum_k M_k (sum_j c_jk X_j)`` over the given rows.
Parameters
----------
moments : numpy.ndarray
``M_k`` from :meth:`moments`.
coefficients : numpy.ndarray
``c_jk``, shape ``(len(values), n_k)`` with ``n_k <= len(moments)``.
values : list of numpy.ndarray
The samples ``X_j``.
dz : float
The step, m.
Returns
-------
numpy.ndarray
The increment.
"""
return self._applied(moments, coefficients, values, dz)
def _applied(
self,
moments: Any,
coefficients: np.ndarray,
values: list[_State],
dz: float,
) -> _State:
"""Return :meth:`apply`'s increment; shared with the coupled passes.
- ``dz``: the step, m.
"""
total = np.zeros_like(values[0])
for k in range(coefficients.shape[1]):
combined = np.zeros_like(values[0])
for weight, value in zip(coefficients[:, k], values, strict=True):
if weight != 0.0:
combined = passes.accumulate(combined, value, float(weight))
total += self._times_moment(moments, k, combined)
return dz * total
[docs]
def predict( # noqa: PLR0917 - buffers are a kernel's parameters
self,
state: _State,
flow: _State,
moments: Complex[np.ndarray, "n_k n_batch n_modes n_pol n_omega"],
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
coefficients: np.ndarray,
dz: float,
*,
factors: _Factors | None = None,
) -> _State:
"""Return the predicted field ``exp(x) u_n + dz sum_k M_k sum_j c_jk F_j``.
Parameters
----------
state : numpy.ndarray
The accepted state at the anchor.
flow : numpy.ndarray
``exp(x)``, the step's exact linear flow.
moments : numpy.ndarray
``M_k`` from :meth:`moments`.
store : numpy.ndarray
The history rows.
slots : numpy.ndarray
The rows the predictor combines, newest first, ``int64``.
coefficients : numpy.ndarray
The predictor's Lagrange coefficients, shape ``(len(slots), len(slots))``.
dz : float
The step, m.
factors : numpy.ndarray or None, optional
A carrier frame's rotation of each row read, ``exp(sigma (1 - theta_j))``
in the slots' order (SEP 35 section 6.2); ``None`` for none.
Returns
-------
numpy.ndarray
The predicted lab-frame field at the end of the step.
"""
return self._predicted(
state, flow, moments, store, slots, coefficients, dz, factors=factors
)
def _predicted( # noqa: PLR0917 - buffers are a kernel's parameters
self,
state: _State,
flow: Any,
moments: Any,
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
coefficients: np.ndarray,
dz: float,
*,
factors: _Factors | None = None,
) -> _State:
"""Return :meth:`predict`'s field; shared with the coupled passes.
- ``dz``: the step, m.
"""
values = _rows_read(store, slots, factors)
return self._times_flow(flow, state) + self._applied(
moments, coefficients, values, dz
)
[docs]
def defect_rows(
self, nodes: list[float], orders: list[int], width: int
) -> np.ndarray:
"""Return the defects' coefficients in the form :meth:`correct` takes.
Parameters
----------
nodes : list of float
The history's normalised positions, newest first.
orders : list of int
The orders scored, the step's own first.
width : int
``1 + max(orders)``.
Returns
-------
numpy.ndarray
The monomial rows of ``_exponential_rows``, shape
``(len(orders), width, width)``.
"""
return _exponential_rows(nodes, orders, width, helpers=self._helpers)
[docs]
def correct( # noqa: PLR0917 - buffers are a kernel's parameters
self,
state: _State,
flow: _State,
moments: Complex[np.ndarray, "n_k n_batch n_modes n_pol n_omega"],
source: _State,
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
corrector: np.ndarray,
rows: np.ndarray,
dz: float,
mask: np.ndarray,
divisor: np.ndarray,
extra: tuple[_State, list[_State]] | None = None,
*,
factors: _Factors | None = None,
) -> tuple[_State, np.ndarray]:
"""Return the corrected field and the scaled norms of the defects.
With ``X = (source, store[slots[0]], ...)``, the field is
``exp(x) u_n + dz sum_k M_k sum_j corrector[j, k] X_j`` and each norm is the
weak norm of ``dz sum_k M_k sum_j rows[r, j, k] X_j`` over the in-band bins;
both are already at the end of the step. A varying operator's compensation
adds its own corrector term to the field and one term to each defect.
Parameters
----------
state, flow, moments : numpy.ndarray
As for :meth:`predict`.
source : numpy.ndarray
The untransported source at the predicted field.
store : numpy.ndarray
The history rows.
slots : numpy.ndarray
The rows after the source, newest first, ``int64``.
corrector : numpy.ndarray
The corrector's coefficients over ``X``, shape ``(width, n_k)``, zero past
its own nodes.
rows : numpy.ndarray
One defect's coefficients over ``X`` per row, shape ``(n_rows, width,
n_k)``.
dz : float
The step, m.
mask : numpy.ndarray
The in-band bins, Boolean.
divisor : numpy.ndarray
The norm's scale per bin.
extra : tuple or None, optional
The compensation's corrector term and its term in each defect, or
``None`` for a constant operator.
factors : numpy.ndarray or None, optional
A carrier frame's rotation of each history row read, in the slots'
order; the new source's is one.
Returns
-------
tuple
The corrected field and one norm per row.
"""
return self._corrected(
state,
flow,
moments,
source,
(store, slots, factors),
(corrector, rows),
dz,
(mask, divisor),
extra,
)
def _corrected( # noqa: PLR0917 - buffers are a kernel's parameters
self,
state: _State,
flow: Any,
moments: Any,
source: _State,
history: tuple[np.ndarray, np.ndarray, _Factors | None],
tables: tuple[np.ndarray, np.ndarray],
dz: float,
scoring: tuple[np.ndarray, np.ndarray],
extra: tuple[_State, list[_State]] | None,
) -> tuple[_State, np.ndarray]:
"""Return :meth:`correct`'s field and norms; shared with the coupled passes.
- ``history``: the store, the rows after the source and their carrier-frame
rotations (``None`` for none).
- ``tables``: the corrector's coefficients and the defect rows.
- ``dz``: the step, m.
- ``scoring``: the in-band mask and the norm's scale per bin.
- Returns: the corrected field and one norm per row.
"""
store, slots, factors = history
corrector, rows = tables
mask, divisor = scoring
values = [source, *_rows_read(store, slots, factors)]
advanced = self._times_flow(flow, state) + self._applied(
moments, corrector, values, dz
)
if extra is not None:
advanced = advanced + extra[0]
norms = np.empty(rows.shape[0])
for index, row in enumerate(rows):
defect = self._applied(moments, row, values, dz)
if extra is not None:
defect = defect + extra[1][index]
norms[index] = float(np.linalg.norm(defect[..., mask] / divisor[mask]))
return advanced, norms
[docs]
class CoupledExponentialKernels(ExponentialKernels):
"""The reference passes beside a coupled flow (SEP 16 phase 5).
A coupled trial's flow and moments are ``_MatrixMoments``
(:func:`coupled_moments`): matrices in the channel axis on the bins that carry a
block, elementwise elsewhere. The passes are :class:`ExponentialKernels`' with
those products.
"""
[docs]
def coupled(self) -> "CoupledExponentialKernels":
"""Return these passes.
Returns
-------
CoupledExponentialKernels
This object.
"""
return self
[docs]
def coupled_moments(
self,
exponent: _State,
count: int,
block: CoupledBlock,
dz: float,
*,
shift: _Shift | None = None,
) -> _MatrixMoments:
"""Return a coupled trial's flow and moments (:func:`coupled_moments`).
Parameters
----------
exponent : numpy.ndarray
``Delta Phi`` over the step.
count : int
The moments formed.
block : CoupledBlock
The step's frozen block.
dz : float
The step, m.
shift : numpy.ndarray or None, optional
A carrier frame's ``sigma``: the shifted block's moments, the true flow.
Returns
-------
_MatrixMoments
The flow and the moments.
"""
return coupled_moments(exponent, count, block, dz, carrier=shift)
def _times_moment(self, moments: Any, k: int, values: _State) -> _State:
"""Return ``M_k values`` through the trial's block.
- ``values``: the array, in the state's layout.
"""
return cast("_MatrixMoments", moments).apply(k, values)
def _times_flow(self, flow: Any, values: _State) -> _State:
"""Return the coupled trial's flow applied to an array.
- ``values``: the array, in the state's layout.
"""
return cast("_MatrixMoments", flow).flow(values)
[docs]
def carry_coupled( # noqa: PLR0917 - called positionally
self,
exponent: _State,
dz: float,
rows: tuple[_StoreRows, _StoreRows, list[np.ndarray | None]],
slots: np.ndarray,
spans: np.ndarray,
block: CoupledBlock,
) -> Complex[np.ndarray, "n_used n_batch n_modes n_pol n_omega"]:
"""Return each sample's compensation carried to the step's end (SEP 16 4.5).
``C_j = (D(z_j) - x / h) u_j + R_s(z_j) u_j``: the diagonal chord's
difference of rates and the frozen-residual's residual relative to the
**step's** section, formed from the sample's own flow weights -- so that a
step in a new section rebuilds the history's residual against its block --
and carried by the reference's forward flow ``exp((1 - theta_j) h B)``.
Parameters
----------
exponent : numpy.ndarray
``x = Delta Phi`` over the step.
dz : float
The step, m.
rows : tuple
The history's rates and states, by slot, and each slot's flow weights.
slots : numpy.ndarray
The samples carried, newest first.
spans : numpy.ndarray
``1 - theta_j`` per sample.
block : CoupledBlock
The step's frozen block.
Returns
-------
numpy.ndarray
One carried row per sample.
"""
rates, states, weights = rows
reference = exponent / dz
carried = []
for slot, span in zip(slots, spans, strict=True):
term = (rates[slot] - reference) * states[slot]
sample_weights = weights[slot]
if block.residual is not None and sample_weights is not None:
term = term + block.residual(states[slot], sample_weights)
if span != 0.0:
flow = self.coupled_moments(span * exponent, 0, block, span * dz)
term = flow.flow(term)
carried.append(term)
return np.stack(carried)
[docs]
def predict( # noqa: PLR0917 - buffers are a kernel's parameters
self,
state: _State,
flow: _State | _MatrixMoments,
moments: Complex[np.ndarray, "n_k n_batch n_modes n_pol n_omega"]
| _MatrixMoments,
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
coefficients: np.ndarray,
dz: float,
*,
factors: _Factors | None = None,
) -> _State:
"""Return the predicted field through the trial's block.
Parameters
----------
state : numpy.ndarray
The accepted state at the anchor.
flow, moments : _MatrixMoments or numpy.ndarray
The coupled trial's flow and moments.
store : numpy.ndarray
The history rows.
slots : numpy.ndarray
The rows the predictor combines, newest first, ``int64``.
coefficients : numpy.ndarray
The predictor's Lagrange coefficients.
dz : float
The step, m.
factors : numpy.ndarray or None, optional
The rows' carrier-frame rotations, in the slots' order.
Returns
-------
numpy.ndarray
The predicted lab-frame field at the end of the step.
"""
return self._predicted(
state, flow, moments, store, slots, coefficients, dz, factors=factors
)
[docs]
def correct( # noqa: PLR0917 - buffers are a kernel's parameters
self,
state: _State,
flow: _State | _MatrixMoments,
moments: Complex[np.ndarray, "n_k n_batch n_modes n_pol n_omega"]
| _MatrixMoments,
source: _State,
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
corrector: np.ndarray,
rows: np.ndarray,
dz: float,
mask: np.ndarray,
divisor: np.ndarray,
extra: tuple[_State, list[_State]] | None = None,
*,
factors: _Factors | None = None,
) -> tuple[_State, np.ndarray]:
"""Return the corrected field and the defects' norms through the block.
Parameters
----------
state : numpy.ndarray
The accepted state at the anchor.
flow, moments : _MatrixMoments or numpy.ndarray
The coupled trial's flow and moments.
source : numpy.ndarray
The untransported source at the predicted field.
store : numpy.ndarray
The history rows.
slots : numpy.ndarray
The rows after the source, newest first, ``int64``.
corrector : numpy.ndarray
The corrector's coefficients.
rows : numpy.ndarray
One defect's coefficients per row.
dz : float
The step, m.
mask : numpy.ndarray
The in-band bins, Boolean.
divisor : numpy.ndarray
The norm's scale per bin.
extra : tuple or None, optional
A varying operator's compensation terms.
factors : numpy.ndarray or None, optional
The history rows' carrier-frame rotations, in the slots' order.
Returns
-------
tuple
The corrected field and one norm per row.
"""
return self._corrected(
state,
flow,
moments,
source,
(store, slots, factors),
(corrector, rows),
dz,
(mask, divisor),
extra,
)
@dataclass
class _ExponentialStep:
"""The last accepted exponential step, for dense observation."""
z: float
dz: float
start: _State
exponent: _State
coefficients: np.ndarray
values: list[_State]
compensation: tuple[list[float], list[_State]] | None = None
shift: np.ndarray | None = None
"""The accepted trial's carrier shift (SEP 35), ``None`` without one."""
@dataclass
class _ExponentialTrial:
"""An exponential trial held between its norms and its acceptance."""
z: float
dz: float
state: _State
exponent: _State
corrector: np.ndarray
source: _State
lab: _State
advanced: _State
slots: np.ndarray
order: int
compensation: "_Compensation | None"
shift: np.ndarray | None = None
"""The trial's carrier shift (SEP 35), ``None`` without one."""
factors: np.ndarray | None = None
"""The rotations of the history rows the trial read, in its slots' order."""
@dataclass
class _FusedTrial:
"""An interaction-formulation trial held between its norms and its acceptance."""
z: float
dz: float
state: _State
propagator: _State
source: _State
advanced: _State
nodes: list[float]
slots: np.ndarray
order: int
@dataclass
class _Compensation:
"""One trial's linear compensation ``C_j = (D(z_j) - B_n) u_j`` (SEP 16 4.2).
The transported treatment integrates it in the reference flow's frame,
``h sum_j a_j exp((1 - theta_j) h B_n) C_j`` with the ordinary Adams weights
``a_j``; every factor is a forward flow over at least the step. The carried rows
are formed once per trial (:meth:`ExponentialKernels.carry`) and every formula is
one weighted sum over them. The untransported treatment, a test witness only
(Q8), interpolates ``C_j`` with the source's exponential weights instead.
"""
exponent: _State
moments: Any
"""The trial's moments, read by the untransported witness only."""
nodes: list[float]
carried: Complex[np.ndarray, "n_used n_batch n_modes n_pol n_omega"]
dz: float
treatment: str
helpers: Helpers
"""The owning backend's implementation of the Adams basis."""
endpoint_rate: _State | None = None
endpoint: _State | None = None
_weights: dict[tuple[float, ...], np.ndarray] = field(default_factory=dict)
def add_endpoint(
self, rate: _State, state: _State, residual: _State | None = None
) -> None:
"""Add the new sample's compensation at the predicted state; no transport.
- ``rate``: ``D(z_n + h)`` from the step's side of the end.
- ``residual``: a frozen-residual flow's ``R_s(z_n + h)`` applied to `state`.
"""
self.endpoint_rate = rate
self.endpoint = (rate - self.exponent / self.dz) * state
if residual is not None:
self.endpoint = self.endpoint + residual
def rotate(self, shift: _Shift) -> None:
"""Rotate the untransported witness's terms by a trial's carrier shift.
The transported treatment's formula is unchanged under a carrier frame (SEP 35
section 4.5): its terms are carried by the reference flow and integrated with
the ordinary Adams weights. The untransported witness integrates its terms
against the trial's moments, which are the shifted ones, so its history terms
take the samples' factors ``exp(sigma (1 - theta_j))``; the endpoint's is one.
"""
if self.treatment == "transported":
return
self.carried = np.stack(
[
np.exp(shift * (1.0 - node)) * row
for node, row in zip(self.nodes, self.carried, strict=True)
]
)
def _sum(self, nodes: list[float], first: _State | None, count: int) -> _State:
"""Return ``h`` times the treatment's integral over the given samples.
- ``nodes``: the formula's nodes, the endpoint's first when `first` is given.
- ``first``: the endpoint sample's compensation, or ``None`` for a predictor.
"""
rows = self.carried[:count]
if self.treatment == "transported":
key = tuple(nodes)
weights = self._weights.get(key)
if weights is None:
weights = integrated_weights(
np.asarray(nodes), check=False, helpers=self.helpers
)
self._weights[key] = weights
total = np.tensordot(weights[len(nodes) - count :], rows, axes=1)
if first is not None:
total = total + weights[0] * first
return self.dz * total
values = ([] if first is None else [first]) + list(rows)
return ExponentialKernels(helpers=self.helpers).apply(
self.moments[: len(nodes)],
lagrange_coefficients(nodes, check=False, helpers=self.helpers),
values,
self.dz,
)
def predictor(self, order: int) -> _State:
"""Return the predictor's compensation over the ``order`` newest samples."""
return self._sum(self.nodes[:order], None, order)
def corrector(self, order: int) -> _State:
"""Return the corrector's, through the endpoint sample."""
return self._sum([1.0, *self.nodes[:order]], self.endpoint, order)
def defects(self, orders: list[int]) -> list[_State]:
"""Return each scored order's corrector less predictor compensation."""
return [self.corrector(k) - self.predictor(k) for k in orders]
def observation(
self, order: int, history: list[_State]
) -> tuple[list[float], list[_State]]:
"""Return the accepted corrector's nodes and untransported compensations.
Dense observation applies its own flow factors ``exp((xi - theta_j) x)``;
the history's terms are formed again from the stored rows by the caller,
rather than carried back by an inverse flow that a damped bin would overflow.
- Returns: the nodes ``(1, theta_0, ...)`` and the matching compensations.
"""
assert self.endpoint is not None
return [1.0, *self.nodes[:order]], [self.endpoint.copy(), *history]
[docs]
class AdamsEngine(Protocol):
"""What an Adams march asks of the engine that holds its history (SEP 16).
The march keeps the controller; an engine keeps the history's rows and does every
state-sized operation, so that a backend can keep them where its state lives.
:class:`HostHistory` is the reference; ``docs/dev/stepper.md`` states the contract.
"""
dense: Any
"""The interaction formulation's last accepted step (a ``_DenseStep`` or ``None``),
for observation."""
compensated: bool
"""Whether trials carry a varying operator's linear compensation (SEP 16 4.2)."""
carrier: CarrierReference | None
"""The run's carrier frame (SEP 35), or ``None`` when off; an exponential trial
removes its phase from the samples it reads."""
[docs]
@property
def n_slots(self) -> int: # numpydoc ignore=RT01
"""The history store's rows."""
...
[docs]
def reset(self) -> None:
"""Forget the last accepted step and any trial in flight."""
...
[docs]
def hold(self) -> object:
"""Return the accepted state, to restore after a dense observation.
Returns
-------
object
A handle :meth:`restore` takes.
"""
...
[docs]
def restore(self, z: float, held: object) -> None:
"""Make a held accepted state current again at `z`.
Parameters
----------
z : float
Its position, m.
held : object
What :meth:`hold` returned.
"""
...
[docs]
def start(self, z: float) -> _State | None:
"""Evaluate the history's first sample at the accepted state into row zero.
Parameters
----------
z : float
The accepted position, m.
Returns
-------
numpy.ndarray or None
The sample on the host, or ``None`` when the rows are resident.
"""
...
[docs]
def value(self, slot: int) -> _State | None:
"""Return a store row on the host, or ``None`` when the rows are resident.
Parameters
----------
slot : int
The row.
Returns
-------
numpy.ndarray or None
The row, or ``None``.
"""
...
[docs]
def fused_trial( # noqa: PLR0917 - called positionally
self,
z: float,
dz: float,
nodes: list[float],
slots: np.ndarray,
order: int,
orders: list[int],
) -> np.ndarray:
"""Form an interaction-formulation trial; return its scored norms.
Parameters
----------
z, dz : float
The anchor and the step, m.
nodes : list of float
The history's normalised positions, newest first.
slots : numpy.ndarray
Their store rows, ``int64``.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
Returns
-------
numpy.ndarray
One scaled norm per scored order.
"""
...
[docs]
def fused_accept(self, keep: np.ndarray, free: int, *, interpolate: bool) -> None:
"""Adopt the held interaction-formulation trial.
Parameters
----------
keep : numpy.ndarray
The history rows kept, ``int64``.
free : int
The row the new sample takes.
interpolate : bool
Whether the step's interpolant is kept for dense observation.
"""
...
[docs]
def exponential_trial( # noqa: PLR0917 - called positionally
self,
z: float,
dz: float,
nodes: list[float],
slots: np.ndarray,
order: int,
orders: list[int],
) -> np.ndarray:
"""Form an exponential trial; return its scored norms.
Parameters
----------
z, dz : float
The anchor and the step, m.
nodes : list of float
The history's normalised positions, newest first.
slots : numpy.ndarray
Their store rows, ``int64``.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
Returns
-------
numpy.ndarray
One scaled norm per scored order.
"""
...
[docs]
def exponential_accept(
self, keep: np.ndarray, free: int, *, interpolate: bool
) -> None:
"""Adopt the held exponential trial.
Parameters
----------
keep : numpy.ndarray
The history rows kept, ``int64``.
free : int
The row the new sample takes.
interpolate : bool
Whether the step's interpolant is kept for dense observation.
"""
...
[docs]
def snapshot_rows(self) -> dict[str, np.ndarray]:
"""Return host copies of the rows a resume needs to continue the history.
Returns
-------
dict
The rows by name, each ``(n_slots, *shape)``, in the engine's own names
(``HostHistory.snapshot_rows`` states the host engine's).
"""
...
[docs]
def snapshot_nbytes(self) -> int:
"""Return the bytes :meth:`snapshot_rows` would copy, copying nothing.
Returns
-------
int
The size of every row.
"""
...
[docs]
def load_rows(self, rows: Mapping[str, np.ndarray], shape: tuple[int, ...]) -> None:
"""Install rows :meth:`snapshot_rows` returned, refusing rows it does not keep.
Parameters
----------
rows : collections.abc.Mapping
The rows, by name.
shape : tuple of int
The state's shape.
Raises
------
BackendError
If the rows are another engine's, missing, or of the wrong shape; nothing
is installed then.
"""
...
[docs]
def observe(self, position: float) -> object:
"""Return the masked field at `position` inside the last accepted step.
Parameters
----------
position : float
The observation, m.
Returns
-------
object
The field, as the backend holds a state.
"""
...
[docs]
class HostHistory:
"""The Adams history and its field arithmetic, on host arrays (SEP 16 phase 3b).
The march (:class:`~solprop.backends.host.adams.HostAdamsMarch`) keeps the
controller -- the orders, the slots, rejection,
restarts, the budget and the ledger -- and hands every state-sized operation to an
engine: the startup sample, a trial's predictor, source, corrector and defect
norms, the adoption and transport of an accepted step, and dense observation. This
engine holds the history in a NumPy store and applies the given kernels; a backend
may supply its own, as the native backend keeps the history resident in its
state (``_ResidentAdams``). Between a trial and its acceptance the engine
holds the trial; nothing else is carried across calls but the store and the last
accepted step for dense observation.
Parameters
----------
backend : solprop.backends.host.driver.HostDriver
The backend whose state, source evaluation and propagators are used.
method : solprop.stepper.multistep.Multistep
The method.
kernels : AdamsKernels
The interaction formulation's fused arithmetic.
exponential : ExponentialKernels or None
The exponential formulation's arithmetic, ``None`` for the interaction one.
compensation : {'transported', 'untransported'}
A varying operator's compensation treatment (SEP 16 Q8).
"""
def __init__(
self,
backend: HostDriver[Any],
method: Multistep,
kernels: AdamsKernels,
exponential: ExponentialKernels | None,
compensation: Literal["transported", "untransported"],
) -> None:
"""Hold the backend, the kernels and an empty store."""
self._backend = backend
self._method = method
self._kernels = kernels
self._exponential = exponential
self._compensation = compensation
linear = backend._problem.linear
# SEP 16 phase 4: a varying operator's linear compensation, integrated in the
# reference flow's frame (Q8); the untransported treatment is a test witness.
# A structurally absent source takes the exact chord instead (section 4.2).
# SEP 16 phase 5: a frozen-residual coupled flow's residual is linear in the
# state and joins the compensation, whatever the diagonal's representation.
flow = linear.coupled_flow
residual = flow is not None and "frozen_residual" in flow.section_methods
self.compensated = exponential is not None and (
residual
or (
linear.phi_representation in ("expansion", "table")
and not source_absent(backend._problem)
)
)
if self.compensated and flow is not None and compensation != "transported":
msg = "the untransported compensation is a witness for diagonal operators"
raise BackendError(msg)
self._weights: list[np.ndarray | None] = []
self._trial_weights: np.ndarray | None = None
self._store: np.ndarray | None = None
self._states: np.ndarray | None = None
self._rates: np.ndarray | None = None
self.dense: Any = None
self._exponential_dense: _ExponentialStep | None = None
self._fused_trial: _FusedTrial | None = None
self._exponential_trial: _ExponentialTrial | None = None
# SEP 35: the carrier phase each exponential trial removes from the samples
# it reads, from a reference fixed for the run; None when off. An inert one
# (sigma = 0) takes the unshifted arithmetic exactly.
self.carrier: CarrierReference | None = (
None if exponential is None else carrier_reference(backend._problem, method)
)
if self.carrier is not None:
logger.info(
"carrier frame %r: %s", self.carrier.policy, self.carrier.reason
)
[docs]
def reset(self) -> None:
"""Forget the last accepted step and any trial in flight."""
self.dense = None
self._exponential_dense = None
self._fused_trial = None
self._exponential_trial = None
[docs]
@property
def n_slots(self) -> int: # numpydoc ignore=RT01
"""The history store's rows: the method's order and two more."""
return self._method.order + 2
# --- the accepted state, held across a dense observation --------------------------
[docs]
def hold(self) -> object:
"""Return the accepted state, to restore after a dense observation.
Returns
-------
object
The backend's host copy of the accepted state.
"""
return self._backend._adams_state()
[docs]
def restore(self, z: float, held: object) -> None:
"""Make a held accepted state current again at `z`.
Parameters
----------
z : float
Its position, m.
held : object
What :meth:`hold` returned.
"""
self._backend._adams_restore(z, cast("_State", held))
# --- startup ----------------------------------------------------------------------
[docs]
def start(self, z: float) -> _State:
"""Evaluate the source at the accepted state, the history's first sample.
The sample goes to the store's row zero.
Parameters
----------
z : float
The accepted position, m.
Returns
-------
numpy.ndarray
The sample: the transported source in the interaction formulation, the
untransported one in the exponential formulation.
"""
backend = self._backend
state = backend._adams_state()
backend._hoist(z)
if self._exponential is not None:
value = backend._adams_lab_source(state, z, purpose="startup")
else:
value, _ = backend._adams_source(state, z, z, purpose="startup")
if self._store is None or self._store.shape[1:] != value.shape:
# Zeroed, not empty: the rows a short history has not reached go into a
# resume snapshot as they are, and a file has to be reproducible.
self._store = np.zeros((self.n_slots, *value.shape), dtype=np.complex128)
if self.compensated:
self._states = np.zeros_like(self._store)
self._rates = np.zeros(self._store.shape, dtype=np.complex128)
self._store[0] = value
if self.compensated:
assert self._states is not None and self._rates is not None
# The startup sample's state and rate, read on the far side of a kink
# that restarted the history.
self._states[0] = state
self._rates[0] = phi_rate(
backend._problem.linear, z, helpers=backend.helpers
)
# A frozen-residual flow's weights at the sample, on the far side too.
start = cast("CoupledBlock | None", backend._adams_block(z, 0.0))
self._weights = [None] * self.n_slots
self._weights[0] = None if start is None else start.end_weights
return self._store[0]
# --- the interaction formulation's fused path -------------------------------------
[docs]
def fused_trial( # noqa: PLR0917 - called positionally
self,
z: float,
dz: float,
nodes: list[float],
slots: np.ndarray,
order: int,
orders: list[int],
) -> np.ndarray:
"""Form one fused trial and return the scaled norms of its scored defects.
The predictor's lab-frame field, the source there (one call, labelled a
predictor), and in one pass the corrected field and each scored order's
defect: its ``k + 1``-node corrector through the new source less its
``k``-node predictor. The trial is held for :meth:`fused_accept`.
Parameters
----------
z, dz : float
The anchor and the step, m.
nodes : list of float
The history's normalised positions, newest first.
slots : numpy.ndarray
Their store rows, ``int64``.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
Returns
-------
numpy.ndarray
One scaled norm per scored order.
"""
backend, kernels = self._backend, self._kernels
store = self._store
assert store is not None
state = backend._adams_state()
backend._hoist(z)
propagator = backend._adams_propagator(z, z + dz)
predictor = kernels.weights(nodes[:order])
lab = kernels.predict(state, propagator, store, slots[:order], predictor, dz)
source, _ = backend._adams_source(lab, z, z + dz, purpose="predictor")
rows, corrector = _defect_rows(kernels, nodes, orders)
width = rows.shape[1]
mask = backend._problem.grids.norm_mask
advanced, norms = kernels.correct(
state,
propagator,
source,
store,
slots[: width - 1],
corrector,
rows,
dz,
mask,
_divisor(backend, state),
)
self._fused_trial = _FusedTrial(
z=z,
dz=dz,
state=state,
propagator=propagator,
source=source,
advanced=advanced,
nodes=nodes,
slots=slots,
order=order,
)
return norms
[docs]
def fused_accept(self, keep: np.ndarray, free: int, *, interpolate: bool) -> None:
"""Adopt the held fused trial: the state, its interpolant, the new sample.
Parameters
----------
keep : numpy.ndarray
The history rows kept, ``int64``; each is re-anchored at the step's end.
free : int
The row the new sample takes.
interpolate : bool
Whether the step's interpolant is kept for dense observation.
"""
trial = self._fused_trial
store = self._store
assert trial is not None and store is not None
self._fused_trial = None
self._backend._adams_adopt(trial.advanced, trial.dz)
order = trial.order
if interpolate:
# Copied before the transport re-anchors the rows in place: the step's own
# interpolant is in its anchor's frame.
slopes = [
trial.source,
*(store[slot].copy() for slot in trial.slots[:order]),
]
self.dense = _DenseStep(
z=trial.z,
dz=trial.dz,
start=trial.state.copy(),
nodes=np.array([1.0, *trial.nodes[:order]]),
slopes=slopes,
)
self._kernels.transport(store, keep, trial.propagator, free, trial.source)
[docs]
def value(self, slot: int) -> _State:
"""Return a store row, for the history entry that names it.
Parameters
----------
slot : int
The row.
Returns
-------
numpy.ndarray
The row itself, not a copy.
"""
assert self._store is not None
return self._store[slot]
# --- persistence across a resume --------------------------------------------------
[docs]
def snapshot_nbytes(self) -> int:
"""Return the bytes :meth:`snapshot_rows` would copy, copying nothing.
Returns
-------
int
The size of every row.
"""
assert self._store is not None
rows = [self._store, *self._weights]
if self.compensated:
assert self._states is not None and self._rates is not None
rows += [self._states, self._rates]
return sum(int(row.nbytes) for row in rows if row is not None)
[docs]
def snapshot_rows(self) -> dict[str, np.ndarray]:
"""Return copies of the rows a resume needs to continue the history.
Returns
-------
dict
``"store"``, the samples; for a compensated march also ``"states"`` and
``"rates"``, and ``"weights_<row>"`` for each row a frozen-residual flow
weights.
"""
assert self._store is not None
rows = {"store": self._store.copy()}
if self.compensated:
assert self._states is not None and self._rates is not None
rows["states"] = self._states.copy()
rows["rates"] = self._rates.copy()
for slot, weights in enumerate(self._weights):
if weights is not None:
rows[f"weights_{slot}"] = np.array(weights, copy=True)
return rows
[docs]
def load_rows(self, rows: Mapping[str, np.ndarray], shape: tuple[int, ...]) -> None:
"""Install rows :meth:`snapshot_rows` returned, for a state of `shape`.
Parameters
----------
rows : collections.abc.Mapping
The rows, by name.
shape : tuple of int
The state's shape.
Raises
------
BackendError
If a row is missing, does not have the store's shape, or is one this
history does not keep -- the native engine's applied frozen-residual
generators (``"applied"``) among them.
"""
expected = (self.n_slots, *shape)
names = ("store", "states", "rates") if self.compensated else ("store",)
# A compensated history keeps a frozen-residual flow's weights per sample.
kept = set(names) | (
{f"weights_{slot}" for slot in range(self.n_slots)}
if self.compensated
else set()
)
foreign = sorted(set(rows) - kept)
if foreign:
name = self._backend.name
if "applied" in foreign:
msg = (
f"the {name} backend cannot continue this Adams history: the "
"snapshot holds the native engine's applied frozen-residual "
"generators ('applied'), and this engine keeps each sample's "
"weights instead"
)
else:
msg = (
f"the {name} backend cannot continue this Adams history: its "
f"engine keeps no {foreign} rows"
)
raise BackendError(msg)
for name in names:
row = rows.get(name)
if row is None or row.shape != expected:
msg = (
f"the snapshot's {name!r} rows do not fit a store of shape "
f"{expected}"
)
raise BackendError(msg)
self.reset()
self._store = np.array(rows["store"], dtype=np.complex128, copy=True)
if self.compensated:
self._states = np.array(rows["states"], dtype=np.complex128, copy=True)
self._rates = np.array(rows["rates"], dtype=np.complex128, copy=True)
self._weights = [
None
if f"weights_{slot}" not in rows
else np.array(rows[f"weights_{slot}"], copy=True)
for slot in range(self.n_slots)
]
# --- the exponential formulation --------------------------------------------------
[docs]
def exponential_trial( # noqa: PLR0917 - called positionally
self,
z: float,
dz: float,
nodes: list[float],
slots: np.ndarray,
order: int,
orders: list[int],
) -> np.ndarray:
"""Form one exponential trial and return the scaled norms of its defects.
The arithmetic of :meth:`fused_trial` with untransported samples: the predictor
and the corrector integrate their interpolants exactly against the step's flow
``exp(x)``, ``x = Phi(z + h) - Phi(z)``; the defects of the neighbouring orders
are scored from the same moments. A varying operator adds the compensation of
its samples (SEP 16 section 4.2). Under a carrier frame the samples read are
rotated and the moments shifted (SEP 35 section 4.1); the flow is unchanged.
The trial is held for :meth:`exponential_accept`.
Parameters
----------
z, dz : float
The anchor and the step, m.
nodes : list of float
The history's normalised positions, newest first.
slots : numpy.ndarray
Their store rows, ``int64``.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
Returns
-------
numpy.ndarray
One scaled norm per scored order.
"""
backend, kernels = self._backend, self._exponential
assert kernels is not None
store = self._store
assert store is not None
state = backend._adams_state()
backend._hoist(z)
width = 1 + max(orders)
# The exponent is the backend's own increment of Phi over the step, so the
# flow below is exactly the propagator the interaction formulation applies.
# Copied: a z table's host evaluator returns its own reused buffer, which the
# source evaluation writes again.
exponent = np.array(backend._delta_phi(backend._stage_scalars(z, z + dz)))
# A coupled section's step (SEP 16 phase 5): its flow and moments are matrix
# functions of the frozen block on the bins that carry one, applied by the
# reference passes.
block = cast("CoupledBlock | None", backend._adams_block(z, dz))
# SEP 35 section 4.1: with F(z_n + s) = exp(-sigma (1 - s/h)) G(s), the
# step interpolates G_j = exp(sigma (1 - theta_j)) F_j and integrates it
# against the moments at x - sigma; exact for any sigma, the flow untouched.
carrier = self.carrier
shift = (
None if carrier is None or not carrier.active else carrier.shift(exponent)
)
if block is not None:
kernels = kernels.coupled()
moments, flow = _trial_propagators(kernels, exponent, width, block, dz, shift)
rows_read = slots[: width - 1]
factors = (
None if shift is None else _carrier_factors(nodes, len(rows_read), shift)
)
rotation = {} if factors is None else {"factors": factors[:order]}
lab = kernels.predict(
state,
flow,
moments,
store,
rows_read[:order],
lagrange_coefficients(nodes[:order], check=False, helpers=backend.helpers),
dz,
**rotation,
)
compensation = None
if self.compensated:
compensation = self._compensation_terms(
exponent, moments, nodes, slots[: width - 1], dz, block
)
if shift is not None:
compensation.rotate(shift)
lab = lab + compensation.predictor(order)
source = backend._adams_lab_source(lab, z + dz, purpose="predictor")
corrector = _exponential_corrector(nodes, order, width, helpers=backend.helpers)
rows = kernels.defect_rows(nodes, orders, width)
extra = None
if compensation is not None:
rate = phi_rate(
backend._problem.linear, z + dz, before=True, helpers=backend.helpers
)
residual = None
if block is not None and block.residual is not None:
assert block.end_weights is not None
residual = block.residual(lab, block.end_weights)
compensation.add_endpoint(rate, lab, residual)
self._trial_weights = None if block is None else block.end_weights
extra = (compensation.corrector(order), compensation.defects(orders))
mask = backend._problem.grids.norm_mask
advanced, norms = kernels.correct(
state,
flow,
moments,
source,
store,
rows_read,
corrector,
rows,
dz,
mask,
_divisor(backend, state),
extra,
**({} if factors is None else {"factors": factors}),
)
self._exponential_trial = _ExponentialTrial(
z=z,
dz=dz,
state=state,
exponent=exponent,
corrector=corrector,
source=source,
lab=lab,
advanced=advanced,
slots=slots,
order=order,
compensation=compensation,
shift=shift,
factors=factors,
)
return norms
[docs]
def exponential_accept(
self, keep: np.ndarray, free: int, *, interpolate: bool
) -> None:
"""Adopt the held exponential trial; the new sample takes row `free`.
Parameters
----------
keep : numpy.ndarray
The history rows kept, ``int64``; nothing moves in this formulation.
free : int
The row the new sample takes.
interpolate : bool
Whether the step's interpolant is kept for dense observation.
"""
del keep # untransported samples are never re-anchored
trial = self._exponential_trial
store = self._store
assert trial is not None and store is not None
self._exponential_trial = None
self._backend._adams_adopt(trial.advanced, trial.dz)
z, dz, order, slots = trial.z, trial.dz, trial.order, trial.slots
compensation = trial.compensation
if interpolate:
# Copied: the store's rows are overwritten by later steps.
self._exponential_dense = _ExponentialStep(
z=z,
dz=dz,
start=trial.state.copy(),
exponent=trial.exponent,
coefficients=trial.corrector[: order + 1, : order + 1].copy(),
# Under a carrier frame, the rotated samples and the accepted
# trial's own shift, held until the next accepted step.
values=[
trial.source.copy(),
*(
store[slot].copy()
if trial.factors is None
else trial.factors[index] * store[slot]
for index, slot in enumerate(slots[:order])
),
],
shift=trial.shift,
compensation=None
if compensation is None
else compensation.observation(
order, self._history_terms(trial.exponent, dz, slots[:order])
),
)
store[free] = trial.source
if compensation is not None:
assert self._states is not None and self._rates is not None
assert compensation.endpoint_rate is not None
self._states[free] = trial.lab
self._rates[free] = compensation.endpoint_rate
# One entry per slot, set with the startup sample or a resumed history.
self._weights[free] = self._trial_weights
def _history_terms(
self, exponent: _State, dz: float, slots: np.ndarray
) -> list[_State]:
"""Return ``(D(z_j) - B_n) u_j`` of history samples, untransported.
- ``exponent``: ``x = h B_n``.
- ``dz``: the step, m.
"""
assert self._states is not None and self._rates is not None
reference = exponent / dz
return [(self._rates[slot] - reference) * self._states[slot] for slot in slots]
def _compensation_terms( # noqa: PLR0917 - called positionally
self,
exponent: _State,
moments: Any,
nodes: list[float],
slots: np.ndarray,
dz: float,
block: CoupledBlock | None = None,
) -> "_Compensation":
"""Return the trial's linear compensation of its history samples.
- ``exponent``: ``x = h B_n``, the step's chord exponent.
- ``nodes``: the history's normalised positions, newest first.
- ``dz``: the step, m.
- ``block``: a coupled step's frozen block: the samples' residuals against it
join their compensation, carried by its forward flow.
"""
assert self._states is not None and self._rates is not None
assert self._exponential is not None
used = list(nodes[: len(slots)])
transported = self._compensation == "transported"
# The untransported witness keeps the samples in their own frame: a span of 0.
spans = np.array([1.0 - node if transported else 0.0 for node in used])
if block is not None:
carried = self._exponential.coupled().carry_coupled(
exponent,
dz,
(self._rates, self._states, self._weights),
slots,
spans,
block,
)
else:
carried = self._exponential.carry(
exponent, dz, self._rates, self._states, slots, spans
)
return _Compensation(
exponent=exponent,
moments=moments,
nodes=used,
carried=carried,
dz=dz,
treatment=self._compensation,
helpers=self._backend.helpers,
)
# --- dense observation ------------------------------------------------------------
[docs]
def observe(self, position: float) -> object:
"""Return the field at `position` inside the last accepted step.
Parameters
----------
position : float
The observation, m, inside the last accepted step.
Returns
-------
object
The masked lab-frame field at `position`, as the backend holds a state:
a host array, or Kokkos's native state loaded with it.
"""
if self._exponential is not None:
return self._observe_exponential(position)
step = cast("_DenseStep | None", self.dense)
assert step is not None
backend = self._backend
theta = (position - step.z) / step.dz
weights = integrated_weights(
step.nodes, theta, check=False, helpers=backend.helpers
)
value = step.start + _combine(weights, step.slopes, step.dz)
# A bin the linear flow changes by more than an e-fold over the step (the
# spectral collar's) takes the anchor's linear flow: no polynomial follows its
# exponentially growing derivative (as for the Runge-Kutta extension).
exponent = backend._delta_phi(backend._stage_scalars(step.z, step.z + step.dz))
value = np.where(np.abs(np.real(exponent)) > 1.0, step.start, value)
return backend._apply_mask(backend._adams_propagator(step.z, position) * value)
def _observe_exponential(self, position: float) -> object:
"""Return the field at `position` inside the last accepted exponential step.
The accepted corrector's interpolant integrated exactly against the flow to the
fraction ``xi`` of the step (SEP 16 section 4.3):
``exp(xi x) u_n + h sum_k k! xi**(k+1) phi_{k+1}(xi x) sum_j c_jk F_j``; under
a carrier frame the rotated samples ``G_j``, the moments at ``xi (x - sigma)``
and the factor ``exp(-sigma (1 - xi))``, with the accepted step's ``sigma``
(SEP 35 section 4.1). Every
factor is a forward flow, so a strongly damped bin needs no guard; no source
is evaluated and nothing enters the history.
- ``position``: the observation, m, inside the last accepted step.
"""
step = self._exponential_dense
assert step is not None
# One evaluation per plane: the reference passes serve every backend.
helpers = self._backend.helpers
kernels = ExponentialKernels(helpers=helpers)
xi = (position - step.z) / step.dz
count = step.coefficients.shape[1]
shift = step.shift
# Under a carrier frame (SEP 35 section 4.1) the accepted step's rotated
# samples, integrated against the moments at xi (x - sigma) and carried back
# by exp(-sigma (1 - xi)); every factor is bounded for an imaginary sigma.
moments = kernels.moments(
xi * (step.exponent if shift is None else step.exponent - shift), count
)
scaled = moments * np.array([xi ** (k + 1) for k in range(count)]).reshape(
(count, *([1] * step.start.ndim))
)
increment = kernels.apply(scaled, step.coefficients, step.values, step.dz)
if shift is not None:
increment = np.exp(-shift * (1.0 - xi)) * increment
value = np.exp(xi * step.exponent) * step.start + increment
if step.compensation is not None:
nodes, terms = step.compensation
weights = integrated_weights(
np.asarray(nodes), xi, check=False, helpers=helpers
)
# e^{(xi - theta_j) x} is a forward flow for every history sample; for
# the endpoint sample it is an inverse one, which a bin the step damps by
# more than an e-fold would amplify: such a bin takes the anchor's flow,
# as the interaction formulation's interpolant does (ADR-039).
for weight, node, term in zip(weights, nodes, terms, strict=True):
value = (
value
+ step.dz * weight * np.exp((xi - node) * step.exponent) * term
)
damped = np.real(step.exponent) < -1.0
value = np.where(damped, np.exp(xi * step.exponent) * step.start, value)
return self._backend._apply_mask(value)
def _trial_propagators( # noqa: PLR0917 - called positionally
kernels: Any,
exponent: Any,
width: int,
block: CoupledBlock | None,
dz: float,
shift: _Shift | None,
) -> tuple[Any, Any]:
"""Return a trial's moments and flow, shifted by `shift` when it is given.
- ``exponent``: the step's increment of Phi, the true linear exponent.
- ``width``: the number of moments, one more than the highest order scored.
- ``block``: the section's frozen block, or None off a coupled section.
- ``shift``: the carrier shift sigma per member, or None when no frame is active.
- Returns: the moments at the shifted exponent and the true flow exp(x).
"""
moments: Any
flow: Any
if block is not None:
# The moments are the shifted block's and the flow exp(sigma) times its
# phi_0, the true exp(h B) (section 4.5): one evaluation, never the
# shifted flow.
moments = (
kernels.coupled_moments(exponent, width, block, dz)
if shift is None
else kernels.coupled_moments(exponent, width, block, dz, shift=shift)
)
flow = moments
else:
moments = (
kernels.moments(exponent, width)
if shift is None
else kernels.moments(exponent, width, shift=shift)
)
# After the moments: a compiled backend forms exp(x) in their pass.
flow = kernels.flow(exponent)
return moments, flow
def _carrier_factors(nodes: list[float], count: int, shift: _Shift) -> _Factors:
"""Return the rotations ``exp(sigma (1 - theta_j))`` of a trial's history rows.
The rotated-sample form of SEP 35 section 6.2: the passes multiply each row by its
factor as they read it, so the store keeps the untransported samples and a
rejection, a resume or a later step with its own shift reads them unchanged.
- ``nodes``: the history's normalised positions ``theta_j``, newest first.
- Returns: one factor per row, ``(count, *shift.shape)``.
"""
return np.stack([np.exp(shift * (1.0 - nodes[index])) for index in range(count)])
def _rows_read(
store: Complex[np.ndarray, "n_slots n_batch n_modes n_pol n_omega"],
slots: np.ndarray,
factors: _Factors | None,
) -> list[_State]:
"""Return the history rows a pass combines, rotated when a carrier frame asks.
- ``factors``: their rotations, in the slots' order; ``None`` for the rows
themselves.
"""
if factors is None:
return [store[slot] for slot in slots]
return [factor * store[slot] for factor, slot in zip(factors, slots, strict=False)]
def _divisor(
backend: HostDriver[Any],
state: _State,
) -> np.ndarray:
"""Return the defect norm's scale per bin, the accepted state's before the step.
The Runge-Kutta march scales by the larger of the states before and after the step.
An Adams step outside its stability region can multiply the state many times over
while its predictor and corrector still agree to a fraction of the *new* state, so
that scale lets a divergence pass as accurate: SEP 13 phase 2 measured PEC4 growing
from 1e7 to 1e236 in five accepted steps with every scaled norm below 0.6. The state
the step starts from is the scale that cannot grow with the step it judges.
- Returns: the scale of every bin, shape ``(n_omega,)``.
"""
mask = backend._problem.grids.norm_mask
controller = backend._stepper.controller
magnitude = float(np.linalg.norm(state[..., mask]))
scale = controller.scale(magnitude, magnitude)
return (
scale
if isinstance(scale, np.ndarray)
else np.full(state.shape[-1], float(scale))
)
def _defect_rows(
kernels: AdamsKernels, nodes: list[float], orders: list[int]
) -> tuple[np.ndarray, np.ndarray]:
"""Return each order's defect weights and the first order's corrector weights.
Over ``(source, newest, ...)``, order ``k``'s defect is its ``k + 1``-node
corrector through the new source less its ``k``-node predictor.
- ``nodes``: the history's normalised positions, newest first.
- Returns: the defect rows, shape ``(len(orders), 1 + max(orders))``, and the step's
own corrector weights, zero past its nodes.
"""
width = 1 + max(orders)
rows = np.zeros((len(orders), width))
corrector = np.zeros(width)
for index, order in enumerate(orders):
stencil = nodes[:order]
through = kernels.weights([1.0, *stencil])
rows[index, : order + 1] = through
rows[index, 1 : order + 1] -= kernels.weights(stencil)
if index == 0:
corrector[: order + 1] = through
return rows, corrector
[docs]
def fused_tables(
nodes: list[float], order: int, orders: list[int], *, helpers: Helpers
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
"""Return a fused trial's coefficient tables, for an engine that applies them.
The interaction formulation's tables of :meth:`HostHistory.fused_trial`, which a
resident engine (ADR-043) receives instead of forming them.
Parameters
----------
nodes : list of float
The history's normalised positions, newest first.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
helpers : solprop._helpers.Helpers
The engine's backend's implementation of the Adams basis.
Returns
-------
tuple
The predictor's integrated weights over the ``order`` newest rows; the
corrector's weights over ``(source, newest, ...)``, zero past its nodes; and
one defect row per scored order over the same samples.
"""
kernels = AdamsKernels(helpers=helpers)
rows, corrector = _defect_rows(kernels, nodes, orders)
return kernels.weights(nodes[:order]), corrector, rows
[docs]
def factored_defect_rows(
nodes: list[float], orders: list[int], width: int
) -> np.ndarray:
"""Return each exponential defect factored: a divided difference and a polynomial.
Order ``k``'s defect, its ``k + 1``-node corrector through the new sample less its
``k``-node predictor, vanishes at the predictor's nodes, so it is
``f[1, theta_0, ..., theta_{k-1}] prod_{j<k} (theta - theta_j)``: one divided
difference of the samples times one polynomial, whose integral against the flow is
one combination of the moments. That is ``2 (k + 1)`` products an element instead
of ``(k + 1) n_k``, the monomial rows of ``_exponential_rows``.
Parameters
----------
nodes : list of float
The history's normalised positions, newest first.
orders : list of int
The orders scored, the step's own first.
width : int
``1 + max(orders)``: the samples and the moments of the trial.
Returns
-------
numpy.ndarray
Shape ``(len(orders), 2 width)``: the divided-difference coefficients over
``(source, newest, ...)``, then the node polynomial's ascending monomial
coefficients.
"""
rows = np.zeros((len(orders), 2 * width))
for index, order in enumerate(orders):
points = [1.0, *nodes[:order]]
row = rows[index]
for t, node in enumerate(points):
denominator = 1.0
for s, other in enumerate(points):
if s != t:
denominator *= node - other
row[t] = 1.0 / denominator
# The node polynomial one linear factor at a time, newest node first, as the
# Adams basis builds its numerators (`_basis_rows`); in plain floats, since
# `numpy.polynomial.polyfromroots` costs four fifths of this function. Every
# node is at or below zero, so every coefficient is positive: no cancellation.
coefficients = [1.0]
for root in nodes[:order]:
product = [0.0, *coefficients]
for power, value in enumerate(coefficients):
product[power] -= root * value
coefficients = product
row[width : width + order + 1] = coefficients
return rows
[docs]
def exponential_tables(
nodes: list[float], order: int, orders: list[int], *, helpers: Helpers
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
"""Return an exponential trial's coefficient tables, for a resident engine.
Parameters
----------
nodes : list of float
The history's normalised positions, newest first.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
helpers : solprop._helpers.Helpers
The engine's backend's implementation of the Adams basis.
Returns
-------
tuple
The predictor's Lagrange coefficients over the ``order`` newest rows, shape
``(order, order)``; the corrector's over ``(source, newest, ...)``, shape
``(width, width)`` with ``width = 1 + max(orders)``; and the factored defect
rows of :func:`factored_defect_rows`.
"""
width = 1 + max(orders)
corrector = _exponential_corrector(nodes, order, width, helpers=helpers)
predictor = lagrange_coefficients(nodes[:order], check=False, helpers=helpers)
return predictor, corrector, factored_defect_rows(nodes, orders, width)
[docs]
def compensation_tables(
nodes: list[float], order: int, orders: list[int], *, helpers: Helpers
) -> tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
"""Return a trial's transported-compensation tables, for a resident engine.
The weights ``_Compensation`` applies (SEP 16 section 4.2, ADR-042), over the
carried rows ``exp((1 - theta_j) x) (D(z_j) - x / h) u_j`` of the history samples
and the endpoint term ``(D(z + h) - x / h) u_pred``, each sum still to be
multiplied by the step.
Parameters
----------
nodes : list of float
The history's normalised positions, newest first.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
helpers : solprop._helpers.Helpers
The engine's backend's implementation of the Adams basis.
Returns
-------
tuple
The spans ``1 - theta_j`` of the ``width - 1`` carried samples; the
predictor's weights over the ``order`` newest of them; the corrector's over
``(endpoint, newest, ...)``; and each scored order's defect weights over the
same, shape ``(len(orders), width)``, with ``width = 1 + max(orders)``.
"""
width = 1 + max(orders)
spans = np.array([1.0 - node for node in nodes[: width - 1]])
# Each count's predictor and corrector once: the step's own pair is also its
# defect's, so a trial forms two weights per scored order, not two more.
weights = {
count: (
integrated_weights(nodes[:count], check=False, helpers=helpers),
integrated_weights([1.0, *nodes[:count]], check=False, helpers=helpers),
)
for count in dict.fromkeys([order, *orders])
}
rows = np.zeros((len(orders), width))
for index, scored in enumerate(orders):
predicted, through = weights[scored]
rows[index, : scored + 1] = through
rows[index, 1 : scored + 1] -= predicted
predictor, corrector = weights[order]
return spans, predictor, corrector, rows
[docs]
def trial_tables(
nodes: list[float],
order: int,
orders: list[int],
*,
compensated: bool,
helpers: Helpers,
) -> tuple[
np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray
]:
"""Return all of an exponential trial's coefficient tables, for a resident engine.
:func:`exponential_tables`' three and, compensated, :func:`compensation_tables`'
four, composed over the record's basis; a record that forms them in one call
(``Helpers.tables``, the native backend's) performs the same operations in the
same order, so the tables are the same, read once a trial instead of from a dozen
small calls.
Parameters
----------
nodes : list of float
The history's normalised positions, newest first.
order : int
The step's order.
orders : list of int
The orders scored, the step's own first.
compensated : bool
Whether the trial carries a varying operator's transported compensation.
helpers : solprop._helpers.Helpers
The engine's backend's implementation of the Adams basis.
Returns
-------
tuple
The predictor, the corrector and the factored defect rows; then the
compensation's spans, predictor, corrector and defect rows, shapes ``(0,)``
and ``(0, 0)`` when not compensated.
"""
one_call = helpers.tables
if one_call is not None:
return one_call(np.asarray(nodes, dtype=np.float64), order, orders, compensated)
predictor, corrector, factored = exponential_tables(
nodes, order, orders, helpers=helpers
)
if not compensated:
return predictor, corrector, factored, *_NO_COMPENSATION
spans, carried, endpoint, defects = compensation_tables(
nodes, order, orders, helpers=helpers
)
return predictor, corrector, factored, spans, carried, endpoint, defects
_NO_COMPENSATION = (np.empty(0), np.empty(0), np.empty(0), np.empty((0, 0)))
"""An uncompensated trial's compensation tables: spans, weights and defect rows."""
def _exponential_rows(
nodes: list[float], orders: list[int], width: int, *, helpers: Helpers
) -> np.ndarray:
"""Return each order's defect coefficients in the monomial basis.
Over ``X = (source, newest, ...)`` in the monomial basis of the moments, order
``k``'s defect is its ``k + 1``-node corrector through the new source less its
``k``-node predictor, as for :func:`_defect_rows`; the weights are the Lagrange
coefficients, which the trial's moments turn into exponential weights.
- ``nodes``: the history's normalised positions, newest first.
- Returns: the defect rows, shape ``(len(orders), width, width)``.
"""
rows = np.zeros((len(orders), width, width))
for index, order in enumerate(orders):
stencil = nodes[:order]
rows[index, : order + 1, : order + 1] = lagrange_coefficients(
[1.0, *stencil], check=False, helpers=helpers
)
rows[index, 1 : order + 1, :order] -= lagrange_coefficients(
stencil, check=False, helpers=helpers
)
return rows
def _exponential_corrector(
nodes: list[float], order: int, width: int, *, helpers: Helpers
) -> np.ndarray:
"""Return the step's own corrector's Lagrange coefficients, padded to `width`.
- ``nodes``: the history's normalised positions, newest first.
- Returns: the coefficients over ``X = (source, newest, ...)``, shape ``(width,
width)``, zero past its nodes.
"""
corrector = np.zeros((width, width))
corrector[: order + 1, : order + 1] = lagrange_coefficients(
[1.0, *nodes[:order]], check=False, helpers=helpers
)
return corrector
def _combine(weights: np.ndarray, values: list[_State], dz: float) -> _State:
"""Return ``dz * sum_j w_j g_j``.
- ``dz``: the step, m.
"""
total = np.zeros_like(values[0])
for weight, value in zip(weights, values, strict=True):
total = passes.accumulate(total, value, float(weight))
return dz * total