562 lines
19 KiB
Python
562 lines
19 KiB
Python
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import numpy as np
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from scipy.linalg import lu_factor, lu_solve
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from scipy.sparse import csc_matrix, issparse, eye
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from scipy.sparse.linalg import splu
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from scipy.optimize._numdiff import group_columns
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from .common import (validate_max_step, validate_tol, select_initial_step,
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norm, num_jac, EPS, warn_extraneous,
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validate_first_step)
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from .base import OdeSolver, DenseOutput
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S6 = 6 ** 0.5
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# Butcher tableau. A is not used directly, see below.
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C = np.array([(4 - S6) / 10, (4 + S6) / 10, 1])
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E = np.array([-13 - 7 * S6, -13 + 7 * S6, -1]) / 3
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# Eigendecomposition of A is done: A = T L T**-1. There is 1 real eigenvalue
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# and a complex conjugate pair. They are written below.
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MU_REAL = 3 + 3 ** (2 / 3) - 3 ** (1 / 3)
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MU_COMPLEX = (3 + 0.5 * (3 ** (1 / 3) - 3 ** (2 / 3))
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- 0.5j * (3 ** (5 / 6) + 3 ** (7 / 6)))
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# These are transformation matrices.
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T = np.array([
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[0.09443876248897524, -0.14125529502095421, 0.03002919410514742],
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[0.25021312296533332, 0.20412935229379994, -0.38294211275726192],
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[1, 1, 0]])
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TI = np.array([
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[4.17871859155190428, 0.32768282076106237, 0.52337644549944951],
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[-4.17871859155190428, -0.32768282076106237, 0.47662355450055044],
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[0.50287263494578682, -2.57192694985560522, 0.59603920482822492]])
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# These linear combinations are used in the algorithm.
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TI_REAL = TI[0]
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TI_COMPLEX = TI[1] + 1j * TI[2]
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# Interpolator coefficients.
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P = np.array([
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[13/3 + 7*S6/3, -23/3 - 22*S6/3, 10/3 + 5 * S6],
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[13/3 - 7*S6/3, -23/3 + 22*S6/3, 10/3 - 5 * S6],
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[1/3, -8/3, 10/3]])
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NEWTON_MAXITER = 6 # Maximum number of Newton iterations.
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MIN_FACTOR = 0.2 # Minimum allowed decrease in a step size.
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MAX_FACTOR = 10 # Maximum allowed increase in a step size.
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def solve_collocation_system(fun, t, y, h, Z0, scale, tol,
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LU_real, LU_complex, solve_lu):
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"""Solve the collocation system.
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Parameters
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----------
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fun : callable
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Right-hand side of the system.
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t : float
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Current time.
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y : ndarray, shape (n,)
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Current state.
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h : float
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Step to try.
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Z0 : ndarray, shape (3, n)
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Initial guess for the solution. It determines new values of `y` at
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``t + h * C`` as ``y + Z0``, where ``C`` is the Radau method constants.
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scale : float
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Problem tolerance scale, i.e. ``rtol * abs(y) + atol``.
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tol : float
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Tolerance to which solve the system. This value is compared with
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the normalized by `scale` error.
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LU_real, LU_complex
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LU decompositions of the system Jacobians.
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solve_lu : callable
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Callable which solves a linear system given a LU decomposition. The
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signature is ``solve_lu(LU, b)``.
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Returns
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-------
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converged : bool
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Whether iterations converged.
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n_iter : int
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Number of completed iterations.
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Z : ndarray, shape (3, n)
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Found solution.
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rate : float
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The rate of convergence.
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"""
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n = y.shape[0]
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M_real = MU_REAL / h
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M_complex = MU_COMPLEX / h
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W = TI.dot(Z0)
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Z = Z0
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F = np.empty((3, n))
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ch = h * C
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dW_norm_old = None
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dW = np.empty_like(W)
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converged = False
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rate = None
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for k in range(NEWTON_MAXITER):
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for i in range(3):
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F[i] = fun(t + ch[i], y + Z[i])
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if not np.all(np.isfinite(F)):
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break
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f_real = F.T.dot(TI_REAL) - M_real * W[0]
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f_complex = F.T.dot(TI_COMPLEX) - M_complex * (W[1] + 1j * W[2])
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dW_real = solve_lu(LU_real, f_real)
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dW_complex = solve_lu(LU_complex, f_complex)
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dW[0] = dW_real
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dW[1] = dW_complex.real
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dW[2] = dW_complex.imag
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dW_norm = norm(dW / scale)
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if dW_norm_old is not None:
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rate = dW_norm / dW_norm_old
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if (rate is not None and (rate >= 1 or
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rate ** (NEWTON_MAXITER - k) / (1 - rate) * dW_norm > tol)):
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break
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W += dW
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Z = T.dot(W)
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if (dW_norm == 0 or
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rate is not None and rate / (1 - rate) * dW_norm < tol):
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converged = True
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break
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dW_norm_old = dW_norm
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return converged, k + 1, Z, rate
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def predict_factor(h_abs, h_abs_old, error_norm, error_norm_old):
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"""Predict by which factor to increase/decrease the step size.
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The algorithm is described in [1]_.
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Parameters
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----------
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h_abs, h_abs_old : float
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Current and previous values of the step size, `h_abs_old` can be None
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(see Notes).
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error_norm, error_norm_old : float
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Current and previous values of the error norm, `error_norm_old` can
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be None (see Notes).
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Returns
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-------
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factor : float
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Predicted factor.
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Notes
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-----
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If `h_abs_old` and `error_norm_old` are both not None then a two-step
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algorithm is used, otherwise a one-step algorithm is used.
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References
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----------
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.. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
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Equations II: Stiff and Differential-Algebraic Problems", Sec. IV.8.
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"""
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if error_norm_old is None or h_abs_old is None or error_norm == 0:
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multiplier = 1
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else:
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multiplier = h_abs / h_abs_old * (error_norm_old / error_norm) ** 0.25
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with np.errstate(divide='ignore'):
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factor = min(1, multiplier) * error_norm ** -0.25
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return factor
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class Radau(OdeSolver):
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"""Implicit Runge-Kutta method of Radau IIA family of order 5.
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The implementation follows [1]_. The error is controlled with a
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third-order accurate embedded formula. A cubic polynomial which satisfies
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the collocation conditions is used for the dense output.
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Parameters
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----------
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fun : callable
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Right-hand side of the system. The calling signature is ``fun(t, y)``.
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Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
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It can either have shape (n,); then ``fun`` must return array_like with
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shape (n,). Alternatively it can have shape (n, k); then ``fun``
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must return an array_like with shape (n, k), i.e., each column
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corresponds to a single column in ``y``. The choice between the two
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options is determined by `vectorized` argument (see below). The
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vectorized implementation allows a faster approximation of the Jacobian
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by finite differences (required for this solver).
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t0 : float
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Initial time.
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y0 : array_like, shape (n,)
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Initial state.
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t_bound : float
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Boundary time - the integration won't continue beyond it. It also
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determines the direction of the integration.
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first_step : float or None, optional
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Initial step size. Default is ``None`` which means that the algorithm
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should choose.
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max_step : float, optional
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Maximum allowed step size. Default is np.inf, i.e., the step size is not
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bounded and determined solely by the solver.
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rtol, atol : float and array_like, optional
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Relative and absolute tolerances. The solver keeps the local error
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estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
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relative accuracy (number of correct digits). But if a component of `y`
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is approximately below `atol`, the error only needs to fall within
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the same `atol` threshold, and the number of correct digits is not
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guaranteed. If components of y have different scales, it might be
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beneficial to set different `atol` values for different components by
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passing array_like with shape (n,) for `atol`. Default values are
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1e-3 for `rtol` and 1e-6 for `atol`.
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jac : {None, array_like, sparse_matrix, callable}, optional
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Jacobian matrix of the right-hand side of the system with respect to
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y, required by this method. The Jacobian matrix has shape (n, n) and
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its element (i, j) is equal to ``d f_i / d y_j``.
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There are three ways to define the Jacobian:
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* If array_like or sparse_matrix, the Jacobian is assumed to
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be constant.
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* If callable, the Jacobian is assumed to depend on both
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t and y; it will be called as ``jac(t, y)`` as necessary.
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For the 'Radau' and 'BDF' methods, the return value might be a
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sparse matrix.
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* If None (default), the Jacobian will be approximated by
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finite differences.
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It is generally recommended to provide the Jacobian rather than
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relying on a finite-difference approximation.
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jac_sparsity : {None, array_like, sparse matrix}, optional
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Defines a sparsity structure of the Jacobian matrix for a
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finite-difference approximation. Its shape must be (n, n). This argument
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is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
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elements in *each* row, providing the sparsity structure will greatly
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speed up the computations [2]_. A zero entry means that a corresponding
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element in the Jacobian is always zero. If None (default), the Jacobian
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is assumed to be dense.
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vectorized : bool, optional
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Whether `fun` is implemented in a vectorized fashion. Default is False.
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Attributes
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----------
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n : int
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Number of equations.
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status : string
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Current status of the solver: 'running', 'finished' or 'failed'.
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t_bound : float
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Boundary time.
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direction : float
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Integration direction: +1 or -1.
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t : float
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Current time.
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y : ndarray
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Current state.
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t_old : float
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Previous time. None if no steps were made yet.
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step_size : float
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Size of the last successful step. None if no steps were made yet.
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nfev : int
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Number of evaluations of the right-hand side.
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njev : int
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Number of evaluations of the Jacobian.
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nlu : int
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Number of LU decompositions.
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References
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----------
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.. [1] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
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Stiff and Differential-Algebraic Problems", Sec. IV.8.
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.. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
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sparse Jacobian matrices", Journal of the Institute of Mathematics
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and its Applications, 13, pp. 117-120, 1974.
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"""
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def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
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rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None,
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vectorized=False, first_step=None, **extraneous):
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warn_extraneous(extraneous)
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super(Radau, self).__init__(fun, t0, y0, t_bound, vectorized)
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self.y_old = None
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self.max_step = validate_max_step(max_step)
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self.rtol, self.atol = validate_tol(rtol, atol, self.n)
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self.f = self.fun(self.t, self.y)
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# Select initial step assuming the same order which is used to control
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# the error.
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if first_step is None:
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self.h_abs = select_initial_step(
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self.fun, self.t, self.y, self.f, self.direction,
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3, self.rtol, self.atol)
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else:
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self.h_abs = validate_first_step(first_step, t0, t_bound)
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self.h_abs_old = None
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self.error_norm_old = None
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self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5))
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self.sol = None
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self.jac_factor = None
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self.jac, self.J = self._validate_jac(jac, jac_sparsity)
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if issparse(self.J):
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def lu(A):
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self.nlu += 1
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return splu(A)
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def solve_lu(LU, b):
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return LU.solve(b)
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I = eye(self.n, format='csc')
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else:
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def lu(A):
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self.nlu += 1
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return lu_factor(A, overwrite_a=True)
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def solve_lu(LU, b):
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return lu_solve(LU, b, overwrite_b=True)
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I = np.identity(self.n)
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self.lu = lu
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self.solve_lu = solve_lu
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self.I = I
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self.current_jac = True
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self.LU_real = None
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self.LU_complex = None
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self.Z = None
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def _validate_jac(self, jac, sparsity):
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t0 = self.t
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y0 = self.y
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if jac is None:
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if sparsity is not None:
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if issparse(sparsity):
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sparsity = csc_matrix(sparsity)
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groups = group_columns(sparsity)
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sparsity = (sparsity, groups)
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def jac_wrapped(t, y, f):
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self.njev += 1
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J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,
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self.atol, self.jac_factor,
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sparsity)
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||
|
|
return J
|
||
|
|
J = jac_wrapped(t0, y0, self.f)
|
||
|
|
elif callable(jac):
|
||
|
|
J = jac(t0, y0)
|
||
|
|
self.njev = 1
|
||
|
|
if issparse(J):
|
||
|
|
J = csc_matrix(J)
|
||
|
|
|
||
|
|
def jac_wrapped(t, y, _=None):
|
||
|
|
self.njev += 1
|
||
|
|
return csc_matrix(jac(t, y), dtype=float)
|
||
|
|
|
||
|
|
else:
|
||
|
|
J = np.asarray(J, dtype=float)
|
||
|
|
|
||
|
|
def jac_wrapped(t, y, _=None):
|
||
|
|
self.njev += 1
|
||
|
|
return np.asarray(jac(t, y), dtype=float)
|
||
|
|
|
||
|
|
if J.shape != (self.n, self.n):
|
||
|
|
raise ValueError("`jac` is expected to have shape {}, but "
|
||
|
|
"actually has {}."
|
||
|
|
.format((self.n, self.n), J.shape))
|
||
|
|
else:
|
||
|
|
if issparse(jac):
|
||
|
|
J = csc_matrix(jac)
|
||
|
|
else:
|
||
|
|
J = np.asarray(jac, dtype=float)
|
||
|
|
|
||
|
|
if J.shape != (self.n, self.n):
|
||
|
|
raise ValueError("`jac` is expected to have shape {}, but "
|
||
|
|
"actually has {}."
|
||
|
|
.format((self.n, self.n), J.shape))
|
||
|
|
jac_wrapped = None
|
||
|
|
|
||
|
|
return jac_wrapped, J
|
||
|
|
|
||
|
|
def _step_impl(self):
|
||
|
|
t = self.t
|
||
|
|
y = self.y
|
||
|
|
f = self.f
|
||
|
|
|
||
|
|
max_step = self.max_step
|
||
|
|
atol = self.atol
|
||
|
|
rtol = self.rtol
|
||
|
|
|
||
|
|
min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)
|
||
|
|
if self.h_abs > max_step:
|
||
|
|
h_abs = max_step
|
||
|
|
h_abs_old = None
|
||
|
|
error_norm_old = None
|
||
|
|
elif self.h_abs < min_step:
|
||
|
|
h_abs = min_step
|
||
|
|
h_abs_old = None
|
||
|
|
error_norm_old = None
|
||
|
|
else:
|
||
|
|
h_abs = self.h_abs
|
||
|
|
h_abs_old = self.h_abs_old
|
||
|
|
error_norm_old = self.error_norm_old
|
||
|
|
|
||
|
|
J = self.J
|
||
|
|
LU_real = self.LU_real
|
||
|
|
LU_complex = self.LU_complex
|
||
|
|
|
||
|
|
current_jac = self.current_jac
|
||
|
|
jac = self.jac
|
||
|
|
|
||
|
|
rejected = False
|
||
|
|
step_accepted = False
|
||
|
|
message = None
|
||
|
|
while not step_accepted:
|
||
|
|
if h_abs < min_step:
|
||
|
|
return False, self.TOO_SMALL_STEP
|
||
|
|
|
||
|
|
h = h_abs * self.direction
|
||
|
|
t_new = t + h
|
||
|
|
|
||
|
|
if self.direction * (t_new - self.t_bound) > 0:
|
||
|
|
t_new = self.t_bound
|
||
|
|
|
||
|
|
h = t_new - t
|
||
|
|
h_abs = np.abs(h)
|
||
|
|
|
||
|
|
if self.sol is None:
|
||
|
|
Z0 = np.zeros((3, y.shape[0]))
|
||
|
|
else:
|
||
|
|
Z0 = self.sol(t + h * C).T - y
|
||
|
|
|
||
|
|
scale = atol + np.abs(y) * rtol
|
||
|
|
|
||
|
|
converged = False
|
||
|
|
while not converged:
|
||
|
|
if LU_real is None or LU_complex is None:
|
||
|
|
LU_real = self.lu(MU_REAL / h * self.I - J)
|
||
|
|
LU_complex = self.lu(MU_COMPLEX / h * self.I - J)
|
||
|
|
|
||
|
|
converged, n_iter, Z, rate = solve_collocation_system(
|
||
|
|
self.fun, t, y, h, Z0, scale, self.newton_tol,
|
||
|
|
LU_real, LU_complex, self.solve_lu)
|
||
|
|
|
||
|
|
if not converged:
|
||
|
|
if current_jac:
|
||
|
|
break
|
||
|
|
|
||
|
|
J = self.jac(t, y, f)
|
||
|
|
current_jac = True
|
||
|
|
LU_real = None
|
||
|
|
LU_complex = None
|
||
|
|
|
||
|
|
if not converged:
|
||
|
|
h_abs *= 0.5
|
||
|
|
LU_real = None
|
||
|
|
LU_complex = None
|
||
|
|
continue
|
||
|
|
|
||
|
|
y_new = y + Z[-1]
|
||
|
|
ZE = Z.T.dot(E) / h
|
||
|
|
error = self.solve_lu(LU_real, f + ZE)
|
||
|
|
scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol
|
||
|
|
error_norm = norm(error / scale)
|
||
|
|
safety = 0.9 * (2 * NEWTON_MAXITER + 1) / (2 * NEWTON_MAXITER
|
||
|
|
+ n_iter)
|
||
|
|
|
||
|
|
if rejected and error_norm > 1:
|
||
|
|
error = self.solve_lu(LU_real, self.fun(t, y + error) + ZE)
|
||
|
|
error_norm = norm(error / scale)
|
||
|
|
|
||
|
|
if error_norm > 1:
|
||
|
|
factor = predict_factor(h_abs, h_abs_old,
|
||
|
|
error_norm, error_norm_old)
|
||
|
|
h_abs *= max(MIN_FACTOR, safety * factor)
|
||
|
|
|
||
|
|
LU_real = None
|
||
|
|
LU_complex = None
|
||
|
|
rejected = True
|
||
|
|
else:
|
||
|
|
step_accepted = True
|
||
|
|
|
||
|
|
recompute_jac = jac is not None and n_iter > 2 and rate > 1e-3
|
||
|
|
|
||
|
|
factor = predict_factor(h_abs, h_abs_old, error_norm, error_norm_old)
|
||
|
|
factor = min(MAX_FACTOR, safety * factor)
|
||
|
|
|
||
|
|
if not recompute_jac and factor < 1.2:
|
||
|
|
factor = 1
|
||
|
|
else:
|
||
|
|
LU_real = None
|
||
|
|
LU_complex = None
|
||
|
|
|
||
|
|
f_new = self.fun(t_new, y_new)
|
||
|
|
if recompute_jac:
|
||
|
|
J = jac(t_new, y_new, f_new)
|
||
|
|
current_jac = True
|
||
|
|
elif jac is not None:
|
||
|
|
current_jac = False
|
||
|
|
|
||
|
|
self.h_abs_old = self.h_abs
|
||
|
|
self.error_norm_old = error_norm
|
||
|
|
|
||
|
|
self.h_abs = h_abs * factor
|
||
|
|
|
||
|
|
self.y_old = y
|
||
|
|
|
||
|
|
self.t = t_new
|
||
|
|
self.y = y_new
|
||
|
|
self.f = f_new
|
||
|
|
|
||
|
|
self.Z = Z
|
||
|
|
|
||
|
|
self.LU_real = LU_real
|
||
|
|
self.LU_complex = LU_complex
|
||
|
|
self.current_jac = current_jac
|
||
|
|
self.J = J
|
||
|
|
|
||
|
|
self.t_old = t
|
||
|
|
self.sol = self._compute_dense_output()
|
||
|
|
|
||
|
|
return step_accepted, message
|
||
|
|
|
||
|
|
def _compute_dense_output(self):
|
||
|
|
Q = np.dot(self.Z.T, P)
|
||
|
|
return RadauDenseOutput(self.t_old, self.t, self.y_old, Q)
|
||
|
|
|
||
|
|
def _dense_output_impl(self):
|
||
|
|
return self.sol
|
||
|
|
|
||
|
|
|
||
|
|
class RadauDenseOutput(DenseOutput):
|
||
|
|
def __init__(self, t_old, t, y_old, Q):
|
||
|
|
super(RadauDenseOutput, self).__init__(t_old, t)
|
||
|
|
self.h = t - t_old
|
||
|
|
self.Q = Q
|
||
|
|
self.order = Q.shape[1] - 1
|
||
|
|
self.y_old = y_old
|
||
|
|
|
||
|
|
def _call_impl(self, t):
|
||
|
|
x = (t - self.t_old) / self.h
|
||
|
|
if t.ndim == 0:
|
||
|
|
p = np.tile(x, self.order + 1)
|
||
|
|
p = np.cumprod(p)
|
||
|
|
else:
|
||
|
|
p = np.tile(x, (self.order + 1, 1))
|
||
|
|
p = np.cumprod(p, axis=0)
|
||
|
|
# Here we don't multiply by h, not a mistake.
|
||
|
|
y = np.dot(self.Q, p)
|
||
|
|
if y.ndim == 2:
|
||
|
|
y += self.y_old[:, None]
|
||
|
|
else:
|
||
|
|
y += self.y_old
|
||
|
|
|
||
|
|
return y
|