EVOLUTION-MANAGER

Edit File: radau.py

from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.linalg import lu_factor, lu_solve
from scipy.sparse import csc_matrix, issparse, eye
from scipy.sparse.linalg import splu
from scipy.optimize._numdiff import group_columns
from .common import (validate_max_step, validate_tol, select_initial_step,
                     norm, num_jac, EPS, warn_extraneous,
                     validate_first_step)
from .base import OdeSolver, DenseOutput

S6 = 6 ** 0.5

# Butcher tableau. A is not used directly, see below.
C = np.array([(4 - S6) / 10, (4 + S6) / 10, 1])
E = np.array([-13 - 7 * S6, -13 + 7 * S6, -1]) / 3

# Eigendecomposition of A is done: A = T L T**-1. There is 1 real eigenvalue
# and a complex conjugate pair. They are written below.
MU_REAL = 3 + 3 ** (2 / 3) - 3 ** (1 / 3)
MU_COMPLEX = (3 + 0.5 * (3 ** (1 / 3) - 3 ** (2 / 3))
              - 0.5j * (3 ** (5 / 6) + 3 ** (7 / 6)))

# These are transformation matrices.
T = np.array([
    [0.09443876248897524, -0.14125529502095421, 0.03002919410514742],
    [0.25021312296533332, 0.20412935229379994, -0.38294211275726192],
    [1, 1, 0]])
TI = np.array([
    [4.17871859155190428, 0.32768282076106237, 0.52337644549944951],
    [-4.17871859155190428, -0.32768282076106237, 0.47662355450055044],
    [0.50287263494578682, -2.57192694985560522, 0.59603920482822492]])
# These linear combinations are used in the algorithm.
TI_REAL = TI[0]
TI_COMPLEX = TI[1] + 1j * TI[2]

# Interpolator coefficients.
P = np.array([
    [13/3 + 7*S6/3, -23/3 - 22*S6/3, 10/3 + 5 * S6],
    [13/3 - 7*S6/3, -23/3 + 22*S6/3, 10/3 - 5 * S6],
    [1/3, -8/3, 10/3]])

NEWTON_MAXITER = 6  # Maximum number of Newton iterations.
MIN_FACTOR = 0.2  # Minimum allowed decrease in a step size.
MAX_FACTOR = 10  # Maximum allowed increase in a step size.

def solve_collocation_system(fun, t, y, h, Z0, scale, tol,
                             LU_real, LU_complex, solve_lu):
    """Solve the collocation system.

Parameters
    ----------
    fun : callable
        Right-hand side of the system.
    t : float
        Current time.
    y : ndarray, shape (n,)
        Current state.
    h : float
        Step to try.
    Z0 : ndarray, shape (3, n)
        Initial guess for the solution. It determines new values of `y` at
        ``t + h * C`` as ``y + Z0``, where ``C`` is the Radau method constants.
    scale : float
        Problem tolerance scale, i.e. ``rtol * abs(y) + atol``.
    tol : float
        Tolerance to which solve the system. This value is compared with
        the normalized by `scale` error.
    LU_real, LU_complex
        LU decompositions of the system Jacobians.
    solve_lu : callable
        Callable which solves a linear system given a LU decomposition. The
        signature is ``solve_lu(LU, b)``.

Returns
    -------
    converged : bool
        Whether iterations converged.
    n_iter : int
        Number of completed iterations.
    Z : ndarray, shape (3, n)
        Found solution.
    rate : float
        The rate of convergence.
    """
    n = y.shape[0]
    M_real = MU_REAL / h
    M_complex = MU_COMPLEX / h

W = TI.dot(Z0)
    Z = Z0

F = np.empty((3, n))
    ch = h * C

dW_norm_old = None
    dW = np.empty_like(W)
    converged = False
    rate = None
    for k in range(NEWTON_MAXITER):
        for i in range(3):
            F[i] = fun(t + ch[i], y + Z[i])

if not np.all(np.isfinite(F)):
            break

f_real = F.T.dot(TI_REAL) - M_real * W[0]
        f_complex = F.T.dot(TI_COMPLEX) - M_complex * (W[1] + 1j * W[2])

dW_real = solve_lu(LU_real, f_real)
        dW_complex = solve_lu(LU_complex, f_complex)

dW[0] = dW_real
        dW[1] = dW_complex.real
        dW[2] = dW_complex.imag

dW_norm = norm(dW / scale)
        if dW_norm_old is not None:
            rate = dW_norm / dW_norm_old

if (rate is not None and (rate >= 1 or
                rate ** (NEWTON_MAXITER - k) / (1 - rate) * dW_norm > tol)):
            break

W += dW
        Z = T.dot(W)

if (dW_norm == 0 or
                rate is not None and rate / (1 - rate) * dW_norm < tol):
            converged = True
            break

dW_norm_old = dW_norm

return converged, k + 1, Z, rate

def predict_factor(h_abs, h_abs_old, error_norm, error_norm_old):
    """Predict by which factor to increase/decrease the step size.

The algorithm is described in [1]_.

Parameters
    ----------
    h_abs, h_abs_old : float
        Current and previous values of the step size, `h_abs_old` can be None
        (see Notes).
    error_norm, error_norm_old : float
        Current and previous values of the error norm, `error_norm_old` can
        be None (see Notes).

Returns
    -------
    factor : float
        Predicted factor.

Notes
    -----
    If `h_abs_old` and `error_norm_old` are both not None then a two-step
    algorithm is used, otherwise a one-step algorithm is used.

References
    ----------
    .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
           Equations II: Stiff and Differential-Algebraic Problems", Sec. IV.8.
    """
    if error_norm_old is None or h_abs_old is None or error_norm == 0:
        multiplier = 1
    else:
        multiplier = h_abs / h_abs_old * (error_norm_old / error_norm) ** 0.25

with np.errstate(divide='ignore'):
        factor = min(1, multiplier) * error_norm ** -0.25

return factor

class Radau(OdeSolver):
    """Implicit Runge-Kutta method of Radau IIA family of order 5.

The implementation follows [1]_. The error is controlled with a
    third-order accurate embedded formula. A cubic polynomial which satisfies
    the collocation conditions is used for the dense output.

Parameters
    ----------
    fun : callable
        Right-hand side of the system. The calling signature is ``fun(t, y)``.
        Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
        It can either have shape (n,); then ``fun`` must return array_like with
        shape (n,). Alternatively it can have shape (n, k); then ``fun``
        must return an array_like with shape (n, k), i.e. each column
        corresponds to a single column in ``y``. The choice between the two
        options is determined by `vectorized` argument (see below). The
        vectorized implementation allows a faster approximation of the Jacobian
        by finite differences (required for this solver).
    t0 : float
        Initial time.
    y0 : array_like, shape (n,)
        Initial state.
    t_bound : float
        Boundary time - the integration won't continue beyond it. It also
        determines the direction of the integration.
    first_step : float or None, optional
        Initial step size. Default is ``None`` which means that the algorithm
        should choose.
    max_step : float, optional
        Maximum allowed step size. Default is np.inf, i.e. the step size is not
        bounded and determined solely by the solver.
    rtol, atol : float and array_like, optional
        Relative and absolute tolerances. The solver keeps the local error
        estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
        relative accuracy (number of correct digits). But if a component of `y`
        is approximately below `atol`, the error only needs to fall within
        the same `atol` threshold, and the number of correct digits is not
        guaranteed. If components of y have different scales, it might be
        beneficial to set different `atol` values for different components by
        passing array_like with shape (n,) for `atol`. Default values are
        1e-3 for `rtol` and 1e-6 for `atol`.
    jac : {None, array_like, sparse_matrix, callable}, optional
        Jacobian matrix of the right-hand side of the system with respect to
        y, required by this method. The Jacobian matrix has shape (n, n) and
        its element (i, j) is equal to ``d f_i / d y_j``.
        There are three ways to define the Jacobian:

* If array_like or sparse_matrix, the Jacobian is assumed to
              be constant.
            * If callable, the Jacobian is assumed to depend on both
              t and y; it will be called as ``jac(t, y)`` as necessary.
              For the 'Radau' and 'BDF' methods, the return value might be a
              sparse matrix.
            * If None (default), the Jacobian will be approximated by
              finite differences.

It is generally recommended to provide the Jacobian rather than
        relying on a finite-difference approximation.
    jac_sparsity : {None, array_like, sparse matrix}, optional
        Defines a sparsity structure of the Jacobian matrix for a
        finite-difference approximation. Its shape must be (n, n). This argument
        is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
        elements in *each* row, providing the sparsity structure will greatly
        speed up the computations [2]_. A zero entry means that a corresponding
        element in the Jacobian is always zero. If None (default), the Jacobian
        is assumed to be dense.
    vectorized : bool, optional
        Whether `fun` is implemented in a vectorized fashion. Default is False.

Attributes
    ----------
    n : int
        Number of equations.
    status : string
        Current status of the solver: 'running', 'finished' or 'failed'.
    t_bound : float
        Boundary time.
    direction : float
        Integration direction: +1 or -1.
    t : float
        Current time.
    y : ndarray
        Current state.
    t_old : float
        Previous time. None if no steps were made yet.
    step_size : float
        Size of the last successful step. None if no steps were made yet.
    nfev : int
        Number of evaluations of the right-hand side.
    njev : int
        Number of evaluations of the Jacobian.
    nlu : int
        Number of LU decompositions.

References
    ----------
    .. [1] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
           Stiff and Differential-Algebraic Problems", Sec. IV.8.
    .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
           sparse Jacobian matrices", Journal of the Institute of Mathematics
           and its Applications, 13, pp. 117-120, 1974.
    """
    def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
                 rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None,
                 vectorized=False, first_step=None, **extraneous):
        warn_extraneous(extraneous)
        super(Radau, self).__init__(fun, t0, y0, t_bound, vectorized)
        self.y_old = None
        self.max_step = validate_max_step(max_step)
        self.rtol, self.atol = validate_tol(rtol, atol, self.n)
        self.f = self.fun(self.t, self.y)
        # Select initial step assuming the same order which is used to control
        # the error.
        if first_step is None:
            self.h_abs = select_initial_step(
                self.fun, self.t, self.y, self.f, self.direction,
                3, self.rtol, self.atol)
        else:
            self.h_abs = validate_first_step(first_step, t0, t_bound)
        self.h_abs_old = None
        self.error_norm_old = None

self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5))
        self.sol = None

self.jac_factor = None
        self.jac, self.J = self._validate_jac(jac, jac_sparsity)
        if issparse(self.J):
            def lu(A):
                self.nlu += 1
                return splu(A)

def solve_lu(LU, b):
                return LU.solve(b)

I = eye(self.n, format='csc')
        else:
            def lu(A):
                self.nlu += 1
                return lu_factor(A, overwrite_a=True)

def solve_lu(LU, b):
                return lu_solve(LU, b, overwrite_b=True)

I = np.identity(self.n)

self.lu = lu
        self.solve_lu = solve_lu
        self.I = I

self.current_jac = True
        self.LU_real = None
        self.LU_complex = None
        self.Z = None

def _validate_jac(self, jac, sparsity):
        t0 = self.t
        y0 = self.y

if jac is None:
            if sparsity is not None:
                if issparse(sparsity):
                    sparsity = csc_matrix(sparsity)
                groups = group_columns(sparsity)
                sparsity = (sparsity, groups)

def jac_wrapped(t, y, f):
                self.njev += 1
                J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,
                                             self.atol, self.jac_factor,
                                             sparsity)
                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