feat: implement FLMM starting weights

Author: alexfikl

docs/references.rst

@@ -22,6 +22,11 @@ References
     Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp. 743-773, 2005,
     `DOI <https://doi.org/10.1016/j.cma.2004.06.006>`__.
 
+.. [Diethelm2006] K. Diethelm, J. M. Ford, N. J. Ford, M. Weilbeer,
+    *Pitfalls in Fast Numerical Solvers for Fractional Differential Equations*,
+    Journal of Computational and Applied Mathematics, Vol. 186, pp. 482--503, 2006,
+    `DOI <https://doi.org/10.1016/j.cam.2005.03.023>`__.
+
 .. [Shen2011] J. Shen, T. Tang, L.-L. Wang,
     *Spectral Methods - Algorithms, Analysis and Applications*,
     Springer, 2011.

src/pycaputo/generating_functions.py

@@ -9,22 +9,160 @@
 
 from pycaputo.utils import Array
 
-# {{{ Lubich1986 weights
+# {{{ FLMM: Starting Weights
 
 
-def lubich_bdf_starting_weights_count(order: int, alpha: float) -> int:
-    """An estimate for the number of starting weights from [Lubich1986]_.
+def lmm_starting_weights(
+    w: Array, sigma: Array, alpha: float, *, atol: float | None = None
+) -> Iterator[tuple[int, Array]]:
+    r"""Constructs starting weights for a given set of weights *w* of a
+    fractional-order linear multistep method (FLMM).
 
-    :arg order: order of the BDF method.
-    :arg alpha: order of the fractional derivative.
+    The starting weights are introduced to handle a lack of smoothness of the
+    function :math:`f(x)` being integrated. They are constructed in such a way
+    that they are exact for a series of monomials, which results in an accurate
+    quadrature for functions of the form :math:`f(x) = f_i x^{\sigma_i} +
+    x^{\beta} g(x)`, where :math:`g` is smooth and :math:`\beta < p`
+    (see Remark 3.6 in [Li2020]_). This ensures that the Taylor expansion near
+    the origin is sufficiently accurate.
+
+    Therefore, they are obtained by solving
+
+    .. math::
+
+        \sum_{k = 0}^s w_{mk} k^{\sigma_q} =
+        \frac{\Gamma(\sigma_q +1)}{\Gamma(\sigma_q + \alpha + 1)}
+        m^{\sigma_q + \alpha} -
+        \sum_{k = 0}^s w_{n - k} k^{\sigma_q}
+
+    for a set of :math:`\sigma_q` powers. In the simplest case, we can just set
+    :math:`\sigma \in \{0, 1, \dots, p - 1\}` and obtain integer powers. Other
+    values can be chosen depending on the behaviour of :math:`f(x)` near the origin.
+
+    Note that the starting weights are only computed for :math:`m \ge s`.
+    The initial :math:`s` steps are expected to be computed in some other
+    fashion.
+
+    :arg w: convolution weights of an FLMM.
+    :arg sigma: an array of monomial powers for which the starting weights are exact.
+    :arg alpha: order of the fractional derivative to approximate.
+    :arg atol: absolute tolerance used for the GMRES solver. If *None*, an
+        exact LU-based solver is used instead (see Section 3.2 in [Diethelm2006]_
+        for a discussion of these methods).
+
+    :returns: the index and starting weights for every point :math:`m \ge s`.
+    """
+
+    from scipy.linalg import lu_factor, lu_solve
+    from scipy.sparse.linalg import gmres
+    from scipy.special import gamma
+
+    s = sigma.size
+    j = np.arange(1, s + 1).reshape(-1, 1)
+
+    A = j**sigma
+    assert A.shape == (s, s)
+    assert np.all(np.isfinite(A))
+
+    if atol is None:
+        lu_p = lu_factor(A)
+
+    for k in range(s, w.size):
+        b = (
+            gamma(sigma + 1) / gamma(sigma + alpha + 1) * k ** (sigma + alpha)
+            - A @ w[k - s : k][::-1]
+        )
+
+        if atol is None:
+            omega = lu_solve(lu_p, b)
+        else:
+            omega, _ = gmres(A, b, atol=atol)
+
+        assert np.all(np.isfinite(omega))
+        yield k, omega
+
+
+def diethelm_starting_powers(order: int, alpha: float) -> Array:
+    r"""Generate monomial powers for the starting weights from [Diethelm2006]_.
+
+    The proposed starting weights are given in Section 3.1 from [Diethelm2006]_ as
+
+    .. math::
+
+        \sigma \in \{i + \alpha j \mid i, j \ge 0, i + \alpha j \le p - 1 \},
+
+    where :math:`p` is the desired order. For certain values of :math:`\alpha`,
+    these monomials can repeat, e.g. for :math:`\alpha = 0.1` we get the same
+    value for :math:`(i, j) = (1, 0)` and :math:`(i, j) = (0, 10)`. This function
+    returns only unique powers.
+
+    :arg order: order of the LMM method.
+    :arg alpha: order of the fractional operator.
+    """
+    from itertools import product
+
+    result = np.array([
+        gamma
+        for i, j in product(range(order), repeat=2)
+        if (gamma := i + abs(alpha) * j) <= order - 1
+    ])
+
+    return np.array(np.unique(result))
+
+
+def lubich_starting_powers(order: int, alpha: float, *, beta: float = 1.0) -> Array:
+    r"""Generate monomial powers for the starting weights from [Lubich1986]_.
+
+    The proposed starting weights are given in Section 4.2 of [Lubich1986]_ as
+
+    .. math::
+
+        \sigma \in \{i + \beta - 1 \mid q \le p - 1\},
+
+    where :math:`\beta` is chosen such that :math:`q + \beta - 1 \le p - 1`
+    and :math:`q + \alpha + \beta - 1 < p`, according to Theorem 2.4 from
+    [Lubich1986]_. In general multiple such :math:`\beta` exist and choosing
+    more can prove beneficial.
+
+    :arg order: order of the LMM method.
+    :arg alpha: order of the fractional operator.
     """
     from math import floor
 
-    return floor(1 / abs(alpha)) - 1
+    # NOTE: trying to satisfy
+    # 0 <= q <= p - \beta and 0 <= q < p + 1 - alpha - beta
+    qmax = floor(max(0, min(order - beta, order - alpha - beta + 1))) + 1
+
+    return np.array([q + beta - 1 for q in range(qmax)])
+
+
+def garrappa_starting_powers(order: int, alpha: float) -> Array:
+    r"""Generate monomial powers for the starting weights from
+    `FLMM2 <https://www.mathworks.com/matlabcentral/fileexchange/47081-flmm2>`__.
+
+    :arg order: order of the LMM method.
+    :arg alpha: order of the fractional operator.
+    """
+    from math import ceil, floor
+
+    if order == 2 and 0 < alpha < 1:
+        # NOTE: this is the estimate from the FLMM2 MATLAB code
+        k = floor(1 / abs(alpha))
+    else:
+        # NOTE: this should be vaguely the cardinality of `diethelm_bdf_starting_powers`
+        k = ceil(order * (order - 1 + 2 * abs(alpha)) / (2 * abs(alpha)))
+
+    return np.arange(k) * abs(alpha)
+
+
+# }}}
+
+
+# {{{ backward differentiation formulas
 
 
 def lubich_bdf_weights(alpha: float, order: int, n: int) -> Array:
-    r"""This function generates the weights for the p-BDF methods of [Lubich1986]_.
+    r"""This function generates the weights for the BDF methods of [Lubich1986]_.
 
     Table 1 from [Lubich1986]_ gives the generating functions. The weights
     constructed here are the coefficients of the Taylor expansion of the
@@ -37,7 +175,7 @@ def lubich_bdf_weights(alpha: float, order: int, n: int) -> Array:
             \left(\sum_{k = 1}^p \frac{1}{k} (1 - \zeta)^k\right)^\alpha.
 
     The corresponding weights also have an explicit formulation based on the
-    recursion formula
+    recursion formula (see Theorem 4 in [Garrappa2015b]_)
 
     .. math::
 
@@ -49,9 +187,11 @@ def lubich_bdf_weights(alpha: float, order: int, n: int) -> Array:
     we restrict here to a maximum order of 6, as the classical BDF methods of
     larger orders are not stable.
 
-    :arg alpha: power of the generating function.
+    :arg alpha: power of the generating function, where a positive value is
+        used to approximate the integral and a negative value is used to
+        approximate the derivative.
     :arg order: order of the method, only :math:`1 \le p \le 6` is supported.
-    :arg n: number of weights.
+    :arg n: number of truncated terms in the power series of :math:`w^\alpha_k(\zeta)`.
     """
     if order <= 0:
         raise ValueError(f"Negative orders are not supported: {order}")
@@ -84,93 +224,123 @@ def lubich_bdf_weights(alpha: float, order: int, n: int) -> Array:
     else:
         raise ValueError(f"Unsupported order '{order}'")
 
-    w = np.empty(n)
-    indices = np.arange(n)
+    import scipy.special as ss
+
+    w = np.empty(n + 1)
+    indices = np.arange(n + 1)
 
-    w[0] = c[0] ** alpha
-    for k in range(1, n):
-        min_j = max(0, k - c.size + 1)
-        max_j = k
+    if order == 1:
+        return np.array(ss.poch(indices, alpha - 1) / ss.gamma(alpha))
+    else:
+        w[0] = c[0] ** -alpha
+        for k in range(1, n + 1):
+            min_j = max(0, k - c.size + 1)
+            max_j = k
 
-        j = indices[min_j:max_j]
-        omega = (alpha * (k - j) - j) * c[1 : k + 1][::-1]
+            j = indices[min_j:max_j]
+            omega = -(alpha * (k - j) + j) * c[1 : k + 1][::-1]
 
-        w[k] = np.sum(omega * w[min_j:max_j]) / (k * c[0])
+            w[k] = np.sum(omega * w[min_j:max_j]) / (k * c[0])
 
     return w
 
 
-def lubich_bdf_starting_weights(
-    w: Array, s: int, alpha: float, *, beta: float = 1.0, atol: float | None = None
-) -> Iterator[Array]:
-    r"""Constructs starting weights for a given set of weights *w* from [Lubich1986]_.
+# }}}
 
-    The starting weights are introduced to handle a lack of smoothness of the
-    functions being integrated. They are constructed in such a way that they
-    are exact for a series of monomials, which results in an accurate
-    quadrature for functions of the form :math:`f(x) = x^{\beta - 1} g(x)`,
-    where :math:`g` is smooth.
 
-    Therefore, they are obtained by solving
+# {{{ trapezoidal formulas
 
-    .. math::
 
-        \sum_{k = 0}^s w_{mk} k^{q + \beta - 1} =
-        \frac{\Gamma(q + \beta)}{\Gamma(q + \beta + \alpha)}
-        m^{q + \beta + \alpha - 1} -
-        \sum_{k = 0}^s w_{n - k} j^{q + \beta - 1}
+def trapezoidal_weights(alpha: float, n: int) -> Array:
+    r"""This function constructions the weights for the trapezoidal method
+    from Section 4.1 in [Garrappa2015b]_.
 
-    where :math:`q \in \{0, 1, \dots, s - 1\}` and :math:`\beta \ne \{0, -1, \dots\}`.
-    In the simplest case, we can take :math:`\beta = 1` and obtain integer
-    powers. Other values can be chosen depending on the behaviour of :math:`f(x)`
-    near the origin.
+    The trapezoidal method has the generating function
 
-    Note that the starting weights are only computed for :math:`m \ge s`.
-    The initial :math:`s` steps are expected to be computed in some other
-    fashion.
+    .. math::
 
-    :arg w: convolution weights defined by [Lubich1986]_.
-    :arg s: number of starting weights.
-    :arg alpha: order of the fractional derivative to approximate.
-    :arg beta: order of the singularity at the origin.
-    :arg atol: absolute tolerance used for the GMRES solver. If *None*, an
-        exact LU-based solver is used instead.
+        w^\alpha(\zeta) \triangleq \left(
+            \frac{1}{2} \frac{1 + \zeta}{1 - \zeta}
+        \right)^\alpha,
 
-    :returns: the starting weights for every point :math:`x_m` starting with
-        :math:`m \ge s`.
+    which expands into an infinite series. This function truncates that series
+    to the first *n* terms, which give the desired weights.
+
+    :arg alpha: power of the generating function, where a positive value is
+        used to approximate the integral and a negative value is used to
+        approximate the derivative.
+    :arg n: number of truncated terms in the power series of :math:`w^\alpha_k(\zeta)`.
     """
 
-    if s <= 0:
-        raise ValueError(f"Negative s is not allowed: {s}")
+    try:
+        from scipy.fft import fft, ifft
+    except ImportError:
+        from numpy.fft import fft, ifft
 
-    if beta.is_integer() and beta <= 0:
-        raise ValueError(f"Values of beta in 0, -1, ... are not supported: {beta}")
+    omega_1 = np.empty(n + 1)
+    omega_2 = np.empty(n + 1)
 
-    from scipy.linalg import lu_factor, lu_solve
-    from scipy.sparse.linalg import gmres
-    from scipy.special import gamma
+    omega_1[0] = omega_2[0] = 1.0
+    for k in range(1, n + 1):
+        omega_1[k] = ((alpha + 1) / k - 1) * omega_1[k - 1]
+        omega_2[k] = (1 + (alpha - 1) / k) * omega_2[k - 1]
 
-    q = np.arange(s) + beta - 1
-    j = np.arange(1, s + 1).reshape(-1, 1)
+    # FIXME: this looks like it can be done faster by using rfft?
+    omega_1_hat = fft(omega_1, 2 * omega_1.size)
+    omega_2_hat = fft(omega_2, 2 * omega_2.size)
+    omega = ifft(omega_1_hat * omega_2_hat)[: omega_1.size].real
 
-    A = j**q
-    assert A.shape == (s, s)
+    return np.array(omega / 2**alpha)
 
-    if atol is None:
-        lu, p = lu_factor(A)
 
-    for k in range(s, w.size):
-        b = (
-            gamma(q + 1) / gamma(q + alpha + 1) * k ** (q + alpha)
-            - A @ w[k - s : k][::-1]
-        )
+# }}}
 
-        if atol is None:
-            omega = lu_solve((lu, p), b)
-        else:
-            omega, _ = gmres(A, b, atol=atol)
 
-        yield omega
+# {{{ Newton-Gregory formulas
+
+
+def newton_gregory_weights(alpha: float, k: int, n: int) -> Array:
+    r"""This function constructions the weights for the Newton-Gregory method
+    from Section 4.2 in [Garrappa2015b]_.
+
+    The Newton-Gregory family of methods have the generating functions
+
+    .. math::
+
+        w^\alpha_k(\zeta) \triangleq
+            \frac{G^\alpha_k(\zeta)}{(1 - \zeta)^\alpha},
+
+    where :math:`G^\alpha_k(\zeta)` is the :math:`k`-term truncated power
+    series expansion of
+
+    .. math::
+
+        G^\alpha(\zeta) = \left(\frac{1 - \zeta}{\log \zeta}\right)^\alpha.
+
+    Note that this is not a classic expansion in the sense of [Lubich1986]_
+    because the :math:`G^\alpha_k(\zeta)` function is first raised to the
+    :math:`\alpha` power and then truncated to :math:`k` terms. In a standard
+    scheme, the generating function :math:`w^\alpha_k(\zeta)` is the one that
+    is truncated.
+
+    :arg alpha: power of the generating function, where a positive value is
+        used to approximate the integral and a negative value is used to
+        approximate the derivative.
+    :arg k: number of truncated terms in the power series of :math:`G^\alpha_k(\zeta)`.
+    :arg n: number of truncated terms in the power series of :math:`w^\alpha_k(\zeta)`.
+    """
+    if k != 1:
+        raise ValueError(f"Only 2-term truncations are supported: '{k + 1}'")
+
+    omega = np.empty(n + 1)
+    omega[0] = 1.0
+    for i in range(1, n + 1):
+        omega[i] = (1 + (alpha - 1) / i) * omega[i - 1]
+
+    omega[1:] = (1 - alpha / 2) * omega[1:] + alpha / 2 * omega[:-1]
+    omega[0] = 1 - alpha / 2.0
+
+    return omega
 
 
 # }}}