forked from 170010011/fr
497 lines
18 KiB
Python
497 lines
18 KiB
Python
|
"""
|
||
|
|
Functions
|
||
|
|
---------
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
fmin_l_bfgs_b
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
## License for the Python wrapper
|
||
|
|
## ==============================
|
||
|
|
|
||
|
|
## Copyright (c) 2004 David M. Cooke <cookedm@physics.mcmaster.ca>
|
||
|
|
|
||
|
|
## Permission is hereby granted, free of charge, to any person obtaining a
|
||
|
|
## copy of this software and associated documentation files (the "Software"),
|
||
|
|
## to deal in the Software without restriction, including without limitation
|
||
|
|
## the rights to use, copy, modify, merge, publish, distribute, sublicense,
|
||
|
|
## and/or sell copies of the Software, and to permit persons to whom the
|
||
|
|
## Software is furnished to do so, subject to the following conditions:
|
||
|
|
|
||
|
|
## The above copyright notice and this permission notice shall be included in
|
||
|
|
## all copies or substantial portions of the Software.
|
||
|
|
|
||
|
|
## THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
|
||
|
|
## IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
|
||
|
|
## FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
|
||
|
|
## AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
|
||
|
|
## LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
||
|
|
## FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
|
||
|
|
## DEALINGS IN THE SOFTWARE.
|
||
|
|
|
||
|
|
## Modifications by Travis Oliphant and Enthought, Inc. for inclusion in SciPy
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
from numpy import array, asarray, float64, zeros
|
||
|
|
from . import _lbfgsb
|
||
|
|
from .optimize import (MemoizeJac, OptimizeResult,
|
||
|
|
_check_unknown_options, _prepare_scalar_function)
|
||
|
|
from ._constraints import old_bound_to_new
|
||
|
|
|
||
|
|
from scipy.sparse.linalg import LinearOperator
|
||
|
|
|
||
|
|
__all__ = ['fmin_l_bfgs_b', 'LbfgsInvHessProduct']
|
||
|
|
|
||
|
|
|
||
|
|
def fmin_l_bfgs_b(func, x0, fprime=None, args=(),
|
||
|
|
approx_grad=0,
|
||
|
|
bounds=None, m=10, factr=1e7, pgtol=1e-5,
|
||
|
|
epsilon=1e-8,
|
||
|
|
iprint=-1, maxfun=15000, maxiter=15000, disp=None,
|
||
|
|
callback=None, maxls=20):
|
||
|
|
"""
|
||
|
|
Minimize a function func using the L-BFGS-B algorithm.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable f(x,*args)
|
||
|
|
Function to minimize.
|
||
|
|
x0 : ndarray
|
||
|
|
Initial guess.
|
||
|
|
fprime : callable fprime(x,*args), optional
|
||
|
|
The gradient of `func`. If None, then `func` returns the function
|
||
|
|
value and the gradient (``f, g = func(x, *args)``), unless
|
||
|
|
`approx_grad` is True in which case `func` returns only ``f``.
|
||
|
|
args : sequence, optional
|
||
|
|
Arguments to pass to `func` and `fprime`.
|
||
|
|
approx_grad : bool, optional
|
||
|
|
Whether to approximate the gradient numerically (in which case
|
||
|
|
`func` returns only the function value).
|
||
|
|
bounds : list, optional
|
||
|
|
``(min, max)`` pairs for each element in ``x``, defining
|
||
|
|
the bounds on that parameter. Use None or +-inf for one of ``min`` or
|
||
|
|
``max`` when there is no bound in that direction.
|
||
|
|
m : int, optional
|
||
|
|
The maximum number of variable metric corrections
|
||
|
|
used to define the limited memory matrix. (The limited memory BFGS
|
||
|
|
method does not store the full hessian but uses this many terms in an
|
||
|
|
approximation to it.)
|
||
|
|
factr : float, optional
|
||
|
|
The iteration stops when
|
||
|
|
``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``,
|
||
|
|
where ``eps`` is the machine precision, which is automatically
|
||
|
|
generated by the code. Typical values for `factr` are: 1e12 for
|
||
|
|
low accuracy; 1e7 for moderate accuracy; 10.0 for extremely
|
||
|
|
high accuracy. See Notes for relationship to `ftol`, which is exposed
|
||
|
|
(instead of `factr`) by the `scipy.optimize.minimize` interface to
|
||
|
|
L-BFGS-B.
|
||
|
|
pgtol : float, optional
|
||
|
|
The iteration will stop when
|
||
|
|
``max{|proj g_i | i = 1, ..., n} <= pgtol``
|
||
|
|
where ``pg_i`` is the i-th component of the projected gradient.
|
||
|
|
epsilon : float, optional
|
||
|
|
Step size used when `approx_grad` is True, for numerically
|
||
|
|
calculating the gradient
|
||
|
|
iprint : int, optional
|
||
|
|
Controls the frequency of output. ``iprint < 0`` means no output;
|
||
|
|
``iprint = 0`` print only one line at the last iteration;
|
||
|
|
``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
|
||
|
|
``iprint = 99`` print details of every iteration except n-vectors;
|
||
|
|
``iprint = 100`` print also the changes of active set and final x;
|
||
|
|
``iprint > 100`` print details of every iteration including x and g.
|
||
|
|
disp : int, optional
|
||
|
|
If zero, then no output. If a positive number, then this over-rides
|
||
|
|
`iprint` (i.e., `iprint` gets the value of `disp`).
|
||
|
|
maxfun : int, optional
|
||
|
|
Maximum number of function evaluations.
|
||
|
|
maxiter : int, optional
|
||
|
|
Maximum number of iterations.
|
||
|
|
callback : callable, optional
|
||
|
|
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
|
||
|
|
current parameter vector.
|
||
|
|
maxls : int, optional
|
||
|
|
Maximum number of line search steps (per iteration). Default is 20.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x : array_like
|
||
|
|
Estimated position of the minimum.
|
||
|
|
f : float
|
||
|
|
Value of `func` at the minimum.
|
||
|
|
d : dict
|
||
|
|
Information dictionary.
|
||
|
|
|
||
|
|
* d['warnflag'] is
|
||
|
|
|
||
|
|
- 0 if converged,
|
||
|
|
- 1 if too many function evaluations or too many iterations,
|
||
|
|
- 2 if stopped for another reason, given in d['task']
|
||
|
|
|
||
|
|
* d['grad'] is the gradient at the minimum (should be 0 ish)
|
||
|
|
* d['funcalls'] is the number of function calls made.
|
||
|
|
* d['nit'] is the number of iterations.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
minimize: Interface to minimization algorithms for multivariate
|
||
|
|
functions. See the 'L-BFGS-B' `method` in particular. Note that the
|
||
|
|
`ftol` option is made available via that interface, while `factr` is
|
||
|
|
provided via this interface, where `factr` is the factor multiplying
|
||
|
|
the default machine floating-point precision to arrive at `ftol`:
|
||
|
|
``ftol = factr * numpy.finfo(float).eps``.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
License of L-BFGS-B (FORTRAN code):
|
||
|
|
|
||
|
|
The version included here (in fortran code) is 3.0
|
||
|
|
(released April 25, 2011). It was written by Ciyou Zhu, Richard Byrd,
|
||
|
|
and Jorge Nocedal <nocedal@ece.nwu.edu>. It carries the following
|
||
|
|
condition for use:
|
||
|
|
|
||
|
|
This software is freely available, but we expect that all publications
|
||
|
|
describing work using this software, or all commercial products using it,
|
||
|
|
quote at least one of the references given below. This software is released
|
||
|
|
under the BSD License.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
* R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound
|
||
|
|
Constrained Optimization, (1995), SIAM Journal on Scientific and
|
||
|
|
Statistical Computing, 16, 5, pp. 1190-1208.
|
||
|
|
* C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B,
|
||
|
|
FORTRAN routines for large scale bound constrained optimization (1997),
|
||
|
|
ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560.
|
||
|
|
* J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B,
|
||
|
|
FORTRAN routines for large scale bound constrained optimization (2011),
|
||
|
|
ACM Transactions on Mathematical Software, 38, 1.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# handle fprime/approx_grad
|
||
|
|
if approx_grad:
|
||
|
|
fun = func
|
||
|
|
jac = None
|
||
|
|
elif fprime is None:
|
||
|
|
fun = MemoizeJac(func)
|
||
|
|
jac = fun.derivative
|
||
|
|
else:
|
||
|
|
fun = func
|
||
|
|
jac = fprime
|
||
|
|
|
||
|
|
# build options
|
||
|
|
if disp is None:
|
||
|
|
disp = iprint
|
||
|
|
opts = {'disp': disp,
|
||
|
|
'iprint': iprint,
|
||
|
|
'maxcor': m,
|
||
|
|
'ftol': factr * np.finfo(float).eps,
|
||
|
|
'gtol': pgtol,
|
||
|
|
'eps': epsilon,
|
||
|
|
'maxfun': maxfun,
|
||
|
|
'maxiter': maxiter,
|
||
|
|
'callback': callback,
|
||
|
|
'maxls': maxls}
|
||
|
|
|
||
|
|
res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds,
|
||
|
|
**opts)
|
||
|
|
d = {'grad': res['jac'],
|
||
|
|
'task': res['message'],
|
||
|
|
'funcalls': res['nfev'],
|
||
|
|
'nit': res['nit'],
|
||
|
|
'warnflag': res['status']}
|
||
|
|
f = res['fun']
|
||
|
|
x = res['x']
|
||
|
|
|
||
|
|
return x, f, d
|
||
|
|
|
||
|
|
|
||
|
|
def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None,
|
||
|
|
disp=None, maxcor=10, ftol=2.2204460492503131e-09,
|
||
|
|
gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000,
|
||
|
|
iprint=-1, callback=None, maxls=20,
|
||
|
|
finite_diff_rel_step=None, **unknown_options):
|
||
|
|
"""
|
||
|
|
Minimize a scalar function of one or more variables using the L-BFGS-B
|
||
|
|
algorithm.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
disp : None or int
|
||
|
|
If `disp is None` (the default), then the supplied version of `iprint`
|
||
|
|
is used. If `disp is not None`, then it overrides the supplied version
|
||
|
|
of `iprint` with the behaviour you outlined.
|
||
|
|
maxcor : int
|
||
|
|
The maximum number of variable metric corrections used to
|
||
|
|
define the limited memory matrix. (The limited memory BFGS
|
||
|
|
method does not store the full hessian but uses this many terms
|
||
|
|
in an approximation to it.)
|
||
|
|
ftol : float
|
||
|
|
The iteration stops when ``(f^k -
|
||
|
|
f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol``.
|
||
|
|
gtol : float
|
||
|
|
The iteration will stop when ``max{|proj g_i | i = 1, ..., n}
|
||
|
|
<= gtol`` where ``pg_i`` is the i-th component of the
|
||
|
|
projected gradient.
|
||
|
|
eps : float or ndarray
|
||
|
|
If `jac is None` the absolute step size used for numerical
|
||
|
|
approximation of the jacobian via forward differences.
|
||
|
|
maxfun : int
|
||
|
|
Maximum number of function evaluations.
|
||
|
|
maxiter : int
|
||
|
|
Maximum number of iterations.
|
||
|
|
iprint : int, optional
|
||
|
|
Controls the frequency of output. ``iprint < 0`` means no output;
|
||
|
|
``iprint = 0`` print only one line at the last iteration;
|
||
|
|
``0 < iprint < 99`` print also f and ``|proj g|`` every iprint iterations;
|
||
|
|
``iprint = 99`` print details of every iteration except n-vectors;
|
||
|
|
``iprint = 100`` print also the changes of active set and final x;
|
||
|
|
``iprint > 100`` print details of every iteration including x and g.
|
||
|
|
callback : callable, optional
|
||
|
|
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
|
||
|
|
current parameter vector.
|
||
|
|
maxls : int, optional
|
||
|
|
Maximum number of line search steps (per iteration). Default is 20.
|
||
|
|
finite_diff_rel_step : None or array_like, optional
|
||
|
|
If `jac in ['2-point', '3-point', 'cs']` the relative step size to
|
||
|
|
use for numerical approximation of the jacobian. The absolute step
|
||
|
|
size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``,
|
||
|
|
possibly adjusted to fit into the bounds. For ``method='3-point'``
|
||
|
|
the sign of `h` is ignored. If None (default) then step is selected
|
||
|
|
automatically.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The option `ftol` is exposed via the `scipy.optimize.minimize` interface,
|
||
|
|
but calling `scipy.optimize.fmin_l_bfgs_b` directly exposes `factr`. The
|
||
|
|
relationship between the two is ``ftol = factr * numpy.finfo(float).eps``.
|
||
|
|
I.e., `factr` multiplies the default machine floating-point precision to
|
||
|
|
arrive at `ftol`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
m = maxcor
|
||
|
|
pgtol = gtol
|
||
|
|
factr = ftol / np.finfo(float).eps
|
||
|
|
|
||
|
|
x0 = asarray(x0).ravel()
|
||
|
|
n, = x0.shape
|
||
|
|
|
||
|
|
if bounds is None:
|
||
|
|
bounds = [(None, None)] * n
|
||
|
|
if len(bounds) != n:
|
||
|
|
raise ValueError('length of x0 != length of bounds')
|
||
|
|
|
||
|
|
# unbounded variables must use None, not +-inf, for optimizer to work properly
|
||
|
|
bounds = [(None if l == -np.inf else l, None if u == np.inf else u) for l, u in bounds]
|
||
|
|
# LBFGSB is sent 'old-style' bounds, 'new-style' bounds are required by
|
||
|
|
# approx_derivative and ScalarFunction
|
||
|
|
new_bounds = old_bound_to_new(bounds)
|
||
|
|
|
||
|
|
# check bounds
|
||
|
|
if (new_bounds[0] > new_bounds[1]).any():
|
||
|
|
raise ValueError("LBFGSB - one of the lower bounds is greater than an upper bound.")
|
||
|
|
|
||
|
|
# initial vector must lie within the bounds. Otherwise ScalarFunction and
|
||
|
|
# approx_derivative will cause problems
|
||
|
|
x0 = np.clip(x0, new_bounds[0], new_bounds[1])
|
||
|
|
|
||
|
|
if disp is not None:
|
||
|
|
if disp == 0:
|
||
|
|
iprint = -1
|
||
|
|
else:
|
||
|
|
iprint = disp
|
||
|
|
|
||
|
|
sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps,
|
||
|
|
bounds=new_bounds,
|
||
|
|
finite_diff_rel_step=finite_diff_rel_step)
|
||
|
|
|
||
|
|
func_and_grad = sf.fun_and_grad
|
||
|
|
|
||
|
|
fortran_int = _lbfgsb.types.intvar.dtype
|
||
|
|
|
||
|
|
nbd = zeros(n, fortran_int)
|
||
|
|
low_bnd = zeros(n, float64)
|
||
|
|
upper_bnd = zeros(n, float64)
|
||
|
|
bounds_map = {(None, None): 0,
|
||
|
|
(1, None): 1,
|
||
|
|
(1, 1): 2,
|
||
|
|
(None, 1): 3}
|
||
|
|
for i in range(0, n):
|
||
|
|
l, u = bounds[i]
|
||
|
|
if l is not None:
|
||
|
|
low_bnd[i] = l
|
||
|
|
l = 1
|
||
|
|
if u is not None:
|
||
|
|
upper_bnd[i] = u
|
||
|
|
u = 1
|
||
|
|
nbd[i] = bounds_map[l, u]
|
||
|
|
|
||
|
|
if not maxls > 0:
|
||
|
|
raise ValueError('maxls must be positive.')
|
||
|
|
|
||
|
|
x = array(x0, float64)
|
||
|
|
f = array(0.0, float64)
|
||
|
|
g = zeros((n,), float64)
|
||
|
|
wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64)
|
||
|
|
iwa = zeros(3*n, fortran_int)
|
||
|
|
task = zeros(1, 'S60')
|
||
|
|
csave = zeros(1, 'S60')
|
||
|
|
lsave = zeros(4, fortran_int)
|
||
|
|
isave = zeros(44, fortran_int)
|
||
|
|
dsave = zeros(29, float64)
|
||
|
|
|
||
|
|
task[:] = 'START'
|
||
|
|
|
||
|
|
n_iterations = 0
|
||
|
|
|
||
|
|
while 1:
|
||
|
|
# x, f, g, wa, iwa, task, csave, lsave, isave, dsave = \
|
||
|
|
_lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr,
|
||
|
|
pgtol, wa, iwa, task, iprint, csave, lsave,
|
||
|
|
isave, dsave, maxls)
|
||
|
|
task_str = task.tobytes()
|
||
|
|
if task_str.startswith(b'FG'):
|
||
|
|
# The minimization routine wants f and g at the current x.
|
||
|
|
# Note that interruptions due to maxfun are postponed
|
||
|
|
# until the completion of the current minimization iteration.
|
||
|
|
# Overwrite f and g:
|
||
|
|
f, g = func_and_grad(x)
|
||
|
|
elif task_str.startswith(b'NEW_X'):
|
||
|
|
# new iteration
|
||
|
|
n_iterations += 1
|
||
|
|
if callback is not None:
|
||
|
|
callback(np.copy(x))
|
||
|
|
|
||
|
|
if n_iterations >= maxiter:
|
||
|
|
task[:] = 'STOP: TOTAL NO. of ITERATIONS REACHED LIMIT'
|
||
|
|
elif sf.nfev > maxfun:
|
||
|
|
task[:] = ('STOP: TOTAL NO. of f AND g EVALUATIONS '
|
||
|
|
'EXCEEDS LIMIT')
|
||
|
|
else:
|
||
|
|
break
|
||
|
|
|
||
|
|
task_str = task.tobytes().strip(b'\x00').strip()
|
||
|
|
if task_str.startswith(b'CONV'):
|
||
|
|
warnflag = 0
|
||
|
|
elif sf.nfev > maxfun or n_iterations >= maxiter:
|
||
|
|
warnflag = 1
|
||
|
|
else:
|
||
|
|
warnflag = 2
|
||
|
|
|
||
|
|
# These two portions of the workspace are described in the mainlb
|
||
|
|
# subroutine in lbfgsb.f. See line 363.
|
||
|
|
s = wa[0: m*n].reshape(m, n)
|
||
|
|
y = wa[m*n: 2*m*n].reshape(m, n)
|
||
|
|
|
||
|
|
# See lbfgsb.f line 160 for this portion of the workspace.
|
||
|
|
# isave(31) = the total number of BFGS updates prior the current iteration;
|
||
|
|
n_bfgs_updates = isave[30]
|
||
|
|
|
||
|
|
n_corrs = min(n_bfgs_updates, maxcor)
|
||
|
|
hess_inv = LbfgsInvHessProduct(s[:n_corrs], y[:n_corrs])
|
||
|
|
|
||
|
|
task_str = task_str.decode()
|
||
|
|
return OptimizeResult(fun=f, jac=g, nfev=sf.nfev,
|
||
|
|
njev=sf.ngev,
|
||
|
|
nit=n_iterations, status=warnflag, message=task_str,
|
||
|
|
x=x, success=(warnflag == 0), hess_inv=hess_inv)
|
||
|
|
|
||
|
|
|
||
|
|
class LbfgsInvHessProduct(LinearOperator):
|
||
|
|
"""Linear operator for the L-BFGS approximate inverse Hessian.
|
||
|
|
|
||
|
|
This operator computes the product of a vector with the approximate inverse
|
||
|
|
of the Hessian of the objective function, using the L-BFGS limited
|
||
|
|
memory approximation to the inverse Hessian, accumulated during the
|
||
|
|
optimization.
|
||
|
|
|
||
|
|
Objects of this class implement the ``scipy.sparse.linalg.LinearOperator``
|
||
|
|
interface.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
sk : array_like, shape=(n_corr, n)
|
||
|
|
Array of `n_corr` most recent updates to the solution vector.
|
||
|
|
(See [1]).
|
||
|
|
yk : array_like, shape=(n_corr, n)
|
||
|
|
Array of `n_corr` most recent updates to the gradient. (See [1]).
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Nocedal, Jorge. "Updating quasi-Newton matrices with limited
|
||
|
|
storage." Mathematics of computation 35.151 (1980): 773-782.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, sk, yk):
|
||
|
|
"""Construct the operator."""
|
||
|
|
if sk.shape != yk.shape or sk.ndim != 2:
|
||
|
|
raise ValueError('sk and yk must have matching shape, (n_corrs, n)')
|
||
|
|
n_corrs, n = sk.shape
|
||
|
|
|
||
|
|
super(LbfgsInvHessProduct, self).__init__(
|
||
|
|
dtype=np.float64, shape=(n, n))
|
||
|
|
|
||
|
|
self.sk = sk
|
||
|
|
self.yk = yk
|
||
|
|
self.n_corrs = n_corrs
|
||
|
|
self.rho = 1 / np.einsum('ij,ij->i', sk, yk)
|
||
|
|
|
||
|
|
def _matvec(self, x):
|
||
|
|
"""Efficient matrix-vector multiply with the BFGS matrices.
|
||
|
|
|
||
|
|
This calculation is described in Section (4) of [1].
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : ndarray
|
||
|
|
An array with shape (n,) or (n,1).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
y : ndarray
|
||
|
|
The matrix-vector product
|
||
|
|
|
||
|
|
"""
|
||
|
|
s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
|
||
|
|
q = np.array(x, dtype=self.dtype, copy=True)
|
||
|
|
if q.ndim == 2 and q.shape[1] == 1:
|
||
|
|
q = q.reshape(-1)
|
||
|
|
|
||
|
|
alpha = np.empty(n_corrs)
|
||
|
|
|
||
|
|
for i in range(n_corrs-1, -1, -1):
|
||
|
|
alpha[i] = rho[i] * np.dot(s[i], q)
|
||
|
|
q = q - alpha[i]*y[i]
|
||
|
|
|
||
|
|
r = q
|
||
|
|
for i in range(n_corrs):
|
||
|
|
beta = rho[i] * np.dot(y[i], r)
|
||
|
|
r = r + s[i] * (alpha[i] - beta)
|
||
|
|
|
||
|
|
return r
|
||
|
|
|
||
|
|
def todense(self):
|
||
|
|
"""Return a dense array representation of this operator.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
arr : ndarray, shape=(n, n)
|
||
|
|
An array with the same shape and containing
|
||
|
|
the same data represented by this `LinearOperator`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho
|
||
|
|
I = np.eye(*self.shape, dtype=self.dtype)
|
||
|
|
Hk = I
|
||
|
|
|
||
|
|
for i in range(n_corrs):
|
||
|
|
A1 = I - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i]
|
||
|
|
A2 = I - y[i][:, np.newaxis] * s[i][np.newaxis, :] * rho[i]
|
||
|
|
|
||
|
|
Hk = np.dot(A1, np.dot(Hk, A2)) + (rho[i] * s[i][:, np.newaxis] *
|
||
|
|
s[i][np.newaxis, :])
|
||
|
|
return Hk
|