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# Copyright 2016 The TensorFlow Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ==============================================================================
"""Solvers for linear equations."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import collections
import tensorflow as tf
from tensorflow.contrib.solvers.python.ops import util
def conjugate_gradient(operator,
rhs,
tol=1e-4,
max_iter=20,
name="conjugate_gradient"):
r"""Conjugate gradient solver.
Solves a linear system of equations `A*x = rhs` for selfadjoint, positive
definite matrix `A` and righ-hand side vector `rhs`, using an iterative,
matrix-free algorithm where the action of the matrix A is represented by
`operator`. The iteration terminates when either the number of iterations
exceeds `max_iter` or when the residual norm has been reduced to `tol`
times its initial value, i.e. \\(||rhs - A x_k|| <= tol ||rhs||\\).
Args:
operator: An object representing a linear operator with attributes:
- shape: Either a list of integers or a 1-D `Tensor` of type `int32` of
length 2. `shape[0]` is the dimension on the domain of the operator,
`shape[1]` is the dimension of the co-domain of the operator. On other
words, if operator represents an N x N matrix A, `shape` must contain
`[N, N]`.
- dtype: The datatype of input to and output from `apply`.
- apply: Callable object taking a vector `x` as input and returning a
vector with the result of applying the operator to `x`, i.e. if
`operator` represents matrix `A`, `apply` should return `A * x`.
rhs: A rank-1 `Tensor` of shape `[N]` containing the right-hand size vector.
tol: A float scalar convergence tolerance.
max_iter: An integer giving the maximum number of iterations.
name: A name scope for the operation.
Returns:
output: A namedtuple representing the final state with fields:
- i: A scalar `int32` `Tensor`. Number of iterations executed.
- x: A rank-1 `Tensor` of shape `[N]` containing the computed solution.
- r: A rank-1 `Tensor` of shape `[M]` containing the residual vector.
- p: A rank-1 `Tensor` of shape `[N]`. `A`-conjugate basis vector.
- gamma: \\(||r||_2^2\\)
"""
# ephemeral class holding CG state.
cg_state = collections.namedtuple("CGState", ["i", "x", "r", "p", "gamma"])
def stopping_criterion(i, state):
return tf.logical_and(i < max_iter, state.gamma > tol)
# TODO(rmlarsen): add preconditioning
def cg_step(i, state):
z = operator.apply(state.p)
alpha = state.gamma / util.dot(state.p, z)
x = state.x + alpha * state.p
r = state.r - alpha * z
gamma = util.l2norm_squared(r)
beta = gamma / state.gamma
p = r + beta * state.p
return i + 1, cg_state(i + 1, x, r, p, gamma)
with tf.name_scope(name):
n = operator.shape[1:]
rhs = tf.expand_dims(rhs, -1)
gamma0 = util.l2norm_squared(rhs)
tol = tol * tol * gamma0
x = tf.expand_dims(tf.zeros(n, dtype=rhs.dtype.base_dtype), -1)
i = tf.constant(0, dtype=tf.int32)
state = cg_state(i=i, x=x, r=rhs, p=rhs, gamma=gamma0)
_, state = tf.while_loop(stopping_criterion, cg_step, [i, state])
return cg_state(
state.i,
x=tf.squeeze(state.x),
r=tf.squeeze(state.r),
p=tf.squeeze(state.p),
gamma=state.gamma)
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