blob: ae8e877470937a77fb85f78f2ceca42f84fcc9b8 [file] [log] [blame]
// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
// http://code.google.com/p/ceres-solver/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: sameeragarwal@google.com (Sameer Agarwal)
//
// A preconditioned conjugate gradients solver
// (ConjugateGradientsSolver) for positive semidefinite linear
// systems.
//
// We have also augmented the termination criterion used by this
// solver to support not just residual based termination but also
// termination based on decrease in the value of the quadratic model
// that CG optimizes.
#include "ceres/conjugate_gradients_solver.h"
#include <cmath>
#include <cstddef>
#include "ceres/fpclassify.h"
#include "ceres/internal/eigen.h"
#include "ceres/linear_operator.h"
#include "ceres/types.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
namespace {
bool IsZeroOrInfinity(double x) {
return ((x == 0.0) || (IsInfinite(x)));
}
// Constant used in the MATLAB implementation ~ 2 * eps.
const double kEpsilon = 2.2204e-16;
} // namespace
ConjugateGradientsSolver::ConjugateGradientsSolver(
const LinearSolver::Options& options)
: options_(options) {
}
LinearSolver::Summary ConjugateGradientsSolver::Solve(
LinearOperator* A,
const double* b,
const LinearSolver::PerSolveOptions& per_solve_options,
double* x) {
CHECK_NOTNULL(A);
CHECK_NOTNULL(x);
CHECK_NOTNULL(b);
CHECK_EQ(A->num_rows(), A->num_cols());
LinearSolver::Summary summary;
summary.termination_type = MAX_ITERATIONS;
summary.num_iterations = 0;
int num_cols = A->num_cols();
VectorRef xref(x, num_cols);
ConstVectorRef bref(b, num_cols);
double norm_b = bref.norm();
if (norm_b == 0.0) {
xref.setZero();
summary.termination_type = TOLERANCE;
return summary;
}
Vector r(num_cols);
Vector p(num_cols);
Vector z(num_cols);
Vector tmp(num_cols);
double tol_r = per_solve_options.r_tolerance * norm_b;
tmp.setZero();
A->RightMultiply(x, tmp.data());
r = bref - tmp;
double norm_r = r.norm();
if (norm_r <= tol_r) {
summary.termination_type = TOLERANCE;
return summary;
}
double rho = 1.0;
// Initial value of the quadratic model Q = x'Ax - 2 * b'x.
double Q0 = -1.0 * xref.dot(bref + r);
for (summary.num_iterations = 1;
summary.num_iterations < options_.max_num_iterations;
++summary.num_iterations) {
VLOG(3) << "cg iteration " << summary.num_iterations;
// Apply preconditioner
if (per_solve_options.preconditioner != NULL) {
z.setZero();
per_solve_options.preconditioner->RightMultiply(r.data(), z.data());
} else {
z = r;
}
double last_rho = rho;
rho = r.dot(z);
if (IsZeroOrInfinity(rho)) {
LOG(ERROR) << "Numerical failure. rho = " << rho;
summary.termination_type = FAILURE;
break;
};
if (summary.num_iterations == 1) {
p = z;
} else {
double beta = rho / last_rho;
if (IsZeroOrInfinity(beta)) {
LOG(ERROR) << "Numerical failure. beta = " << beta;
summary.termination_type = FAILURE;
break;
}
p = z + beta * p;
}
Vector& q = z;
q.setZero();
A->RightMultiply(p.data(), q.data());
double pq = p.dot(q);
if ((pq <= 0) || IsInfinite(pq)) {
LOG(ERROR) << "Numerical failure. pq = " << pq;
summary.termination_type = FAILURE;
break;
}
double alpha = rho / pq;
if (IsInfinite(alpha)) {
LOG(ERROR) << "Numerical failure. alpha " << alpha;
summary.termination_type = FAILURE;
break;
}
xref = xref + alpha * p;
// Ideally we would just use the update r = r - alpha*q to keep
// track of the residual vector. However this estimate tends to
// drift over time due to round off errors. Thus every
// residual_reset_period iterations, we calculate the residual as
// r = b - Ax. We do not do this every iteration because this
// requires an additional matrix vector multiply which would
// double the complexity of the CG algorithm.
if (summary.num_iterations % options_.residual_reset_period == 0) {
tmp.setZero();
A->RightMultiply(x, tmp.data());
r = bref - tmp;
} else {
r = r - alpha * q;
}
// Quadratic model based termination.
// Q1 = x'Ax - 2 * b' x.
double Q1 = -1.0 * xref.dot(bref + r);
// For PSD matrices A, let
//
// Q(x) = x'Ax - 2b'x
//
// be the cost of the quadratic function defined by A and b. Then,
// the solver terminates at iteration i if
//
// i * (Q(x_i) - Q(x_i-1)) / Q(x_i) < q_tolerance.
//
// This termination criterion is more useful when using CG to
// solve the Newton step. This particular convergence test comes
// from Stephen Nash's work on truncated Newton
// methods. References:
//
// 1. Stephen G. Nash & Ariela Sofer, Assessing A Search
// Direction Within A Truncated Newton Method, Operation
// Research Letters 9(1990) 219-221.
//
// 2. Stephen G. Nash, A Survey of Truncated Newton Methods,
// Journal of Computational and Applied Mathematics,
// 124(1-2), 45-59, 2000.
//
double zeta = summary.num_iterations * (Q1 - Q0) / Q1;
VLOG(3) << "Q termination: zeta " << zeta
<< " " << per_solve_options.q_tolerance;
if (zeta < per_solve_options.q_tolerance) {
summary.termination_type = TOLERANCE;
break;
}
Q0 = Q1;
// Residual based termination.
norm_r = r. norm();
VLOG(3) << "R termination: norm_r " << norm_r
<< " " << tol_r;
if (norm_r <= tol_r) {
summary.termination_type = TOLERANCE;
break;
}
}
return summary;
};
} // namespace internal
} // namespace ceres