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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// 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
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// used to endorse or promote products derived from this software without
// specific prior written permission.
//
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//
// 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/stringprintf.h"
#include "ceres/types.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
namespace {
bool IsZeroOrInfinity(double x) {
return ((x == 0.0) || (IsInfinite(x)));
}
} // 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 = LINEAR_SOLVER_NO_CONVERGENCE;
summary.message = "Maximum number of iterations reached.";
summary.num_iterations = 0;
const int num_cols = A->num_cols();
VectorRef xref(x, num_cols);
ConstVectorRef bref(b, num_cols);
const double norm_b = bref.norm();
if (norm_b == 0.0) {
xref.setZero();
summary.termination_type = LINEAR_SOLVER_SUCCESS;
summary.message = "Convergence. |b| = 0.";
return summary;
}
Vector r(num_cols);
Vector p(num_cols);
Vector z(num_cols);
Vector tmp(num_cols);
const 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 (options_.min_num_iterations == 0 && norm_r <= tol_r) {
summary.termination_type = LINEAR_SOLVER_SUCCESS;
summary.message =
StringPrintf("Convergence. |r| = %e <= %e.", norm_r, tol_r);
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) {
// 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)) {
summary.termination_type = LINEAR_SOLVER_FAILURE;
summary.message = StringPrintf("Numerical failure. rho = r'z = %e.", rho);
break;
}
if (summary.num_iterations == 1) {
p = z;
} else {
double beta = rho / last_rho;
if (IsZeroOrInfinity(beta)) {
summary.termination_type = LINEAR_SOLVER_FAILURE;
summary.message = StringPrintf(
"Numerical failure. beta = rho_n / rho_{n-1} = %e, "
"rho_n = %e, rho_{n-1} = %e", beta, rho, last_rho);
break;
}
p = z + beta * p;
}
Vector& q = z;
q.setZero();
A->RightMultiply(p.data(), q.data());
const double pq = p.dot(q);
if ((pq <= 0) || IsInfinite(pq)) {
summary.termination_type = LINEAR_SOLVER_NO_CONVERGENCE;
summary.message = StringPrintf(
"Matrix is indefinite, no more progress can be made. "
"p'q = %e. |p| = %e, |q| = %e",
pq, p.norm(), q.norm());
break;
}
const double alpha = rho / pq;
if (IsInfinite(alpha)) {
summary.termination_type = LINEAR_SOLVER_FAILURE;
summary.message =
StringPrintf("Numerical failure. alpha = rho / pq = %e, "
"rho = %e, pq = %e.", alpha, rho, pq);
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.
const 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.
//
const double zeta = summary.num_iterations * (Q1 - Q0) / Q1;
if (zeta < per_solve_options.q_tolerance &&
summary.num_iterations >= options_.min_num_iterations) {
summary.termination_type = LINEAR_SOLVER_SUCCESS;
summary.message =
StringPrintf("Iteration: %d Convergence: zeta = %e < %e. |r| = %e",
summary.num_iterations,
zeta,
per_solve_options.q_tolerance,
r.norm());
break;
}
Q0 = Q1;
// Residual based termination.
norm_r = r. norm();
if (norm_r <= tol_r &&
summary.num_iterations >= options_.min_num_iterations) {
summary.termination_type = LINEAR_SOLVER_SUCCESS;
summary.message =
StringPrintf("Iteration: %d Convergence. |r| = %e <= %e.",
summary.num_iterations,
norm_r,
tol_r);
break;
}
if (summary.num_iterations >= options_.max_num_iterations) {
break;
}
}
return summary;
}
} // namespace internal
} // namespace ceres