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+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 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: keir@google.com (Keir Mierle)
+
+#ifndef CERES_INTERNAL_CGNR_LINEAR_OPERATOR_H_
+#define CERES_INTERNAL_CGNR_LINEAR_OPERATOR_H_
+
+#include <algorithm>
+#include "ceres/linear_operator.h"
+#include "ceres/internal/scoped_ptr.h"
+#include "ceres/internal/eigen.h"
+
+namespace ceres {
+namespace internal {
+
+class SparseMatrix;
+
+// A linear operator which takes a matrix A and a diagonal vector D and
+// performs products of the form
+//
+//   (A^T A + D^T D)x
+//
+// This is used to implement iterative general sparse linear solving with
+// conjugate gradients, where A is the Jacobian and D is a regularizing
+// parameter. A brief proof that D^T D is the correct regularizer:
+//
+// Given a regularized least squares problem:
+//
+//   min  ||Ax - b||^2 + ||Dx||^2
+//    x
+//
+// First expand into matrix notation:
+//
+//   (Ax - b)^T (Ax - b) + xD^TDx
+//
+// Then multiply out to get:
+//
+//   = xA^TAx - 2b^T Ax + b^Tb + xD^TDx
+//
+// Take the derivative:
+//
+//   0 = 2A^TAx - 2A^T b + 2 D^TDx
+//   0 = A^TAx - A^T b + D^TDx
+//   0 = (A^TA + D^TD)x - A^T b
+//
+// Thus, the symmetric system we need to solve for CGNR is
+//
+//   Sx = z
+//
+// with S = A^TA + D^TD
+//  and z = A^T b
+//
+// Note: This class is not thread safe, since it uses some temporary storage.
+class CgnrLinearOperator : public LinearOperator {
+ public:
+  CgnrLinearOperator(const LinearOperator& A, const double *D)
+      : A_(A), D_(D), z_(new double[A.num_rows()]) {
+  }
+  virtual ~CgnrLinearOperator() {}
+
+  virtual void RightMultiply(const double* x, double* y) const {
+    std::fill(z_.get(), z_.get() + A_.num_rows(), 0.0);
+
+    // z = Ax
+    A_.RightMultiply(x, z_.get());
+
+    // y = y + Atz
+    A_.LeftMultiply(z_.get(), y);
+
+    // y = y + DtDx
+    if (D_ != NULL) {
+      int n = A_.num_cols();
+      VectorRef(y, n).array() += ConstVectorRef(D_, n).array().square() *
+                                 ConstVectorRef(x, n).array();
+    }
+  }
+
+  virtual void LeftMultiply(const double* x, double* y) const {
+    RightMultiply(x, y);
+  }
+
+  virtual int num_rows() const { return A_.num_cols(); }
+  virtual int num_cols() const { return A_.num_cols(); }
+
+ private:
+  const LinearOperator& A_;
+  const double* D_;
+  scoped_array<double> z_;
+};
+
+}  // namespace internal
+}  // namespace ceres
+
+#endif  // CERES_INTERNAL_CGNR_LINEAR_OPERATOR_H_