text: // Ceres Solver - A fast non-linear least squares minimizer
  // Copyright 2015 Google Inc. All rights reserved.
  // http://ceres-solver.org/
  //
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  //
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  //   this list of conditions and the following disclaimer.
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  //   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
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  //
  // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
  // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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  // 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_
  