1 // Ceres Solver - A fast non-linear least squares minimizer
2 // Copyright 2012 Google Inc. All rights reserved.
3 // http://code.google.com/p/ceres-solver/
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29 // Author: keir@google.com (Keir Mierle)
30 
31 #ifndef CERES_INTERNAL_CGNR_LINEAR_OPERATOR_H_
32 #define CERES_INTERNAL_CGNR_LINEAR_OPERATOR_H_
33 
34 #include <algorithm>
35 #include "ceres/linear_operator.h"
36 #include "ceres/internal/scoped_ptr.h"
37 #include "ceres/internal/eigen.h"
38 
39 namespace ceres {
40 namespace internal {
41 
42 class SparseMatrix;
43 
44 // A linear operator which takes a matrix A and a diagonal vector D and
45 // performs products of the form
46 //
47 //   (A^T A + D^T D)x
48 //
49 // This is used to implement iterative general sparse linear solving with
50 // conjugate gradients, where A is the Jacobian and D is a regularizing
51 // parameter. A brief proof that D^T D is the correct regularizer:
52 //
53 // Given a regularized least squares problem:
54 //
55 //   min  ||Ax - b||^2 + ||Dx||^2
56 //    x
57 //
58 // First expand into matrix notation:
59 //
60 //   (Ax - b)^T (Ax - b) + xD^TDx
61 //
62 // Then multiply out to get:
63 //
64 //   = xA^TAx - 2b^T Ax + b^Tb + xD^TDx
65 //
66 // Take the derivative:
67 //
68 //   0 = 2A^TAx - 2A^T b + 2 D^TDx
69 //   0 = A^TAx - A^T b + D^TDx
70 //   0 = (A^TA + D^TD)x - A^T b
71 //
72 // Thus, the symmetric system we need to solve for CGNR is
73 //
74 //   Sx = z
75 //
76 // with S = A^TA + D^TD
77 //  and z = A^T b
78 //
79 // Note: This class is not thread safe, since it uses some temporary storage.
80 class CgnrLinearOperator : public LinearOperator {
81  public:
CgnrLinearOperator(const LinearOperator & A,const double * D)82   CgnrLinearOperator(const LinearOperator& A, const double *D)
83       : A_(A), D_(D), z_(new double[A.num_rows()]) {
84   }
~CgnrLinearOperator()85   virtual ~CgnrLinearOperator() {}
86 
RightMultiply(const double * x,double * y)87   virtual void RightMultiply(const double* x, double* y) const {
88     std::fill(z_.get(), z_.get() + A_.num_rows(), 0.0);
89 
90     // z = Ax
91     A_.RightMultiply(x, z_.get());
92 
93     // y = y + Atz
94     A_.LeftMultiply(z_.get(), y);
95 
96     // y = y + DtDx
97     if (D_ != NULL) {
98       int n = A_.num_cols();
99       VectorRef(y, n).array() += ConstVectorRef(D_, n).array().square() *
100                                  ConstVectorRef(x, n).array();
101     }
102   }
103 
LeftMultiply(const double * x,double * y)104   virtual void LeftMultiply(const double* x, double* y) const {
105     RightMultiply(x, y);
106   }
107 
num_rows()108   virtual int num_rows() const { return A_.num_cols(); }
num_cols()109   virtual int num_cols() const { return A_.num_cols(); }
110 
111  private:
112   const LinearOperator& A_;
113   const double* D_;
114   scoped_array<double> z_;
115 };
116 
117 }  // namespace internal
118 }  // namespace ceres
119 
120 #endif  // CERES_INTERNAL_CGNR_LINEAR_OPERATOR_H_
121