Remove operator^ from Matrix and SubgraphPreconditioner

Eigen recently added an operator^ for coefficient-wise bitwise xor,
which conflicts with gtsam's operator^. Since gtsam's operator^ isn't
xor, I opted to remove it.

Fixes #1951.

gtsam/base/Matrix.cpp

@@ -126,17 +126,6 @@ bool linear_dependent(const Matrix& A, const Matrix& B, double tol) {
 }
 
 /* ************************************************************************* */
-Vector operator^(const Matrix& A, const Vector & v) {
-  if (A.rows()!=v.size()) {
-    throw std::invalid_argument("Matrix operator^ : A.m(" + std::to_string(A.rows()) + ")!=v.size(" +
-                                std::to_string(v.size()) + ")");
-  }
-//  Vector vt = v.transpose();
-//  Vector vtA = vt * A;
-//  return vtA.transpose();
-  return A.transpose() * v;
-}
-
 const Eigen::IOFormat& matlabFormat() {
   static const Eigen::IOFormat matlab(
     Eigen::StreamPrecision, // precision

gtsam/base/Matrix.h

@@ -132,12 +132,6 @@ GTSAM_EXPORT bool linear_independent(const Matrix& A, const Matrix& B, double to
  */
 GTSAM_EXPORT bool linear_dependent(const Matrix& A, const Matrix& B, double tol = 1e-9);
 
-/**
- * overload ^ for trans(A)*v
- * We transpose the vectors for speed.
- */
-GTSAM_EXPORT Vector operator^(const Matrix& A, const Vector & v);
-
 /** products using old-style format to improve compatibility */
 template<class MATRIX>
 inline MATRIX prod(const MATRIX& A, const MATRIX&B) {

gtsam/linear/SubgraphPreconditioner.cpp

@@ -149,20 +149,6 @@ void SubgraphPreconditioner::multiplyInPlace(const VectorValues& y, Errors& e) c
   Ab2_.multiplyInPlace(x, ei);                  // use iterator version
 }
 
-/* ************************************************************************* */
-// Apply operator A', A'*e = [I inv(R1')*A2']*e = e1 + inv(R1')*A2'*e2
-VectorValues SubgraphPreconditioner::operator^(const Errors& e) const {
-
-  Errors::const_iterator it = e.begin();
-  VectorValues y = zero();
-  for(auto& key_value: y) {
-    key_value.second = *it;
-    ++it;
-  }
-  transposeMultiplyAdd2(1.0, it, e.end(), y);
-  return y;
-}
-
 /* ************************************************************************* */
 // y += alpha*A'*e
 void SubgraphPreconditioner::transposeMultiplyAdd

gtsam/linear/SubgraphPreconditioner.h

@@ -125,9 +125,6 @@ namespace gtsam {
     /** Apply operator A in place: needs e allocated already */
     void multiplyInPlace(const VectorValues& y, Errors& e) const;
 
-    /** Apply operator A' */
-    VectorValues operator^(const Errors& e) const;
-
     /**
     * Add A'*e to y
     *  y += alpha*A'*[e1;e2] = [alpha*e1; alpha*inv(R1')*A2'*e2]

gtsam/linear/iterative.h

@@ -59,11 +59,6 @@ namespace gtsam {
     /** Access b vector */
     const Vector& b() const { return b_; }
 
-    /** Apply operator A'*e */
-    Vector operator^(const Vector& e) const {
-      return A_ ^ e;
-    }
-
     /**
      * Print with optional string
      */
@@ -71,7 +66,7 @@ namespace gtsam {
 
     /** gradient of objective function 0.5*|Ax-b_|^2 at x = A_'*(Ax-b_) */
     Vector gradient(const Vector& x) const {
-      return A() ^ (A() * x - b());
+      return A().transpose() * (A() * x - b());
     }
 
     /** Apply operator A */