Eigen  3.3.3
Eigen::FullPivHouseholderQR< _MatrixType > Class Template Reference

Detailed Description

template<typename _MatrixType>
class Eigen::FullPivHouseholderQR< _MatrixType >

Householder rank-revealing QR decomposition of a matrix with full pivoting.

Template Parameters:
_MatrixTypethe type of the matrix of which we are computing the QR decomposition

This class performs a rank-revealing QR decomposition of a matrix A into matrices P, P', Q and R such that

\[ \mathbf{P} \, \mathbf{A} \, \mathbf{P}' = \mathbf{Q} \, \mathbf{R} \]

by using Householder transformations. Here, P and P' are permutation matrices, Q a unitary matrix and R an upper triangular matrix.

This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also:
MatrixBase::fullPivHouseholderQr()

List of all members.

Public Member Functions

MatrixType::RealScalar absDeterminant () const
const PermutationTypecolsPermutation () const
template<typename InputType >
FullPivHouseholderQRcompute (const EigenBase< InputType > &matrix)
Index dimensionOfKernel () const
 FullPivHouseholderQR ()
 Default Constructor.
 FullPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
template<typename InputType >
 FullPivHouseholderQR (const EigenBase< InputType > &matrix)
 Constructs a QR factorization from a given matrix.
template<typename InputType >
 FullPivHouseholderQR (EigenBase< InputType > &matrix)
 Constructs a QR factorization from a given matrix.
const HCoeffsType & hCoeffs () const
const Inverse
< FullPivHouseholderQR
inverse () const
bool isInjective () const
bool isInvertible () const
bool isSurjective () const
MatrixType::RealScalar logAbsDeterminant () const
MatrixQReturnType matrixQ (void) const
const MatrixType & matrixQR () const
RealScalar maxPivot () const
Index nonzeroPivots () const
Index rank () const
const IntDiagSizeVectorTyperowsTranspositions () const
FullPivHouseholderQRsetThreshold (const RealScalar &threshold)
FullPivHouseholderQRsetThreshold (Default_t)
template<typename Rhs >
const Solve
< FullPivHouseholderQR, Rhs > 
solve (const MatrixBase< Rhs > &b) const
RealScalar threshold () const

Constructor & Destructor Documentation

template<typename _MatrixType>
Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( ) [inline]

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).

template<typename _MatrixType>
Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( Index  rows,
Index  cols 
) [inline]

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also:
FullPivHouseholderQR()
template<typename _MatrixType>
template<typename InputType >
Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( const EigenBase< InputType > &  matrix) [inline, explicit]

Constructs a QR factorization from a given matrix.

This constructor computes the QR factorization of the matrix matrix by calling the method compute(). It is a short cut for:

 FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
 qr.compute(matrix);
See also:
compute()
template<typename _MatrixType>
template<typename InputType >
Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( EigenBase< InputType > &  matrix) [inline, explicit]

Constructs a QR factorization from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See also:
FullPivHouseholderQR(const EigenBase&)

Member Function Documentation

template<typename MatrixType >
MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType >::absDeterminant ( ) const
Returns:
the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.
Note:
This is only for square matrices.
Warning:
a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.
See also:
logAbsDeterminant(), MatrixBase::determinant()
template<typename _MatrixType>
const PermutationType& Eigen::FullPivHouseholderQR< _MatrixType >::colsPermutation ( ) const [inline]
Returns:
a const reference to the column permutation matrix
template<typename MatrixType >
template<typename InputType >
FullPivHouseholderQR< MatrixType > & Eigen::FullPivHouseholderQR< MatrixType >::compute ( const EigenBase< InputType > &  matrix)

Performs the QR factorization of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

See also:
class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)
template<typename _MatrixType>
Index Eigen::FullPivHouseholderQR< _MatrixType >::dimensionOfKernel ( ) const [inline]
Returns:
the dimension of the kernel of the matrix of which *this is the QR decomposition.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename _MatrixType>
const HCoeffsType& Eigen::FullPivHouseholderQR< _MatrixType >::hCoeffs ( ) const [inline]
Returns:
a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

template<typename _MatrixType>
const Inverse<FullPivHouseholderQR> Eigen::FullPivHouseholderQR< _MatrixType >::inverse ( ) const [inline]
Returns:
the inverse of the matrix of which *this is the QR decomposition.
Note:
If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.
template<typename _MatrixType>
bool Eigen::FullPivHouseholderQR< _MatrixType >::isInjective ( ) const [inline]
Returns:
true if the matrix of which *this is the QR decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename _MatrixType>
bool Eigen::FullPivHouseholderQR< _MatrixType >::isInvertible ( ) const [inline]
Returns:
true if the matrix of which *this is the QR decomposition is invertible.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename _MatrixType>
bool Eigen::FullPivHouseholderQR< _MatrixType >::isSurjective ( ) const [inline]
Returns:
true if the matrix of which *this is the QR decomposition represents a surjective linear map; false otherwise.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename MatrixType >
MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType >::logAbsDeterminant ( ) const
Returns:
the natural log of the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.
Note:
This is only for square matrices.
This method is useful to work around the risk of overflow/underflow that's inherent to determinant computation.
See also:
absDeterminant(), MatrixBase::determinant()
template<typename MatrixType >
FullPivHouseholderQR< MatrixType >::MatrixQReturnType Eigen::FullPivHouseholderQR< MatrixType >::matrixQ ( void  ) const [inline]
Returns:
Expression object representing the matrix Q
template<typename _MatrixType>
const MatrixType& Eigen::FullPivHouseholderQR< _MatrixType >::matrixQR ( ) const [inline]
Returns:
a reference to the matrix where the Householder QR decomposition is stored
template<typename _MatrixType>
RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::maxPivot ( ) const [inline]
Returns:
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U.
template<typename _MatrixType>
Index Eigen::FullPivHouseholderQR< _MatrixType >::nonzeroPivots ( ) const [inline]
Returns:
the number of nonzero pivots in the QR decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.
See also:
rank()
template<typename _MatrixType>
Index Eigen::FullPivHouseholderQR< _MatrixType >::rank ( ) const [inline]
Returns:
the rank of the matrix of which *this is the QR decomposition.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename _MatrixType>
const IntDiagSizeVectorType& Eigen::FullPivHouseholderQR< _MatrixType >::rowsTranspositions ( ) const [inline]
Returns:
a const reference to the vector of indices representing the rows transpositions
template<typename _MatrixType>
FullPivHouseholderQR& Eigen::FullPivHouseholderQR< _MatrixType >::setThreshold ( const RealScalar &  threshold) [inline]

Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero. This is not used for the QR decomposition itself.

When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.

Parameters:
thresholdThe new value to use as the threshold.

A pivot will be considered nonzero if its absolute value is strictly greater than $ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert $ where maxpivot is the biggest pivot.

If you want to come back to the default behavior, call setThreshold(Default_t)

template<typename _MatrixType>
FullPivHouseholderQR& Eigen::FullPivHouseholderQR< _MatrixType >::setThreshold ( Default_t  ) [inline]

Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

You should pass the special object Eigen::Default as parameter here.

 qr.setThreshold(Eigen::Default); 

See the documentation of setThreshold(const RealScalar&).

template<typename _MatrixType>
template<typename Rhs >
const Solve<FullPivHouseholderQR, Rhs> Eigen::FullPivHouseholderQR< _MatrixType >::solve ( const MatrixBase< Rhs > &  b) const [inline]

This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition.

Parameters:
bthe right-hand-side of the equation to solve.
Returns:
the exact or least-square solution if the rank is greater or equal to the number of columns of A, and an arbitrary solution otherwise.

This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use MatrixBase::isApprox() directly, for instance like this:

 bool a_solution_exists = (A*result).isApprox(b, precision); 

This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get inf or nan values.

If there exists more than one solution, this method will arbitrarily choose one.

Example:

Matrix3f m = Matrix3f::Random();
Matrix3f y = Matrix3f::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
Matrix3f x;
x = m.fullPivHouseholderQr().solve(y);
assert(y.isApprox(m*x));
cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;

Output:

Here is the matrix m:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Here is the matrix y:
  0.108   -0.27   0.832
-0.0452  0.0268   0.271
  0.258   0.904   0.435
Here is a solution x to the equation mx=y:
 0.609   2.68   1.67
-0.231  -1.57 0.0713
  0.51   3.51   1.05
template<typename _MatrixType>
RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::threshold ( ) const [inline]

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).


The documentation for this class was generated from the following file:
 All Classes Functions Variables Typedefs Enumerations Enumerator Friends