Public Types | Public Member Functions | Protected Attributes
ColPivHouseholderQR< _MatrixType > Class Template Reference

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

#include <ColPivHouseholderQR.h>

List of all members.

Public Types

enum  {
  RowsAtCompileTime,
  ColsAtCompileTime,
  Options,
  MaxRowsAtCompileTime,
  MaxColsAtCompileTime
}
typedef
internal::plain_diag_type
< MatrixType >::type 
HCoeffsType
typedef HouseholderSequence
< MatrixType, HCoeffsType >
::ConjugateReturnType 
HouseholderSequenceType
typedef MatrixType::Index Index
typedef
internal::plain_row_type
< MatrixType, Index >::type 
IntRowVectorType
typedef Matrix< Scalar,
RowsAtCompileTime,
RowsAtCompileTime, Options,
MaxRowsAtCompileTime,
MaxRowsAtCompileTime
MatrixQType
typedef _MatrixType MatrixType
typedef PermutationMatrix
< ColsAtCompileTime,
MaxColsAtCompileTime
PermutationType
typedef
internal::plain_row_type
< MatrixType, RealScalar >
::type 
RealRowVectorType
typedef MatrixType::RealScalar RealScalar
typedef
internal::plain_row_type
< MatrixType >::type 
RowVectorType
typedef MatrixType::Scalar Scalar

Public Member Functions

MatrixType::RealScalar absDeterminant () const
 ColPivHouseholderQR ()
 Default Constructor.
 ColPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 ColPivHouseholderQR (const MatrixType &matrix)
Index cols () const
const PermutationTypecolsPermutation () const
ColPivHouseholderQRcompute (const MatrixType &matrix)
Index dimensionOfKernel () const
const HCoeffsTypehCoeffs () const
HouseholderSequenceType householderQ (void) const
const internal::solve_retval
< ColPivHouseholderQR,
typename
MatrixType::IdentityReturnType > 
inverse () const
bool isInjective () const
bool isInvertible () const
bool isSurjective () const
MatrixType::RealScalar logAbsDeterminant () const
const MatrixTypematrixQR () const
RealScalar maxPivot () const
Index nonzeroPivots () const
Index rank () const
Index rows () const
ColPivHouseholderQRsetThreshold (const RealScalar &threshold)
ColPivHouseholderQRsetThreshold (Default_t)
template<typename Rhs >
const internal::solve_retval
< ColPivHouseholderQR, Rhs > 
solve (const MatrixBase< Rhs > &b) const
RealScalar threshold () const

Protected Attributes

PermutationType m_colsPermutation
RealRowVectorType m_colSqNorms
IntRowVectorType m_colsTranspositions
Index m_det_pq
HCoeffsType m_hCoeffs
bool m_isInitialized
RealScalar m_maxpivot
Index m_nonzero_pivots
RealScalar m_prescribedThreshold
MatrixType m_qr
RowVectorType m_temp
bool m_usePrescribedThreshold

Detailed Description

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

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

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, Q and R such that

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

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

This decomposition performs column pivoting in order to be rank-revealing and improve numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.

See also:
MatrixBase::colPivHouseholderQr()

Member Typedef Documentation

typedef internal::plain_diag_type<MatrixType>::type HCoeffsType
typedef MatrixType::Index Index
typedef internal::plain_row_type<MatrixType, Index>::type IntRowVectorType
typedef _MatrixType MatrixType
typedef internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType
typedef MatrixType::RealScalar RealScalar
typedef internal::plain_row_type<MatrixType>::type RowVectorType
typedef MatrixType::Scalar Scalar

Member Enumeration Documentation

anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Constructor & Destructor Documentation

ColPivHouseholderQR ( ) [inline]

Default Constructor.

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

ColPivHouseholderQR ( 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:
ColPivHouseholderQR()
ColPivHouseholderQR ( const MatrixType matrix) [inline]

Member Function Documentation

MatrixType::RealScalar 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()

References abs().

Index cols ( void  ) const [inline]
const PermutationType& colsPermutation ( ) const [inline]
Index 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&).

References ColPivHouseholderQR< _MatrixType >::cols(), ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::rank().

const HCoeffsType& hCoeffs ( ) const [inline]
Returns:
the matrix Q as a sequence of householder transformations
const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> 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.

References ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.

bool 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&).

References ColPivHouseholderQR< _MatrixType >::cols(), ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::rank().

Referenced by ColPivHouseholderQR< _MatrixType >::isInvertible().

bool 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&).

References ColPivHouseholderQR< _MatrixType >::isInjective(), ColPivHouseholderQR< _MatrixType >::isSurjective(), and ColPivHouseholderQR< _MatrixType >::m_isInitialized.

bool 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&).

References ColPivHouseholderQR< _MatrixType >::m_isInitialized, ColPivHouseholderQR< _MatrixType >::rank(), and ColPivHouseholderQR< _MatrixType >::rows().

Referenced by ColPivHouseholderQR< _MatrixType >::isInvertible().

MatrixType::RealScalar 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()
const MatrixType& matrixQR ( ) const [inline]
Returns:
a reference to the matrix where the Householder QR decomposition is stored

References ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.

RealScalar maxPivot ( ) const [inline]
Returns:
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of R.

References ColPivHouseholderQR< _MatrixType >::m_maxpivot.

Index 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()

References ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots.

Index rank ( ) const [inline]
Index rows ( void  ) const [inline]
ColPivHouseholderQR& 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)

References ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold, ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold, and ColPivHouseholderQR< _MatrixType >::threshold().

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&).

References ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.

const internal::solve_retval<ColPivHouseholderQR, Rhs> 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, if any exists.

Parameters:
bthe right-hand-side of the equation to solve.
Returns:
a solution.
Note:
The case where b is a matrix is not yet implemented. Also, this code is space inefficient.

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.colPivHouseholderQr().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

References ColPivHouseholderQR< _MatrixType >::m_isInitialized.

RealScalar threshold ( ) const [inline]

Member Data Documentation

Index m_det_pq [protected]
HCoeffsType m_hCoeffs [protected]
bool m_isInitialized [protected]
RealScalar m_maxpivot [protected]
Index m_nonzero_pivots [protected]
MatrixType m_qr [protected]
RowVectorType m_temp [protected]

The documentation for this class was generated from the following file: