Eigen: Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex

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#include <Eigen/src/QR/ColPivHouseholderQR.h>

Detailed Description

template<typename MatrixType_, typename PermutationIndex_>
class Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >

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

Template Parameters

MatrixType_ the 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.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::colPivHouseholderQr()

Inheritance diagram for Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >:

Public Member Functions
MatrixType::RealScalar	absDeterminant () const

	ColPivHouseholderQR ()
	Default Constructor.

template<typename InputType>
	ColPivHouseholderQR (const EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix.

template<typename InputType>
	ColPivHouseholderQR (EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix.

	ColPivHouseholderQR (Index rows, Index cols)
	Default Constructor with memory preallocation.

const PermutationType &	colsPermutation () const

template<typename InputType>
ColPivHouseholderQR< MatrixType, PermutationIndex > &	compute (const EigenBase< InputType > &matrix)

MatrixType::Scalar	determinant () const

Index	dimensionOfKernel () const

const HCoeffsType &	hCoeffs () const

HouseholderSequenceType	householderQ () const

ComputationInfo	info () const
	Reports whether the QR factorization was successful.

Inverse< ColPivHouseholderQR >	inverse () const

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

MatrixType::RealScalar	logAbsDeterminant () const

const MatrixType &	matrixQR () const

const MatrixType &	matrixR () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

RealScalar	pivotCoeff (Index i) const

Index	rank () const

ColPivHouseholderQR &	setThreshold (const RealScalar &threshold)

ColPivHouseholderQR &	setThreshold (Default_t)

MatrixType::Scalar	signDeterminant () const

template<typename Rhs>
Solve< ColPivHouseholderQR, Rhs >	solve (const MatrixBase< Rhs > &b) const

RealScalar	threshold () const

Public Member Functions inherited from Eigen::SolverBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >
const AdjointReturnType	adjoint () const

constexpr ColPivHouseholderQR< MatrixType_, PermutationIndex_ > &	derived ()

constexpr const ColPivHouseholderQR< MatrixType_, PermutationIndex_ > &	derived () const

Solve< ColPivHouseholderQR< MatrixType_, PermutationIndex_ >, Rhs >	solve (const MatrixBase< Rhs > &b) const

	SolverBase ()=default

const ConstTransposeReturnType	transpose () const

Public Member Functions inherited from Eigen::EigenBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >
constexpr Index	cols () const noexcept

constexpr ColPivHouseholderQR< MatrixType_, PermutationIndex_ > &	derived ()

constexpr const ColPivHouseholderQR< MatrixType_, PermutationIndex_ > &	derived () const

constexpr Index	rows () const noexcept

constexpr Index	size () const noexcept

Public Member Functions inherited from Eigen::RankRevealingBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >
Index	dimensionOfKernel () const

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

Index	rank () const

ColPivHouseholderQR< MatrixType_, PermutationIndex_ > &	setThreshold (const RealScalar &threshold)

ColPivHouseholderQR< MatrixType_, PermutationIndex_ > &	setThreshold (Default_t)

RealScalar	threshold () const

Additional Inherited Members
Public Types inherited from Eigen::EigenBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >
using	Index
	The interface type of indices.

Constructor & Destructor Documentation

◆ ColPivHouseholderQR() [1/4]

template<typename MatrixType_, typename PermutationIndex_>

Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::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() [2/4]

template<typename MatrixType_, typename PermutationIndex_>

Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::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() [3/4]

template<typename MatrixType_, typename PermutationIndex_>

template<typename InputType>

Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::ColPivHouseholderQR ( const EigenBase< InputType > & matrix )

inlineexplicit

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:

ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());

qr.compute(matrix);

Eigen::ColPivHouseholderQR::ColPivHouseholderQR

ColPivHouseholderQR()

Default Constructor.

Definition ColPivHouseholderQR.h:107

See also: compute()

◆ ColPivHouseholderQR() [4/4]

template<typename MatrixType_, typename PermutationIndex_>

template<typename InputType>

Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::ColPivHouseholderQR ( EigenBase< InputType > & matrix )

inlineexplicit

Constructs a QR factorization from a given matrix.

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

See also: ColPivHouseholderQR(const EigenBase&)

Member Function Documentation

◆ absDeterminant()

template<typename MatrixType, typename PermutationIndex>

MatrixType::RealScalar Eigen::ColPivHouseholderQR< MatrixType, PermutationIndex >::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: determinant(), logAbsDeterminant(), MatrixBase::determinant()

◆ colsPermutation()

template<typename MatrixType_, typename PermutationIndex_>

const PermutationType & Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::colsPermutation ( ) const

inline

Returns: a const reference to the column permutation matrix

◆ compute()

template<typename MatrixType_, typename PermutationIndex_>

template<typename InputType>

ColPivHouseholderQR< MatrixType, PermutationIndex > & Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::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 ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)

◆ determinant()

template<typename MatrixType, typename PermutationIndex>

MatrixType::Scalar Eigen::ColPivHouseholderQR< MatrixType, PermutationIndex >::determinant ( ) const

Returns: 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: absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()

◆ dimensionOfKernel()

template<typename MatrixType_, typename PermutationIndex_>

Index Eigen::RankRevealingBase< ColPivHouseholderQR >::dimensionOfKernel ( ) const

inline

Returns: the dimension of the kernel of the matrix of which *this is the 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&).

◆ hCoeffs()

template<typename MatrixType_, typename PermutationIndex_>

const HCoeffsType & Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::hCoeffs ( ) const

inline

Returns: a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

◆ householderQ()

template<typename MatrixType, typename PermutationIndex>

ColPivHouseholderQR< MatrixType, PermutationIndex >::HouseholderSequenceType Eigen::ColPivHouseholderQR< MatrixType, PermutationIndex >::householderQ ( ) const

Returns: the matrix Q as a sequence of householder transformations. You can extract the meaningful part only by using:
qr.householderQ().setLength(qr.nonzeroPivots())

◆ info()

template<typename MatrixType_, typename PermutationIndex_>

ComputationInfo Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::info ( ) const

inline

Reports whether the QR factorization was successful.

Note: This function always returns Success. It is provided for compatibility with other factorization routines.

Returns: Success

◆ inverse()

template<typename MatrixType_, typename PermutationIndex_>

Inverse< ColPivHouseholderQR > Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::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.

◆ isInjective()

template<typename MatrixType_, typename PermutationIndex_>

bool Eigen::RankRevealingBase< ColPivHouseholderQR >::isInjective ( ) const

inline

Returns: true if the matrix of which *this is the 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&).

◆ isInvertible()

template<typename MatrixType_, typename PermutationIndex_>

bool Eigen::RankRevealingBase< ColPivHouseholderQR >::isInvertible ( ) const

inline

Returns: true if the matrix of which *this is the 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&).

◆ isSurjective()

template<typename MatrixType_, typename PermutationIndex_>

bool Eigen::RankRevealingBase< ColPivHouseholderQR >::isSurjective ( ) const

inline

Returns: true if the matrix of which *this is the 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&).

◆ logAbsDeterminant()

template<typename MatrixType, typename PermutationIndex>

MatrixType::RealScalar Eigen::ColPivHouseholderQR< MatrixType, PermutationIndex >::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: determinant(), absDeterminant(), MatrixBase::determinant()

◆ matrixQR()

template<typename MatrixType_, typename PermutationIndex_>

const MatrixType & Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixQR ( ) const

inline

Returns: a reference to the matrix where the Householder QR decomposition is stored

◆ matrixR()

template<typename MatrixType_, typename PermutationIndex_>

const MatrixType & Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixR ( ) const

inline

Returns: a reference to the matrix where the result Householder QR is stored

Warning: The strict lower part of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use
matrixR().template triangularView<Upper>()

Eigen::ColPivHouseholderQR::matrixR
const MatrixType & matrixR() const
Definition ColPivHouseholderQR.h:194

For rank-deficient matrices, use
matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Eigen::ColPivHouseholderQR::rank
Index rank() const
Definition RankRevealingBase.h:95

◆ maxPivot()

template<typename MatrixType_, typename PermutationIndex_>

RealScalar Eigen::RankRevealingBase< ColPivHouseholderQR >::maxPivot ( ) const

inline

Returns: the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U (or R).

◆ nonzeroPivots()

template<typename MatrixType_, typename PermutationIndex_>

Index Eigen::RankRevealingBase< ColPivHouseholderQR >::nonzeroPivots ( ) const

inline

Returns: the number of nonzero pivots in the 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()

◆ pivotCoeff()

template<typename MatrixType_, typename PermutationIndex_>

RealScalar Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::pivotCoeff ( Index i ) const

inline

Returns: the absolute value of the i-th pivot coefficient (for RankRevealingBase).

◆ rank()

template<typename MatrixType_, typename PermutationIndex_>

Index Eigen::RankRevealingBase< ColPivHouseholderQR >::rank ( ) const

inline

Returns: the rank of the matrix of which *this is the 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&).

◆ setThreshold() [1/2]

template<typename MatrixType_, typename PermutationIndex_>

ColPivHouseholderQR & Eigen::RankRevealingBase< ColPivHouseholderQR >::setThreshold ( const RealScalar & threshold )

inline

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

threshold The 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)

◆ setThreshold() [2/2]

template<typename MatrixType_, typename PermutationIndex_>

ColPivHouseholderQR & Eigen::RankRevealingBase< ColPivHouseholderQR >::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.

dec.setThreshold(Eigen::Default);

See the documentation of setThreshold(const RealScalar&).

◆ signDeterminant()

template<typename MatrixType, typename PermutationIndex>

MatrixType::Scalar Eigen::ColPivHouseholderQR< MatrixType, PermutationIndex >::signDeterminant ( ) const

Returns: the sign 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: determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()

◆ solve()

template<typename MatrixType_, typename PermutationIndex_>

template<typename Rhs>

Solve< ColPivHouseholderQR, Rhs > Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::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

b	the right-hand-side of the equation to solve.

Returns: a solution.

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:

// SPDX-FileCopyrightText: The Eigen Authors
// SPDX-License-Identifier: MPL-2.0
 
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

◆ threshold()

template<typename MatrixType_, typename PermutationIndex_>

RealScalar Eigen::RankRevealingBase< ColPivHouseholderQR >::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 files:

/builds/libeigen/eigen/Eigen/src/Core/util/ForwardDeclarations.h
/builds/libeigen/eigen/Eigen/src/QR/ColPivHouseholderQR.h

Detailed Description

Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation

◆ ColPivHouseholderQR() [1/4]

◆ ColPivHouseholderQR() [2/4]

◆ ColPivHouseholderQR() [3/4]

◆ ColPivHouseholderQR() [4/4]

Member Function Documentation

◆ absDeterminant()

◆ colsPermutation()

◆ compute()

◆ determinant()

◆ dimensionOfKernel()

◆ hCoeffs()

◆ householderQ()

◆ info()

◆ inverse()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQR()

◆ matrixR()

◆ maxPivot()

◆ nonzeroPivots()

◆ pivotCoeff()

◆ rank()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ signDeterminant()

◆ solve()

◆ threshold()