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 Eigen  3.3.9
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
 _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, 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.

MatrixBase::fullPivHouseholderQr()

## Public Member Functions

MatrixType::RealScalar absDeterminant () const

const PermutationTypecolsPermutation () const

template<typename InputType >
FullPivHouseholderQR< MatrixType > & compute (const EigenBase< InputType > &matrix)

Index dimensionOfKernel () const

FullPivHouseholderQR ()
Default Constructor. More...

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

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

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

const HCoeffsType & hCoeffs () const

const Inverse< FullPivHouseholderQRinverse () 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

## ◆ FullPivHouseholderQR() [1/4]

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&).

## ◆ FullPivHouseholderQR() [2/4]

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.

FullPivHouseholderQR()

## ◆ FullPivHouseholderQR() [3/4]

template<typename _MatrixType >
template<typename InputType >
 Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( 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:

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

## ◆ FullPivHouseholderQR() [4/4]

template<typename _MatrixType >
template<typename InputType >
 Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( 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.

FullPivHouseholderQR(const EigenBase&)

## ◆ absDeterminant()

template<typename MatrixType >
 MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType >::absDeterminant
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.
logAbsDeterminant(), MatrixBase::determinant()

## ◆ colsPermutation()

template<typename _MatrixType >
 const PermutationType& Eigen::FullPivHouseholderQR< _MatrixType >::colsPermutation ( ) const
inline
Returns
a const reference to the column permutation matrix

## ◆ compute()

template<typename _MatrixType >
template<typename InputType >
 FullPivHouseholderQR& 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.

class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)

## ◆ dimensionOfKernel()

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&).

## ◆ hCoeffs()

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.

## ◆ inverse()

template<typename _MatrixType >
 const Inverse 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.

## ◆ isInjective()

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&).

## ◆ isInvertible()

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&).

## ◆ isSurjective()

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&).

## ◆ logAbsDeterminant()

template<typename MatrixType >
 MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType >::logAbsDeterminant
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.
absDeterminant(), MatrixBase::determinant()

## ◆ matrixQ()

template<typename MatrixType >
 FullPivHouseholderQR< MatrixType >::MatrixQReturnType Eigen::FullPivHouseholderQR< MatrixType >::matrixQ ( void ) const
inline
Returns
Expression object representing the matrix Q

## ◆ matrixQR()

template<typename _MatrixType >
 const MatrixType& Eigen::FullPivHouseholderQR< _MatrixType >::matrixQR ( ) const
inline
Returns
a reference to the matrix where the Householder QR decomposition is stored

## ◆ maxPivot()

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.

## ◆ nonzeroPivots()

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.
rank()

## ◆ 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&).

## ◆ rowsTranspositions()

template<typename _MatrixType >
 const IntDiagSizeVectorType& Eigen::FullPivHouseholderQR< _MatrixType >::rowsTranspositions ( ) const
inline
Returns
a const reference to the vector of indices representing the rows transpositions

## ◆ setThreshold() [1/2]

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
 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 >
 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&).

## ◆ solve()

template<typename _MatrixType >
template<typename Rhs >
 const Solve 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
 b the 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:
-1 -0.0827  -0.906
-0.737  0.0655   0.358
0.511  -0.562   0.359
Here is the matrix y:
0.869  0.662 0.0594
-0.233 -0.931  0.342
0.0388 -0.893 -0.985
Here is a solution x to the equation mx=y:
-0.117  0.626 -0.278
-0.667   1.18   1.56
-0.77  -1.53 0.0985


## ◆ threshold()

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:
Eigen::DenseBase::Random
static const RandomReturnType Random()
Definition: Random.h:113