Eigen  3.4.90 (git rev 67eeba6e720c5745abc77ae6c92ce0a44aa7b7ae)
Eigen::HouseholderQR< MatrixType_ > Class Template Reference

## Detailed Description

### template<typename MatrixType_> class Eigen::HouseholderQR< MatrixType_ >

Householder QR decomposition of a matrix.

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

This class performs a QR decomposition of a matrix A into matrices Q and R such that

$\mathbf{A} = \mathbf{Q} \, \mathbf{R}$

by using Householder transformations. Here, Q a unitary matrix and R an upper triangular matrix. The result is stored in a compact way compatible with LAPACK.

Note that no pivoting is performed. This is not a rank-revealing decomposition. If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.

This Householder QR decomposition is faster, but less numerically stable and less feature-full than FullPivHouseholderQR or ColPivHouseholderQR.

This class supports the inplace decomposition mechanism.

MatrixBase::householderQr()
Inheritance diagram for Eigen::HouseholderQR< MatrixType_ >:

## Public Member Functions

MatrixType::RealScalar absDeterminant () const

const HCoeffsType & hCoeffs () const

HouseholderSequenceType householderQ () const

HouseholderQR ()
Default Constructor. More...

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

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

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

MatrixType::RealScalar logAbsDeterminant () const

const MatrixType & matrixQR () const

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

Public Member Functions inherited from Eigen::SolverBase< HouseholderQR< MatrixType_ > >

HouseholderQR< MatrixType_ > & derived ()

const HouseholderQR< MatrixType_ > & derived () const

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

SolverBase ()

const ConstTransposeReturnType transpose () const

Public Member Functions inherited from Eigen::EigenBase< Derived >
EIGEN_CONSTEXPR Index cols () const EIGEN_NOEXCEPT

Derived & derived ()

const Derived & derived () const

EIGEN_CONSTEXPR Index rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index size () const EIGEN_NOEXCEPT

## Protected Member Functions

void computeInPlace ()

Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index Index
The interface type of indices. More...

## ◆ HouseholderQR() [1/4]

template<typename MatrixType_ >
 Eigen::HouseholderQR< MatrixType_ >::HouseholderQR ( )
inline

Default Constructor.

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

## ◆ HouseholderQR() [2/4]

template<typename MatrixType_ >
 Eigen::HouseholderQR< MatrixType_ >::HouseholderQR ( 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.

HouseholderQR()

## ◆ HouseholderQR() [3/4]

template<typename MatrixType_ >
template<typename InputType >
 Eigen::HouseholderQR< MatrixType_ >::HouseholderQR ( 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:

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

## ◆ HouseholderQR() [4/4]

template<typename MatrixType_ >
template<typename InputType >
 Eigen::HouseholderQR< MatrixType_ >::HouseholderQR ( 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.

HouseholderQR(const EigenBase&)

## ◆ absDeterminant()

template<typename MatrixType >
 MatrixType::RealScalar Eigen::HouseholderQR< 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()

## ◆ computeInPlace()

template<typename MatrixType >
 void Eigen::HouseholderQR< MatrixType >::computeInPlace
protected

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 HouseholderQR, HouseholderQR(const MatrixType&)

## ◆ hCoeffs()

template<typename MatrixType_ >
 const HCoeffsType& Eigen::HouseholderQR< MatrixType_ >::hCoeffs ( ) const
inline
Returns
a const reference to the vector of Householder coefficients used to represent the factor Q.

## ◆ householderQ()

template<typename MatrixType_ >
 HouseholderSequenceType Eigen::HouseholderQR< MatrixType_ >::householderQ ( void ) const
inline

This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.

The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object. Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:

Example:

A.setRandom();
HouseholderQR<MatrixXf> qr(A);
Q = qr.householderQ();
thinQ = qr.householderQ() * thinQ;
std::cout << "The complete unitary matrix Q is:\n" << Q << "\n\n";
std::cout << "The thin matrix Q is:\n" << thinQ << "\n\n";
static const RandomReturnType Random()
Definition: Random.h:114
static const IdentityReturnType Identity()
Definition: CwiseNullaryOp.h:801
Matrix< float, Dynamic, Dynamic > MatrixXf
Dynamic×Dynamic matrix of type float.
Definition: Matrix.h:500

Output:

The complete unitary matrix Q is:
-0.676   0.0793    0.713  -0.0788   -0.147
-0.221   -0.322    -0.37   -0.366   -0.759
-0.353   -0.345   -0.214    0.841  -0.0518
0.582   -0.462    0.555    0.176   -0.329
-0.174   -0.747 -0.00907   -0.348    0.539

The thin matrix Q is:
-0.676   0.0793    0.713
-0.221   -0.322    -0.37
-0.353   -0.345   -0.214
0.582   -0.462    0.555
-0.174   -0.747 -0.00907



## ◆ logAbsDeterminant()

template<typename MatrixType >
 MatrixType::RealScalar Eigen::HouseholderQR< 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()

## ◆ matrixQR()

template<typename MatrixType_ >
 const MatrixType& Eigen::HouseholderQR< MatrixType_ >::matrixQR ( ) const
inline
Returns
a reference to the matrix where the Householder QR decomposition is stored in a LAPACK-compatible way.

## ◆ solve()

template<typename MatrixType_ >
template<typename Rhs >
 const Solve Eigen::HouseholderQR< 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, 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:

typedef Matrix<float,3,3> Matrix3x3;
Matrix3x3 m = Matrix3x3::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
x = m.householderQr().solve(y);
assert(y.isApprox(m*x));
cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;
Matrix< float, 3, 3 > Matrix3f
3×3 matrix of type float.
Definition: Matrix.h:500

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


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