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Eigen  3.2.5
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HouseholderQR< _MatrixType > Class Template Reference
[QR module]

Householder QR decomposition of a matrix. More...

List of all members.

Public Member Functions

MatrixType::RealScalar absDeterminant () const
HouseholderQRcompute (const MatrixType &matrix)
const HCoeffsType & hCoeffs () const
HouseholderSequenceType householderQ () const
 HouseholderQR (const MatrixType &matrix)
 Constructs a QR factorization from a given matrix.
 HouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 HouseholderQR ()
 Default Constructor.
MatrixType::RealScalar logAbsDeterminant () const
const MatrixType & matrixQR () const
template<typename Rhs >
const internal::solve_retval
< HouseholderQR, Rhs > 
solve (const MatrixBase< Rhs > &b) const

Detailed Description

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

Householder QR decomposition of a matrix.

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.

See also:
MatrixBase::householderQr()

Constructor & Destructor Documentation

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 ( 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:
HouseholderQR()
HouseholderQR ( const MatrixType &  matrix  )  [inline]

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);
See also:
compute()

References HouseholderQR< _MatrixType >::compute().


Member Function Documentation

MatrixType::RealScalar absDeterminant (  )  const [inline]
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()
HouseholderQR< MatrixType > & compute ( const MatrixType &  matrix  )  [inline]

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 HouseholderQR, HouseholderQR(const MatrixType&)

Referenced by HouseholderQR< _MatrixType >::HouseholderQR().

const HCoeffsType& hCoeffs (  )  const [inline]
Returns:
a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

HouseholderSequenceType 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:

MatrixXf A(MatrixXf::Random(5,3)), thinQ(MatrixXf::Identity(5,3)), Q;
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";

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

MatrixType::RealScalar logAbsDeterminant (  )  const [inline]
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 in a LAPACK-compatible way.
const internal::solve_retval<HouseholderQR, 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:
b the 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:

typedef Matrix<float,3,3> Matrix3x3;
Matrix3x3 m = Matrix3x3::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.householderQr().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

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