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Eigen
3.4.0
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Householder QR decomposition of a matrix.
_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.
Public Member Functions | |
MatrixType::RealScalar | absDeterminant () const |
const HCoeffsType & | hCoeffs () const |
HouseholderSequenceType | householderQ () const |
HouseholderQR () | |
Default Constructor. | |
template<typename InputType> | |
HouseholderQR (const EigenBase< InputType > &matrix) | |
Constructs a QR factorization from a given matrix. | |
template<typename InputType> | |
HouseholderQR (EigenBase< InputType > &matrix) | |
Constructs a QR factorization from a given matrix. | |
HouseholderQR (Index rows, Index cols) | |
Default Constructor with memory preallocation. | |
MatrixType::RealScalar | logAbsDeterminant () const |
const MatrixType & | matrixQR () const |
template<typename Rhs> | |
const Solve< HouseholderQR, Rhs > | solve (const MatrixBase< Rhs > &b) const |
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AdjointReturnType | adjoint () const |
HouseholderQR< _MatrixType > & | derived () |
const HouseholderQR< _MatrixType > & | derived () const |
const Solve< HouseholderQR< _MatrixType >, Rhs > | solve (const MatrixBase< Rhs > &b) const |
SolverBase () | |
ConstTransposeReturnType | transpose () const |
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EIGEN_CONSTEXPR Index | cols () const EIGEN_NOEXCEPT |
HouseholderQR< _MatrixType > & | derived () |
const HouseholderQR< _MatrixType > & | derived () const |
EIGEN_CONSTEXPR Index | rows () const EIGEN_NOEXCEPT |
EIGEN_CONSTEXPR Index | size () const EIGEN_NOEXCEPT |
Protected Member Functions | |
void | computeInPlace () |
Additional Inherited Members | |
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typedef Eigen::Index | Index |
The interface type of indices. | |
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inline |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via HouseholderQR::compute(const MatrixType&).
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inline |
Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
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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:
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inlineexplicit |
Constructs a QR factorization from a given matrix.
This overloaded constructor is provided for inplace decomposition when MatrixType
is a Eigen::Ref.
MatrixType::RealScalar Eigen::HouseholderQR< MatrixType >::absDeterminant | ( | ) | const |
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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.
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inline |
Q
.For advanced uses only.
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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:
Output:
The complete unitary matrix Q is: -0.21 0.239 -0.639 0.217 -0.666 0.0208 -0.579 0.242 -0.489 -0.606 0.702 0.575 0.0784 -0.356 -0.206 0.646 -0.43 -0.0648 0.62 -0.0938 0.211 -0.304 -0.723 -0.45 0.372 The thin matrix Q is: -0.21 0.239 -0.639 0.0208 -0.579 0.242 0.702 0.575 0.0784 0.646 -0.43 -0.0648 0.211 -0.304 -0.723
MatrixType::RealScalar Eigen::HouseholderQR< MatrixType >::logAbsDeterminant | ( | ) | const |
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inline |
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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.
b | the right-hand-side of the equation to solve. |
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:
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:
Output:
Here is the matrix m: 0.196 0.68 0.597 -0.823 -0.211 0.823 -0.192 0.566 -0.605 Here is the matrix y: -0.33 0.108 -0.27 0.536 -0.0452 0.0268 -0.444 0.258 0.904 Here is a solution x to the equation mx=y: -0.256 -0.131 -0.899 -0.619 0.293 0.507 0.237 -0.11 -0.736