Householder QR decomposition of a matrix. More...
#include <HouseholderQR.h>
Public Member Functions | |
MatrixType::RealScalar | absDeterminant () const |
HouseholderQR () | |
Default Constructor. | |
HouseholderQR (Index rows, Index cols) | |
Default Constructor with memory preallocation. | |
MatrixType::RealScalar | logAbsDeterminant () const |
const MatrixType & | matrixQR () const |
template<typename Rhs > | |
const internal::solve_retval < HouseholderQR, Rhs > | solve (const MatrixBase< Rhs > &b) const |
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
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.
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.
MatrixType::RealScalar absDeterminant | ( | ) | const |
References abs().
MatrixType::RealScalar logAbsDeterminant | ( | ) | const |
const MatrixType& matrixQR | ( | ) | const [inline] |
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.
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:
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