Householder rank-revealing QR decomposition of a matrix with column-pivoting. More...
Public Member Functions | |
MatrixType::RealScalar | absDeterminant () const |
ColPivHouseholderQR () | |
Default Constructor. | |
ColPivHouseholderQR (Index rows, Index cols) | |
Default Constructor with memory preallocation. | |
Index | dimensionOfKernel () const |
HouseholderSequenceType | householderQ (void) const |
const internal::solve_retval < ColPivHouseholderQR, typename MatrixType::IdentityReturnType > | inverse () const |
bool | isInjective () const |
bool | isInvertible () const |
bool | isSurjective () const |
MatrixType::RealScalar | logAbsDeterminant () const |
const MatrixType & | matrixQR () const |
RealScalar | maxPivot () const |
Index | nonzeroPivots () const |
Index | rank () const |
ColPivHouseholderQR & | setThreshold (Default_t) |
ColPivHouseholderQR & | setThreshold (const RealScalar &threshold) |
template<typename Rhs > | |
const internal::solve_retval < ColPivHouseholderQR, Rhs > | solve (const MatrixBase< Rhs > &b) const |
RealScalar | threshold () const |
Householder rank-revealing QR decomposition of a matrix with column-pivoting.
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, Q and R such that
by using Householder transformations. Here, P is a permutation matrix, Q a unitary matrix and R an upper triangular matrix.
This decomposition performs column pivoting in order to be rank-revealing and improve numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
ColPivHouseholderQR | ( | ) | [inline] |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
ColPivHouseholderQR | ( | 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 |
Index dimensionOfKernel | ( | ) | const [inline] |
ColPivHouseholderQR< MatrixType >::HouseholderSequenceType householderQ | ( | void | ) | const |
const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> inverse | ( | ) | const [inline] |
bool isInjective | ( | ) | const [inline] |
bool isInvertible | ( | ) | const [inline] |
bool isSurjective | ( | ) | const [inline] |
MatrixType::RealScalar logAbsDeterminant | ( | ) | const |
const MatrixType& matrixQR | ( | ) | const [inline] |
RealScalar maxPivot | ( | ) | const [inline] |
Index nonzeroPivots | ( | ) | const [inline] |
Index rank | ( | ) | const [inline] |
ColPivHouseholderQR& 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&).
ColPivHouseholderQR& 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.
threshold | The new value to use as the threshold. |
A pivot will be considered nonzero if its absolute value is strictly greater than where maxpivot is the biggest pivot.
If you want to come back to the default behavior, call setThreshold(Default_t)
const internal::solve_retval<ColPivHouseholderQR, 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:
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.colPivHouseholderQr().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
RealScalar threshold | ( | ) | const [inline] |
Returns the threshold that will be used by certain methods such as rank().
See the documentation of setThreshold(const RealScalar&).