class Eigen::FullPivHouseholderQR< MatrixType >
Householder rank-revealing QR decomposition of a matrix with full 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 a very prudent full pivoting in order to be rank-revealing and achieve optimal numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
- See Also
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)
References FullPivHouseholderQR< MatrixType >::threshold().
This method finds a solution x to the equation Ax=b, where A is the matrix of which
*this is the QR decomposition.
|b||the right-hand-side of the equation to solve.|
- the exact or least-square solution if the rank is greater or equal to the number of columns of A, and an arbitrary solution otherwise.
- 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
If there exists more than one solution, this method will arbitrarily choose one.
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
x = m.fullPivHouseholderQr().solve(y);
cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;
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