Eigen::CompleteOrthogonalDecomposition

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via CompleteOrthogonalDecomposition::compute(const* MatrixType&).

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also

CompleteOrthogonalDecomposition()

Constructs a complete orthogonal decomposition from a given matrix.

This constructor computes the complete orthogonal decomposition of the matrix matrix by calling the method compute(). The default threshold for rank determination will be used. It is a short cut for:

CompleteOrthogonalDecomposition<MatrixType> cod(matrix.rows(),
                                                matrix.cols());
cod.setThreshold(Default);
cod.compute(matrix);

See also

compute()

Constructs a complete orthogonal decomposition from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See also

CompleteOrthogonalDecomposition(const EigenBase&)

Returns

the absolute value of the determinant of the matrix of which *this is the complete orthogonal decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the complete orthogonal 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()

Overwrites rhs with \( \mathbf{Z}^* * \mathbf{rhs} \).

Overwrites rhs with \( \mathbf{Z} * \mathbf{rhs} \) or \( \mathbf{\overline Z} * \mathbf{rhs} \) if Conjugate is set to true.

Returns

a const reference to the column permutation matrix

Performs the complete orthogonal decomposition of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

See also

class CompleteOrthogonalDecomposition, CompleteOrthogonalDecomposition(const MatrixType&)

Returns

the dimension of the kernel of the matrix of which *this is the complete orthogonal decomposition.

Note

This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Returns

a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

Returns

the matrix Q as a sequence of householder transformations

Reports whether the complete orthogonal decomposition was successful.

Note

This function always returns Success. It is provided for compatibility with other factorization routines.

Returns

Success

Returns

true if the matrix of which *this is the decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.

Note

This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Returns

true if the matrix of which *this is the complete orthogonal decomposition is invertible.

Note

This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Returns

true if the matrix of which *this is the decomposition represents a surjective linear map; false otherwise.

Note

This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Returns

the natural log of the absolute value of the determinant of the matrix of which *this is the complete orthogonal decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the complete orthogonal 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()

Returns

a reference to the matrix where the complete orthogonal decomposition is stored

Returns

a reference to the matrix where the complete orthogonal decomposition is stored.

Warning

The strict lower part and

cols() - rank() 

right columns of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use

matrixT().template triangularView<Upper>() 

For rank-deficient matrices, use

matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Returns

the matrix Z.

Returns

the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of R.

Returns

the number of nonzero pivots in the complete orthogonal decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.

See also

rank()

Returns

the pseudo-inverse of the matrix of which *this is the complete orthogonal decomposition.

Warning

: Do not compute this->pseudoInverse()*rhs to solve a linear systems. It is more efficient and numerically stable to call this->solve(rhs).

Returns

the rank of the matrix of which *this is the complete orthogonal decomposition.

Note

This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

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. Most be called before calling compute().

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.

Parameters

threshold

The new value to use as the threshold.

A pivot will be considered nonzero if its absolute value is strictly greater than \( \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \) where maxpivot is the biggest pivot.

If you want to come back to the default behavior, call setThreshold(Default_t)

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&).

This method computes the minimum-norm solution X to a least squares problem

\[\mathrm{minimize} \|A X - B\|, \]

where A is the matrix of which *this is the complete orthogonal decomposition.

Parameters

b	the right-hand sides of the problem to solve.

Returns

a solution.

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).

Returns

a const reference to the vector of Householder coefficients used to represent the factor Z.

For advanced uses only.