Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via ColPivHouseholderQR::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

ColPivHouseholderQR()

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:

ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
qr.compute(matrix);

See also

compute()

Constructs a QR factorization from a given matrix.

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

See also

ColPivHouseholderQR(const EigenBase&)

Returns

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

Returns

a const reference to the column permutation matrix

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.

See also

class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)

Returns

the dimension of the kernel of the matrix of which *this is the QR 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. You can extract the meaningful part only by using:

qr.householderQ().setLength(qr.nonzeroPivots()) 

Reports whether the QR factorization was successful.

Note

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

Returns

Success

Returns

the inverse of the matrix of which *this is the QR decomposition.

Note

If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.

Returns

true if the matrix of which *this is the QR 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 QR 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 QR 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 QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR 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 Householder QR decomposition is stored

Returns

a reference to the matrix where the result Householder QR is stored

Warning

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

matrixR().template triangularView<Upper>() 

For rank-deficient matrices, use

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

Returns

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

Returns

the number of nonzero pivots in the QR 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 rank of the matrix of which *this is the QR 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. 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.

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 finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition, if any exists.

Parameters

b	the right-hand-side of the equation to solve.

Returns

a solution.

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

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

See the documentation of setThreshold(const RealScalar&).