Default Constructor.

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

FullPivLU()

Constructor.

Parameters

matrix

the matrix of which to compute the LU decomposition. It is required to be nonzero.

Constructs a LU factorization from a given matrix.

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

See also

FullPivLU(const EigenBase&)

Computes the LU decomposition of the given matrix.

Parameters

matrix

the matrix of which to compute the LU decomposition. It is required to be nonzero.

Returns

a reference to *this

Returns

the determinant of the matrix of which *this is the LU decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the LU decomposition has already been computed.

Note

This is only for square matrices.

For fixed-size matrices of size up to 4, MatrixBase::determinant() offers optimized paths.

Warning

a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow.

See also

MatrixBase::determinant()

Returns

the dimension of the kernel of the matrix of which *this is the LU 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

the image of the matrix, also called its column-space. The columns of the returned matrix will form a basis of the image (column-space).

Parameters

originalMatrix

the original matrix, of which *this is the LU decomposition. The reason why it is needed to pass it here, is that this allows a large optimization, as otherwise this method would need to reconstruct it from the LU decomposition.

Note

If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.

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&).

Example:

Matrix3d m;
m << 1,1,0,
     1,3,2,
     0,1,1;
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Notice that the middle column is the sum of the two others, so the "
     << "columns are linearly dependent." << endl;
cout << "Here is a matrix whose columns have the same span but are linearly independent:"
     << endl << m.fullPivLu().image(m) << endl;

Output:

Here is the matrix m:
1 1 0
1 3 2
0 1 1
Notice that the middle column is the sum of the two others, so the columns are linearly dependent.
Here is a matrix whose columns have the same span but are linearly independent:
1 1
3 1
1 0

See also

kernel()

Returns

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

Note

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

See also

MatrixBase::inverse()

Returns

true if the matrix of which *this is the LU 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 LU 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 LU 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 kernel of the matrix, also called its null-space. The columns of the returned matrix will form a basis of the kernel.

Note

If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.

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&).

Example:

MatrixXf m = MatrixXf::Random(3,5);
cout << "Here is the matrix m:" << endl << m << endl;
MatrixXf ker = m.fullPivLu().kernel();
cout << "Here is a matrix whose columns form a basis of the kernel of m:"
     << endl << ker << endl;
cout << "By definition of the kernel, m*ker is zero:"
     << endl << m*ker << endl;

Output:

Here is the matrix m:
   0.68   0.597   -0.33   0.108   -0.27
 -0.211   0.823   0.536 -0.0452  0.0268
  0.566  -0.605  -0.444   0.258   0.904
Here is a matrix whose columns form a basis of the kernel of m:
 -0.219   0.763
0.00335  -0.447
      0       1
      1       0
 -0.145  -0.285
By definition of the kernel, m*ker is zero:
 7.45e-09  1.49e-08
-1.86e-09 -4.05e-08
        0 -2.98e-08

See also

image()

Returns

the LU decomposition matrix: the upper-triangular part is U, the unit-lower-triangular part is L (at least for square matrices; in the non-square case, special care is needed, see the documentation of class FullPivLU).

See also

matrixL(), matrixU()

Returns

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

Returns

the number of nonzero pivots in the LU 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 permutation matrix P

See also

permutationQ()

Returns

the permutation matrix Q

See also

permutationP()

Returns

the rank of the matrix of which *this is the LU 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

an estimate of the reciprocal condition number of the matrix of which *this is the LU decomposition.

Returns

the matrix represented by the decomposition, i.e., it returns the product: \( P^{-1} L U Q^{-1} \). This function is provided for debug purposes.

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 LU 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.

lu.setThreshold(Eigen::Default); 

See the documentation of setThreshold(const RealScalar&).

Returns

a solution x to the equation Ax=b, where A is the matrix of which *this is the LU decomposition.

Parameters

b	the right-hand-side of the equation to solve. Can be a vector or a matrix, the only requirement in order for the equation to make sense is that b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.

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. If you need a complete analysis of the space of solutions, take the one solution obtained by this method and add to it elements of the kernel, as determined by kernel().

Example:

Matrix<float,2,3> m = Matrix<float,2,3>::Random();
Matrix2f y = Matrix2f::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
Matrix<float,3,2> x = m.fullPivLu().solve(y);
if((m*x).isApprox(y))
{
  cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;
}
else
  cout << "The equation mx=y does not have any solution." << endl;

Output:

Here is the matrix m:
  0.68  0.566  0.823
-0.211  0.597 -0.605
Here is the matrix y:
 -0.33 -0.444
 0.536  0.108
Here is a solution x to the equation mx=y:
     0      0
 0.291 -0.216
  -0.6 -0.391

See also

TriangularView::solve(), kernel(), inverse()

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

See the documentation of setThreshold(const RealScalar&).