Default Constructor.

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

PartialPivLU()

Constructor.

Parameters

matrix

the matrix of which to compute the LU decomposition.

Warning

The matrix should have full rank (e.g. if it's square, it should be invertible). If you need to deal with non-full rank, use class FullPivLU instead.

Constructor for inplace decomposition

Parameters

matrix

the matrix of which to compute the LU decomposition.

Warning

The matrix should have full rank (e.g. if it's square, it should be invertible). If you need to deal with non-full rank, use class FullPivLU instead.

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

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

Warning

The matrix being decomposed here is assumed to be invertible. If you need to check for invertibility, use class FullPivLU instead.

See also

MatrixBase::inverse(), LU::inverse()

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 permutation matrix P.

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. This function is provided for debug purpose.

This method returns the 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

the solution.

Example:

MatrixXd A = MatrixXd::Random(3,3);
MatrixXd B = MatrixXd::Random(3,2);
cout << "Here is the invertible matrix A:" << endl << A << endl;
cout << "Here is the matrix B:" << endl << B << endl;
MatrixXd X = A.lu().solve(B);
cout << "Here is the (unique) solution X to the equation AX=B:" << endl << X << endl;
cout << "Relative error: " << (A*X-B).norm() / B.norm() << endl;

Output:

Here is the invertible matrix A:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Here is the matrix B:
  0.108   -0.27
-0.0452  0.0268
  0.258   0.904
Here is the (unique) solution X to the equation AX=B:
 0.609   2.68
-0.231  -1.57
  0.51   3.51
Relative error: 3.28e-16

Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution theoretically exists and is unique regardless of b.

See also

TriangularView::solve(), inverse(), computeInverse()