Default Constructor.

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

LLT()

Constructs a LLT factorization from a given matrix.

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

See also

LLT(const EigenBase&)

Returns

the adjoint of *this, that is, a const reference to the decomposition itself as the underlying matrix is self-adjoint.

This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:

x = decomposition.adjoint().solve(b) 

Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of matrix

Returns

a reference to *this

Example:

#include <iostream>
#include <Eigen/Dense>
 
using namespace std;
using namespace Eigen;
 
int main()
{
   Matrix2f A, b;
   LLT<Matrix2f> llt;
   A << 2, -1, -1, 3;
   b << 1, 2, 3, 1;
   cout << "Here is the matrix A:\n" << A << endl;
   cout << "Here is the right hand side b:\n" << b << endl;
   cout << "Computing LLT decomposition..." << endl;
   llt.compute(A);
   cout << "The solution is:\n" << llt.solve(b) << endl;
   A(1,1)++;
   cout << "The matrix A is now:\n" << A << endl;
   cout << "Computing LLT decomposition..." << endl;
   llt.compute(A);
   cout << "The solution is now:\n" << llt.solve(b) << endl;
}

Output:

Here is the matrix A:
 2 -1
-1  3
Here is the right hand side b:
1 2
3 1
Computing LLT decomposition...
The solution is:
1.2 1.4
1.4 0.8
The matrix A is now:
 2 -1
-1  4
Computing LLT decomposition...
The solution is now:
    1  1.29
    1 0.571

Reports whether previous computation was successful.

Returns

Success if computation was successful, NumericalIssue if the matrix.appears not to be positive definite.

Returns

a view of the lower triangular matrix L

Returns

the LLT decomposition matrix

TODO: document the storage layout

Returns

a view of the upper triangular matrix U

Performs a rank one update (or dowdate) of the current decomposition. If A = LL^* before the rank one update, then after it we have LL^* = A + sigma * v v^* where v must be a vector of same dimension.

Returns

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

Returns

the matrix represented by the decomposition, i.e., it returns the product: L L^*. This function is provided for debug purpose.

Returns

the solution x of \( A x = b \) using the current decomposition of A.

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

Example:

typedef Matrix<float,Dynamic,2> DataMatrix;
// let's generate some samples on the 3D plane of equation z = 2x+3y (with some noise)
DataMatrix samples = DataMatrix::Random(12,2);
VectorXf elevations = 2*samples.col(0) + 3*samples.col(1) + VectorXf::Random(12)*0.1;
// and let's solve samples * [x y]^T = elevations in least square sense:
Matrix<float,2,1> xy
 = (samples.adjoint() * samples).llt().solve((samples.adjoint()*elevations));
cout << xy << endl;

Output:

2.02
2.97

See also

solveInPlace(), MatrixBase::llt(), SelfAdjointView::llt()