For direct methods, the solution is computed at the machine precision.

Sometimes, the solution need not be too accurate. In this case, the iterative methods are more suitable and the desired accuracy can be set before the solve step using setTolerance.

Ordering methods for sparse matrices. They are typically used to reduce the number of elements during the sparse matrix decomposition (LLT, LU, QR). Precisely, in a preprocessing step, a permutation matrix P is computed using those ordering methods and applied to the columns of the matrix. Using for instance the sparse Cholesky decomposition, it is expected that the nonzeros elements in LLT(A*P) will be much smaller than that in LLT(A).

A conjugate gradient solver for sparse self-adjoint problems.

This class allows to solve for A.x = b sparse linear problems using a conjugate gradient algorithm. The sparse matrix A must be selfadjoint.

The maximal number of iterations and tolerance value can be controlled via the setMaxIterations and setTolerance methods. The defaults are the size of the problem for the maximal number of iterations and epsilon for the tolerance

A bi conjugate gradient stabilized solver for sparse square problems.

This class allows to solve for A.x = b sparse linear problems using a bi conjugate gradient stabilized algorithm. The vectors x and b can be either dense or sparse.

The maximal number of iterations and tolerance value can be controlled via the setMaxIterations and setTolerance methods. The defaults are the size of the problem for the maximal number of iterations and epsilon for the tolerance

Sparse supernodal LU factorization for general matrices.

This class implements the supernodal LU factorization for general matrices. It uses the main techniques from the sequential SuperLU package. It handles transparently real and complex arithmetics with single and double precision, depending on the scalar type of your input matrix. The code has been optimized to provide BLAS-3 operations during supernode-panel updates. It benefits directly from the built-in high-performant Eigen BLAS routines. Moreover, when the size of a supernode is very small, the BLAS calls are avoided to enable a better optimization from the compiler. For best performance, you should compile it with NDEBUG flag to avoid the numerous bounds checking on vectors.

An important parameter of this class is the ordering method. It is used to reorder the columns (and eventually the rows) of the matrix to reduce the number of new elements that are created during numerical factorization. The cheapest method available is COLAMD. See OrderingMethods module for the list of built-in and external ordering methods.

Sparse left-looking rank-revealing QR factorization.

This class implements a left-looking rank-revealing QR decomposition of sparse matrices. When a column has a norm less than a given tolerance it is implicitly permuted to the end. The QR factorization thus obtained is given by A*P = Q*R where R is upper triangular or trapezoidal.

P is the column permutation which is the product of the fill-reducing and the rank-revealing permutations.

Q is the orthogonal matrix represented as products of Householder reflectors.

R is the sparse triangular or trapezoidal matrix. The later occurs when A is rank-deficient.

Initializes the iterative solver for the sparsity pattern of the matrix A for further solving Ax=b problems.

Initializes the iterative solver with the numerical values of the matrix A for further solving Ax=b problems.

Initializes the iterative solver with the matrix A for further solving Ax=b problems.

The compute method is equivalent to calling both analyzePattern and factorize.

An expression of the solution x of Ax=b using the current decomposition of A.

• Success if the iterations converged or computation was succesful
• NumericalIssue if the factorization reports a numerical problem
• NoConvergence if the iterations are not converged
• InvalidInput if the input matrix is invalid

The tolerance threshold used by the stopping criteria.

Sets the tolerance threshold used by the stopping criteria.

This value is used as an upper bound to the relative residual error: |Ax-b|/|b|. The default value is the machine precision given by epsilon

The max number of iterations. It is either the value setted by setMaxIterations or, by default, twice the number of columns of the matrix.

Sets the max number of iterations. Default is twice the number of columns of the matrix.

The tolerance error reached during the last solve. It is a close approximation of the true relative residual error |Ax-b|/|b|.

The number of iterations performed during the last solve

Returns the b sparse upper triangular matrix R of the QR factorization.

Returns the matrix Q as products of sparse Householder reflectors.

Returns the number of non linearly dependent columns as determined by the pivoting threshold.

Sets the threshold that is used to determine linearly dependent columns during the factorization.

In practice, if during the factorization the norm of the column that has to be eliminated is below this threshold, then the entire column is treated as zero, and it is moved at the end.

Indicate that the pattern of the input matrix is symmetric

Returns the matrix L

Returns the matrix U

The determinant of the matrix.

The absolute value of the determinant of the matrix of which *this is the QR decomposition.

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.

A number representing the sign of the determinant

The natural log of the absolute value of the determinant of the matrix of which this is the QR decomposition

This method is useful to work around the risk of overflow/underflow that's inherent to the determinant computation.