Linalg.hpp File Reference

Linear algebra operations (solve, inverse, determinant, eigendecomposition, norms). More...

#include "metis/core/MetisConcepts.hpp"
#include "metis/core/MetisError.hpp"
#include "metis/core/MetisTypes.hpp"
#include "metis/math/Arithmetic.hpp"
#include "metis/math/Logic.hpp"
#include "metis/math/Trig.hpp"
#include <Eigen/Dense>
#include <Eigen/IterativeLinearSolvers>
#include <Eigen/SparseCholesky>
#include <Eigen/SparseLU>
#include <Eigen/SparseQR>
#include <algorithm>
#include <array>
#include <casadi/casadi.hpp>
#include <cmath>
#include <complex>
#include <functional>
#include <limits>
#include <numeric>
#include <string>
#include <unsupported/Eigen/IterativeSolvers>
#include <vector>

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Classes
struct	metis::LinearSolvePolicy
	Configuration for linear system solve backend and algorithm. More...
class	metis::detail::FunctionalPreconditioner
struct	metis::EigenDecomposition< Scalar >
	Result of eigendecomposition: eigenvalues and eigenvectors. More...

Namespaces
namespace	metis
namespace	metis::detail
	Smooth approximation of ReLU function: softplus(x) = (1/beta) * log(1 + exp(beta * x)).

Enumerations
enum class	metis::LinearSolveBackend { metis::Dense , metis::SparseDirect , metis::IterativeKrylov }
	Backend selection for linear system solves. More...
enum class	metis::DenseLinearSolver {   metis::ColPivHouseholderQR , metis::PartialPivLU , metis::FullPivLU , metis::LLT ,   metis::LDLT }
	Dense linear solver algorithm. More...
enum class	metis::SparseDirectLinearSolver { metis::SparseLU , metis::SparseQR , metis::SimplicialLLT , metis::SimplicialLDLT }
	Sparse direct solver algorithm. More...
enum class	metis::IterativeKrylovSolver { metis::BiCGSTAB , metis::GMRES }
	Iterative Krylov solver algorithm. More...
enum class	metis::IterativePreconditioner { metis::None , metis::Diagonal }
	Preconditioner for iterative solvers. More...
enum class	metis::NormType { metis::L1 , metis::L2 , metis::Inf , metis::Frobenius }
	Norm type selection. More...

Functions
void	metis::detail::validate_linear_solve_dims (Eigen::Index a_rows, Eigen::Index a_cols, Eigen::Index b_rows, const std::string &context)
void	metis::detail::validate_square_required (Eigen::Index rows, Eigen::Index cols, const std::string &context, const std::string &solver_name)
void	metis::detail::validate_iterative_policy (const LinearSolvePolicy &policy, const std::string &context)
std::function< NumericVector(const NumericVector &)>	metis::detail::make_preconditioner (const SparseMatrix &A, const LinearSolvePolicy &policy)
template<typename MatrixLike>
SparseMatrix	metis::detail::dense_to_sparse (const MatrixLike &A)
template<typename DerivedB>
auto	metis::detail::solve_sparse_direct_numeric (const SparseMatrix &A, const Eigen::MatrixBase< DerivedB > &b, const LinearSolvePolicy &policy)
template<typename Solver, typename DerivedB>
auto	metis::detail::solve_iterative_with_solver (Solver &solver, const Eigen::MatrixBase< DerivedB > &b)
template<typename DerivedB>
auto	metis::detail::solve_iterative_numeric (const SparseMatrix &A, const Eigen::MatrixBase< DerivedB > &b, const LinearSolvePolicy &policy)
template<typename DerivedA, typename DerivedB>
auto	metis::detail::solve_dense_numeric (const Eigen::MatrixBase< DerivedA > &A, const Eigen::MatrixBase< DerivedB > &b, const LinearSolvePolicy &policy)
template<typename DerivedA, typename DerivedB>
auto	metis::solve (const Eigen::MatrixBase< DerivedA > &A, const Eigen::MatrixBase< DerivedB > &b)
	Solves linear system Ax = b using the default backend policy.
template<typename DerivedA, typename DerivedB>
auto	metis::solve (const Eigen::MatrixBase< DerivedA > &A, const Eigen::MatrixBase< DerivedB > &b, const LinearSolvePolicy &policy)
	Solves linear system Ax = b using an explicit backend policy.
template<typename DerivedB>
auto	metis::solve (const SparseMatrix &A, const Eigen::MatrixBase< DerivedB > &b)
	Solve a numeric sparse linear system with a sparse-aware default backend.
template<typename DerivedB>
auto	metis::solve (const SparseMatrix &A, const Eigen::MatrixBase< DerivedB > &b, const LinearSolvePolicy &policy)
	Solve a numeric sparse linear system using an explicit backend policy.
template<typename DerivedX, typename DerivedY>
auto	metis::outer (const Eigen::MatrixBase< DerivedX > &x, const Eigen::MatrixBase< DerivedY > &y)
	Computes outer product x * y^T.
template<typename DerivedA, typename DerivedB>
auto	metis::dot (const Eigen::MatrixBase< DerivedA > &a, const Eigen::MatrixBase< DerivedB > &b)
	Computes dot product of two vectors.
template<typename DerivedA, typename DerivedB>
auto	metis::cross (const Eigen::MatrixBase< DerivedA > &a, const Eigen::MatrixBase< DerivedB > &b)
	Computes 3D cross product.
template<typename Derived>
auto	metis::inv (const Eigen::MatrixBase< Derived > &A)
	Computes matrix inverse.
template<typename Derived>
auto	metis::det (const Eigen::MatrixBase< Derived > &A)
	Computes matrix determinant.
template<typename DerivedA, typename DerivedB>
auto	metis::inner (const Eigen::MatrixBase< DerivedA > &a, const Eigen::MatrixBase< DerivedB > &b)
	Computes inner product of two vectors (dot product).
template<typename Derived>
auto	metis::pinv (const Eigen::MatrixBase< Derived > &A)
	Computes Moore-Penrose pseudo-inverse.
template<typename Derived>
auto	metis::norm (const Eigen::MatrixBase< Derived > &x, NormType type=NormType::L2)
	Computes vector/matrix norm.
template<typename Scalar>
MetisVector< Scalar >	metis::detail::normalize_vector (const MetisVector< Scalar > &v)
template<typename Scalar>
MetisVector< Scalar >	metis::detail::best_eigenvector_candidate (const std::array< MetisVector< Scalar >, 3 > &cands)
template<typename Scalar>
MetisVector< Scalar >	metis::detail::symmetric_eigenvector_2x2 (const MetisMatrix< Scalar > &A, const Scalar &lambda)
template<typename Scalar>
MetisVector< Scalar >	metis::detail::symmetric_eigenvector_3x3 (const MetisMatrix< Scalar > &A, const Scalar &lambda)
template<typename Scalar>
Scalar	metis::detail::determinant_3x3 (const MetisMatrix< Scalar > &A)
template<typename Scalar>
void	metis::detail::sort_eigenpairs (MetisVector< Scalar > &eigenvalues, MetisMatrix< Scalar > &eigenvectors)
template<typename Scalar>
EigenDecomposition< Scalar >	metis::detail::eig_symmetric_symbolic (const MetisMatrix< Scalar > &A)
template<typename Derived>
auto	metis::eig (const Eigen::MatrixBase< Derived > &A)
	Computes the eigendecomposition of a square matrix with a real spectrum.
template<typename Derived>
auto	metis::eig_symmetric (const Eigen::MatrixBase< Derived > &A)
	Computes the eigendecomposition of a symmetric matrix.
template<typename T>
std::tuple< T, T, T, T, T, T >	metis::inv_symmetric_3x3_explicit (const T &m11, const T &m22, const T &m33, const T &m12, const T &m23, const T &m13)
	Explicit inverse of a symmetric 3x3 matrix.
SparseMatrix	metis::sparse_from_triplets (int rows, int cols, const std::vector< SparseTriplet > &triplets)
	Create sparse matrix from triplets.
SparseMatrix	metis::to_sparse (const NumericMatrix &dense, double tol=0.0)
	Convert dense matrix to sparse.
NumericMatrix	metis::to_dense (const SparseMatrix &sparse)
	Convert sparse matrix to dense.
SparseMatrix	metis::sparse_identity (int n)
	Create identity sparse matrix.

Variables
template<typename DerivedA>
constexpr bool	metis::detail::dense_solver_square_compatible
template<typename Scalar>
constexpr bool	metis::is_numeric_scalar_v = std::is_floating_point_v<Scalar> || std::is_integral_v<Scalar>
	Compile-time check for numeric scalar types.

Detailed Description

Linear algebra operations (solve, inverse, determinant, eigendecomposition, norms).

See also

Arithmetic.hpp