Tensorium — Core Module

This directory contains the core linear algebra structures and operations of the Tensorium library. It provides the foundational vector, matrix, tensor, and derivative tools used across all higher-level modules, including symbolic computation, spectral methods, and numerical relativity.

Purpose

The module offers:

• Generic, aligned data structures for Vector, Matrix, and Tensor with SIMD support
• Fundamental linear algebra operations (addition, scaling, dot product, etc.)
• Matrix multiplication, transposition, inversion, and LU decomposition
• Generalized tensors of arbitrary rank with contraction and tensor product
• First and fourth-order derivative schemes (finite difference and spectral)

Structure

Core/
├── Vector.hpp         // Fixed-size aligned vectors with SIMD operations
├── Matrix.hpp         // Dense matrix with optimized arithmetic and inversion
├── Tensor.hpp         // Arbitrary-rank tensor with contraction and layout
├── LinearSolver.hpp   // Gaussian, Jacobi and LU solvers
├── Derivate.hpp       // Centered finite difference and spectral derivatives

Features

Vector

• Generic Vector<T> with aligned storage
• Operations: add, subtract, scale, dot product, cross product (3D), norm
• Support for SIMD acceleration (SSE/AVX/AVX512)

Matrix

• Matrix<T> with dense layout and cache-aware optimizations
• SIMD-aware matrix multiplication with unrolling and OpenMP support
• LU decomposition, determinant, trace, inverse, transpose

Tensor

• Tensor<T, Rank> with full support for arbitrary-rank tensor algebra
• Compile-time rank enforcement
• Operations: contraction, tensor product, slicing, transposition

Derivative

• Derivate<T> and DerivateND<T, Rank> structures
• First-order and fourth-order centered finite difference derivatives
• Optional support for spectral methods via Spectral.hpp

Linear Solvers

• Solvers for linear systems: Gaussian elimination, Jacobi iterations
• Designed to work with both matrices and tensors

Internal Dependencies

• Depends on SIMD/ for vectorization and aligned allocation
• No external linear algebra or math libraries
• Compatible with Symbolics/ and DiffGeometry/ layers

Example Usage

tensorium::Matrix<double> A(3, 3);
tensorium::Vector<double> b(3);
 
// Fill A and b...
 
auto x = tensorium::gauss_solve(A, b);
 
tensorium::Matrix<double> B = A.transpose();
auto det = tensorium::det_mat(A);
 
tensorium::Tensor<double, 3> T({4, 4, 4});
auto T2 = tensorium::contract_tensor<0, 2>(T);

Status

The module is stable and used across all parts of the Tensorium library. It is designed for performance-critical numerical code and supports both analytical and simulation backends.

Notes

• Matrix sizes are expected to fit in cache (optimized kernels exist for 4×4, 8×8, 16×16).
• Tensor operations are layout-aware for compatibility with differential geometry modules.