Lingebra

Lingebra is a high-performance, lightweight linear algebra library written in C++20. It features custom memory management, AVX-friendly data structures, and optimized algorithms for matrix operations.

Key Features

• High-Performance Matrix Multiplication: Implements the IKJ loop interchange algorithm to maximize CPU cache locality, achieving $O(N^3)$ performance.
• Optimized Memory Layout:
  • Uses a flat 1D array representation for matrices to ensure spatial locality.
  • Custom Vector container uses 32-byte memory alignment (via aligned_alloc), enabling the compiler to auto-vectorize loops (AVX/SIMD).
• Fast Inversion & Determinants: Replaces naive $O(N!)$ Laplace expansion with LU Decomposition ( $O(N^3)$ ) for instant calculations on large matrices.
• Modern C++ Design:
  • Implements the Rule of Five for efficient resource management (Move Semantics).
  • RAII-compliant memory safety.
  • No external dependencies (Standard Library only).

Build & Installation

Lingebra is a source-based library. To use it, simply include the headers and compile the source files with your project.

Requirements

• C++ Compiler supporting C++17 or later (e.g., GCC, Clang, MSVC).
• Optimization flags (e.g., -O3) are highly recommended to enable vectorization.

Compilation:

To compile a project using Lingebra:

# Recommended: Enable O3 optimization for SIMD/AVX support
g++ main.cpp src/matrix.cpp src/vector.cpp -O3 -o my_app

API Reference

Class: Lingebra::Matrix

Method / Operator	Signature	Description
Constructor	Matrix(size_t r, size_t c)	Initializes a matrix of dimensions r × c filled with zeros.
Constructor	Matrix({ {v1, v2}, ... })	Initializes a matrix from a nested initializer list (e.g., {{1,2}, {3,4}}).
Shape	Vector shape()	Returns a vector containing {nrows, ncols}.
Multiplication	Matrix matmul(const Matrix& B)	Performs matrix multiplication (A × B) using the optimized IKJ algorithm.
Determinant	double determinant()	Calculates the determinant using LU Decomposition (O(N³))
Inverse	Matrix inverse()	Calculates the inverse matrix using LU Decomposition and Back Substitution.
Transpose	Matrix transpose()	Returns a new matrix that is the transpose of the current one.
Cofactors	Matrix cofactors()	Returns the cofactor matrix (derived efficiently via Inverse).
Access	double& operator()(i, j)	Logical element access (row i, column j).
Raw Access	double* data_ptr()	Returns a raw pointer to the underlying flat 1D array (useful for optimizations).
Arithmetic	+, -, +=, -=	Standard element-wise matrix addition and subtraction.
Scaling	*, *=	Scalar multiplication (Matrix * double).

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Performance

Operation	Naive Implementation	Lingebra Implementation
Multiplication	O(N³) (Cache Miss Heavy)	O(N³) (Cache Friendly / IKJ)
Determinant	O(N!) (Laplace Expansion)	O(N³) (LU Decomposition)
Inverse	O(N!) (Cofactors)	O(N³) (LU + Substitution)
Memory Access	Pointer-to-Pointer (double**)	Contiguous Aligned Flat Array