LAPACK Linear Algebra Library

LAPACK (Linear Algebra PACKage) is a standard software library for numerical linear algebra, providing routines for solving systems of linear equations, linear least squares, eigenvalue problems, and singular value decomposition. Originally written in FORTRAN 77 as a successor to LINPACK and EISPACK in 1992, LAPACK was designed to exploit cache-based memory hierarchies and level-3 BLAS operations for high performance on modern architectures. Key features: matrix factorizations including LU, Cholesky, QR, and LQ decompositions with pivoting strategies for numerical stability. Linear system solving for general, symmetric, symmetric positive definite, triangular, banded, and tridiagonal matrices. Least squares problems including over-determined, under-determined, and constrained least squares with rank-deficient handling via complete orthogonal factorization. Eigenvalue and eigenvector computation for standard and generalized eigenvalue problems including symmetric, non-symmetric, and band matrices. Singular Value Decomposition (SVD) for general and bidiagonal matrices, including divide-and-conquer and Jacobi methods. Matrix condition number estimation and componentwise error bounds for assessing solution accuracy. Iterative refinement for improving solution accuracy. BLAS (Basic Linear Algebra Subprograms) integration with Level 1 (vector-vector), Level 2 (matrix-vector), and Level 3 (matrix-matrix) operations. Reference implementation in Fortran with optimized vendor implementations from Intel MKL, AMD AOCL, Apple Accelerate, NVIDIA cuSOLVER, and OpenBLAS. LAPACKE C interface for calling Fortran routines from C and C++. Python bindings via NumPy and SciPy. Used in MATLAB, R, Julia, and virtually all scientific computing environments.