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| Cholesky Decomposition | |
|---|---|
| Name | Cholesky Decomposition |
| Type | Matrix factorization |
| Inventor | André-Louis Cholesky |
| Field | Numerical linear algebra |
| Firstpublished | 20th century |
Cholesky Decomposition is a matrix factorization technique for converting a positive-definite Hermitian matrix into the product of a lower triangular matrix and its conjugate transpose. It is widely used across scientific computing, engineering, economics, and statistics for solving linear systems, optimizing quadratic forms, and performing probabilistic simulations. Developed in the early 20th century, the method underpins many numerical libraries and software packages used in research at institutions and projects worldwide.
The decomposition is named after André-Louis Cholesky, who applied it to problems arising in geodesy and cartography that concerned agencies such as the French Army and institutions like the École Polytechnique. Later adoption in computational science connected the method to efforts at laboratories such as Los Alamos National Laboratory and organizations including Bell Labs and IBM during the mid-20th century. Its role became central in algorithms developed at universities such as Massachusetts Institute of Technology, Stanford University, University of Cambridge, and École Normale Supérieure, and in software efforts by groups like the Netlib community and commercial vendors including Intel and NVIDIA.
For a complex Hermitian positive-definite matrix A, there exists a unique lower triangular matrix L with strictly positive diagonal entries such that A = L L*, where L* denotes the conjugate transpose; for real symmetric positive-definite matrices, this reduces to A = L L^T. Fundamental properties connect to results established by mathematicians at institutions such as University of Göttingen and University of Paris, with links to theory developed by figures like Carl Friedrich Gauss, David Hilbert, and John von Neumann. The decomposition implies that determinants satisfy det(A) = (prod_i L_ii)^2 and that principal minors are positive, which relates to criteria studied by Augustin-Louis Cauchy and Peter Gustav Lejeune Dirichlet in matrix analysis. Spectral considerations tie the factorization to theorems by Eugène Beltrami and modern expositions from researchers at Princeton University and University of California, Berkeley.
Practical computation uses variants such as the standard Cholesky algorithm, the blocked Cholesky method for cache efficiency, and the sparse Cholesky factorization for large-scale problems. Implementations in numerical linear algebra libraries originate from projects like LAPACK, BLAS, and the ScaLAPACK parallel library, and are optimized for hardware by companies such as Intel Corporation, AMD, and NVIDIA Corporation. Sparse direct solvers leveraging elimination trees and symbolic factorization were advanced by research groups at University of Manchester, INRIA, and Courant Institute; examples include multifrontal and supernodal methods used in packages like SuiteSparse and MUMPS. Parallelization strategies connect to architectures developed at Cray Research and modern supercomputers at centers such as Oak Ridge National Laboratory.
The factorization is numerically stable for positive-definite matrices when implemented with appropriate pivoting, roundoff control, and scaling strategies studied by numerical analysts at Stanford University, University of California, Los Angeles, and ETH Zurich. Conditioning analysis uses concepts linked to work by Alan Turing, James Wilkinson, and Gene H. Golub; the condition number of A governs sensitivity of solutions and relates to backward error bounds derived in the literature from groups at National Institute of Standards and Technology and academic centers like University of Illinois Urbana-Champaign. For indefinite or nearly singular problems, modifications such as diagonal pivoting and LDL^T factorization connect to contributions from researchers at Argonne National Laboratory and University of Oxford.
Cholesky-based methods are applied in multivariate statistics at institutions like Harvard University and Columbia University for computing Mahalanobis distances, in finance models developed on Wall Street and in firms such as Goldman Sachs and JPMorgan Chase, and in machine learning frameworks produced by companies like Google and Facebook. In engineering, they are used in finite element analysis in projects at Siemens and General Electric, and in geophysics workflows associated with United States Geological Survey and California Institute of Technology. Other domains include computational chemistry at Lawrence Berkeley National Laboratory, signal processing in companies like Siemens AG, and control systems research from NASA and European Space Agency programs.
Generalizations include the LDL^T (Bunch–Kaufman) factorization for symmetric indefinite matrices, the pivoted Cholesky algorithm for low-rank approximations explored by teams at Microsoft Research and Google Research, and the block and hierarchical matrix variants (H-matrices) developed in collaborations between Max Planck Institute and universities such as Technical University of Munich. Extensions to kernel methods in machine learning relate to work at Carnegie Mellon University and Imperial College London, while randomized and communication-avoiding versions connect to projects at Lawrence Livermore National Laboratory and initiatives funded by the National Science Foundation.
Computational complexity for dense n×n matrices is O(n^3/3) arithmetic operations for the standard algorithm, with blocked and parallel implementations reducing wall-clock time on multicore and distributed systems designed by vendors like Intel Corporation and NVIDIA Corporation. Sparse factorizations depend on fill-in patterns determined by graph ordering heuristics such as minimum degree and nested dissection, developed by teams at Princeton University and University of British Columbia, and implemented in solvers used in large-scale simulations at Argonne National Laboratory and high-performance computing centers including National Energy Research Scientific Computing Center. Practical implementations balance floating-point performance, memory usage, and numerical robustness across software ecosystems like MATLAB, R (programming language), and scientific Python distributions maintained by communities around SciPy and NumPy.
Category:Matrix decompositions