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| LU Decomposition | |
|---|---|
| Name | LU Decomposition |
| Other names | LU Factorization |
| Field | Numerical Linear Algebra |
| Introduced | 19th century |
| Applications | Scientific computing, Engineering, Economics |
LU Decomposition
LU Decomposition is a matrix factorization that expresses a square matrix as the product of a lower triangular matrix and an upper triangular matrix. Developed in the context of numerical analysis and matrix theory, it underpins algorithms used by institutions such as IBM, Bell Labs, Los Alamos National Laboratory, NASA and influenced software projects like LAPACK, BLAS, MATLAB, Scilab. Historical work by figures associated with Gauss, Crout, Doolittle, Turing, Von Neumann and contributions from researchers at Cambridge University, Princeton University, Harvard University shaped its modern role in computational science.
Existence and uniqueness results for triangular factorizations trace to classical linear algebra theorems proved by mathematicians connected to Cauchy, Gauss, Gauß–Jordan, Cholesky and modern expositors at Courant Institute, ETH Zurich, University of Oxford. A nonsingular matrix with all leading principal minors nonzero admits a unique LU factorization with unit diagonal in the lower factor, a condition discussed alongside concepts from Sylvester and Hadamard. Singular or rank-deficient matrices lead to factorizations studied by researchers affiliated with University of Cambridge, Max Planck Society, Imperial College London which often require permutation matrices related to work by Fisher, Wright and others to secure existence.
Algorithms for computing LU factors evolved from hand computations associated with the schools of Gauss and Crout to modern implementations influenced by projects at Bell Laboratories, Argonne National Laboratory, Oak Ridge National Laboratory and the National Institute of Standards and Technology. Classical Doolittle and Crout methods appear in texts from Cambridge University Press and Springer Verlag; block algorithms and tiled factorizations were advanced in collaborations involving Intel Corporation, NVIDIA and research groups at University of Illinois Urbana–Champaign. Recursive variants draw on ideas from Strassen and blocked implementations in LAPACK exploit caches in processors designed by AMD and Intel.
Pivoting strategies — partial, complete, rook and threshold pivoting — were developed in response to stability issues analyzed by Turing, Von Neumann, Goldstine, Higham and others at institutions like University of Manchester, Columbia University, University of California, Berkeley. Partial pivoting with row permutations is the pragmatic standard used in LINPACK and LAPACK; complete pivoting, studied by scholars at Stanford University and Princeton University, offers stronger growth factor bounds at higher cost. Backward stability proofs connect to research by Wilkinson and modern treatments in monographs from SIAM and Cambridge University Press.
LU-based solvers are central to boundary-value problems in computational work performed by teams at NASA, European Space Agency, CERN and industry players like Siemens and General Electric. In structural engineering designs associated with Bechtel and Arup Group, LU factorization accelerates finite element analysis; in quantitative finance models used at Goldman Sachs and J.P. Morgan Chase it speeds valuation routines. Signal processing algorithms referenced in contributions from Bell Labs and MIT rely on triangular solves; control systems developed at Honeywell and Rockwell International exploit LU for Kalman filtering and state estimation.
The arithmetic cost of LU factorization for an n×n dense matrix is O(n^3), a fact established in literature from Knuth and computational complexity studies at MIT and Harvard University. Optimized implementations in LAPACK, ScaLAPACK and vendor libraries from Intel (MKL) and AMD achieve significant speedups on architectures produced by Intel, AMD and accelerators from NVIDIA by employing blocking, cache-aware kernels and parallelization strategies explored at Argonne National Laboratory and Lawrence Berkeley National Laboratory. Distributed-memory factorizations leverage message-passing standards like MPI with scalability experiments reported by teams at Oak Ridge National Laboratory and NERSC.
Variants include Cholesky factorization for symmetric positive-definite matrices, QR factorization developed in work by Householder and Givens, and generalized Schur forms tied to studies at Bell Labs and RIKEN. Rank-revealing LU, sparse LU factorizations researched at Sandia National Laboratories and Lawrence Livermore National Laboratory, and hierarchically semiseparable approaches from groups at EPFL extend applicability to large-scale problems. Recent generalizations incorporate randomized algorithms influenced by research at MIT, Stanford University, UC Berkeley and industrial labs like Google and Microsoft Research.