LU-decomposition of a rectangular MxN matrix A is a representation of A as a product of permutation matrix P, lower unitriangular matrix L and upper triangular matrix U: A = PLU.
ALGLIB package implements recursive version of LU decomposition. Algorithm splits matrix into two approximately equal parts, decomposes first part, updates second one and builds its decomposition. Recursion is stopped when matrix size becomes small enough to fit in the smallest L1 cache of the modern CPUs. At this point, non-recursive version is called. This approach has the following advantages:
These features allows to achieve the maximum performance possible with your compiler/language. However, different compilers and different programming languages have different limitations. As for linear algebra algorithms, C++ version of ALGLIB is the most efficient implementation. Implementations in other languages have significantly lower performance.
LU decomposition is calculated byand subroutines. Permutatuion matrix P is stored in separate array, L and U replace A. For example, for 4x3 matrix we will get:
A11 A12 A13 => U11 U12 U13 A21 A22 A23 => L21 U22 U23 A31 A32 A33 => L31 L32 U33 A41 A42 A43 => L41 L42 L43
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C++ source. MPFR/GMP is used.
GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.
Python version (CPython and IronPython are supported).