## Quick Reference

A method used in numerical linear algebra in order to solve a set of linear equations,

*Ax*= *b*

where *A* is a square matrix and *b* is a column vector. In this method, a lower triangular matrix *L* and an upper triangular matrix *U* are sought such that

*LU*= *A*

For definiteness, the diagonal elements of *L* may be taken to be 1. The elements of successive rows of *U* and *L* may easily be calculated from the defining equations.

Once *L* and *U* have been determined, so that

*LUx*= *b*,

the equation

*Ly*= *b*

is solved by **forward substitution**. Thereafter the equation

*Ux*= *y*

is solved for *x* by **backward substitution**. *x* is then the solution to the original problem.

A variant of the method, the method of LDU decomposition, seeks lower and upper triangular matrices with unit diagonal and a diagonal matrix *D*, such that

*A*= *LDU*

if the matrix *A* is symmetric and positive definite, there is an advantage in finding a lower triangular matrix *L* such that

*A*= *LL*^{T}

(see transpose). This method is known as **Cholesky decomposition**; the diagonal elements of *L* are not, in general, unity.

*Subjects:*
Computing.

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