In numerical analysis and linear algebra, LU decomposition (where 'LU' stands for 'lower upper', and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. The LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix. The LU decomposition was introduced by mathematician Banachiewicz in 1938.
Contents
- Definitions
- LU factorization with Partial Pivoting
- LU factorization with full pivoting
- LDU decomposition
- Example
- Square matrices
- Symmetric positive definite matrices
- General matrices
- Algorithms
- Closed formula
- Doolittle algorithm
- Crout and LUP algorithms
- Randomized Algorithm
- Theoretical complexity
- Sparse matrix decomposition
- Solving linear equations
- Inverting a matrix
- Computing the determinant
- References
Definitions
Let A be a square matrix. An LU factorization refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors, a lower triangular matrix L and an upper triangular matrix U,
In the lower triangular matrix all elements above the diagonal are zero, in the upper triangular matrix, all the elements below the diagonal are zero. For example, for a 3-by-3 matrix A, its LU decomposition looks like this:
Without a proper ordering or permutations in the matrix, the factorization may fail to materialize. For example, it is easy to verify (by expanding the matrix multiplication) that
LU factorization with Partial Pivoting
It turns out that a proper permutation in rows (or columns) is sufficient for the LU factorization. The LU factorization with Partial Pivoting (LUP) refers often to the LU factorization with row permutations only,
where L and U are again lower and upper triangular matrices, and P is a permutation matrix which, when left-multiplied to A, reorders the rows of A. It turns out that all square matrices can be factorized in this form, and the factorization is numerically stable in practice. This makes LUP decomposition a useful technique in practice.
LU factorization with full pivoting
An LU factorization with full pivoting involves both row and column permutations,
where L, U and P are defined as before, and Q is a permutation matrix that reorders the columns of A.
LDU decomposition
An LDU decomposition is a decomposition of the form
where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U are one.
Above we required that A be a square matrix, but these decompositions can all be generalized to rectangular matrices as well. In that case, L and D are square matrices both of which have the same number of rows as A, and U has exactly the same dimensions as A. Upper triangular should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner.
Example
We factorize the following 2-by-2 matrix:
One way to find the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. Expanding the matrix multiplication gives
This system of equations is underdetermined. In this case any two non-zero elements of L and U matrices are parameters of the solution and can be set arbitrarily to any non-zero value. Therefore, to find the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can conveniently require the lower triangular matrix L to be a unit triangular matrix (i.e. set all the entries of its main diagonal to ones). Then the system of equations has the following solution:
Substituting these values into the LU decomposition above yields
Square matrices
Any square matrix
If a square, invertible matrix has an LDU factorization with all diagonal entries of L and U equal to 1, then the factorization is unique. In that case, the LU factorization is also unique if we require that the diagonal of
Symmetric positive definite matrices
If A is a symmetric (or Hermitian, if A is complex) positive definite matrix, we can arrange matters so that U is the conjugate transpose of L. That is, we can write A as
This decomposition is called the Cholesky decomposition. The Cholesky decomposition always exists and is unique — provided the matrix is positive definite. Furthermore, computing the Cholesky decomposition is more efficient and numerically more stable than computing some other LU decompositions.
General matrices
For a (not necessarily invertible) matrix over any field, the exact necessary and sufficient conditions under which it has an LU factorization are known. The conditions are expressed in terms of the ranks of certain submatrices. The Gaussian elimination algorithm for obtaining LU decomposition has also been extended to this most general case.
Algorithms
The LU decomposition is basically a modified form of Gaussian elimination. We transform the matrix A into an upper triangular matrix U by eliminating the entries below the main diagonal. The Doolittle algorithm does the elimination column by column starting from the left, by multiplying A to the left with atomic lower triangular matrices. It results in a unit lower triangular matrix and an upper triangular matrix. The Crout algorithm is slightly different and constructs a lower triangular matrix and a unit upper triangular matrix.
Computing the LU decomposition using either of these algorithms requires 2n3 / 3 floating point operations, ignoring lower order terms. Partial pivoting adds only a quadratic term; this is not the case for full pivoting.
Closed formula
When an LDU factorization exists and is unique there is a closed (explicit) formula for the elements of L, D, and U in terms of ratios of determinants of certain submatrices of the original matrix A. In particular,
Doolittle algorithm
Given an N × N matrix
we define
We eliminate the matrix elements below the main diagonal in the n-th column of A(n − 1) by adding to the i-th row of this matrix the n-th row multiplied by
for
We set
After N − 1 steps, we eliminated all the matrix elements below the main diagonal, so we obtain an upper triangular matrix A(N − 1). We find the decomposition
Denote the upper triangular matrix A(N − 1) by U, and
We obtain
It is clear that in order for this algorithm to work, one needs to have
Crout and LUP algorithms
The LUP decomposition algorithm by Cormen et al. generalizes Crout matrix decomposition. It can be described as follows.
- If
A has a nonzero entry in its first row, then take a permutation matrixP 1 A P 1 P 1 A 1 = A P 1 - Let
A 2 A 1 A 2 = L 2 U 2 P 2 L fromL 2 A 1 - Make
U 3 U 2 P 3 P 2 A 3 = A 1 / P 3 = A P 1 / P 3 P be the inverse ofP 1 / P 3 - At this point,
A 3 L U 3 A is zero, thenA 3 = L U 3 A = L U 3 P follows, as desired. Otherwise,A 3 L U 3 A 3 = L U 3 U 1 U 1 U 1 L U 3 A 3 A = L U 3 U 1 P is a decomposition of the desired form.
Randomized Algorithm
It is possible to find a low rank approximation to the LU decomposition using a randomized algorithm. Given an input matrix
Theoretical complexity
If two matrices of order n can be multiplied in time M(n), where M(n) ≥ na for some a>2, then the LU decomposition can be computed in time O(M(n)). This means, for example, that an O(n2.376) algorithm exists based on the Coppersmith–Winograd algorithm.
Sparse matrix decomposition
Special algorithms have been developed for factorizing large sparse matrices. These algorithms attempt to find sparse factors L and U. Ideally, the cost of computation is determined by the number of nonzero entries, rather than by the size of the matrix.
These algorithms use the freedom to exchange rows and columns to minimize fill-in (entries which change from an initial zero to a non-zero value during the execution of an algorithm).
General treatment of orderings that minimize fill-in can be addressed using graph theory.
Solving linear equations
Given a system of linear equations in matrix form
we want to solve the equation for x given A and b. Suppose we have already obtained the LUP decomposition of A such that
In this case the solution is done in two logical steps:
- First, we solve the equation
L y = P b for y; - Second, we solve the equation
U x = y for x.
Note that in both cases we are dealing with triangular matrices (L and U) which can be solved directly by forward and backward substitution without using the Gaussian elimination process (however we do need this process or equivalent to compute the LU decomposition itself).
The above procedure can be repeatedly applied to solve the equation multiple times for different b. In this case it is faster (and more convenient) to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b, rather than using Gaussian elimination each time. The matrices L and U could be thought to have "encoded" the Gaussian elimination process.
The cost of solving a system of linear equations is approximately
Inverting a matrix
When solving systems of equations, b is usually treated as a vector with a length equal to the height of matrix A. Instead of vector b, we have matrix B, where B is an n-by-p matrix, so that we are trying to find a matrix X (also a n-by-p matrix):
We can use the same algorithm presented earlier to solve for each column of matrix X. Now suppose that B is the identity matrix of size n. It would follow that the result X must be the inverse of A. An implementation of this methodology in the C programming language can be found here.
Computing the determinant
Given the LUP decomposition
The second equation follows from the fact that the determinant of a triangular matrix is simply the product of its diagonal entries, and that the determinant of a permutation matrix is equal to (−1)S where S is the number of row exchanges in the decomposition.
In the case of LU decomposition with full pivoting,
The same method readily applies to LU decomposition by setting P equal to the identity matrix.