LU decomposition
Encyclopedia
In linear algebra
, LU decomposition (also called LU factorization) is a matrix decomposition
which writes a matrix
as the product of a lower triangular matrix
and an upper triangular matrix. The product sometimes includes a permutation matrix
as well. This decomposition is used in numerical analysis
to solve systems of linear equations or calculate the determinant
of a matrix. LU decomposition can be viewed as a matrix form of Gaussian elimination
. LU decomposition was introduced by mathematician Alan Mathison Turing
where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal.
For a
matrix, this becomes:
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.
An LUP decomposition (also called a LU decomposition with partial pivoting
) is a decomposition of the form
where L and U are again lower and upper triangular matrices and P is a permutation matrix
, i.e., a matrix of zeros and ones that has exactly one entry 1 in each row and column.
An LU decomposition with full pivoting (Trefethen and Bau) takes the form
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 P are square matrices which each have the same number of rows as A, while U is exactly the same shape as A. Upper triangular should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner.
all its leading principal minors
are non-zero. The factorization is unique if we require that the diagonal of L (or U) consist of ones. The matrix has a unique LDU factorization under the same conditions.
If the matrix is singular, then an LU factorization may still exist. In fact, a square matrix of rank
k has an LU factorization if the first k leading principal minors are non-zero, although the converse is not true.
The exact necessary and sufficient conditions under which a not necessarily invertible matrix over any field 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 .
Every invertible matrix A admits an LUP factorization.
, then we can arrange matters so that U is the conjugate transpose
of L. In this case, we have written A as
This decomposition is called the Cholesky decomposition
. The Cholesky decomposition always exists and is unique. Furthermore, computing the Cholesky decomposition is more efficient and numerically more stable
than computing some other LU decompositions.
and for 
, 
is the ratio of the 
principal submatrix to the 
principal submatrix.
. 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.
we define
and then we iterate n = 1,...,N-1 as follows.
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
. This can be done by
multiplying A(n-1) to the left with the
lower triangular matrix
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
. Because the inverse of a lower triangular matrix Ln is again a lower triangular matrix, and the multiplication of two lower triangular matrices is again a lower triangular matrix, it follows that L is a lower triangular matrix.
Moreover, it can be seen that
We obtain
.
It is clear that in order for this algorithm to work, one needs to have
at each step (see the definition of
). If this assumption fails at some point, one needs to interchange n-th row with another row below it before continuing. This is why the LU decomposition in general looks like 
.
. It can be described as follows.
.
One way of finding the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. You know that:
Such a 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 require the lower triangular matrix L to be a unit one (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:
. 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
.
we want to solve the equation for x given A and b. In this case the solution is done in two logical steps:
Note that in both cases we have triangular matrices (lower and upper) which can be solved directly using forward and backward substitution without using the Gaussian elimination
process (however we need this process or equivalent to compute the LU decomposition itself). Thus the LU decomposition is computationally efficient only when we have to solve a matrix equation multiple times for different b; it is faster in this case to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b, than to use Gaussian elimination each time.
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.
and 
can be used to compute the determinant
of the matrix
very quickly, because det(A) = det(L) det(U) and the determinant of a triangular matrix is simply the product of its diagonal entries. In particular, if L is a unit triangular matrix, then
The same approach can be used for LUP decompositions. The determinant of the permutation matrix P is (−1)S, where
is the number of row exchanges in the decomposition.
Computer code
Online resources
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, LU decomposition (also called LU factorization) is a matrix decomposition
Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...
which writes a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
as the product of a lower triangular matrix
Triangular matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...
and an upper triangular matrix. The product sometimes includes a permutation matrix
Permutation matrix
In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
as well. This decomposition is used in numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
to solve systems of linear equations or calculate the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of a matrix. LU decomposition can be viewed as a matrix form of Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
. LU decomposition was introduced by mathematician Alan Mathison Turing
Definitions
Let A be a square matrix. An LU decomposition is a decomposition of the formwhere L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal.
For a
matrix, this becomes:An LDU decomposition is a decomposition of the form
where D is a diagonal matrix
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U are one.
An LUP decomposition (also called a LU decomposition with partial pivoting
Pivot element
The pivot or pivot element is the element of a matrix, an array, or some other kind of finite set, which is selected first by an algorithm , to do certain calculations...
) is a decomposition of the form
where L and U are again lower and upper triangular matrices and P is a permutation matrix
Permutation matrix
In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
, i.e., a matrix of zeros and ones that has exactly one entry 1 in each row and column.
An LU decomposition with full pivoting (Trefethen and Bau) takes the form
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 P are square matrices which each have the same number of rows as A, while U is exactly the same shape as A. Upper triangular should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner.
Existence and uniqueness
An invertible matrix admits an LU factorization if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
all its leading principal minors
Minor (linear algebra)
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns...
are non-zero. The factorization is unique if we require that the diagonal of L (or U) consist of ones. The matrix has a unique LDU factorization under the same conditions.
If the matrix is singular, then an LU factorization may still exist. In fact, a square matrix of rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
k has an LU factorization if the first k leading principal minors are non-zero, although the converse is not true.
The exact necessary and sufficient conditions under which a not necessarily invertible matrix over any field 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 .
Every invertible matrix A admits an LUP factorization.
Positive definite matrices
If the matrix A is Hermitian and positive definitePositive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
, then we can arrange matters so that U is the conjugate transpose
Conjugate transpose
In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
of L. In this case, we have written A as
This decomposition is called the Cholesky decomposition
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...
. The Cholesky decomposition always exists and is unique. Furthermore, computing the Cholesky decomposition is more efficient and numerically more stable
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
than computing some other LU decompositions.
Explicit formulation
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 (Householder 1975). In particular, and for , is the ratio of the principal submatrix to the principal submatrix.Algorithms
The LU decomposition is basically a modified form of Gaussian eliminationGaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
. 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
Pivot element
The pivot or pivot element is the element of a matrix, an array, or some other kind of finite set, which is selected first by an algorithm , to do certain calculations...
adds only a quadratic term; this is not the case for full pivoting.
Doolittle algorithm
Given an N × N matrixwe define
and then we iterate n = 1,...,N-1 as follows.
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
. This can be done bymultiplying A(n-1) to the left with the
lower triangular matrix
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
. Because the inverse of a lower triangular matrix Ln is again a lower triangular matrix, and the multiplication of two lower triangular matrices is again a lower triangular matrix, it follows that L is a lower triangular matrix.Moreover, it can be seen that
We obtain
.It is clear that in order for this algorithm to work, one needs to have
at each step (see the definition of
). If this assumption fails at some point, one needs to interchange n-th row with another row below it before continuing. This is why the LU decomposition in general looks like .Crout and LUP algorithms
The LUP decomposition algorithm by Cormen et al. generalizes Crout matrix decompositionCrout matrix decomposition
In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix , an upper triangular matrix and, although not always needed, a permutation matrix ....
. It can be described as follows.
- If
has a nonzero entry in its first row, then take a permutation matrix 
such that 
has a nonzero entry in its upper left corner. Otherwise, take for 
the identity matrix. Let 
. - Let
be the matrix that one gets from 
by deleting both the first row and the first column. Decompose 
recursively. Make 
from 
by first adding a zero row above and then adding the first column of 
at the left. - Make
from 
by first adding a zero row above and a zero column at the left and then replacing the upper left entry (which is 0 at this point) by 1. Make 
from 
in a similar manner and define 
. Let 
be the inverse of 
. - At this point,
is the same as 
, except (possibly) at the first row. If the first row of 
is zero, then 
, since both have first row zero, and 
follows, as desired. Otherwise, 
and 
have the same nonzero entry in the upper left corner, and 
for some upper triangular square matrix 
with ones on the diagonal (
clears entries of 
and adds entries of 
by way of the upper left corner). Now 
is a decomposition of the desired form.
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 algorithmCoppersmith–Winograd algorithm
In the mathematical discipline of linear algebra, the Coppersmith–Winograd algorithm, named after Don Coppersmith and Shmuel Winograd, is the asymptotically fastest known algorithm for square matrix multiplication. It can multiply two n \times n matrices in O time...
.
Small example
One way of finding the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. You know that:
Such a 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 require the lower triangular matrix L to be a unit one (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:
Sparse matrix decomposition
Special algorithms have been developed for factorizing large sparse matricesSparse matrix
In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....
. 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
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
.
Solving linear equations
Given a matrix equationwe want to solve the equation for x given A and b. In this case the solution is done in two logical steps:
- First, we solve the equation
for y - Second, we solve the equation
for x.
Note that in both cases we have triangular matrices (lower and upper) which can be solved directly using forward and backward substitution without using the Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
process (however we need this process or equivalent to compute the LU decomposition itself). Thus the LU decomposition is computationally efficient only when we have to solve a matrix equation multiple times for different b; it is faster in this case to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b, than to use Gaussian elimination each time.
Inverse 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.
Determinant
The matrices and can be used to compute the determinantDeterminant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the matrix
very quickly, because det(A) = det(L) det(U) and the determinant of a triangular matrix is simply the product of its diagonal entries. In particular, if L is a unit triangular matrix, thenThe same approach can be used for LUP decompositions. The determinant of the permutation matrix P is (−1)S, where
is the number of row exchanges in the decomposition.See also
- Block LU decompositionBlock LU decompositionIn linear algebra, a Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U...
- Cholesky decompositionCholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...
- Matrix decompositionMatrix decompositionIn the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...
- QR decompositionQR decompositionIn linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...
- LU ReductionLU reductionLU reduction is an algorithm related to LU decomposition. This term is usually used in the context of super computing and highly parallel computing. In this context it is used as a benchmarking algorithm, i.e. to provide a comparative measurement of speed for different computers...
External links
References- LU decomposition on MathWorld.
- LU decomposition on Math-Linux.
- Module for LU Factorization with Pivoting, Prof. J. H. Mathews, California State University, FullertonCalifornia State University, FullertonCalifornia State University, Fullerton is a public university located in Fullerton, California. It is the largest institution in the CSU System by enrollment, it offers long-distance education and adult-degree programs...
- LU decomposition at Holistic Numerical Methods Institute
Computer code
- LAPACK is a collection of FORTRAN subroutines for solving dense linear algebra problems
- ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, etc.
- C++ code, Prof. J. Loomis, University of DaytonUniversity of DaytonThe University of Dayton is a private Roman Catholic university operated by the Society of Mary located in Dayton, Ohio...
- C code, Mathematics Source Library
- Pseudo-code: Gaussian elimination approach, Prof. P. Wapperom, Virginia Tech
- Pseudo-code: Doolittle algorithm, Prof. P. Wapperom, Virginia Tech
- LU in X10
Online resources
- WebApp descriptively solving systems of linear equations with LU Decomposition
- Matrix Calculator, bluebit.gr
- LU Decomposition Tool, uni-bonn.de
- LU Decomposition by Ed Pegg, Jr.Ed Pegg, Jr.Ed Pegg, Jr. is an expert on mathematical puzzles and is a self-described recreational mathematician. He creates puzzles for the Mathematical Association of America online at Ed Pegg, Jr.'s Math Games. His puzzles have also been used by Will Shortz on the puzzle segment of NPR's Weekend Edition...
, The Wolfram Demonstrations Project, 2007.