LU decomposition

In numerical analysis and linear algebra, lower–upper (LU) decomposition or 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. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using 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 the Polish mathematician Tadeusz Banachiewicz in 1938.
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 × 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 formula_3. If formula_4, then at least one of formula_5 and formula_6 has to be zero, which implies that either "L" or "U" is singular. This is impossible if "A" is nonsingular (invertible). This is a procedural problem. It can be removed by simply reordering the rows of "A" so that the first element of the permuted matrix is nonzero. The same problem in subsequent factorization steps can be removed the same way; see the basic procedure below.
LU factorization with partial pivoting.
It turns out that a proper permutation in rows (or columns) is sufficient for LU factorization. LU factorization with partial pivoting (LUP) refers often to 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".
Lower-diagonal-upper (LDU) decomposition.
An Lower-diagonal-upper (LDU) decomposition is a decomposition of the form
where "D" is a diagonal matrix, and "L" and "U" are unitriangular matrices, meaning that all the entries on the diagonals of "L" and "U" are one.
Rectangular matrices.
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. Similarly, the more precise term for "U" is that it is the "row echelon form" of the matrix "A".
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
Existence and uniqueness.
Square matrices.
Any square matrix formula_14 admits "LUP" and "PLU" factorizations. If formula_14 is invertible, then it admits an "LU" (or "LDU") factorization if and only if all its leading principal minors are nonzero. If formula_14 is a singular matrix of rank formula_17, then it admits an "LU" factorization if the first formula_17 leading principal minors are nonzero, although the converse is not true.
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 formula_19 (or formula_20) consists of ones.
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.
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, formula_22, and for formula_23, formula_24 is the ratio of the formula_25-th principal submatrix to the formula_26-th principal submatrix. Computation of the determinants is computationally expensive, so this explicit formula is not used in practice.
Using Gaussian elimination.
The following algorithm is essentially a modified form of Gaussian elimination. Computing an LU decomposition using this algorithm requires formula_27 floating-point operations, ignoring lower-order terms. Partial pivoting adds only a quadratic term; this is not the case for full pivoting.
Given an "N" × "N" matrix formula_28, define formula_29.
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
This can be done by multiplying "A"("n" − 1) to the left with the lower triangular matrix
that is, the "N" × "N" identity matrix with its "n"-th column replaced by the vector formula_32
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 formula_35. Because the inverse of a lower triangular matrix "L""n" 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 formula_37
It is clear that in order for this algorithm to work, one needs to have formula_38 at each step (see the definition of formula_39). If this assumption fails at some point, one needs to interchange "n"-th row with another row below it before continuing. This is why an LU decomposition in general looks like formula_40.
Note that the decomposition obtained through this procedure is a "Doolittle decomposition": the main diagonal of "L" is composed solely of "1"s. If one would proceed by removing elements "above" the main diagonal by adding multiples of the "columns" (instead of removing elements "below" the diagonal by adding multiples of the "rows"), we would obtain a "Crout decomposition", where the main diagonal of "U" is of "1"s.
Another (equivalent) way of producing a Crout decomposition of a given matrix "A" is to obtain a Doolittle decomposition of the transpose of "A". Indeed, if formula_41 is the LU-decomposition obtained through the algorithm presented in this section, then by taking formula_42 and formula_43, we have that formula_44 is a Crout decomposition.
Through recursion.
Cormen et al. describe a recursive algorithm for LUP decomposition.
Given a matrix "A", let "P1" be a permutation matrix such that
where formula_46, if there is a nonzero entry in the first column of "A"; or take "P1" as the identity matrix otherwise. Now let formula_47, if formula_46; or formula_49 otherwise. We have
Now we can recursively find an LUP decomposition formula_51. Let formula_52. Therefore
which is an LUP decomposition of "A".
Randomized algorithm.
It is possible to find a low rank approximation to an LU decomposition using a randomized algorithm. Given an input matrix formula_54 and a desired low rank formula_55, the randomized LU returns permutation matrices formula_56 and lower/upper trapezoidal matrices formula_57 of size formula_58 and formula_59 respectively, such that with high probability formula_60, where formula_61 is a constant that depends on the parameters of the algorithm and formula_62 is the formula_63-th singular value of the input matrix formula_54.
Theoretical complexity.
If two matrices of order "n" can be multiplied in time "M"("n"), where "M"("n") ≥ "n""a" for some "n" > 2, then an LU decomposition can be computed in time O("M"("n")). This means, for example, that an O("n"2.376) 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 that 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.
Applications.
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 formula_66, so formula_67.
In this case the solution is done in two logical steps:
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 formula_70 floating-point operations if the matrix formula_54 has size formula_72. This makes it twice as fast as algorithms based on QR decomposition, which costs about formula_73 floating-point operations when Householder reflections are used. For this reason, LU decomposition is usually preferred.
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". In matrix inversion however, 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".
Computing the determinant.
Given the LUP decomposition formula_75 of a square matrix "A", the determinant of "A" can be computed straightforwardly as
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, formula_77 also equals the right-hand side of the above equation, if we let "S" be the total number of row and column exchanges.
The same method readily applies to LU decomposition by setting "P" equal to the identity matrix.
Code examples.
C code example.
/* INPUT: A - array of pointers to rows of a square matrix having dimension N
 * Tol - small tolerance number to detect failure when the matrix is near degenerate
 * OUTPUT: Matrix A is changed, it contains a copy of both matrices L-E and U as A=(L-E)+U such that P*A=L*U.
 * The permutation matrix is not stored as a matrix, but in an integer vector P of size N+1 
 * containing column indexes where the permutation matrix has "1". The last element P[N]=S+N, 
 * where S is the number of row exchanges needed for determinant computation, det(P)=(-1)^S 
int LUPDecompose(double **A, int N, double Tol, int *P) {
 int i, j, k, imax; 
 double maxA, *ptr, absA;
 for (i = 0; i <= N; i++)
 P[i] = i; //Unit permutation matrix, P[N] initialized with N
 for (i = 0; i < N; i++) {
 maxA = 0.0;
 imax = i;
 for (k = i; k < N; k++)
 if ((absA = fabs(A[k][i])) > maxA) { 
 maxA = absA;
 imax = k;
 if (maxA < Tol) return 0; //failure, matrix is degenerate
 if (imax != i) {
 //pivoting P
 j = P[i];
 P[i] = P[imax];
 P[imax] = j;
 //pivoting rows of A
 ptr = A[i];
 A[i] = A[imax];
 A[imax] = ptr;
 //counting pivots starting from N (for determinant)
 P[N]++;
 for (j = i + 1; j < N; j++) {
 A[j][i] /= A[i][i];
 for (k = i + 1; k < N; k++)
 A[j][k] -= A[j][i] * A[i][k];
 return 1; //decomposition done 
/* INPUT: A,P filled in LUPDecompose; b - rhs vector; N - dimension
 * OUTPUT: x - solution vector of A*x=b
void LUPSolve(double **A, int *P, double *b, int N, double *x) {
 for (int i = 0; i < N; i++) {
 x[i] = b[P[i]];
 for (int k = 0; k < i; k++)
 x[i] -= A[i][k] * x[k];
 for (int i = N - 1; i >= 0; i--) {
 for (int k = i + 1; k < N; k++)
 x[i] -= A[i][k] * x[k];
 x[i] /= A[i][i];
/* INPUT: A,P filled in LUPDecompose; N - dimension
 * OUTPUT: IA is the inverse of the initial matrix
void LUPInvert(double **A, int *P, int N, double **IA) {
 for (int j = 0; j < N; j++) {
 for (int i = 0; i < N; i++) {
 IA[i][j] = P[i] == j ? 1.0 : 0.0;
 for (int k = 0; k < i; k++)
 IA[i][j] -= A[i][k] * IA[k][j];
 for (int i = N - 1; i >= 0; i--) {
 for (int k = i + 1; k < N; k++)
 IA[i][j] -= A[i][k] * IA[k][j];
 IA[i][j] /= A[i][i];
/* INPUT: A,P filled in LUPDecompose; N - dimension. 
 * OUTPUT: Function returns the determinant of the initial matrix
double LUPDeterminant(double **A, int *P, int N) {
 double det = A[0][0];
 for (int i = 1; i < N; i++)
 det *= A[i][i];
 return (P[N] - N) % 2 == 0 ? det : -det
C# code example.
public class SystemOfLinearEquations
 public double[] SolveUsingLU(double[,] matrix, double[] rightPart, int n)
 // decomposition of matrix
 double[,] lu = new double[n, n];
 double sum = 0;
 for (int i = 0; i < n; i++)
 for (int j = i; j < n; j++)
 sum = 0;
 for (int k = 0; k < i; k++)
 sum += lu[i, k] * lu[k, j];
 lu[i, j] = matrix[i, j] - sum;
 for (int j = i + 1; j < n; j++)
 sum = 0;
 for (int k = 0; k < i; k++)
 sum += lu[j, k] * lu[k, i];
 lu[j, i] = (1 / lu[i, i]) * (matrix[j, i] - sum);
 // lu = L+U-I
 // find solution of Ly = b
 double[] y = new double[n];
 for (int i = 0; i < n; i++)
 sum = 0;
 for (int k = 0; k < i; k++)
 sum += lu[i, k] * y[k];
 y[i] = rightPart[i] - sum;
 // find solution of Ux = y
 double[] x = new double[n];
 for (int i = n - 1; i >= 0; i--)
 sum = 0;
 for (int k = i + 1; k < n; k++)
 sum += lu[i, k] * x[k];
 x[i] = (1 / lu[i, i]) * (y[i] - sum);
 return x;
MATLAB code example.
function LU = LUDecompDoolittle(A)
 n = length(A);
 LU = A;
 % decomposition of matrix, Doolittle’s Method
 for i = 1:1:n
 for j = 1:(i - 1)
 LU(i,j) = (LU(i,j) - LU(i,1:(j - 1))*LU(1:(j - 1),j)) / LU(j,j);
 end
 j = i:n;
 LU(i,j) = LU(i,j) - LU(i,1:(i - 1))*LU(1:(i - 1),j);
 end
 %LU = L+U-I
end
function x = SolveLinearSystem(LU, B)
 n = length(LU);
 y = zeros(size(B));
 % find solution of Ly = B
 for i = 1:n
 y(i,:) = B(i,:) - LU(i,1:i)*y(1:i,:);
 end
 % find solution of Ux = y
 x = zeros(size(B));
 for i = n:(-1):1
 x(i,:) = (y(i,:) - LU(i,(i + 1):n)*x((i + 1):n,:))/LU(i, i);
 end 
end
A = [ 4 3 3; 6 3 3; 3 4 3 ]
LU = LUDecompDoolittle(A)
B = [ 1 2 3; 4 5 6; 7 8 9; 10 11 12 ]'
x = SolveLinearSystem(LU, B)
A * x
External links.
References
Computer code
Online resources