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numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
, lower–upper (LU) decomposition or factorization factors a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
as the product of a lower
triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal ar ...
and an upper triangular matrix (see matrix decomposition). The product sometimes includes a
permutation matrix In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, wh ...
as well. LU decomposition can be viewed as the matrix form of
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
. 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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
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'': : A = LU. 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: : \begin a_ & a_ & a_ \\ a_ & a_ & a_ \\ a_ & a_ & a_ \end = \begin \ell_ & 0 & 0 \\ \ell_ & \ell_ & 0 \\ \ell_ & \ell_ & \ell_ \end \begin u_ & u_ & u_ \\ 0 & u_ & u_ \\ 0 & 0 & u_ \end. 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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
) that a_ = \ell_ u_. If a_ = 0, then at least one of \ell_ and u_ has to be zero, which implies that either ''L'' or ''U'' is
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar ...
. 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: : PA = LU, where ''L'' and ''U'' are again lower and upper triangular matrices, and ''P'' is a
permutation matrix In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, wh ...
, which, when left-multiplied to ''A'', reorders the rows of ''A''. It turns out that all square matrices can be factorized in this form,, Corollary 3. 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: : PAQ = LU, 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

A Lower-diagonal-upper (LDU) decomposition is a decomposition of the form : A = LDU, where ''D'' is a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...
, 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: : \begin 4 & 3 \\ 6 & 3 \end = \begin \ell_ & 0 \\ \ell_ & \ell_ \end \begin u_ & u_ \\ 0 & u_ \end. 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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
gives :\begin \ell_ \cdot u_ + 0 \cdot 0 &= 4 \\ \ell_ \cdot u_ + 0 \cdot u_ &= 3 \\ \ell_ \cdot u_ + \ell_ \cdot 0 &= 6 \\ \ell_ \cdot u_ + \ell_ \cdot u_ &= 3. \end 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: :\begin \ell_ = \ell_ &= 1 \\ \ell_ &= 1.5 \\ u_ &= 4 \\ u_ &= 3 \\ u_ &= -1.5 \end Substituting these values into the LU decomposition above yields : \begin 4 & 3 \\ 6 & 3 \end = \begin 1 & 0 \\ 1.5 & 1 \end \begin 4 & 3 \\ 0 & -1.5 \end.


Existence and uniqueness


Square matrices

Any square matrix A admits ''LUP'' and ''PLU'' factorizations. If A is
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
, then it admits an ''LU'' (or ''LDU'') factorization
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
all its leading principal minors are nonzero, Corollary 3.5.5 (for example \begin 0 & 1 \\ 1 & 0 \end does not admit an ''LU'' or ''LDU'' factorization). If A is a singular matrix of rank k , then it admits an ''LU'' factorization if the first k 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 L (or U ) consists of ones. In general, any square matrix A_ could have one of the following: # a unique LU factorization (as mentioned above) # infinitely many LU factorizations if two or more of any first (''n''−1) columns are linearly dependent or any of the first (''n''−1) columns are 0, then A has infinitely many LU factorizations. # no LU factorization if the first (''n''−1) columns are non-zero and linearly independent and at least one leading principal minor is zero. In Case 3, one can approximate an LU factorization by changing a diagonal entry a_ to a_ \pm \varepsilon to avoid a zero leading principal minor.


Symmetric positive-definite matrices

If ''A'' is a symmetric (or Hermitian, if ''A'' is complex)
positive-definite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, ...
, we can arrange matters so that ''U'' is the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
of ''L''. That is, we can write ''A'' as : A = LL^*. \, 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, D_1 = A_, and for i = 2, \ldots, n, D_i is the ratio of the i-th principal submatrix to the (i - 1)-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 In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
. Computing an LU decomposition using this algorithm requires \tfrac n^3 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 A = (a_)_, define A^ as the matrix A in which the necassary rows have been swapped for the 1st column, where the parenthetical superscript (e.g., (0)) is the version of the matrix. So, A^ is the original, unmodified version of the A matrix. The matrix A^ is the A matrix in which the elements below the
main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix ...
have already been eliminated to 0 through Gaussian elimination for the first n columns, and the necassary rows have been swapped for the (n+1)^ column. We perform the operation row_i=row_i-(\ell_)\cdot row_n for each row i with elements (labelled as a_^ where i = n+1, \dotsc, N) below the main diagonal in the ''n''-th column of A^. For this operation, :\ell_ := \frac. We perform these row operations to eliminate the elements a_^ to zero. Once we have subtracted these rows, we may swap rows to provide the desired conditions for the (n+1)^ column. We may swap rows here to perform partial pivoting, or because the element a_ on the main diagonal is zero (and therefore cannot be used to implement Gaussian elimination). We define the final
permutation matrix In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, wh ...
P as the identity matrix which has all the same rows swapped in the same order as the A matrix. Once we have performed the row operations for the first N-1 columns, we have obtained an upper triangular matrix A^ which is denoted by U. We can also calculate the lower triangular matrix denoted denoted as L, such that PA = LU, by directly inputting the values of values of \ell_ via the formula below. :L = \begin 1 & & & & & \\ \ell_ & \ddots & & & & \\ & \ddots & 1 & & & \\ \vdots & \cdots & \ell_ & 1 & & \\ & & \vdots & \ddots & \ddots & \\ \ell_ & \cdots & \ell_ & \cdots & \ell_ & 1 \end If we did not swap rows at all during this process, we can perform the row operations simultaneously for each column n by setting A^ := L_n A^, where L_n is the ''N'' × ''N'' identity matrix with its ''n''-th column replaced by the
transposed In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
vector \begin0 & \dotsm & 0 & 1 & -\ell_ & \dotsm & -\ell_ \end^\textsf. In other words, the lower triangular matrix :L_n = \begin 1 & & & & & \\ & \ddots & & & & \\ & & 1 & & & \\ & & -\ell_ & & & \\ & & \vdots & & \ddots & \\ & & -\ell_ & & & 1 \end. Performing all the row operations for the first N-1 columns using the A^ := L_n A^ formula is equivalent to finding the decomposition :A = L_1^ L_1 A^ = L_1^ A^ = L_1^ L_2^ L_2 A^ = L_1^ L_2^ A^ = \dotsm = L_1^ \dotsm L_^ A^. Denote L = L_1^ \dotsm L_^ so that A=LA^=LU. Now let's compute the sequence of L_1^ \dotsm L_^. We know that L_^ has the following formula. :L_n^ = \begin 1 & & & & & \\ & \ddots & & & & \\ & & 1 & & & \\ & & \ell_ & & & \\ & & \vdots & & \ddots & \\ & & \ell_ & & & 1 \end If there are two lower triangular matrices with 1s in the main diagonal, and neither have a non-zero item below the main diagonal in the same column as the other, then we can include all non-zero items at their same location in the product of the two matrices. For example: \left(\begin 1 & 0 & 0 & 0 & 0 \\ 77 & 1 & 0 & 0 & 0 \\ 12 & 0 & 1 & 0 & 0 \\ 63 & 0 & 0 & 1 & 0 \\ 7 & 0 & 0 & 0 & 1 \end\right)\left(\begin 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 22 & 1 & 0 & 0 \\ 0 & 33 & 0 & 1 & 0 \\ 0 & 44 & 0 & 0 & 1 \end\right)=\left(\begin 1 & 0 & 0 & 0 & 0 \\ 77 & 1 & 0 & 0 & 0 \\ 12 & 22 & 1 & 0 & 0 \\ 63 & 33 & 0 & 1 & 0 \\ 7 & 44 & 0 & 0 & 1 \end\right) Finally, multiply L_^ together and generate the fused matrix denoted as L (as previously mentioned). Using the matrix L, we obtain A = LU. It is clear that in order for this algorithm to work, one needs to have a_^ \neq 0 at each step (see the definition of \ell_). 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 P^A = L U . 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 A^\textsf = L_0 U_0 is the LU-decomposition obtained through the algorithm presented in this section, then by taking L = U_0^\textsf and U = L_0^\textsf, we have that A = LU 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 : P_1 A = \left( \begin a & & w^\textsf & \\ \hline & & & \\ v & & A' & \\ & & & \end \right) , where a \neq 0, if there is a nonzero entry in the first column of ''A''; or take ''P1'' as the identity matrix otherwise. Now let c = 1/a, if a \neq 0; or c = 0 otherwise. We have : P_1 A = \left( \begin 1 & & 0 & \\ \hline & & & \\ cv & & I_ & \\ & & & \end \right) \left( \begin a & w^\textsf \\ \hline & \\ 0 & A'-cvw^\textsf \\ & \end \right) . Now we can recursively find an LUP decomposition P' \left(A' - cvw^\textsf\right) = L' U'. Let v' = P'v. Therefore : \left( \begin 1 & & 0 & \\ \hline & & & \\ 0 & & P' & \\ & & & \end \right) P_1 A = \left( \begin 1 & & 0 & \\ \hline & & & \\ cv' & & L' & \\ & & & \end \right) \left( \begin a & & w^\textsf & \\ \hline & & & \\ 0 & & U' & \\ & & & \end \right) , 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 A and a desired low rank k, the randomized LU returns permutation matrices P, Q and lower/upper trapezoidal matrices L, U of size m \times k and k \times n respectively, such that with high probability \left\, PAQ-LU \right\, _2 \le C\sigma_, where C is a constant that depends on the parameters of the algorithm and \sigma_ is the (k+1)-th singular value of the input matrix A.


Theoretical complexity

If two matrices of order ''n'' can be multiplied in time ''M''(''n''), where ''M''(''n'') ≥ ''n''''a'' for some ''a'' > 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 In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
.


Applications


Solving linear equations

Given a system of linear equations in matrix form :A\mathbf x = \mathbf b, we want to solve the equation for x, given ''A'' and b. Suppose we have already obtained the LUP decomposition of ''A'' such that PA = LU, so LU \mathbf x = P \mathbf b. In this case the solution is done in two logical steps: # First, we solve the equation L \mathbf y = P \mathbf b for y. # Second, we solve the equation U \mathbf x = \mathbf y for x. 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 In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
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 \frac n^3 floating-point operations if the matrix A has size n. This makes it twice as fast as algorithms based on
QR decomposition In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthogonal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...
, which costs about \frac n^3 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): :AX = LUX = B. 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''., p. 121


Computing the determinant

Given the LUP decomposition A = P^ LU of a square matrix ''A'', the determinant of ''A'' can be computed straightforwardly as :\det(A) = \det\left(P^\right) \det(L) \det(U) = (-1)^S \left( \prod_^n l_ \right) \left( \prod_^n u_ \right) . 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, \det(A) 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 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) /* 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) /* 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) /* 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)


C# code example

public class SystemOfLinearEquations


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


See also

* Block LU decomposition *
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the princip ...
* Cholesky decomposition * Crout matrix decomposition * Incomplete LU factorization * LU Reduction * Matrix decomposition *
QR decomposition In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthogonal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...


Notes


References

* . * . * . * . See Section 3.5. ''N'' − 1 * . * . * . * . * . *


External links

References
LU decomposition
on ''MathWorld''.
LU decomposition
on ''Math-Linux''.

at ''Holistic Numerical Methods Institute''

MATLAB reference. 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.

Prof. J. Loomis,
University of Dayton The University of Dayton (UD) is a private, Catholic research university in Dayton, Ohio. Founded in 1850 by the Society of Mary, it is one of three Marianist universities in the nation and the second-largest private university in Ohio. The uni ...

C code
Mathematics Source Library
Rust code

LU in X10
Online resources
WebApp descriptively solving systems of linear equations with LU Decomposition

Matrix Calculator with steps, including LU decompostion

LU Decomposition Tool
uni-bonn.de
LU Decomposition
by
Ed Pegg, Jr. Edward Taylor Pegg Jr. (born December 7, 1963) is an expert on mathematical puzzles and is a self-described recreational mathematician. He wrote an online puzzle column called Ed Pegg Jr.'s ''Math Games'' for the Mathematical Association of Amer ...
, The Wolfram Demonstrations Project, 2007. {{Numerical linear algebra Matrix decompositions Numerical linear algebra Articles with example C code Articles with example C Sharp code de:Gaußsches Eliminationsverfahren#LR-Zerlegung