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Topic: LU factorization

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  PlanetMath: LU decomposition
The LU factorization is closely related to the row reduction algorithm.
The key idea behind LU factorization is that one does not need to employ row scalings to do row reduction until the second half (the back-substitution phase) of the algorithm.
This is version 6 of LU decomposition, born on 2002-01-04, modified 2006-09-11.
planetmath.org /encyclopedia/LUDecomposition2.html   (315 words)

 LU decomposition - Wikipedia, the free encyclopedia
In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix.
In fact, a square matrix of rank k has an LU factorization if the first k principal minors are non-zero.
The LU decomposition is basically a modified form of Gaussian elimination.
en.wikipedia.org /wiki/LU_decomposition   (794 words)

 LU Factorization Case Study Using FAST: Dataflow Parallelism with the Forte Application Scalability Tool
Starting with an overview of the LU Factorization algorithm, the problems with using traditional methods for parallelizing this algorithm are discussed.
The majority of the computation in the LU Factorization is performed in the matrix multiply and the triangular solve.
From the second factorization node in the graph, there is only a data dependency on one of the preceding matrix multiply nodes shown as blue.
developers.sun.com /prodtech/cc/articles/FAST/lu_content.html   (2737 words)

 3.2 dgemm and block LU factorization
Performances for dgemm are important because it's a building block for block LU factorization.
LU factorization is used to solve linear system like A.
The second solution let the LU task be a sequential bottleneck and use parallelism only on level 3 operations.
www.usenix.org /publications/library/proceedings/als00/2000papers/papers/full_papers/thomas/thomas_html/node7.html   (545 words)

 Matrix Decompositions
The LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation.
The first element is a combination of upper and lower triangular matrices, the second element is a vector specifying rows used for pivoting (a permutation vector which is equivalent to the permutation matrix), and the third element is an estimate of the condition number.
This factorization has a number of uses, one of which is that, because it is a triangular factorization, it can be used to solve systems of equations involving symmetric positive definite matrices.
documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/4.5.html   (2005 words)

 LU Factorization
The LU factorization applies a sequence of Gaussian eliminations to form
Figure 3 shows a snapshot of the block LU factorization.
Figure 3: A snapshot of block LU factorization.
www.netlib.org /utk/papers/factor/node7.html   (304 words)

 NMath Core User's Guide - 6.3 Using LU Factorizations
Once an LU factorization is constructed from a matrix (see Section 6.2), it can be reused to solve for different right hand sides, to compute inverses, to compute condition numbers, and so on.
Note that the length of vector v must be equal to the number of rows in the factored matrix A or a MismatchedSizeException is thrown.
You can use an LU factorization to compute inverses using the Inverse() method, and determinants using the Determinant() method.
www.centerspace.net /doc/NMath/Core/user/linearsolve4.html   (427 words)

 lu (MATLAB Function Reference)
The factorization is often called the LU, or sometimes the LR, factorization.
Most of the algorithms for computing LU factorization are variants of Gaussian elimination.
An upper triangular matrix that is a factor of
www.cs.berkeley.edu /titan/sww/software/matlab/techdoc/ref/lu.html   (246 words)

 dgesvx(l): use LU factorization to compute ... - Linux man page
Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
The system of equations is solved for X using the factored form of A. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.
www.die.net /doc/linux/man/manl/dgesvx.l.html   (1303 words)

 Class jnt.linear_algebra.LU
A - (in) the matrix to associate with this factorization.
The U factor is stored in the upper triangular portion, and the L factor is stored in the lower triangular portion.
LU - (in) the factored matrix in LU form.
math.nist.gov /jnt/api/jnt.linear_algebra.LU.html   (237 words)

 LU Factorization Example
The algorithm for LU factorization is given in Figure 1.4 and the graphical representation of the algorithm is given in Figure 1.5, as it would look part way through the computation.
The first operation in the LU factorization is to find the maximum element in the column.
The next operation in the LU factorization is to swap the current row with the row that has the maximum element.
www.osl.iu.edu /research/mtl/tutorial.php3   (956 words)

 LU (JMSL Numerical Library)
The LU factorization is done using scaled partial pivoting.
Scaled partial pivoting differs from partial pivoting in that the pivoting strategy is the same as if each row were scaled to have the same infinity norm.
Solve ax=b for x using the LU factorization of a.
www.vni.com /products/imsl/jmsl/v30/api/com/imsl/math/LU.html   (520 words)

 Gauss Transformation and LU-Factorization
If the LU factorization exists and A is regular, then the LU factorization is unique and
As it is known from the main course of algebra, the direct application of the Gaussian elimination, therefore also the direct realization of the LU factorization, fails if at least one of the principal minors is singular.
It turns out that for a regular matrix it is possible after an appropriate interchange of matrix rows to find the LU factotization.
www.cs.ut.ee /~toomas_l/linalg/lin2/node4.html   (456 words)

 PA = LU Factorization with Pivoting
It turns out that this factorization (when it exists) is not unique.
A sufficient condition for the factorization to exist is that all principal minors of A are nonsingular.
We have seen in Example 3 an example of a nonsingular matrix A could not be directly factored as A = LU.
math.fullerton.edu /mathews/n2003/LUFactorMod.html   (469 words)

 5.2 LU Factorization
If A is a square matrix, it has an LU factorization with a permutation matrix P, an upper triangular matrix U, and a lower triangular matrix L, such that A = PLU.
The basic idea is that an LU factorization is actually implemented as a class in the Math.h++ class library.
This approach can result in great time savings because calculating the LU factorization takes most of the time, but need be done only once.
www.roguewave.com /support/docs/hppdocs/mthug/5-2.html   (406 words)

 LU Factorization
Dense matrix computations, such as LU factorization, have important applications, as discussed in a recent survey by Edelman [28].
The dense linear systems generated are commonly solved using LU factorization.
Boundary element methods are also used in the study of fluid flows, and here the variant of the boundary element method used is called the panel method [36, 37].
www.netlib.org /utk/papers/siam-review93/node12.html   (244 words)

 3 LU Factorization (with pivoting)
  The implementation of the LU factorization in parallel using PLAPACK closely follows the blocked algorithm discussed in the last section.
With these points in mind, we present the algorithm for the parallel LU factorization side by side with the PLAPACK code implementation.
The code for the column-by-column factorization of the panel as multivector is not presented.
www.cs.utexas.edu /users/plapack/icpp98/node3.html   (254 words)

 LU Factorization of A Square Matrix and Stability Calculation
LU Factorization of A Square Matrix and Stability Calculation
Computes an LU factorization of the matrix A where A is a square matrix.
If ratio is too large (the meaning of too large is up to you), the current pivots do not yield a stable LU factorization of A.
www.coin-or.org /CppAD/Doc/luratio.htm   (616 words)

 2 Cholesky Factorization
Given the derivation of the basic algorithm given above, it is not difficult to derive a blocked (matrix-matrix operation based) algorithm, as is shown in Fig.
Thus, the calling the calling sequence for a Cholesky factorization need only have one parameter, the object that describes the matrix.
This is an operation that is very much in the critical path, and thus contributes considerably to the overall time required for the Cholesky factorization.
www.cs.utexas.edu /users/plapack/icpp98/node2.html   (1281 words)

 11 - C H A P T E R -
The LU factorization routine uses a parallel, block-partitioned algorithm based on the ScaLAPACK implementation.
It can be used in calls to other LU routines to reference the computed LU factors.
A = LU implies that AX = B is equivalent to C = L
www.sun.com /products-n-solutions/hardware/docs/html/817-0086-10/prog-gaussian-elimination.html   (1619 words)

The library is written in C and is callable from either C or Fortran.
The LU factorization routines can handle non-square matrices but the triangular solves are performed only for square matrices.
The matrix columns may be preordered (before factorization) either through library or user supplied routines.
crd.lbl.gov /~xiaoye/SuperLU   (565 words)

 LU Factorization of a Matrix   (Site not responding. Last check: 2007-10-31)
The LU Factorization of a Matrix is performed.
A linear system is then solved using the factorization.
The inverse, determinant, and condition number of the input matrix are also computed.
www.vni.com /products/imsl/jmsl/v30/api/com/imsl/math/LUEx1.html   (82 words)

 Sparse LU Factorization on The Cray T3D - Zapata, factorization, CRAY, High, Networking (ResearchIndex)   (Site not responding. Last check: 2007-10-31)
Sparse LU Factorization on The Cray T3D (1995)
18.1%: Sparse LU Factorization on The Cray T3D - Asenjo, Zapata (1995)
0.2: Parallel Pivots LU Algorithm on the Cray T3E - Rafael Asenjo (1998)
citeseer.ifi.unizh.ch /94870.html   (507 words)

 UMFPACK: unsymmetric multifrontal sparse LU factorization package
Appears as a built-in routine (for lu, backslash, and forward slash) in MATLAB.
UMFPACK Version 1.0 is described in T. Davis and I. Duff, "An unsymmetric-pattern multifrontal method for sparse LU factorization", SIAM J. Matrix Analysis and Applications, vol.
An unsymmetric-pattern multifrontal method for sparse LU factorization, T.
www.cise.ufl.edu /research/sparse/umfpack   (540 words)

 Keyword Index for the NAG Library Manual, Mark 21 : factorization   (Site not responding. Last check: 2007-10-31)
LU factorization of real sparse matrix with known sparsity pattern
LU factorization of complex m by n matrix
LU factorization of complex m by n band matrix
www.nag.co.uk /numeric/fl/manual/html/indexes/kwic/fl_factorization.html   (737 words)

 MAT 200 Lecture Notes -- The lu factorization, how to find it
Go forward to The lu factorization: how to use it.
We could figure out what to do to U by some rather complicated arguments about bonds and portfolios, but it is easier in this case to think in terms of matrices.
I will describe in the next section how to use the lu factorization to find implied discount factors or an actual arbitrage opportunity.
www.math.princeton.edu /~stalker/200s00/lu.html   (983 words)

 Lab for the LU Factorization of A
Part II involves solving LU = B. A = LU Factorization of the matrix A. If row interchanges are not needed to solve the linear system AX = B, then A has the LU factorization (illustrated with 4x4 matrices)
Exercises 1-2 involve the A = LU Factorization of the matrix A. Exercise 1.
Find the LU factorization of the matrix A.
math.fullerton.edu /mathews/numerical/lu.htm   (168 words)

 AMTH247 Lecture 8 Linear Equations II
The standard method of implementing Gaussian Elimination by computer is LU factorization.
To obtain the lower triangular part of LU factorization we form a matrix from the multipliers, i.e.
This is the upper triangular part of the LU factorization we found earlier.
turing.une.edu.au /~amth247/Lectures_2003/Lecture_08/lecture   (648 words)

 Optimization of an LU Factorization Routine Using Communication/Computation Overlap (ResearchIndex)   (Site not responding. Last check: 2007-10-31)
Abstract: This paper presents some works on the LU factorization from the ScaLAPACK library.
1 Introduction The LU factorization is the kernel...
7 Performance Complexity of LU Factorization with Efficient Pi..
citeseer.ifi.unizh.ch /92847.html   (320 words)

 Matrix Factorization and Matrix Norms
  Note that a unit lower triangular matrix is just a lower triangular matrix with the elements along the main diagonal equal to one (factorizations are not unique).
The details of performing the factorization are presented in the lecture.
  You are asked to perform a factorization by hand in the homework.
www.ee.ucla.edu /~brien/Rec5_MatrixFactorizationAndNorms.htm   (718 words)

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