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Topic: LU decomposition

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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.
	www.planetmath.org /encyclopedia/LUDecomposition.html (305 words)

	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/LUDecomposition.html (305 words)

	LU Decomposition
		LU decomposition is used to find the solutions of linear systems.
		Notice that this is a square matrix and is not yet augmented with the vector of constants from the equations.
		There is an adjustment that can be made in order to accomplish an LU decomposition is a row interchange is necessary though we will not discuss it here.
	www.aspire.cs.uah.edu /textbook/LU.html (650 words)

	LU decomposition -- CFD-Wiki, the free CFD reference
		LU decomposition essentially stores the operations of Gaussian elimination in "higher-level" form (see Golub and Van Loan), so repeated solutions using the same left-hand side are computed without repetition of operations that are independent of the right-hand side.
		As with Gaussian elimination, LU decomposition is probably best used for relatively small, relatively non-sparse systems of equations (with small and non-sparse open to some interpretation).
		If one has a single lefthand-side matrix and many right-hand side vectors, then LU decomposition would be a good solution procedure to consider.
	www.cfd-online.com /Wiki/LU_decomposition (289 words)

	LU Decomposition
		A barrier is placed after the distributed loop and is used to synchronize between iterations of the outer sequential loop.
		The computation decomposition algorithm minimizes true-sharing by assigning all operations on the same column of data to the same processor.
		For load balance, the columns and operations on the columns are distributed across the processor in a cyclic manner.
	suif.stanford.edu /papers/anderson95/node21.html (367 words)

	LU Decomposition - GNU Scientific Library -- Reference Manual
		For square matrices this decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = P b, U x = y), which can be solved by forward and back-substitution.
		Note that the LU decomposition is valid for singular matrices.
		It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtain the same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for details).
	www.gnu.org /software/gsl/manual/html_node/LU-Decomposition.html (606 words)

	First Steps in Numerical Analysis
		Another general approach to solving Ax = b is known as the method of LU decomposition, which provides new insights into matrix algebra and has many theoretical and practical uses.
		For an LU decomposition of a given n x n matrix A, we seek a lower triangular matrix L and an upper triangular matrix U (both of order n x n) such that A = LU.
		Another technique that may be used to find an LU decomposition of an n x n matrix is by direct decomposition.
	mpec.sc.mahidol.ac.th /numer/STEP15.HTM (795 words)

	HPL References
		LU Factorization and the LINPACK Benchmark on the Intel Paragon, D. Womble, D. Greenberg, D. Wheat and S. Riesen, Sandia Technical Report, 1994.
		LU Factorization Algorithms on Distributed-Memory Multiprocessor Architectures, G. Geist and C. Romine, SIAM Journal on Scientific and Statistical Computing, Vol.
		Locality of Reference in LU Decomposition with partial pivoting, S. Toledo, SIAM Journal on Matrix.
	www.netlib.org /benchmark/hpl/references.html (885 words)

	Computational Linear Algebra
		Through the decomposition process, the coupled equations are decoupled and the solution can be obtained with much less effort.
		Thus, the system of linear algebraic equations can be decomposed to two systems of linear algebraic equations which can be solved directly.
		The coefficient matrices of the system can then be moved to the right hand side of the equal sign.
	www.efunda.com /math/num_linearalgebra/num_linearalgebra.cfm (277 words)

	Laboratory Products & Instrumentation - Australia and New Zealand -
		In combination with special halogen-resistant decomposition vessels of the C5012 series, quantitative decompositions can be performed in parallel to the analysis of gross calorific value to determine halogen and sulfur content.
		In contrast to conventional calorimeters which measure the released reaction heat in water, determination of the combustion value in the case of the C7000 is based on temperature measurement in the decomposition vessel itself.
		The system is ideally suited for decomposition of solid and liquid samples for further determination of all halogens as well as for sulfur.
	www.protechgroup.com.au /ika_werkes.htm (921 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.
		This decomposition is used in numerical analysis to solve systems of linear equations or find the inverse of a matrix.
		The LU decomposition is basically a modified form of Gaussian elimination.
	en.wikipedia.org /wiki/LU_decomposition (751 words)

	Block LU decomposition - Wikipedia, the free encyclopedia
		In linear algebra, a Block LU decomposition is a decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U.
		This decomposition is used in numerical analysis to reduce the complexity of the block matrix formula.
		and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition.
	en.wikipedia.org /wiki/Block_LU_decomposition (134 words)

	AP-931 Streaming SIMD Extensions - LU Decomposition
		This application note describes LU Decomposition of matrices with arbitrary dimensions using Intel?s Streaming SIMD Extensions.
		The performance of the code, which uses the Streaming SIMD Extensions for LU Decomposition, is approximately 2.6x times faster (for 15 x 15 matrices) than a generic C code implementation (See section 5.1).
		These measurements are based on tests run on a 450MHz Pentium®
	developer.intel.com /design/pentiumiii/sml/245046.htm (72 words)

	[No title]
		No ongoing effort is required, but plan on minimal to medium effort to turn the compost depending on your desire for speedy decomposition.
		With some systems, you can add a second bin next to the first to transform the single holding bin into a multi-bin system to make turning easier.
		Low to high maintenance, depending on desired rate of decomposition.
	www.compostinfo.com /tutorial/Bins.htm (755 words)

	CS267: Notes for Lecture 14, Feb 29, 1996
		The result of LU decomposition on A is as follows, where fl(a op b) is the floating point result of a op b:
		For LU decomposition, this means that we will process the matrix in blocks of b columns at a time, rather than one column at a time.
		As soon as the leftmost b columns of A(end+1:n,end+1:n) are updated, their LU factorization may begin while the remaining columns of the green submatrix are being updated by other processors.
	www.cs.berkeley.edu /~demmel/cs267/lecture12/lecture12.html (8014 words)

	comp 391-079 Numerical Analysis
		LU decomposition is useful when many b’s have to be solved for one A, where Ax=b.
		LU decomposition is an alternative method to the Gaussian elimination solving the set of linear equations.
		The matrix inverse is equivalent to solving d= {a,b,c} for Ad={1,0,0}, and d for the columns in I, where d is the columns of the inverse matrix of A. very cool, I will never do a matrix inverse another way.
	www.cs.unc.edu /~dorianm/academics/comp391079/linear (487 words)

	Matrix Decomposition Methods
		Performs LU decomposition on a square matrix, i.e., the matrix A is decomposed as,
		On decomposition, the matrix A is replaced by both L and U. The new changed row index is returned in the valarray ri for performing the back substitution later.
		After successful decomposition, the original matrix A is replaced by the orthogonal matrix Q, and the upper triangular matrix R is returned in r.
	www.techsoftpl.com /matrix/doc/matdcmp.htm (995 words)

	TooN: TemplateTooN::LU< Size > class Reference
		Performs LU decomposition and back substitutes to solve equations.
		The LU decomposition is the fastest way of solving the equation
		The convention LU<> (=LU<-1>) is used to create an LU decomposition whose size is determined at runtime.
	svr-www.eng.cam.ac.uk /~twd20/TooNhtml/classTooN_1_1LU.html (231 words)

	PA = LU Factorization with Pivoting
		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.
		If row interchanges are permitted then a factorization of a "permuted matrix" will be obtained.
	math.fullerton.edu /mathews/n2003/LUFactorMod.html (469 words)

	LU Decomposition
		We shall describe its modern form, which is the decomposition of
		This decomposition is useful, because the solution of triangular matrices is easily accomplished by successive substitution in the corresponding linear equations (starting with the corner of the triangle corresponding to a single non-zero term on the left side of the equation).
		We begin by observing that the decomposition (1) is not unique, since the matrices
	www.dean.usma.edu /MATH/courses/ma371/ma371/LU_Decomposition.html (196 words)

	[No title] (Site not responding. Last check: )
		Vsip_lud_d performs a LU decomposition and vsip_lusol_d performs the related back substitution.
		Finally we call the ESSL routine dgef which performs the LU decomposition.
		Note that we have sufficient information to call many different LU decomposition routines.
	www.sdsc.edu /PET/NAVO/reports/vsip/ludecomp.html (900 words)

	LU-GPU: Efficient Algorithms for Solving Dense Linear Systems on Graphics Hardware
		We reduce matrix decomposition and row operations to a series of rasterization problems on the GPU architecure.
		Average running time of LU matrix decomposition without pivoting, for different number of parallel fragment processors disabled in a NVidia 6800 Ultra GPU, in function of matrix size.
		We map the problem to the GPU architecure based on the observation that the fundamental operations in matrix decomposition are elementary row operations.
	gamma.cs.unc.edu /LU-GPU (586 words)

	Gauss Elimination and LU Decomposition
		I claim that the matrix product LU is equal to the original coefficient matrix for my equations.
		You use LU decomposition to do most of the work up front then additional equations are relatively cheap.
		Let's check my claim that the product of L and U is equal to the original coefficient matrix for the linear equations, and at the same time clearly define matrix multiplication.
	www.personal.psu.edu /faculty/j/h/jhm/f90/lectures/lu.html (1116 words)

	Essays/LU Decomposition - J Wiki
		A recursive computations of the LU decomposition reveals itself by considering
		as a 2-row matrix of matrices and substituting matrix operations for scalar and vector ones in the LU decomposition of a 2-row matrix of scalars.
		The algorithm is described in section 6.4 of Aho, Hopcroft, and Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.
	www.jsoftware.com /jwiki/Essays/LU_Decomposition (137 words)

	LU decomposition - math-linux.com
		Given a matrix A, the aim is to build a lower triangular matrix L and an upper triangular matrix which has the following property: diagonal elements of L are unity and A=LU.
		The resolution is facilitated by the triangular shape of the matrices.
		admits an LU factorization, the LU Decomposition algorithm is:
	www.math-linux.com /spip.php?article51 (183 words)

	AP-931 Streaming SIMD Extensions - LU Decomposition
		This application note describes LU Decomposition of matrices with arbitrary dimensions using Intel?s Streaming SIMD Extensions.
		The performance of the code, which uses the Streaming SIMD Extensions for LU Decomposition, is approximately 2.6x times faster (for 15 x 15 matrices) than a generic C code implementation (See section 5.1).
		These measurements are based on tests run on a 450MHz Pentium®
	www.intel.co.jp /design/pentiumiii/sml/245046.htm (72 words)

	IntelÂ® Software Network - determinant by LU-decomposition with ?getrf (Site not responding. Last check: )
		I want to calculate the determinant of a matrix by LU-decomposition with ?getrf.
		Since ?getrf does not compute the decomposition of the original matrix but of a matrix obtained by row permutations of the original one, the sign of the determinant depends on the number of those permutations.
		The MKL reference tells me that this information is contained in the array 'ipiv' where the pivot indices mean: row i was interchanged with row ipiv(i).
	softwareforums.intel.com /ISN/Community/en-US/forums/thread/5460425.aspx (515 words)

	GNU Scientific Library -- Reference Manual: LU Decomposition
		The algorithm used in the decomposition is Gaussian Elimination with partial pivoting (Golub & Van Loan,
		These functions solve the system A x = b using the LU decomposition of A into (
		These functions solve the system A x = b in-place using the LU decomposition of A into (
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_203.html (400 words)

	Yn-Yz
		In the solution procedure pivotal interchanges are used in an attempt to preserve both the stability of the computations and the sparsity of the original matrix.
		A decomposition LU=PAQ is usually calculated where P and Q are permutation matrices, L is a lower triangular matrix, and U is an upper triangular matrix.
		The right hand side vector is also modified during the decomposition so the vector c=L*(-1)Pb is available after the decomposition* stage.
	stommel.tamu.edu /~baum/linuxlist/linuxlist/node56.html (1220 words)

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