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Topic: Block LU decomposition

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	[No title] (Site not responding. Last check: 2007-11-06)
		Ultimately DGEMM is a set of blocks of the matrix A multiplied by blocks of the matrix B, with careful attention given to cache management.
		We use a block size of 16x32 for the A matrix, where 16 is the blocking size for rows of C and 32 is the inner product size.
		Row pivoting is done in accordance with the block LU found in LAPACK[11].
	www.cs.utk.edu /~ghenry/sc97/paper.htm (5858 words)

	Landscaping Block (Site not responding. Last check: 2007-11-06)
		Block LU decomposition - 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.
		Block cipher - In cryptography, a block cipher is a symmetric key cipher which operates on fixed-length groups of bits, termed blocks, with an unvarying transformation.
		Gifford's landscapes are known for their emphasis on light and soft atmospheric effects, and he is regarded as a practicioner of Luminism, an offshoot style of the Hudson...
	grass.vvvvvv3.com /landscapingblock.html (900 words)

	Directives are needed for Parallel Execution
		The BLOCK keyword for the second dimension means that for any given row, the array elements are distributed over each processor in large blocks.
		If you are familiar with the low-level details of parallel programming, you may wonder how any speed-up is achieved with the LU decomposition algorithm, because the sub-matrix modification appears to require a separate communication for each element in the submatrix.
		Even though LU decomposition is not an embarrassingly parallel algorithm, parallel speed-up for this algorithm with the Digital Fortran 90 compiler is excellent.
	www.mun.ca /hpc/hpf_pse/manual/hpf0009.htm (1321 words)

	Advanced Topic: Matrix Decompositions
		QR decomposition is often used in solving least-squares fitting problems, and is typically faster than singular value decomposition.
		Schur decomposition is often used in evaluating functions of matrices.
		LU decomposition is convenient for preprocessing matrices that will appear repeatedly in collections of linear equations, as discussed in Section 3.7.8.
	documents.wolfram.com /v4/MainBook/3.7.10.html (439 words)

	Randomized Gaussian Elimination
		Block matrix decompositions are used heavily in theory, but less in practice, since even when a matrix is nondegenerate (has full rank) its block submatrices can be degenerate.
		Recursive block decomposition algorithms (also known as quadtree algorithms when the blocks are all square) have been proposed to solve well-known problems such as matrix addition, multiplication, inversion, determinant computation, block LDU decomposition, discrete Fourier transform, and Cholesky and QR factorization.
		Specifically, recursive block decomposition methods are impractical for matrix inversion and Gaussian elimination, since both require leading submatrices of the input matrix to be nonsingular (which is not always guaranteed).
	www.cs.ucla.edu /~stott/ge (1816 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 (823 words)

	CS267: Notes for Lecture 13, Feb 28, 1995
		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.
		Now consider the red line, which is the speed of LU decomposition divided by the speed of matrix-multiply.
	www.cs.berkeley.edu /~demmel/cs267/lecture12/lecture13.html (1981 words)

	LU Decomposition
		The most efficient way to solve a set of linear equations is to use an LU decomposition, since then one can solve for multiple right-hand sides with little extra work.
		This decomposition is essentially a Gaussian elimination, arranged for maximum efficiency.
		When solving several simultaneous equations it is important to arrange the unknowns in a particular way to achieve the block tri-diagonal form of the matrix and the resulting efficiencies.
	faculty.washington.edu /finlayso/ebook/algebraic/advanced/LUDecomp.htm (520 words)

	Technical Report: An Alternative to Block LU Schur Complements
		The Schur complement is typically formed from permuted systems for which a decomposition may be formed in parallel, and, for symmetric positive definite systems, is at least as well conditioned.
		We consider an iteration matrix of the same dimension as the Schur complement, which may be efficiently decomposed in parallel, which exists for all nonsigular systems, improves conditioning in every non-trivial case, and is more readily preconditioned.
		This iteration matrix decomposition is derived from a block QR, or block LP, rather than a block LU decomposition, and forms normal equations equal to a Schur complement of the normal equations of the original matrix.
	www.cs.umn.edu /research/technical_reports.php?page=report&report_id=99-026 (285 words)

	A plea for domain decomposition
		Note that domain decomposition provides a unifying framework both for preprocessing (reordering) and preconditioning, in the sense that the block structure of the matrix obtained with domain decomposition reordering leads naturally to a preconditioner.
		The top diagonal block of D is itself a block diagonal matrix, with each block being an interface preconditioner for a corresponding edge.
		The bottom block of D is the discrete form of the PDE operator on the grid defined by the crosspoints.
	filebox.vt.edu /users/gmateesc/dd.html (1828 words)

	Matrix Factorizations - Untitled
		The Hessenberg decomposition is usually used as the first step in an eigenvalue computation, but has other applications as well (see Golub, Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979).
		The Hessenberg-triangular decomposition is the first step in Moler and Stewart's QZ decomposition algorithm.
		The Schur decomposition is used to compute eigenvalues of a square matrix, and has applications in the solution of algebraic Riccati equations in control (see
	www.gnu.org /software/octave/doc/interpreter/Matrix-Factorizations.html (601 words)

	Dr. Dobb's \| Why C++ Will Replace Fortran \| July 22, 2001
		The LU decomposition of a matrix consists of finding two matrices such that A = LU, where L is a lower-triangular matrix, and U is an upper-triangular matrix.
		Example 6: (a) Using a matrix decomposition class; (b) using LU decomposition to solve five different sets of equations; (c) requesting the inverse of the original matrix and letting type conversion do the work; (d) specifying conversion explicitly.
		In Example 6(a), the LU decomposition was created first, then used to calculate, say, the inverse of the matrix.
	www.ddj.com /184408915?pgno=3 (4788 words)

	CS267: Notes for Lecture 14, Feb 29, 1996
		The Block Skewed Layout is not useful for Gaussian elimination, but can be useful in a variety of other matrix operations, so we mention it here.
		The block size b in the algorithm and the block sizes brow and bcol in the layout satisfy b=brow=bcol.
		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)

	Sequential Out-Of-Core LU Factorization (Site not responding. Last check: 2007-11-06)
		A crucial factor for performance is the choice of the blocksize, k (i.e., the column width) of the second block column.
		The shaded parts indicate the matrix elements accessed in forming a block row or column, and the darker shading indicates the block row or column being modified.
		The right-looking variant (the familiar recursive algorithm) computes a block row and column at each step and uses them to update the trailing submatrix.
	www.netlib.org /utk/papers/outofcore/node2.html (307 words)

	6.2 Gaussian Elimination on Distributed Memory Machines (Site not responding. Last check: 2007-11-06)
		Some theoretical analyses of this sort for special cases may be found in [30] and the references therein.
		As an example of the performance that can be attained in practice, on an Intel Delta with 512 processors the speed of LU ranged from a little over 1 gigaflop for n=2000 to nearly 12 gigaflops for n=25000.
		Even if the layout is not block scatter as described so far, essentially the same algorithm may be used.
	www.phy.ornl.gov /csep/CSEP/LA/NODE10A.html (263 words)

	LU Decomposition (Site not responding. Last check: 2007-11-06)
		The code is shown in Figure 5 and the speedups for each version of LU decomposition are displayed in Figure 6 for two different data set sizes (
		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)

	Linear Algebra - Preliminary lectures
		A block matrix is a matrix which can be divided into similar parts i.e.
		Cholesky decomposition is based on the following extension of this property.
		Similarly to the LU decomposition, the Gram-Schmidt orthogonalisation technique is used for calculating
	vision.unige.ch /~marchand/teaching/linalg (1819 words)

	cm conference abstract: Jun ZHANG
		We first search for a block independent set among the vertex set such that the nodes between the blocks are not coupled, but the nodes within each block may be coupled.
		Due to the block structure of the upper part submatrix this part of the ILUT factorization can be performed in parallel with respect to individual blocks.
		Thus, the reduced (coarse level) system is then defined as the approximate Schur complement associated with the block independent set partitioning and it is obtained implicitly as a byproduct of the partial ILUT factorization with respect to the complement of the independent set without explicit matrix-matrix multiplications.
	www.mgnet.org /mgnet/Conferences/CopperMtn99/abs/zhang.html (678 words)

	Cholesky Factorization (DSP Blockset)
		is the Hermitian (complex conjugate) transpose of L. The block outputs a matrix with lower triangle elements from L and upper triangle elements from L
		Hermitian -- The block does not check whether the input is Hermitian; it uses only the diagonal and upper triangle of the input to compute the output.
		Positive definite -- Set the block to notify you when the input is not positive definite as described in Response to Non-Positive Definite Input.
	www.weizmann.ac.il /matlab/toolbox/dspblks/choleskyfactorization.html (327 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)

	Looping Constructs
		In the method of LU decomposition, matrix "A" is decomposed into a product of lower and upper triangular matrices.
		Matrix functions are provided for the LU decomposition and forward/backward substitution.
		In the method of LU decomposition, solutions to multiple right-hand vectors can be computed once the matrix [A] has been decomposed.
	www.isr.umd.edu /~austin/aladdin.d/matrix-solve.html (682 words)

	cm conference abstract: Yousef Saad and Brian Suchomel
		The construction phase of the preconditioner involves computing the restriction and prolongation operators between levels, and an incomplete LU decomposition of the last reduced system.
		This matrix is again sparse because of the block-diagonal structure of the (1,1) block.
		Thus, the solution process is implemented similar to a classic V or W-cycle, except that the number of visits to the coarse levels is not necessarily predetermined.
	www.mgnet.org /mgnet/Conferences/CopperMtn99/abs/saad.html (751 words)

	BYTE.com
		The specific benchmark compon ent involved was LU Decomposition, one of the three members of the floating-point portion of the BYTEmark.
		Such was the case with the LU Decomposition test.
		The first suspect was a temporary memory buffer allocated and released within the LU Decomposition test.
	www.byte.com /art/9603/sec14/art3.htm (1696 words)

	Articles in journals and book chapters
		Revisiting the Decomposition of Karp, Miller and Winograd.
		Block algorithms for linear algebra on the IBM 3090 vector multiprocessor.
		Block LU decomposition of a band matrix on a systolic array.
	graal.ens-lyon.fr /biblio/Category/articles.html (2300 words)

	Algorithm
		A standard method for solving this is the method of LU decomposition, based on the fact that any non-singular square matrix
		to perform an LU decomposition of the matrix, then either computing the determinant from the diagonal elements, or using
		The routines that compute the determinant will also handle any ``singular matrix'' error code returned by the LU decomposition routines, returning zero as the determinant.
	www.lsc-group.phys.uwm.edu /lal/lsd/node1527.html (287 words)

	Matrix Manual: Matrix Decompositions
		This decomposition is the standard way of solving the simultaneous equations Ax = b.
		TQ where each block of T is either a 1#1 matrix or a 2#2 matrix having complex conjugate eigenvalues.
		TV where each block of S and T is either a 1#1 matrix or a 2#2 matrix having complex conjugate eigenvalues.
	www.psi.toronto.edu /matrix/decomp.html (694 words)

	Continued Studies with LU (Site not responding. Last check: 2007-11-06)
		We've looked at timings and counts of a LU Decomposition HPF program written only in data parallel intrinsic operations.
		We will use the same intrinsic LU as well as an in-place F77 style of LU.
		Designate who should run the sequential uniprocessor LU experiments and coordinate results for composite graphs as before.
	www.cs.albany.edu /~csi668/hw2/node2.html (201 words)

	A 4x4 Matrix Inverse
		The fundamental principal behind branch elimination is that expressing a value as a simple function of its inputs (a single basic block) is often more efficient than selecting a result through a change in control flow (branching).
		Inversion using LU decomposition: The description of the inverse calculation can be found in "Numerical Recipes in C" [library.cornell.edu] chapter 2.3.
		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.
	www.cellperformance.com /articles/2006/06/a_4x4_matrix_inverse_1.html (3286 words)

	[No title]
		These techniques— scalar replacement and unroll-and-jam (sometimes called register blocking), were developed in collaboration with John Cocke and David Callahan.
		With David Callahan and Alan Porterfield, he also developed standard compiler strategies for compiler cache management, including work on cache blocking with alignment with Porterfield and one of the seminal papers on software prefetching of cache blocks, establishing that the technique could be extremely effective in practice.
		A particularly important result of this work is a new strategy for multilevel blocking on irregular problems.
	www.cs.rice.edu /~ken/kennedy-techbio.html (3004 words)

	Lab 4: Advanced Programming in SA-C (Site not responding. Last check: 2007-11-06)
		Consider the system of equations for LU Decomposition given in the Notes 1 as the initial specification.
		Your program should have four functions, three of which correspond to the equations for forward substitution, backward substitution and LU Decomposition respectively.
		It decomposes the matrix A using the routine for LU decomposition, then calls the backward and the forward substitutions on either of the L and U components and returns a vector x, which is the solution for the equation Ax=b.
	www.cs.colostate.edu /~cs560/Lab4.html (576 words)

	A Spectral Element Semi-Lagrangian (SESL) Method for the Spherical Shallow Water Equations (Site not responding. Last check: 2007-11-06)
		The equations are discretized fully-implicitly in time in order to avoid having to interpolate derivatives at departure points.
		By incorporating the Coriolis terms into the Lagrangian derivative, the block LU decomposition of the equations results in a symmetric positive-definite pseudo-Helmholtz operator which we solve using the generalized minimum residual method (GMRES) with a fast projection method \cite{fischer}.
		Results for eight test cases are presented to confirm the accuracy and stability of the method.
	www.nrlmry.navy.mil /~giraldo/papers/sesl_sphere.html (269 words)

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