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Topic: Gaussian elimination

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	Structured Gaussian elimination
		The basic idea of structured Gaussian elimination is to declare some columns (those with the largest number of non-zero elements) as heavy, and to work only on preserving the sparsity of the remaining light columns.
		Those experiments indicated that structured Gaussian elimination ought to be very successful, and that to achieve big reductions in the size of the matrix that has to be solved, the original matrix ought to be kept very sparse, which has implications for the choices of parameters in factorization and discrete logarithm algorithms.
		Structured Gaussian elimination was very successful on data set K, since it reduced it to set L very quickly (in about 20 minutes for reading the data, roughly the same amount of time for the basic run, and then under an hour to produce the dense set of equations that form set L).
	www.farcaster.com /papers/crypto-solve/node5.html (3236 words)

	Gaussian elimination - Wikipedia, the free encyclopedia
		Gaussian elimination is, however, a good method for systems of equations over a field where computations are exact, such as finite fields.
		The strategy is as follows: eliminate x from all but the first equation, eliminate y from all but the second equation, and then eliminate z from all but the third equation.
		Gaussian elimination amounts to using the matrix operations to obtain a matrix in RREF.
	www.wikipedia.org /wiki/Gauss-Jordan_elimination (1427 words)

	Encyclopedia: Gaussian elimination (Site not responding. Last check: 2007-11-02)
		Gaussian elimination is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix.
		The Gaussian elimination algorithm can be applied to any m -by- n matrix A. The three basic operations used in the Gaussian elimination (multiplying rows, switching rows, and adding multiples of rows to other rows) amount to multiplying the original matrix A with invertible m -by- m matrices from the left.
		Remember: For a system of equations with a 3x3 matrix of coefficients, the goal of the process of Gaussian Elimination is to create (at least) a triangle of zeros in the lower left hand corner of the matrix below the diagonal.
	www.nationmaster.com /encyclopedia/Gaussian-elimination (467 words)

	PlanetMath: row reduction
		Row reduction, also commonly known as Gaussian elimination, is an algorithm for solving a system of linear equations
		A variant of Gaussian elimination is Gauss-Jordan elimination.
		In essence, Gauss-Jordan elimination performs the back substitution; the values of the unknowns can be read off directly from the terminal augmented matrix.
	planetmath.org /encyclopedia/GaussianElimination.html (513 words)

	Parallel Gaussian Elimination: Description (Site not responding. Last check: 2007-11-02)
		Gaussian elimination converts our equation Ax = b into Ux = y, where U is an Upper Triangle Matrix and y is a new vector of equation values.
		The difficulty in parallelizing Gaussian elimination is that calculating a new value in the upper triangle regions requires that all previous values of the upper triangle matrix be known.
		Thus the difficulty in parallelizing Gaussian elimination is that the calculation of each row requires the calculation of all the rows that have come before it.
	www-cse.ucsd.edu /classes/fa98/cse164b/Projects/PastProjects/LU/description.html (324 words)

	Randomized Gaussian Elimination
		Gaussian elimination is an important example of an algorithm affected by the possibility of degeneracy.
		Gaussian elimination is complicated by pivoting, which handles degenerate matrices having zero elements on the diagonal.
		In this paper we resurrect a characterization of the behavior of Gaussian Elimination, namely that every one of its intermediate results is a ratio of determinants of submatrices of the input matrix, and show its equivalence to another characterization in terms of Schur complements.
	www.cs.ucla.edu /~stott/ge (1816 words)

	15.5 Important Observations about Gaussian Elimination (Site not responding. Last check: 2007-11-02)
		Gaussian elimination is practical, under most circumstances, for finding the inverse to matrices involving thousands of equations and variables.
		Gaussian elimination is, however, an extremely boring an repetitive procedure, not well suited for human beings.
		Gaussian elimination provides a straightforward way to evaluate the determinant of a matrix: the product of all the quantities divided by in the row reduction is the magnitude of the determinant of the matrix.
	www-math.mit.edu /~djk/18_022/chapter15/section05.html (381 words)

	\bf Gaussian Elimination with scaled partial pivoting
		Row 2 is the pivot row that will be used to eliminate the first variable from equations 1,3, and 4.
		Row 4 is the pivot row that will be used to eliminate the second variable from equations 1 and 3.
		Row 3 is the pivot row that will be used to eliminate the third variable from equation 1.
	www.csc.uvic.ca /~fmilinaz/csc340/notes/gaussScPPInPlace.html (204 words)

	Gaussian Elimination
		One of the most common techniques is called Gaussian elimination.
		Gaussian elimination proceeds according to the following steps: 1.
		Note that this set of equations has a special form: it has a triangle in the lower left corner in which all coefficients are zero.
	www.owlnet.rice.edu /~comp210/99fall/Lectures/22.shtml (693 words)

	lufact.html (Site not responding. Last check: 2007-11-02)
		Gaussian Elimination with good record keeping gives A = LU Recall an elimination matrix is like the identity matrix except it has one nonzero entry off the main diagonal.
		In order to eliminate the entry A[2,1], which is -35, of A (turn it into 0), 1st column, we multiply the top row of A by -A[2,1]/A[1,1] and add it to the 2nd row.
		Observe that the product of the elimination matrices in reverse order is just the lower triangular matrix L whose entries below the diagonal are the negatives of the multiplier entries of the elimination matrices below the diagonal.
	www.msc.uky.edu /carl/ma322/spr2001/html/lufact1.html (863 words)

	Gaussian Elimination
		Gaussian Elimination is considered the workhorse of computational science for the solution of a system of linear equations.
		Gaussian Elimination is a systematic application of elementary row operations to a system of linear equations in order to convert the system to upper triangular form.
		As we go through the steps of Gaussian Elimination with our 3 x 3 example system, keep in mind that although the numbers in the augmented matrix may change significantly after each elementary row operation, our solution set has not changed.
	ceee.rice.edu /Books/CS/chapter2/linear43.html (621 words)

	Balancing Reaction Equations using Gaussian Elimination
		Gaussian elimination does not (in general) give you a solution in lowest terms, so we need to handle that separately: First we do the Gaussian elimination, then afterward we rewrite the answer in lowest terms.
		Gaussian elimination follows a simple pattern: Basically you work your way down, cleaning out the columns one by one until the matrix is in upper-triangular form.
		Practically speaking, Gaussian elimination is often the most efficient way to determine if a given system of equations is underdetermined or not.
	www.av8n.com /physics/gauss-elim.htm (3748 words)

	Gaussian Elimination (Site not responding. Last check: 2007-11-02)
		The above is known as the elimination phase, the aim is to turn all the elements below the diagonal into zeros.
		The Gaussian elimination process consists of two steps, first reducing the elements below the diagonal to 0 and second, back substituting to find the solutions.
		In order to eliminate the appropriate element it is not sufficient to simply use a ratio based automatically on the values on the elements column as this may result in a divide by zero.
	astronomy.swin.edu.au /~pbourke/analysis/gausselim (488 words)

	Gaussian elimination -- CFD-Wiki, the free CFD reference
		Gaussian elimination is best used for relatively small, relatively full systems of equations.
		For sparse systems, the use of Gaussian elimination is complicated by the possible introduction of more nonzero entries (fill-in).
		In any case, it is important to keep in mind that the basic algorithm is vulnerable to accuracy issues, including (but not limited to) the distinct possibility of division by zero at various places in the solution process.
	www.cfd-online.com /Wiki/Gaussian_elimination (234 words)

	Gaussian Elimination (Site not responding. Last check: 2007-11-02)
		This module illustrates LU factorization of a matrix using Gaussian elimination with partial pivoting.
		The successive steps of Gaussian elimination are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step.
		When Gaussian elimination is complete, the resulting L and U factors are displayed as separate lower and upper triangular matrices.
	www.cse.uiuc.edu /eot/modules/linear_equations/gaussian_elimination (210 words)

	Gaussian elimination procedure
		Under Gaussian Elimination Procedure the augmented matrix becomes "triangular" in the sense that coefficients in the lower left corner are zeroes.
		After the first step of Gaussian Elimination Procedure we make sure that the leading element in the first row is not zero.
		Gaussian Elimination Procedure gives us a simple and effective way of dealing with a linear system.
	lagrange.la.asu.edu /VirtualClass/Algebra/1stCourseLinAlg/node1.html (858 words)

	Gaussian Elimination: a case study in efficient genericity with MetaOCaml
		The Gaussian Elimination algorithm is in fact an algorithm family - common implementations contain at least 6 (mostly independent) ``design choices''.
		Using MetaOCaml's staging facilities, we show how we can produce a natural and type-safe implementation of Gaussian Elimination which exposes its design choices at code-generation time, so that these choices can effectively be specialized away, and where the resulting code is quite efficient.
		The code from figure 1 in the text (also as text), a generic version from the Domains package using fraction-free elimination (also as text), and a modular version working over finite fields (also as text).
	www.metaocaml.org /examples/gausselim (188 words)

	GaussianElimination.nb
		The first is Gaussian Elimination with nonzero column pivoting.
		The second is Gaussian Elimination with partial pivoting.
		The third is Gaussian Elimination with scaled column pivoting.
	banach.millersville.edu /~bob/math375/GaussianElimination (36 words)

	Gaussian Elimination: A Case Study (Site not responding. Last check: 2007-11-02)
		The Gaussian elimination stages of the algorithm comprises N-1 steps.
		In the basic algorithm, the ith step eliminates nonzero subdiagonal elements in column i by subtracting the ith row from each row j in the range [i+1, n], in each case scaling the ith row by the factor Aji/Aii so as to make the element Aji zero.
		For numerical stability, this basic algorithm is modified so that instead of stepping through rows in order, it selects in step i the row in the range [i, n] with the largest element in column i.
	hpc.doc.ic.ac.uk /PPS/PPS99/sld086.htm (200 words)

	M439 Gaussian Eliminatino and Pivoting
		The matrix form for a linear system of m equations in n variables is Ax = b, where A is the mxn matrix of coefficients, x is the nx1 column vector whose elements are the variables, and b is the mx1 column vector whose elements are the right hand sides of the system of equations.
		Gaussian Elimination: Row-reduce the augmented matrix to upper echelon form -- called upper triangular form for square matrices -- using elementary row operations.
		In order to reduce errors in the reduction of a matrix to upper triangular form, partial pivoting is frequently used in Gaussian elimination..
	www.saintjoe.edu /~karend/m439/m439-gauss.html (1092 words)

	kbAlertz: (51605) - This article explains the purpose of Gaussian elimination and gives a code example. In Microsoft ...
		Gaussian elimination reduces the augmented matrix [the combination of a(m,n) and b(m)] to a matrix of reduced row-echelon form, which looks like a square identity matrix attached to a 1 by m vector [ b() ].
		The Gaussian elimination functions MatSEqnS%, MatSEqnD%, and MatSEqnC% accept a square matrix a() and a vector b() as input arguments (together composing the input-augmented matrix), and give the solution in the one-dimensional array b().
		After you invoke the function, a() is replaced with the identity matrix, and the solution values overwrite the input arguments that you had placed in b().
	www.kbalertz.com /kb_51605.aspx (900 words)

	UCES Methods and Analysis Chap. 1.6: Gaussian Elimination and Steady State Heat Conduction
		The general Gaussian elimination method requires the augmented matrix, forward sweep to convert the problem to upper triangular form, and the backward sweep to solve this upper triangular system.
		Maple has a number of intrinsic procedures which are useful for illustration of Gaussian elimination.
		The procedure gausselim does the forward elimination sweep which is the row operations to get zeros in the lower triangular part of the augmented matrix.
	www.krellinst.org /UCES/archive/classes/CNA/dir1.6/uces1.6.html (1347 words)

	ENCH250, Spring 2004 (Site not responding. Last check: 2007-11-02)
		The Gaussian elimination procedure provides more than just the solution to a system of linear equations - the forward elimination procedure gives a clear indication of the number of truely independent equations in the set.
		The forward elimination procedure is an effective means of removing redundant equations from a set of linear equations.
		Pivoting is a procedure that increases the numerical accuracy of the Gaussian elimination procedure by reducing roundoff errors.
	www.isr.umd.edu /~adomaiti/ench250/notes/linearsystems.html (198 words)

	4.5 Example: Gaussian Elimination (Site not responding. Last check: 2007-11-02)
		This particular example is chosen because of the near-universal familiarity with Gaussian elimination, so that maximum attention can be paid to the data parallel techniques with a minimum of distraction from becoming familiar with the problem.
		The effective cost of a parallel operation, in terms of a scalar operation, currently varies widely from system to system, but the trend appears to be (and certainly this is not inconsistent with theoretical possibility and the inexorable march of technology) asymtotic toward scalar operation costs.
		Viewed in these terms, the data-parallel version of Gaussian elimination is indeed attractive.
	csep1.phy.ornl.gov /pl/node19.html (527 words)

	No Title
		Count how many short operations and long operations are involved in the Gaussian elimination with scaled row pivoting.
		To check your counting it is suggested that you put counters in your code and compare your estimate to the one given in the code for many different values of n, the matrix size.
		Both the basic Gaussian elimination and and the Gaussian elimination with scaled row pivoting are important subjects for this course.
	www.physics.arizona.edu /~restrepo/475A/hw10g/hw10g.html (497 words)

	7.8 Case Study: Gaussian Elimination
		The problem considered is the Gaussian elimination method used to solve a system of linear equations
		Figure 7.10: The i th step of the Gaussian elimination algorithm in which nonzero subdiagonal elements in column i are eliminated by subtracting appropriate multiples of the pivot row.
		In the basic algorithm, the i th step eliminates nonzero subdiagonal elements in column i by subtracting the i th row from each row j in the range [i+1,n], in each case scaling the i th row by the factor
	www-unix.mcs.anl.gov /dbpp/text/node90.html (841 words)

	CS 130, Gaussian Elimination
		If the number of equations and variables in a linear system are equal it frequently has a unique solution, exactly one set of values that satisfy all equations.
		In Gaussian reduction a system is represented by its rectangular array (matrix) of coefficients.
		The method of solution is to use equivalence preserving row operations on the rows of the matrix so as to generate a sub-matrix equivalence to the identity matrix.
	students.cs.byu.edu /~cs130ta/labs/block_3/block3_1.html (785 words)

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