Gauss-Jordan elimination

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	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.
		This algorithm differs slightly from the one discussed earlier, because before eliminating a variable, it first exchanges rows to move the entry with the largest absolute value to the "pivot position".
	www.wikipedia.org /wiki/Gauss-Jordan_elimination (1383 words)

	Gauss-Jordan Elimination
		In Gauss-Jordan Elimination, the goal is to transform the coefficient matrix into a diagonal matrix, and the zeros are introduced into the matrix one column at a time.
		Gauss-Jordan Elimination is a variant of Gaussian Elimination.
		However, we will show later that Gauss-Jordan elimination involves slightly more work than does Gaussian elimination, and thus it is not the method of choice for solving systems of linear equations on a computer.
	ceee.rice.edu /Books/CS/chapter2/linear44.html (439 words)

	Bicycle
		While there are many methods for solving sets of simultaneous equations, one of the better ways of doing so is to set-up an augmented matrix based on the coefficients of your variables and then to perform Gauss Elimination or Gauss-Jordan Elimination.
		The process of Gauss elimination involves a series of matrix operations that reduce an augmented matrix into simpler forms from which the solution set of a system of equations can be more easily determined.
		Gauss elimination leaves a matrix in lower triangular form, with ones (1's) on the diagonal.
	links.math.rpi.edu /devmodules/bicycle/html/gauss.html (331 words)

	Calculating Matrix Determinant Using Gauss-Jordan Elimination
		Full Gauss-Jordan elimination requires both upper and lower triangles be zeroes.
		This elimination comes into play due to the fact that if the elements in lower triangle of the matrix are all zeroes, the determinant is simply the product of the diagonals.
		However, the full-fledged version of the elimination is not needed.
	www.geocities.com /SiliconValley/Park/3230/misc/misc20011214-0000.html (254 words)

	PlanetMath: row reduction
		In essence, Gauss-Jordan elimination performs the back substitution; the values of the unknowns can be read off directly from the terminal augmented matrix.
		The row reduction algorithm (also commonly known as Gaussian elimination) is used to solve a system of linear equations
		In this variation we reduce to echelon form, and then if the system proves to be consistent, continue to apply the elementary row operations until the augmented matrix is in reduced echelon form.
	planetmath.org /encyclopedia/GaussJordanElimination.html (471 words)

	Cost of Solving a System of Linear Equations Using Gauss-Jordan Elimination
		To calculate the cost of Gauss-Jordan Elimination, we begin by counting the number of arithmetic operations necessary to convert the coefficient matrix of the original system to a diagonal matrix.
		Now let us consider solving a general n x n system using Gauss-Jordan Elimination.
		As before, we can use the expression for the sum of the first n integers to rewrite the number of multiplications required for Gauss-Jordan Elimination as
	ceee.rice.edu /Books/CS/chapter5/cost2.html (450 words)

	Gauss-Jordan Elimination
		When Gauss-Jordan elimination is complete, the remaining diagonal matrix on the left is scaled to produce an identity matrix, at which point the matrix on the right will be the inverse of the original matrix.
		Each Gauss-Jordan elimination step is also applied to an auxiliary matrix on the right, which is initially the identity matrix.
		The successive steps of Gauss-Jordan elimination are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step.
	www.cse.uiuc.edu /eot/modules/linear_equations/gauss_jordan (233 words)

	Gauss, Carl Friedrich -- Britannica Concise Encyclopedia - The online encyclopedia you can trust!
		He published over 150 works and made such important contributions as the fundamental theorem of algebra (in his doctoral dissertation), the least squares method, Gauss-Jordan elimination (for solving matrix equations), and the bell curve, or Gaussian error curve (see normal distribution).
		Gauss made important contributions to physics and astronomy and pioneered the application of mathematics to gravitation, electricity, and magnetism.
		One gauss corresponds to the magnetic flux density that will induce an electromotive force of one abvolt (10-8 volt) in each linear centimetre of a wire moving laterally at one centimetre per second at right angles to a magnetic flux.
	www.britannica.com /ebc/article?eu=390629 (867 words)

	Gauss elimination
		However, Gauss-Jordan elimination is less efficient in usage of computer resources than Gaussian elimination.
		An alternative procedure, Gauss-Jordan elimination, uses elementary row operations to both zero the elements below the diagonal and above.
		Gaussian elimination solves this linear system of equations by converting the original equations, using elementary row operations, to a simpler form that allows a simple substitution process.
	home.att.net /~srschmitt/script_gauss_elimination3.html (307 words)

	Note on the stability of Gauss-Jordan elimination for diagonally dominant matrices (ResearchIndex)
		Gauss-Jordan elimination is backward stable for matrices diagonally dominant by rows and not backward stable for matrices diagonally dominant by columns.
		Gauss-Jordan elimination (GJE) reduces a matrix A 2 R n\Thetan to diagonal form in n successive...
		25.5%: Note on the stability of Gauss-Jordan elimination for diagonally..
	citeseer.ist.psu.edu /120498.html (300 words)

	lecture04
		The underlying assumption of Gauss-Jordan elimination is that the coefficient array, A, can be factored into two matrices, L and U, where L is a lower triangular matrix and U is an upper triangular matrix.
		Compared to the brute force calculation of the matrix inverse, Gauss-Jordan elimination on sparse matrices saves time by avoiding repeated multiplying by zeros and saves storage by taking advantage of sparsity.
		In the system of equations shown below, A is the square (sparse) coefficient array, x is the vector of unknowns, and b is the vector of knowns.
	rock.uta.edu /dillon/ee5301/lecture04.htm (840 words)

	Abstraction from Gauss-Jordan Elimination Method
		The method we applied in Section 1.2 to solve the linear system given in Section 1.1.2 is called the Gauss-Jordan Elimination method.
		Eliminating as many entries of the coefficient matrix as possible.
		Given an arbitrary augmented matrix, the reason we try to apply the above two row operations to simplify it to a row echelon matrix is: ``For a row echelon matrix, we may simply read out the solution set for the linear system it represents."
	amath.nchu.edu.tw /~hsu/run/node11.html (259 words)

	Learn more about Numerical analysis in the online encyclopedia.
		The linear systems that come form discretized Partial Differential Equations can then be solved by a variant of Gauss-Jordan elimination, by some Iterative method such as Conjugate Gradients, or by Multigrid.
		For very large problems, the partial differential equation can be split into smaller subproblems and solved in parallel, as in domain decomposition methods.
	www.onlineencyclopedia.org /n/nu/numerical_analysis_1.html (718 words)

	Reduced Row-Echelon Form
		Gauss-Jordan Elimination Use ERO to reduce augumented matrix to RREF and then use back substitution to solve equivalent system.
		Gaussian Elimination Use ERO to reduce augumented matrix to REF and then use back substitution to solve equivalent system.
		Continue until the entire matrix is in REF. This procedure is called Gaussian Elimination.
	www.math.ucdavis.edu /~schan/22A/SS99/rref/rref.html (293 words)

	Row Reduced Echelon Form of a Matrix
		Definition 4.3 (Basic, Non-Basic Variables) In the Gauss elimination procedure, the variables corresponding to the leading columns are called the basic variables.
	home.iitk.ac.in /~arlal/book/nptel/la/node18.html (180 words)

	6
		Calculate the inverse by the Gauss-Jordan elimination or state that it does not exist.
	www.math.fsu.edu /~fusaro/EngMath/Ch6/hw6.7/6.7hw.htm (14 words)

	Gauss-Jordan Elimination
		The Gauss-Jordan elimination method is the "heuristic" scheme found in most linear algebra textbooks.
		Use the improved Gauss-Jordan elimination subroutine with row interchanges to solve
		Use the Gauss-Jordan elimination method to solve the linear system
	www.ecs.fullerton.edu /~mathews/n2003/GaussianJordanMod.html (470 words)

	supplement.txt
		Use Gauss-Jordan elimination (Section 2.4) to find the inverse of [ 1 1 1 ] A = [ 0 1 1 ].
		Use Gauss-Jordan elimination (Section 2.4) to find the inverse of [ 1 3 0 ] A = [ 2 5 1 ].
		Find the symmetric factorization L D L^{T} for A^{-1}, where the matrix A is given in Problem 6a.
	www.caam.rice.edu /~zhang/caam310/supplement.txt (1301 words)

	matrixlnk19.html
		However, Gauss-Jordan elimination is the way most students have learned to solve matrix equations by hand, it is quite a stable method if implemented with full pivoting, and it is always useful to have alternative ways of doing something.
		To invert a matrix, Gauss-Jordan elimination is a good method.
		In addition to solving sets of matrix equations with one or more righthand sides b, Gauss-Jordan elimination finds the matrix inverse A-1.
	eyrie.shef.ac.uk /mtica/matrix/matrixlnk19.html (304 words)

	Systems of Linear Equations solved by Block Gauss Jordan Method using a Transputer Cube
		The method chosen is the Gauss-Jordan elimination method for full, general matrices.
		The method is a block version of the Gauss Jordan method.
		Thus it is important that we have efficient methods for linear equations, and one of the means of achieving this is parallel programming.
	www.imm.dtu.dk /documents/ftp/tr95/tr08_95.abstract.html (331 words)

	Matrix Analysis
		Gauss elimination can be used to decompose [A] into [L] and [u].
		This is very similar to what we do in Gauss elimination.
		This will give the you original matrix [A] when multiplied by the upper triangular matrix results from forward elimination.
	www.phys.subr.edu /physics/course/phy542/sysofequs/sysofequs.htm (906 words)

	Math Applications - Linear Systems
		In the case where there is a unique solution, Gauss-Jordan elimination is basically the process of reducing the coefficient matrix to an identity matrix by modifying the constant matrix.
		If your calculator does not, you can use this Gauss-Jordan Elimination Spreadsheet to solve systems of three equations and three variables.
		This has been further streamlined by reducing the equations to matrices of coefficients (numbers multiplying the variables) and constants (numbers without variables on the opposite side of the "=" sign) and doing Gauss-Jordan elimination.
	www.people.vcu.edu /~ldwibber/tutoring/appsystems.html (632 words)

	Linear Equations -- JHU MATLAB Help Page
		Gauss-Jordan Elimination is a technique for solving systems of linear equations.
		Since all the rows have the same dimensions (all are single rows with the same amount of columns), we can use our techniques for addition (and subtraction) of matrices.
		Step 3 - Eliminate all other entries in the cursor column, by subtracting suitable multiples of the cursor row from all other rows.
	mathnt.mat.jhu.edu /matlab/1-6.html (614 words)

	Precalculus with Limits: A Graphing Approach- ACE Practice Tests
		Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
		use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
	college.hmco.com /cgi-bin/SaCGI.cgi/ace1app.cgi?FNC=AcePresent__Apresent_html___mathematics_larson_precalculus_limits_aga_3e_08-01 (185 words)

	linalg::gaussJordan -- Gauss-Jordan elimination
		If we consider the matrix from example 1 as an integer matrix and apply the Gauss-Jordan elimination we get the following matrix:
		We apply Gauss-Jordan elimination to the following matrix:
		Because the determinant of a matrix is only defined for square matrices, the third element of the returned list is the value
	www.sciface.com /STATIC/DOC25/eng/linalg/gaussJordan.shtml (226 words)

	Lab for the Gauss-Jordan Method for Linear Systems
		Then perform Gaussian elimination as you did before, showing all the intermediate computations.
		Construct the solution to AX = B, by using Gaussian elimination.
		Use above Gaussian elimination method to solve the linear system AX = B:
	math.fullerton.edu /mathews/numerical/gj.htm (331 words)

	Linear Algebra Lecture Notes, 01/28/03
		Recall from Section 1.2 the method of solving a linear system by Gauss-Jordan elimination: we construct the augmented matrix of the system, then apply row operations to construct leading 1's, and when the augmented matrix is in reduced row-echelon form we re-introduce the variables and read the solution of the system.
		If the system is inconsistent then Gauss-Jordan elimination will give us a numerical equation that is always false, and the system has no solution.
		has a leading 1 after we've completed Gauss-Jordan elimination, then the system has exactly one solution.
	www.assumption.edu /alfano/MAT203-SP03/Notes/012803.html (566 words)

	The Gauss-Jordan Elimination Method - Preview
		The elimination method of Gauß and Jordan transforms a given system of linear equations in one which has the same solutions as the original one but which is so simple that one can read off its solutions right away.
		- For example, Gauß-Jordan elimination transforms the system
		You can check that each of these number triples also solves the first system.
	www.ualberta.ca /dept/math/gauss/fcm/LinAlg/InRn/LnrEqtn/GJelmntn_prvw.htm (163 words)

	gj.m
		% x = GJ(A, b, PIV) solves Ax = b by Gauss-Jordan elimination, % where A is a square, nonsingular matrix.
		function x = gj(A, b, piv) %GJ Gauss-Jordan elimination to solve Ax = b.
	www.cs.unc.edu /~dm/UNC/COMP236/Program/Matrix/gj.m (131 words)

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