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Topic: Crout matrix decomposition

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Cholesky decomposition

LU factorization

LU decomposition

	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 Crout algorithm is slightly different and constructs a lower triangular matrix and a unit upper triangular matrix.
	en.wikipedia.org /wiki/LU_decomposition (823 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		The Crout rule says to copy the 2 to the left factor in the matching position.
		The orthogonal matrix on the left is obtained by permuting the 2nd and 3rd rows of the unit matrix.
		More generally, to get the matrix that does any desired permutation of rows, we start with the unit matrix and apply the desired permutation to its rows.
	www.physics.utah.edu /~detar/phycs6720/handouts/crout.txt (517 words)

	PDL::MatrixOps -- Some Useful Matrix Operations
		Matrix manipulation, particularly with large matrices, is a challenging field and no one algorithm is suitable in all cases.
		If you want to use threading on a matrix that's less than, say, 10x10, and might be near singular, then you might want to use determinant, which is a more robust (but slower) determinant finder, instead.
		LU decomposition is the answer to a lot of matrix questions, including inversion and determinant-finding, and lu_decomp is used by inverse.
	pdl.sourceforge.net /PDLdocs/MatrixOps.html (2506 words)

	Method and apparatus for solving simultaneous linear equations - Patent 6826585
		In the conventional numerical solving methods for the simultaneous linear equations, in the case of solving the simultaneous linear equations the coefficient matrix of which is the sparse matrix, the selection of the optimum solving method has necessitated the empirical and professional knowledge.
		In the present invention, all the elements of the coefficient matrix elements of the sparse matrix including the zero elements, and all the elements of the right-side vector elements are stored into an array in much the same way as the case of the dense matrix.
		In the decomposition processing section 112 in the present embodiment, processings at steps 602 to 608 are repeated, while incrementing K from 1 up to N on a one-by-one basis at a step 601.
	www.freepatentsonline.com /6826585.html?highlight=5905666 (5150 words)

	ipedia.com: LU decomposition Article (Site not responding. Last check: 2007-10-11)
		In linear algebra, a LU decomposition, or LUP decomposition or Doolittle decomposition is a matrix decomposition of a matrix into a lower triangular matrix L, an upper-triangular matrix U and a permut...
		In linear algebra, a LU decomposition, or LUP decomposition or Doolittle decomposition is a matrix decomposition of a matrix into a lower triangular matrix L, an upper-triangular matrix U and a permutation matrix P.
		This decomposition is used in numerical analysis to solve a system of linear equations.
	www.ipedia.com /lu_decomposition.html (353 words)

	LINEAR SYSTEMS OF EQUATIONS
		(matrix of 0s and 1s with exactly one 1 in each row and column), a unit lower triangular
		These formulas (known as the Crout or Doolittle algorithm) require exactly the same computational effort as Gaussian elimination, but are inappropriate unless it is known a priori that pivoting is not necessary.
		A Householder reflection is a matrix of the form
	filebox.vt.edu /users/whorsley/NM/web/notes/linsys/linsys.html (271 words)

	Math Library - Source Code
		Most generally a matrix is simply a rectangular array of entities also called the components of the matrix.
		Depending on the context in which the matrix comes into existence, the entities themselves may be elements of number field, such as the field of real numbers, complex numbers, or the finite fields or a ring, such as the integers, ring of functions, or polynomials over a field.
		matrix addition may be defined component-wise using the addition operation of the ring or matrix multiplication may be defined using the addition operation and multiplication operation defined on the ring.
	mymathlib.webtrellis.net /matrices.html (1091 words)

	The Algorithm (Site not responding. Last check: 2007-10-11)
		matrix, and the matrix equation do not need to be solved again, because only the right-hand side was changed.
		That is, a new LU decomposition is not needed.
		Thus the forward substitution (as described in section 14.1.4) and the backward substitution (as described in section 14.1.4) must be slightly modified.
	qucs.sourceforge.net /tech/node24.html (708 words)

	Matrix Inversion examples
		The value // passed to this decomposition, is the data member fTol of the matrix.
		If something goes wrong because the matrix is (near) singular, you have // overwritten your original matrix and no factorization is available anymore to get more // information like condition number or change the tolerance number.
		// Inverting a matrix is nothing else than solving a set of equations where the rhs is given // by the unit matrix, so the steps to take are identical to those solving a linear equation : // cout << "3.
	root.cern.ch /root/html/examples/invertMatrix.C.html (771 words)

	Numerical Computation in Science and Engineering
		Crout decomposition of a matrix, determinant and inverse.
		Doolittle decomposition of a matrix, determinant and inverse.
		QR decomposition of a matrix by three methods: Gram-Schmidt, rotation, reflection.
	dehesa.freeshell.org /NCSE (1060 words)

	February 1999/LUP Matrix Decomposition Using Templates (Site not responding. Last check: 2007-10-11)
		Crout's factorization results in a U matrix which has 1s along the main diagonal, and Doolittle's factorization results in an L matrix with ones along the the main diagonal.
		The LUP code requires the Matrix class to have a constructor of the form Matrix(unsigned int number_of_rows, unsigned int number_of_columns) and an access operator of the form T andoperator()(unsigned int row, unsigned int col).
		Listing 6 displays a sample program which calculates the LUP decomposition for a matrix, then uses the results to calculate the X vector for a given Y vector, and calculates the inverse matrix.
	www.tcnj.edu /~hernande/cujv5/html/17.02/rumore/rumore.htm (1082 words)

	TDecompLU
		For a matrix A(m,m), its inverse A_inv is defined as A * A_inv = A_inv * A = unit (m x m) Ainv is returned.
		For a matrix A(m,n), its inverse A_inv is defined as A * A_inv = A_inv * A = unit (n x m) Ainv is returned.
		The decomposition is stored in fLU: U is explicit in the upper triag and L is in multiplier form in the subdiagionals.
	root.cern.ch /root/html404/TDecompLU.html (411 words)

	Numerical Methods Lecture Notes: linear
		In partial or column pivoting, we rearrange the rows of the matrix and the right-hand side to bring the numerically largest value in the column onto the diagonal.
		Note that this decomposition is underspecified in that we may choose the relative scale of the two matrices arbitrarily.
		matrix is scaled to have a leading diagonal of unit values.
	www.damtp.cam.ac.uk /user/fdl/people/sd/lectures/nummeth98/linear.htm (1744 words)

	KPG1_LUDCx - Performs an LU decomposition of a square matrix
		The number of rows and columns in the matrix to be decomposed.
		On exit it is the LU decomposed form of the rowwise-permuted input ARRAY, with the diagonals being part of the upper triangular matrix.
		The matrix and the workspace should have this data type as well.
	star-www.rl.ac.uk /star/dvi/sun238.htx/node487.html (278 words)

	vtkMath class Reference
		Diagonalize a symmetric 3x3 matrix and return the eigenvalues in w and the eigenvectors in the columns of V. The matrix V will have a positive determinant, and the three eigenvectors will be aligned as closely as possible with the x, y, and z axes.
		Input is square matrix A, integer array of pivot indices index[0->n-1], and size of square matrix n.
		Square nxn matrix a; size of matrix in n; output eigenvalues in w; and output eigenvectors in v.
	www.rhichome.bnl.gov /People/johannes/VTK/html/classvtkMath.html (1857 words)

	Citations: of John Hopkins series in the mathematical sciences - Golub, Van Loan, computations (ResearchIndex)
		) In Figure 4 the size of matrix A is not a power of two, but it is for B and C. The size of A has been chosen so that if there is interference due the reference on A, it does not happen very often.
		....matrix given in [3] Proposition 1 The following three conditions are equivalent: i) E is an essential matrix; ii) E has one zero singular value and two non zero and equal singular values; iii) EE T = I Gamma bb T.
		If M is the matrix from Equation 6.5, any linear combination of these vectors can be chosen as a set of weights.
	citeseer.ist.psu.edu /context/370091/0 (1525 words)

	Springer Online Reference Works
		There are a number of variants of this method, mostly based on practical implementation considerations (like the methods of Crout and Doolittle) or efficiency (like the method of Choleski for symmetric systems).
		In the Western literature, the notions of LU-decomposition, forward elimination and back substitution are often associated with Gauss' method (which is also called the Gaussian elimination method).
		If the absolutely largest element in the entire matrix (or submatrix at later stages) is used as a pivot, this is being called complete pivoting.
	eom.springer.de /G/g043480.htm (625 words)

	gang::DblMatrix Class Reference
		find the determinent of the matrix in this code, the Gaussian elimination with full pivoting note: if you are dealing with positive definite matrices, you might want to use Cholesky decomposition to find out the determinant as well.
		is the output Cholesky matrix with diagonal 1
		a matrix whose elements are the sub of the two
	students.washington.edu /~gji/sources/doc/classgang_1_1DblMatrix.html (173 words)

	Newmat10 documentation
		The files consist of a large number of matrix formulae all of which evaluate to zero (except the first one which is used to check that we are detecting non-zero matrices).
		This is not a matrix thing at all but is included because it is a useful thing and because it is a simpler version of the technique used in the non-linear least squares.
		Each matrix object contains the basic information such as the number of rows and columns, the amount of memory used, a status variable and a pointer to the data array which is on the heap.
	www.robertnz.net /nm10.htm (13815 words)

	BYTE.com
		You therefore construct a matrix; machines are the rows, jobs are the col umns, and the [i,j] element of the array is the cost of doing the jth job on the ith machine.
		The authors are approving of LU decomposition as a means of solving linear equations, pointing out that their version (which makes use of what we would have to call "Crout's method with partial implicit pivoting") is a factor of 3 better than one of their Gauss-Jordan routines, a factor of 1.5 better than another.
		The LU Decomposition benchmark considers a single iteration to be the solution of one set of equations (the size of A is fixed at 101 x 101 elements).
	www.byte.com /bmark/bdoc.htm (12353 words)

	GAUSS: SOME BASICS
		If the matrix is too large, all the data will be loaded and then GAUSS will return to the first element to keep loading data until the matrix is full.
		are conformable in the sense of matrix algebra.
		Be careful not to lose track of the last matrix passed to the decomposition routines if you use this.
	www.american.edu /econ/notes/gauss.htm (3940 words)

	Matrix Umd
		Dependency Structure Matrix - A Dependency Structure Matrix, or DSM (also referred to as Dependency Structure Method, Design Structure Matrix, Problem Solving Matrix (PSM), incidence matrix, N-square matrix or Design Precedence Matrix), is a compact, matrix representation of a system or project.
		Block matrix - In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks.
		Crout matrix decomposition - In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix (P).
	ps97.triptych03.com /matrixumd.html (1428 words)

	Software Download
		It is compatible with Win2000/NT/XP and consists of a set of more than 90 functions for arithmetic, trigonometric, logarithm, exponential and matrix calculus.
		Matrix Factorization: LU, QR, QH, SVD and Cholesky decomposition.
		In general, EULER is a numerical matrix system.
	digilander.libero.it /foxes/SoftwareDownload.htm (1121 words)

	[No title]
		Write the equations in the matrix form: EMBED Equation b.
		Use Crout decomposition to transform the matrix of coefficients into the product of a lower triangular matrix and an upper triangular matrix (1’s on the diagonal) in the form: EMBED Equation .
		Multiply the lower and upper matrices to show that the original coefficient matrix is obtained.
	www.engr.udayton.edu /faculty/fbogner/cie320/h11.doc (135 words)

	Command Summary
		DELIF Deletes rows from a matrix using a logical expression.
		X.' Bookkeeping transpose of matrix X. The functions RESHAPE, VEC, VECR and the dot transpose operator (.') change the shape of matrices, while REV, ROTATER and SHIFTR move elements in the matrix, but retain the structure of the matrix.
		DUMMY Creates a dummy matrix, expanding values in a vector to rows with ones in columns corresponding to true categories and zeros elsewhere.
	faculty.washington.edu /ezivot/cmdsum.htm (3269 words)

	Algorithm
		LU decomposition is performed by Crout's algorithm, described in Sec.
		1], there is a distinction between between a manifestly singular matrix (where an entire row of the matrix is zero) and a numerically singular matrix (if a diagonal element in the decomposed matrix turns out to be zero).
		In the former case, they raise an error signal; in the latter, they replace the offending element with a ``tiny'' but nonzero number and continue.
	www.lsc-group.phys.uwm.edu /lal/lsd/node1534.html (185 words)

	Compute some Eigenvalues first thing in the morning, and nothing worse will happen to you all day.
		As I had stated, The HP-15C's L matrix is a unit lower-triangular matrix with ones on the main diagonal and also the lower-triangular elements normalized between -1 and +1.
		The LU decomposition is the preferred method used by modern numerical software, AFAIK, for computing the determinant and inverse, and for solving linear systems (other than over- and under-determined systems), because it is numerically favorable.
		The LU decomposition on the HP48 has a unit U matrix (see the AUR), rather than the unit L matrix on the HP15, a result of its use of the Crout algorithm.
	www.hpmuseum.org /cgi-sys/cgiwrap/hpmuseum/archv016.cgi?read=96055 (4386 words)

	ding: [ding_math] Linear Algebra
		The reason to have the solver constructed outside of this function is to avoid repeating the same matrix decomposition when it is called for multiple times.
		By setting this matrix to an identity matrix, this function can be used to refine the inverse computation.
		The input matrix must be the same as the source matrix fed into the solver prior to this call, and
	mywebpages.comcast.net /ylding/c/ding/group__LinearAlgebra.html (2128 words)

	Cholesky, Doolittle and Crout Factorization
		The nonsingular matrix A has an LU-factorization if it can be expressed as the product of a lower-triangular matrix L and an upper triangular matrix U:
		If U has 1's on its diagonal, then it is called a Crout factorization.
		Since we require the computation to be as exact as possible and an exact formula is known for the inverse of the Hilbert matrix, this is an example where an inverse matrix comes in handy.
	math.fullerton.edu /mathews/n2003/CholeskyMod.html (339 words)

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