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Topic: Hessenberg matrix

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	Matrix Manual: Special Matrices
		[22]: A 22 matrix is a Givens reflection iff it is a Householder matrix.
		A symmetric or Hermitian Hessenberg matrix is tridiagonal.
		A is symmetric iff it is congruent to a diagonal matrix.
	www.psi.toronto.edu /matrix/special.html (2192 words)

	Matrix Reference Manual: Special Matrices
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		A Rotation matrix is orthogonal with a determinant of +1.
		A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4470 words)

	GLOSSARY (Site not responding. Last check: 2007-11-04)
		A band matrix is one whose entries are all zero except for the diagonal and a few of the immediately adjacent diagonals, whose entries may be nonzero.
		An elementary matrix is one which, when pre-multiplying another matrix, interchanges two rows, or multiplies one row by a nonzero constant, or adds a multiple of one row to another.
		The transpose of a matrix is the matrix obtained by switching the pairs of entries A(I,J) and A(J,I).
	www.psc.edu /general/software/packages/eispack/glossary.html (995 words)

	gallery (MATLAB Functions)
		The eigenvector matrix of the Chebyshev spectral differentiation matrix is ill-conditioned.
		A circulant matrix has the property that each row is obtained from the previous one by cyclically permuting the entries one step forward.
		The diagonal matrix of eigenvalues is initially subjected to a random orthogonal similarity transformation, and then a sequence of Givens rotations is applied (default).
	www-rohan.sdsu.edu /doc/matlab/techdoc/ref/gallery.html (1559 words)

	Hessenberg matrix - Wikipedia, the free encyclopedia
		In linear algebra, a Hessenberg matrix is one that is "almost" triangular.
		To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal.
		When the constraints of a linear algebra problem don't allow a general matrix to be reduced to a triangular one, reduction to Hessenberg form is often the next best thing.
	en.wikipedia.org /wiki/Hessenberg_matrix (137 words)

	Tridiagonal matrix - Wikipedia, the free encyclopedia
		In linear algebra, a tridiagonal matrix is one that is "almost" diagonal.
		Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have one of these properties.
		A tridiagonal matrix can also be stored more efficiently than a general matrix by using a special storage scheme.
	en.wikipedia.org /wiki/Tridiagonal_matrix (372 words)

	PlanetMath: Hessenberg matrix
		An upper Hessenberg matrix is of the form
		and a lower Hessenberg matrix is of the form
		This is version 2 of Hessenberg matrix, born on 2002-01-14, modified 2006-10-28.
	planetmath.org /encyclopedia/HessenbergMatrix.html (77 words)

	Matrix Market: Glossary
		A defective matrix cannot be transformed to a diagonal matrix using similarity transformations.
		A Hessenberg matrix is `almost' triangular, that is, it is (upper or lower) triangular with one additional off-diagonal band (immediately adjacent to the main diagonal).
		An ill-conditioned matrix is one where the solution to Ax=b is overly sensitive to perturbations in A or b.
	math.nist.gov /MatrixMarket/glossary.html (756 words)

	The Finite Element Method (Partial Differential Equation Toolbox)
		Note that in the most common case, when the function d(x) is positive, the mass matrix M is positive definite symmetric.
		Some of the eigenvalues of this Hessenberg matrix Hj,j eventually give good approximations to the eigenvalues of the original pencil (K,M) when the basis grows in dimension j, and less and less of the eigenvector is hidden in the residual matrix Ej.
		This way, the algorithm is applied to a projected matrix, and picks up a second copy of any double eigenvalue there may be in the interval.
	www-rohan.sdsu.edu /doc/matlab/toolbox/pde/4fem6.html (1090 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-11-04)
		A border banded matrix is a 2 by 2 block matrix comprising a (large) leading block which is a square banded matrix, two dense rectangular side strips, and a (small) trailing block which is a square dense matrix.
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13553 words)

	Downdating of Szego polynomials
		For example, when the implicitly shifted QR algorithm is applied to find the spectral decomposition of a unitary Hessenberg matrix, a sequence of intermediate unitary Hessenberg matrices is generated that converges to a diagonal matrix.
		In [5], the QR algorithm for unitary Hessenberg matrices is formulated in terms of the Schur parameters of the intermediate matrices.
		In [10], the approximation problem (1.6) is solved using the IUQR algorithm to construct the Schur parameters of the unitary Hessenberg matrix H.
	www.math.niu.edu /~ammar/downdate/node2.html (1438 words)

	Documentation for EISPACK
		COMBAK Forms eigenvectors of complex general matrix from eigenvectors of upper Hessenberg matrix output from COMHES.
		CORTB Forms eigenvectors of complex general matrix from eigenvectors of upper Hessenberg matrix output from CORTH CORTH Reduces complex general matrix to complex upper Hessenberg using unitary similarity transformations.
		Accepts an upper Hessenberg and an upper triangular matrix and reduces the former to quasi-triangular form while preserving the form of the latter.
	orion.math.iastate.edu /docs/cmlib/eispack.html (1079 words)

	Guide and Reference
		are overwritten by the corresponding elements of the upper Hessenberg matrix H.
		The global general matrix A must be distributed using a square block-cyclic distribution; that is, MB_A = NB_A.
		Example: This example shows the reduction of a general matrix of order 3 to upper Hessenberg form using a 2 × 2 process grid.
	www.navo.hpc.mil /usersupport/IBM/PESSL/pessl209.html (1134 words)

	Guide and Reference
		The upper triangle and the first subdiagonal of A are overwritten by the corresponding elements of the upper Hessenberg matrix H.
		Example: This example shows the reduction of a general matrix of order 3 to upper Hessenberg form.
		As in "Example", array data for A is block-cyclically distributed using a 2 × 2 process grid, with &tau.
	www.navo.hpc.mil /usersupport/IBM/PESSL/pessl266.html (634 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 matrix is not required to be square.
		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)

	A -step maintains the Hessenberg-like structure
		Theorem 61 Suppose we have a Hessenberg-like matrix
		Theorem 63 Suppose we have a Hessenberg-like matrix
		performed on a generator representable Hessenberg-like matrix we get again a generator representable Hessenberg-like matrix, or we will have convergence to one eigenvalue in the lower right corner, while the remaining part in the upper left corner will again be a generator representable Hessenberg-like matrix.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node81.html (1202 words)

	Physics at Minnesota:
		A Hessenberg matrix is zero below the first subdiagonal.
		If the matrix is symmetric or Hermitian, the form is tridiagonal.
		This matrix has the same eigenvalues as the original, but less computation is needed to reveal them.
	www.physics.umn.edu /support/doc/matlab/techdoc/ref/hess.html (228 words)

	[No title]
		We will utilize a modular algorithm to compute $c(x)$ by computing the characteristic polynomial of $\mathbf{A}$ modulo a sequence of primes $p_1, p_2, p_3,...$ using the Hessenberg matrix algorithm, then use the Chinese remaindering algorithm to reconstruct $c(x).$ The cost of the Hessenberg approach is $O(n^3)$ arithmetic operations in $\mathbb{Z}_p$ for each prime $p$.
		An alternative to the Hessenberg matrix approach would be a Krylov approach which has the same asymptotic complexity.
		Therefore if the matrix is sparse, one needs to implement more efficient procedures for extracting the column or row indices of the nonzero entries of a matrix.
	www.cecm.sfu.ca /CAG/papers/CPpaper.txt (2027 words)

	Hessenberg Matrix and Factorization
		can be used to reduce it to a similar symmetric tridiagonal matrix.
		For complex matrices, a unitary matrix is a matrix
		This is the case that is easiest to illustrate in a first course in numerical methods.
	math.fullerton.edu /mathews/n2003/HessenbergMod.html (222 words)

	Alphabetical list of EISPACK routines (Site not responding. Last check: 2007-11-04)
		CH Eigenvalues and eigenvectors of a complex Hermitian matrix.
		COMQR Eigenvalues of a complex matrix in upper Hessenberg form.
		IMTQL2 All eigenvalues/vectors of a symmetric tridiagonal matrix.
	www.psc.edu /general/software/packages/eispack/subroutines.html (444 words)

	CHSEQR
		CHSEQR - compute the eigenvalues of a complex upper Hessenberg Àix H, and, optionally, the matrices T and Z from the Schur Sðmposition H = Z T ZH, where T is an upper triangular matrix T@ Schur form), and Z is the unitary matrix** of Schur vectors
		CHSEQR computes the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T ZH, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix** of Schur vectors.
		= ’I’: Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned; = ’V’: Z must contain an unitary matrix Q on entry, and the product Q*Z is returned.
	www.math.ucla.edu /computing/docindex/lapack-manpages-man-115.html (512 words)

	NAG C Library, Mark 7 : f08psc
		Also, nag_zgebak (f08nwc) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix.
		On entry: the stride separating matrix row or column elements (depending on the value of order) in the array h.
		On entry: the stride separating matrix row or column elements (depending on the value of order) in the array z.
	www.nag.co.uk /numeric/CL/manual/xhtml/F08/f08psc.xml (661 words)

	Timing the Nonsymmetric Eigenproblem
		INVIT (EISPACK computation of the right eigenvectors of an upper Hessenberg matrix using inverse iteration, to be compared to SHSEIN(R)).
		A variety of matrix types is allowed because the number of iterations to compute the eigenvalues, and hence the timing, can depend on the type of matrix whose eigendecomposition is desired.
		The matrix 85#85 is upper triangular with random 124#124 entries in the strict upper triangle and has on its diagonal
	www.netlib.org /lapack/lawn41/node101.html (697 words)

	The QR and QL Algorithms
		L is the lower triangular matrix (the left part) that includes the diagonal, and
		A Hessenberg matrix is a matrix with everything under the diagonal and under the one line that is under the diagonal (like in a tridiagonal matrix) set to zero.
		QR transformations is such an efficient method for finding eigenvalues of a symmetric matrix.
	beige.ucs.indiana.edu /B673/node35.html (376 words)

	An implicit -step on the Hessenberg-like matrix
		can be performed immediately on both sides of the matrix at the same time.
		block, with the remaining part of the matrix still of the correct form.
		We will now design a chasing technique which will remove the disturbance in the matrix.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node93.html (133 words)

	Finding Evalues of a Hessenberg Matrix (Site not responding. Last check: 2007-11-04)
		Because almost all elements below the diagonal are 0, we can find a simple recurrence for the determinant, and therefore the characteristic polynomial p(l), of an nn Hessenberg* matrix H: h[i][j] =0 if i>j+1.
		The formula is more complex than for a symmetric tridiagonal matrix, but is similar in form.
		We can assume that subdiagonal elements h[i+1,i] are nonzero since if one is zero, the matrix is Block Triangular and the evalue problem decomposes into solving eigenvalue problems for the two diagonal blocks, which are themselves Hessenberg.
	www.cs.colorado.edu /~mcbryan/3656.04/mail/56.htm (220 words)

	Rational Krylov Subspace Method A. Ruhe
		as a value in the complex plane where we are interested in studying the behavior of the matrix pencil (8.17).
		is used throughout and the pencil (8.17) is transformed into two Hessenberg matrices.
		We have found that this new formulation gives a more natural way to continue with a new shift and signal when we risk losing accuracy due to linear dependence.
	www.cs.utk.edu /~dongarra/etemplates/node295.html (712 words)

	NAG C Library, Mark 7 : f08pec
		nag_dhseqr (f08pec) computes all the eigenvalues, and optionally the Schur factorization, of a real Hessenberg matrix or a real general matrix which has been reduced to Hessenberg form.
		Also, nag_dgebak (f08njc) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix.
		See also the example for nag_dorghr (f08nfc), which illustrates the use of this function to compute the Schur factorization of a general matrix.
	www.nag.co.uk /numeric/CL/manual/xhtml/F08/f08pec.xml (720 words)

	EISPACK: Eigenvalue Calculations
		EISPACK is a library of routines for calculating the eigenvalues and eigenvectors of a matrix.
		A variety of options are available for special matrix formats.
		INVIT computes eigenvectors given eigenvalues, for a real upper Hessenberg matrix.
	orion.math.iastate.edu /burkardt/f_src/eispack/eispack.html (750 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		IT IS A MODIFIED VERSION OF DGEFA IN WHICH THE C MATRIX A HAS THE STRUCTURE OF AN UPPER TRIANGULAR MATRIX PLUS C SD NON-ZERO SUBDIAGONALS.
		TO SAVE SPACE THE TWO-DIMENSIONAL C MATRIX A IS REPRESENTED AS A ONE-DIMENSIONAL ARRAY AV.
		C THE INPUT MATRIX IS IN THE FORM OF AN UPPER TRIANGULAR MATRIX C PLUS SD NON-ZERO SUBDIAGONALS.
	www.unc.edu /~ewanders/ahms/algorithms/lib4.f (239 words)

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