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Topic: Givens rotation

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QR decomposition

Householder transformation

Orthogonal matrix

Numerical stability

QR algorithm

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	Rotation Diet -- Recommendations and Resources (Site not responding. Last check: 2007-10-18)
		One consequence of the rotation of a planet is the phenomenon of precession.
		The rotational delay is the delay caused by having to wait for the portion of the disk (or drum or whatever) to rotate round so that the data you want to access is readable by the read/write head.
		Maximum rotational delay is the time it takes to do a full rotation (as the relevant part of the disk may have just passed the head when you wanted the data from it).
	www.becomingapediatrician.com /health/127/rotation-diet.html (831 words)

	QR decomposition - Wikipedia, the free encyclopedia
		There are several methods for actually computing the QR decomposition, such as by means of Givens rotations, Householder transformations, or the Gram-Schmidt decomposition.
		A Givens rotation procedure is used instead which does the equivalent of the sparse Givens matrix multiplication, without the extra work of handling the sparse elements.
		The Givens rotation procedure is useful in situations where only a relatively few off diagonal elements need to be zeroed, and is more easily parallelized than Householder transformations.
	en.wikipedia.org /wiki/QR_decomposition (849 words)

	srotmg - Compaq Tru64 UNIX
		This modification eliminates the square root from the construction of the plane rotation and reduces the operation count when the modified Givens rotation, rather than the standard Givens rotations are applied.
		Given real a and b in factored form: a = sqrt(d1) * x(1) b = sqrt(d2) * y(1) SROTMG and DROTMG construct the modified Givens plane rotation, d1', d2' and H as H(11) H(12) H(21) H(22) such that - -
		The modified Givens rotations reduce the operation count of constructing and applying the rotations at the cost of increased storage to represent the rotations.
	www.fhi-berlin.mpg.de /th/locserv/alphas/libs/cxml/srotmg.3dxml.html (326 words)

	Physics at Minnesota:
		The elements of the rotated vector y are y(i) = -sx(i) + cy(i).
		The Givens plane rotation for CROT and ZROT follows: x(i) = cx(i) + sy(i) y(i) = -conjugate(s)x(i) + cy(i) The elements of the rotated vector x are x(i) = cx(i) + sy(i).
		The elements of the rotated vector y are y(i) = -conjugate(s)x(i) + cy(i).
	www.physics.umn.edu /support/doc/cxml/zrot.3dxml.html?printer=yes (586 words)

	BLAS - Documentation (Site not responding. Last check: 2007-10-18)
		On exit, a is overwritten with the rotated element r.
		On exit, c is overwritten with the first rotation ele- ment, that is, the cosine of the angle of rotation.
		On exit, s is overwritten with the second rotation ele- ment, that is, the sine of the angle of rotation.
	www.physik.tu-muenchen.de /~gammel/matpack/html/Software/Blas/blas1_drotg.html (384 words)

	Matrix Reference Manual: Special Matrices
		A 2#2 orthogonal matrix is either a Givens rotation or a Givens reflection according to whether it is proper or improper.
		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.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4204 words)

	World Intellectual Property Organization (Site not responding. Last check: 2007-10-18)
		To determine rotation, BLISS seeks a maximum in an objective function based upon aligning of an estimated joint probability density function with an axis of a coordinate system in which it is plotted: this finds the required rotation explicitly.
		Step a) may comprise determining delay and rotation parameters which implement at least one elementary paraunitary matrix providing for rotation of a pair of input signals and relative delay of the or as the case may be each other input signal.
		On leaving the switches 32, an upper channel signal is multiplied by Givens rotation parameters-so and co at amplifiers 33 and 34 respectively.
	www.wipo.int /ipdl/IPDL-CIMAGES/view/pct/getbykey5?KEY=03/73612.030904&ELEMENT_SET=DECL (8007 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		c q itself is not given, rather the information to recover the c gv, gw rotations is supplied.
		v(i) must contain the c information necessary to recover the givens rotation gv(i) c described above.
		more c c ******** integer i,j,nmj,nm1 real cos,one,sin,temp data one /1.0e0/ c c apply the first set of givens rotations** to a.
	elib.zib.de /netlib/sminpack/r1mpyq.f (301 words)

	Givens QR Factorization (Site not responding. Last check: 2007-10-18)
		A Givens rotation is then determined that annihilates the chosen entry.
		The Givens rotation matrix is displayed on the right, and the cosine, sine, and angle of rotation (in radians) are shown in text boxes.
		The Givens rotation is applied to the relevant portion of the matrix, and then the process is repeated with another matrix entry.
	www.cse.uiuc.edu /eot/modules/least_squares/givensQR (227 words)

	The CORDIC Algorithm
		It is a class of shift-add algorithms for rotating vectors in a plane.
		In a nutshell, the CORDIC rotator performs a rotation using a series of specific incremental rotation angles selected so that each is performed by a shift and add operation.
		Rotation of unit vectors provides us with a way to accurately compute trig functions, as well as a mechanism for computing the magnitude and phase angle of an input vector.
	www.andraka.com /cordic.htm (476 words)

	Givens rotation - Wikipedia, the free encyclopedia
		In mathematics, a Givens rotation is a matrix of the form
		x represents a counter-clockwise rotation of the vector x in the (i,k) plane about θ radians, hence the name Givens rotation.
		The main use of Givens rotations in numerical linear algebra is to introduce zeros in vectors/matrices.
	en.wikipedia.org /wiki/Givens_rotation (146 words)

	JSpline+ API Specification: Class Rotation
		This fact is useful for storing of rotation coefficients in place of annuled matrix entries in rotation based factorization methods.
		The construction of a rotation coefficients is implemented in a number of construct methods by the efficiency reason (to provide reusing of the class instance).
		Calculates rotation coefficients to annigilate the beginning entry of the second vector, does the rotation on entries of both vectors with the
	www.excelsior-usa.com /doc/jspline/api/ru/sscc/util/Rotation.html (454 words)

	Java Mandala Applet (Site not responding. Last check: 2007-10-18)
		Rotations around the origin can be done by representing the point as a vector of two elements, and multiplying by a 2 x 2 matrix like this:
		Suppose the rotation angle is an even fraction of 360 degrees.
		Another question I have is this: given an arbitrary rotation approximation of this kind, are all points on cycles, or are there paths that go out to infinity in both directions?
	www.tiac.net /~sw/2005/03/Mandala (1332 words)

	Linear Algebra Glossary
		A Givens rotation is a linear transformation applied to two vectors, or two rows or columns of a matrix, which can be interpreted as a coordinate axis rotation.
		A Givens rotation is similar to the elementary row operation that adds a multiple of one row to another, but because a Givens rotation is an orthogonal similarity transformation, it offers greater stability and easy invertibility.
		It is possible to zero out entries of a matrix, one by one, using Givens rotations, similar to the way that Householder matrices are used, to reduce a matrix to a simpler form.
	orion.math.iastate.edu /burkardt/papers/linear_glossary.html (13535 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Self-Scaling fast rotations are shown to be essentially as accurate as slow rotations and at least as efficient as standard fast rotations.
		Algorithms are presented which apply self-scaling fast plane rotations to the QR factorization for stiff least squares problems.
		Self-Scaling fast plane rotation algorithms, having competitive computational complexities, must therefore be the method of choice for the QR factorization of stiff matrices.
	www-users.cs.umn.edu /~anda/CV/AAA-abstract.text (341 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		SROT.........Apply Givens plane rotation to a single precision vector.
		SROTG........Construct Givens plane rotation of single precision matrix.
		SROTMG.......Construct modified Givens plane rotation of single precision matrix.
	www.umbc.edu /doc/cmlib/doc/blas/Summary.html (352 words)

	Computer Science Technical Report 1995-10
		Givens and Householder Reductions for Linear Least Squares on a Cluster of Workstations
		We report on the properties of implementations of fast-Givens rotation and Householder reflector based parallel algorithms for the solution of linear least squares problems on a cluster of workstations.
		It is shown that the Givens rotations enable communication hiding and take greater advantage of parallelism than Householder reflectors, provided the matrices are sufficiently large.
	www.cs.ucsb.edu /research/trcs/abstracts/1995-10.shtml (94 words)

	Ordering Givens Rotations for Sparse $QR$ Factorization
		The $QR$ factorization of a large, sparse matrix $A$ is frequently computed using Givens rotations.
		The precise order in which the rotations are applied can affect the amount of storage required.
		We present an ordering for the Givens rotations that, when $A$ has the Hall property, is optimal with regard to storage for $Q$ (a so-called "tight" ordering) and that preserves sparsity by restricting fill to those locations in $R$ that are necessarily nonzero.
	epubs.siam.org /sam-bin/dbq/article/25334 (191 words)

	[No title]
		The standard Givens rotation algorithm takes more operations than the Householder algorithm for doing a QR factorization.
		The fast Givens algorithm tries to speed this procedure by reducing the number of flops.
		The relationship between A, M and D is given by Q = MD^(-1/2) = M * diag(1/sqrt(di)); This algorithm is given below.
	www.cs.utexas.edu /~hmliu/project/report/progress_report.txt (1459 words)

	HPC - CXML, the Extended Math Library: Blas Level 1
		Calculates, in double-precision arithmetic, the square root of the sum of the squares of the elements of a complex vector.
		Applies a complex Givens plane rotation to two complex, single-precision vectors.
		Generates the elements for a Givens plane rotation.
	www.hp.com /techservers/software/blas1.html (517 words)

	PlanetMath: Givens rotation
		If one wants to clear parts of a matrix one element at a time, one can use Givens rotation, which is particularly practical for parallel implementation.
		The elements can be zeroed column by column from the bottom up in the following order:
		This is version 3 of Givens rotation, born on 2002-01-04, modified 2002-03-08.
	planetmath.org /encyclopedia/GivensRotation.html (150 words)

	AHPCRC Preprint Abstracts (Site not responding. Last check: 2007-10-18)
		Spheres dropped at a vertical wall in Newtonian liquids are forced into anomalous rotation and are pushed away from the wall, while in viscoelastic liquids they are forced into anomalous rotation, but are pushed towards the wall.
		The hydrodynamic mechanisms underway in the settling of circular particles in a Newtonian fluid at a vertical wall are revealed by an exact numerical simulation based on a finite-element solution of the Navier-Stokes equations and Newton's equations of motion for a rigid body.
		As opposed to previous approaches, the schemes for approximating the desired values of transformation parameters are not dictated by the function relating the effect of transformation parameters on the reduction in the off-norm, and this leads to simpler and efficient VLSI implementations.
	www.ahpcrc.org /publications/preprints/abstracts93.html (18448 words)

	deal.II Linear Algebra Classes (LAC) library: SolverGMRES< VECTOR > Class Template Reference
		You have to give the maximum number of temporary vectors to the constructor which are to be used to do the orthogonalization.
		If the number of iterations needed to solve the problem to the given criterion, an intermediate solution is computed and a restart is performed.
		Note the subtraction, which is due to the fact that three vectors are used for other purposes, so the number of iterations before a restart occurs is less by three than the total number of temporary vectors.
	www.dealii.org /developer/doxygen/lac/classSolverGMRES.html (551 words)

	Rotations
		The 2x2 matrix G(a) is called a Givens rotation and it has a number of important properties.
		A Givens rotation is the special case where the orthonormal matrix is 2x2.
		This is helpful because the derotation by the angle (-a) requires only transposing the matrix used for the rotation, e.g.
	web.usna.navy.mil /~niewoehn/ea301/givens2.htm (437 words)

	Rechenzentrum Uni KA - Beschreibung der einzelnen Routinen
		Construct a Givens plane rotation in single precision.
		Construct a modified Givens plane rotation in single precision.
		Construct a Givens plane rotation in double precision.
	www.rz.uni-karlsruhe.de /produkte/1700.php (2812 words)

	hpux srot.3x
		The vectors may be stored in one-dimensional arrays or in either rows or columns of two-dimensional arrays, and the indexing through the arrays may be either forward or backward.
		ARGUMENTS Input n Number of elements of vectors x and y to be used in the Givens rotation.
		The modified Givens subprograms are a little more difficult to use, but are more efficient.
	www.informatik.uni-frankfurt.de /doc/man/hpux/srot.3x.html (491 words)

	dlals0 the right hand side matrix B in solving the least squares problem using th... (Site not responding. Last check: 2007-10-18)
		DLALS0 applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach.
		For the left singular vector matrix, three types of orthogonal matrices are involved: (1L) Givens rotations: the number of such rotations is GIVPTR; the pairs of columns/rows they were applied to are stored in GIVCOL; and the C- and S-values of these rotations are stored in GIVNUM.
		GIVCOL (input) INTEGER array, dimension (LDGCOL, 2) Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.
	www.uni-kiel.de /rz/nvv/altix-doc/man_html/man3/dlals0.3s.html (760 words)

	Computer Science: Publication: Algorithm 830: Another Visit With Standard and Modified Givens Transformations and A ... (Site not responding. Last check: 2007-10-18)
		First we report on a correction and improvement to the Level 1 Blas routine \texttt{srotmg} for computing the Modified Givens Transformation (MG).
		We then, in the light of the performance of the code on modern compiler/hardware combinations, reconsider the strategy of supplying separate routines to compute and apply the transformation.
		Finally, we show that the apparent savings in multiplies obtained by using MG rather than the Standard Givens Transformation (SG) do not always translate into reductions in execution time.
	www.cs.kent.ac.uk /pubs/2004/1867 (175 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		DROTI - Applies a Givens rotation to a sparse vector x stored in compressed form and another vector y in full storage form do i = 1, n temp = -s * x(i) + c * y(indx(i)) x(i) = c * x(i) + s * y(indx(i)) y(indx(i)) = temp enddo
		On exit, only the elements corresponding to the indices in INDX have been modified.
		C (input) Scalar defining the Givens rotation S (input) Scalar defining the Givens rotation
	docs.sun.com /source/817-0934/droti.html (139 words)

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