Hyperbolic Rotations Research Articles

Hyperbolic rotation surfaces of constant mean curvature in 3-de Sitter space

Hyperbolic rotation surfaces of constant mean curvature in 3-de Sitter space

A fast sliding window QRD-RLS algorithm

A fast sliding window QRD-RLS algorithm is proposed. “Fast” means that for the pth-order system we need only O( p) operations at each iteration. This is derived by using Proudler's method of relating an adaptive filter problem to two linear prediction problems and transforming the corresponding signal-flow-graphs. A “sliding window” is used to restrict the size of the data matrix. To do so, we delete the oldest data after attaching a new one. For these two steps the Givens rotations and the stabilized hyperbolic rotation are used. Finally, it is shown by computer simulations that this new algorithm has high performance for convergence and good stability in case where the system is rapidly changing.

Lattices in the Pseudoeuclidean Plane

Because of the metric structure of the pseudoeuclidean plane, lattices have some remarkable features, e.g., concerning their symmetry groups and concerning isotropic, space- and time-like elements. Especially (p,q,s)-lattices, which are characterized by the fact that their symmetry groups contain hyperbolic rotations, are showing some interesting structures and substructures.

Sliding window adaptive fast QR and QR-lattice algorithms

Sliding window formulations of the fast QR and fast QR-lattice algorithms are presented. The derivations are based on the partial triangularization of raw data matrices. Three methods for window downdating are discussed: the method of plane hyperbolic rotations, the Chambers' method, and the LINPACK algorithm. A numerically ill-conditioned stationary signal and a speech signal are used in finite wordlength simulations of the full QR (nonfast), fast QR, and QR-lattice algorithms. All algorithms are observed to be numerically stable over billions of iterations for double-precision mantissas (53 bits), but as the number of bits is decreased in the mantissa, the algorithms exhibit divergent behavior. Hence, practically, the algorithms can de regarded as numerically stable for long wordlengths.

Stabilizing the Generalized Schur Algorithm

This paper provides a detailed analysis that shows how to stabilize the {\em generalized} Schur algorithm, which is a fast procedure for the Cholesky factorization of positive-definite structured matrices $R$ that satisfy displacement equations of the form $R-FRF^T=GJG^T$, where $J$ is a $2\times 2$ signature matrix, $F$ is a stable lower-triangular matrix, and $G$ is a generator matrix. In particular, two new schemes for carrying out the required hyperbolic rotations are introduced and special care is taken to ensure that the entries of a Blaschke matrix are computed to high relative accuracy. Also, a condition on the smallest eigenvalue of the matrix, along with several computational enhancements, is introduced in order to avoid possible breakdowns of the algorithm by assuring the positive-definiteness of the successive Schur complements. We use a perturbation analysis to indicate the best accuracy that can be expected from {\em any} finite-precision algorithm that uses the generator matrix as the input data. We then show that the modified Schur algorithm proposed in this work essentially achieves this bound when coupled with a scheme to control the generator growth. The analysis further clarifies when pivoting strategies may be helpful and includes illustrative numerical examples. For all practical purposes, the major conclusion of the analysis is that the modified Schur algorithm is backward stable for a large class of structured matrices.

A novel implementation of CORDIC algorithm using backward angle recoding (BAR)

We propose a backward angle recoding (BAR) method to eliminate redundant CORDIC elementary rotations and hence expedite the CORDIC rotation computation. We prove that for each of the linear, circular, and hyperbolic CORDIC rotations, the use of BAR guarantees more than 50% reduction of elementary CORDIC rotations provided the scaling factor needs not be kept constant. The proposed BAR algorithm is simple, and amenable to VLSI implementation. Taking practical applications into consideration, we discuss how to incorporate convergence range enhancement procedure with BAR, and how easy it is to devise a constant-scaling-factor BAR algorithm while still enjoying 25% reduction of CORDIC elementary rotations.

Fast residual computation for sliding window recursive least squares methods

A new algorithm which directly computes a component of the residual vector without finding the weight (solution) vector is introduced for the recursive least squares (RLS) problem with sliding window method using the QR decomposition. This algorithm uses stabilized hyperbolic rotations for downdating. An algorithm associated with the LINPACK downdating scheme is also presented for comparison. The algorithm based on stabilized hyperbolic rotations is faster than the one based on LINPACK downdating algorithm and achieves at least the same accuracy as the LINPACK algorithm. Moreover, the same systolic array for exponential window method can be used for sliding window method when the downdating is done with stabilized hyperbolic rotations.

A floating-point advanced cordic processor

In this paper, a novel architecture of a floating-point digital signal processor is presented. It introduces a single hardware structure with a full set of elementary arithmetic functions which includessin, cos, tan, arctanh, circular rotation andvectoring, sinh, cosh, tanh, arctanh, hyperbolic rotation andvectoring, square root, logarithm, exponential as well asaddition, multiplication anddivision. The architecture of the processor is based on the COordinate Rotation DIgital Computer (CORDIC) and the Convergence Computing Method (CCM) algorithms for computing arithmetic functions and it is fully parallel and pipelined. Its advanced functionality is achieved without significant increase in hardware, in comparison to ordinary CORDIC processor, and makes it an ideal processing element in high speed multiprocessor applications, e.g. real time Digital Signal Processing (DSP) and computer graphics.

Two-fold normalized square-root Schur RLS adaptive filter

A square-root Schur RLS adaptive filter with two-fold (input and residual) normalization is presented. The algorithm has several attractive features such as a fully systolic structure based on elementary hyperbolic plane rotations. All internal variables are bounded in the unit interval and fully utilize it in successive stages due to an inherent autoscaling property of the algorithm. This work extends the previous work of the author in the area of Schur RLS adaptive filtering. >

A Jacobi eigenreduction algorithm for definite matrix pairs

We propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairsA, J of small to medium-size real symmetric matrices withJ 2=I,J diagonal (neitherJ norA itself need be definite). Our Jacobi reduction works only on one matrix and usesJ-orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process. For the rotation parameters only the pivotal elements of the current matrix are needed which facilitates parallelization. We prove the global convergence of the method; the quadratic convergence was observed in all experiments. We apply our method in two situations: (i) eigenreducing a single real symmetric matrix and (ii) eigenreducing an overdamped quadratic matrix pencil. In both cases our method is preceded by a symmetric indefinite decomposition and performed in its "one-sided" variant on the thus obtained factors. Our method outdoes the standard methods like standard Jacobi orqr/ql in accuracy in spite of the use of hyperbolic transformations which are not orthogonal (a theoretical justification of this behaviour is made elsewhere). The accuracy advantage of our method can be particularly drastic if the eigenvalues are of different order. In addition, in working with quadratic pencils our method is shown to either converge or to detect non-overdampedness.

Block RLS using row householder reflections

Householder reflections applied from the left are generally used to zero a contiguous sequence of entries in a column of a matrix A. Our purpose in this paper is to introduce new row Householder and row hyperbolic Householder reflections which are also applied from the left, but now zero a contiguous sequence of entries in a row of A. We then show how these row Householder reflections can be used to design efficient sliding-data-window recursive least-squares (RLS) covariance algorithms, which are based upon rank- k modifications to the inverse Cholesky factor R -1 of the covariance matrix. The algorithms are rich in matrix-matrix BLAS-3 computations, making them efficient on vector and parallel architectures. Preliminary numerical experiments are reported, comparing these row Householder-based rank- k modification schemes with k applications of the classical updating and downdating covariance schemes which use Givens and hyperbolic rotations.

Crystallographic symmetries of quasicrystals

Abstract The crystallographic symmetries of atomic positions in a quasicrystal are considered within a direct-superspace approach involving acceptance regions which conceptually correspond to crystal forms in space. The symmetries of the embedded quasicrystal structure are elements of an n-dimensional multi-metrical space group. The corresponding rotational parts are either Euclidean rotations, or hyperbolic rotations or a product of both (multimetrical, thus) giving rise in space to combined rotational and scaling symmetries. For the example of the decagonal phase Al 78 Mn 22, structural relations are shown to exist in space which are due to multimetrical symmetries in superspace.

CORDIC-based VLSI architectures for digital signal processing

The evolution of CORDIC, an iterative arithmetic computing algorithm capable of evaluating various elementary functions using a unified shift-and-add approach, and of CORDIC processors is reviewed. A method to utilize a CORDIC processor array to implement digital signal processing algorithms is presented. The approach is to reformulate existing DSP algorithms so that they are suitable for implementation with an array performing circular or hyperbolic rotation operations. Three categories of algorithm are surveyed: linear transformations, digital filters, and matrix-based DSP algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Dual-state systolic architectures for up/downdating RLS adaptive filtering

A dual-state systolic structure to perform joint up/down-dating operations encountered in windowed recursive least-squares, (RLS) estimation problems is proposed. It is based on successively performing Givens rotations for updating and hyperbolic rotations for downdating. Due to data independence, a series of Givens and hyperbolic rotations can be interleaved and parallel processing can be achieved by alternatively performing updating and downdating both in time and space. This flip-flop nature of up/down-dating characterizes the feature of the dual-state systolic triarray. Efficient implementation on the evaluation of optimal residuals is also considered. This systolic architecture is promising for the VLSI implementation of fixed size sliding-window recursive least-squares estimations. >

Which symmetry will an ideal quasicrystal admit?

The crystallographic nature of a quasicrystal structure is expressed in terms of the possibility of labeling `translationally' equivalent atomic positions by a set of n integers. The corresponding position vectors are integral linear combinations of n basic ones generating a vector module M of rank n and dimension m. Because of the aperiodic nature of the quasicrystal, n is larger than m. Typical values observed in nature are m = 3 and n = 5 or n = 6. Lattice symmetry is recovered by embedding the quasicrystal in an n-dimensional space (the superspace) in such a way that M is the projection of a lattice 2. The rotational symmetries of the quasicrystal are included in those of the vector module M and, after embedding, appear as n-dimensional rotations leaving Σ invariant and a corresponding Euclidean metric. Scaling symmetries are also possible in the atomic point-like approximation of a quasicrystal. In that case, enlarging by a given constant factor all the distances between `translationally' equivalent atoms, the `inflated' pattern still belongs to the original one: the occupied atomic positions in space are transformed into other ones also occupied in the original structure. This is called inflation procedure (of a scaling invariant pattern), the reverse transformation being a deflation. The module M is then invariant with respect to such discrete dilatations. In the superspace these correspond to crystallographic point-group transformations leaving the lattice Σ and an indefinite metric invariant. Scaling symmetries in space appear as hyperbolic rotations in the superspace. In these non-Euclidean rotations the improper ones are included. The compatibility between the two types of n-dimensional point-group symmetries (Euclidean and non-Euclidean rotations) is discussed both at the level of the quasicrystal structure and of that of the double metrical nature of the translational lattice 2. For a characterization of the symmetry of the quasicrystal, one eventually arrives at the concept of the scale-space group, which includes as its Euclidean subgroup an n-dimensional space group (the super-space group). Examples are taken from aperiodic tilings admitting inflation–deflation symmetry. The vertices of these tilings are supposed to represent `translationally' equivalent atomic positions. A number of basic concepts not expected to be familiar to crystallographers, even if explained in the text, are also listed and defined in an Appendix.

Generalized Apollonian packings

In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements ofSL(2,ℂ) as generators:R andT that are hyperbolic rotations of 2π/3 and 2π/N (N=2,3,4....), around two distinct points. The limit set of the discrete group generated byR andT provides, forN=7,8,9,.... a generalization of the Apollonian packing (which is itself recovered forN=∞). The valuesN=2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube forN=2 and 4, and the icosahedron forN=3 and 5. ForN=6 the group is no longer discrete. To further analyze this structure forN≥7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.

Hyperbolic Householder Algorithms for Factoring Structured Matrices

Efficient algorithms for computing triangular decompositions of Hermitian matrices with small displacement rank using hyperbolic Householder matrices are derived. These algorithms can be both vectorized and parallelized. Implementations along with performance results on an Alliant FX/80, Cray X-MP/48, and Cray-2 are discussed. The use of Householder-type transformations is shown to improve performance for problems with nontrivial displacement ranks. In special cases, the general algorithm reduces to the well-known Schur algorithm for factoring Toeplitz matrices and Elden’s algorithm for solvig structured regularization problems. It gives a Householder formulation to the class of algorithms based on hyperbolic rotations studied by Kailath, Lev-Ari, Chun, and their colleagues for Hermitian matrices with small displacement structure. In addition, an extension to the efficient factorization of indefinite systems is described.

From Bareiss' algorithm to the stable computation of partial correlations

This paper presents a new algorithm for the stable computation of sample partial correlation coefficients. We start by generalizing Bareiss' algorithm for the solution of linear systems of equations with (non-symmetric) Toeplitz coefficient matrix to matrices that are not Toeplitz, and show that it computes their LU and UL factorizations. For symmetric positive-definite matrices B, the normalized version of Bareiss' algorithm is the Hyperbolic Cholesky algorithm, which computes the upper and lower triangular Cholesky factors U and L of B by means of 2×2 hyperbolic rotations. If B is a convariance matrix, certain partial correlation coefficients may be obtained as the geometric means of the multiplier pairs in Bareiss' algorithm or as the hyperbolic tangents of the hyperbolic rotations. The partial correlation coefficients for a given data matrix A may suffer from serious loss of numerical accuracy when computed from the covariance matrix B = A T A. While the data flow graph of Bareiss' algorithm suggests a method, due to Cybenko, for computing partial correlation coefficients directly from the columns of A, the graph of Hyperbolic Cholesky suggests an alternative that requires fewer operations nd exhibits a higher degree of parallelism. In that method one computes the QR decomposition A = QU of the data matrix and applies orthogonal rotations to transform U to L; the sines of the rotation angles constitute a set of partial correlation coefficients. We extend this approach to determine arbitrary partial correlation coefficients at little extra computational cost.

A vector majorization method for solving a nonlinear programming problem

In the rounding error analysis of a so-called hyperbolic rotation algorithm applied to downdating the Cholesky factor, a sharp upper bound is needed for a product ▪ where a = (a 1,…,a n T ϵ R n, ∥a∥ ⩽ θ < 1 , and θ is fixed. It is shown that ▪ In the original proof by Pan and Sigmon, the principle of Lagrange multipliers was used. A second proof by using vector majorization is presented here, which provides insights into this rather strange-looking bound. Moreover, the new proof can be readily generalized to solve a large class of nonlinear programming problems.

The square-root overdetermined recursive instrumental variable algorithm

A square-root version of the overdetermined recursive instrumental variable (ORIV) algorithm is derived. This version improves the numerical stability of the algorithm, and avoids the problem of positive-definiteness of the inverse covariance matrix. The algorithm uses square-root arrays, and both orthogonal and hyperbolic rotations, as necessitated by the nature of the ORIV algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>