Computing the complete CS decomposition

An algorithm is developed to compute the complete CS decomposition (CSD) of a partitioned unitary matrix. Although the existence of the CSD has been recognized since 1977, prior algorithms compute only a reduced version (the 2-by-1 CSD) that is equivalent to two simultaneous singular value decompositions. The algorithm presented here computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical features.Comment: New in v3: additional discussion on efficiency, Wilkinson shifts, connection with tridiagonal QR iteration. New in v2: additional figures and a reorganization of the tex

Improving Academic Performance Through the Enhancement of Teacher/Student Relationships: The Relationship Teaching Model

The authors present their case for the development of strong and appropriate relationships with students as a key for success in college teaching. The model of Relationship Teaching includes a wide and varied agenda of techniques and commitments with which to strengthen the interpersonal relationships present in the educational environment

A Backward Stable Algorithm for Computing the CS Decomposition via the Polar Decomposition

We introduce a backward stable algorithm for computing the CS decomposition of a partitioned $2n \times n$ matrix with orthonormal columns, or a rank-deficient partial isometry. The algorithm computes two $n \times n$ polar decompositions (which can be carried out in parallel) followed by an eigendecomposition of a judiciously crafted $n \times n$ Hermitian matrix. We prove that the algorithm is backward stable whenever the aforementioned decompositions are computed in a backward stable way. Since the polar decomposition and the symmetric eigendecomposition are highly amenable to parallelization, the algorithm inherits this feature. We illustrate this fact by invoking recently developed algorithms for the polar decomposition and symmetric eigendecomposition that leverage Zolotarev's best rational approximations of the sign function. Numerical examples demonstrate that the resulting algorithm for computing the CS decomposition enjoys excellent numerical stability

The Rocketdyne Multifunction Tester. Part 1: Test Method

The Rocketdyne Multifunction Tester is a general purpose test apparatus which utilizes axial and radial magnetic bearings as shaft excitation devices. The tester is modular in design so that different seal and bearing packages can be tested on the same test stand. The tester will be used for rotordynamic coefficient extraction, as well as life and fluid/material compatibility evaluations. Use of a magnetic bearing as a shaft excitation device opens up many possibilities for shaft excitation and rotordynamic coefficient extraction. In addition to describing the basic apparatus, some of the excitation and extraction methods are described. Some of the excitation methods to be discussed include random, aperiodic, harmonic, impulse and chirp

Asp-120 Locates Zn2 for Optimal Metallo-β-lactamase Activity

Metallo-β-lactamases are zinc-dependent hydrolases that inactivate β-lactam antibiotics, rendering bacteria resistant to them. Asp-120 is fully conserved in all metallo-β-lactamases and is central to catalysis. Several roles have been proposed for Asp-120, but so far there is no agreed consensus. We generated four site-specifically substituted variants of the enzyme BcII from Bacillus cereus as follows: D120N, D120E, D120Q, and D120S. Replacement of Asp-120 by other residues with very different metal ligating capabilities severely impairs the lactamase activity without abolishing metal binding to the mutated site. A kinetic study of these mutants indicates that Asp-120 is not the proton donor, nor does it play an essential role in nucleophilic activation. Spectroscopic and crystallographic analysis of D120S BcII, the least active mutant bearing the weakest metal ligand in the series, reveals that this enzyme is able to accommodate a dinuclear center and that perturbations in the active site are limited to the Zn2 site. It is proposed that the role of Asp-120 is to act as a strong Zn2 ligand, locating this ion optimally for substrate binding, stabilization of the development of a partial negative charge in the β-lactam nitrogen, and protonation of this atom by a zinc-bound water molecule

Weighted p-bits for FPGA implementation of probabilistic circuits

Probabilistic spin logic (PSL) is a recently proposed computing paradigm based on unstable stochastic units called probabilistic bits (p-bits) that can be correlated to form probabilistic circuits (p-circuits). These p-circuits can be used to solve problems of optimization, inference and also to implement precise Boolean functions in an "inverted" mode, where a given Boolean circuit can operate in reverse to find the input combinations that are consistent with a given output. In this paper we present a scalable FPGA implementation of such invertible p-circuits. We implement a "weighted" p-bit that combines stochastic units with localized memory structures. We also present a generalized tile of weighted p-bits to which a large class of problems beyond invertible Boolean logic can be mapped, and how invertibility can be applied to interesting problems such as the NP-complete Subset Sum Problem by solving a small instance of this problem in hardware