Degrees of freedom and modular structure in matrix multiplication
Matrix multiplication algorithm based on degrees of freedom analysis for various transformations and spectral analysis
Andrews, H. C.↗
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Parallel matrix multiplication on the Connection Machine
Matrix multiplication is a computation and communication intensive problem. Six parallel algorithms for matrix multiplication on the Connection Machine are presented and compared with respect to their performance and processor usage. For n by n matrices, the algorithms have theoretical running times of O(n to the 2nd power log n), O(n log n), O(n), and O(log n), and require n, n to the 2nd power, n to the 2nd power, and n to the 3rd power processors, respectively. With careful attention to communication patterns, the theoretically predicted runtimes can indeed be achieved in practice. The parallel algorithms illustrate the tradeoffs between performance, communication cost, and processor usage.
Tichy, Walter F.↗
E-beam generated holographic masks for optical vector-matrix multiplication
An optical vector matrix multiplication scheme that encodes the matrix elements as a holographic mask consisting of linear diffraction gratings is proposed. The binary, chrome on glass masks are fabricated by e-beam lithography. This approach results in a fairly simple optical system that promises both large numerical range and high accuracy. A partitioned computer generated hologram mask was fabricated and tested. This hologram was diagonally separated outputs, compact facets and symmetry about the axis. The resultant diffraction pattern at the output plane is shown. Since the grating fringes are written at 45 deg relative to the facet boundaries, the many on-axis sidelobes from each output are seen to be diagonally separated from the adjacent output signals.
Arnold, S. M.↗
A family of new efficient arrays for matrix multiplication
The authors present a regular iterative algorithm for matrix multiplication and show that several well-known matrix multiplication arrays are directly obtained from it, differing only in the choice of iteration vector. They then present a regular iterative algorithm for matrix multiplication using the method of Winograd (1968) and show in detail how to derive one array from this algorithmic description. Other arrays in the same family can similarly be obtained for different choices of the iteration space. The new arrays compute the product of two matrices faster than available conventional arrays and use a smaller number of processor cells.
Jagadish, H. V.↗
Integrated-optical approaches to matrix multiplication
The solution of matrix equations is essential to carrying out a large variety of control algorithms and to reducing certain types of data such as the output of a multispectral sensor array. Optical techniques and, in particular, integrated-optical circuits (IOC's) can provide compact, low-power devices for performing the mitrix multiplications necessary for the solution of these problems. A specific IOC for performing vector-matrix multiplication and several approaches to the design of IOC's for matrix-matrix multiplication will be discussed.
Verber, C. M.↗
Software for Fault-Tolerant Matrix Multiplication
Formal Linear Algebra Recovery Environment is a computer program for high-performance, fault-tolerant matrix multiplication. The program is based on an extension of the prior theory and practice of fault-tolerant matrix matrix multiplication of the form C = AB. This extension provides low-overhead methods for detecting errors, not only in C, but also in A and/or B. These methods enable the detection of all errors as long as, in a given case, only one entry in A, B, or C is corrupted. The program also provides for following a low-overhead rollback approach to correct errors once detected. Results of computational experiments have demonstrated that the methods implemented in this program work well in practice while imposing an acceptably low level of overhead, relative to high-performance matrix-multiplication methods that do not afford fault tolerance.
Katz, Daniel↗
The use of the Winograd matrix multiplication algorithm in digital multispectral processing
The Winograd procedure for matrix multiplication provides a method whereby general matrix products may be computed more efficiently than the normal method. The algorithm and the time savings that can be effected are described. A FORTRAN program is provided which performs a general matrix multiply according to this algorithm. A variation of this procedure that may be used to calculate Gaussian probability density functions is also described. It is shown how a time savings can be effected in this calculation. The extension of this method to other similar calculations should yield similar savings.
Vanrooy, D. L.↗
Optical matrix-matrix multiplication method demonstrated by the use of a multifocus hololens
A method of optical matrix-matrix multiplication is presented. The feasibility of the method is also experimentally demonstrated by the use of a dichromated-gelatin multifocus holographic lens (hololens). With the specific values of matrices chosen, the average percentage error between the theoretical and experimental data of the elements of the output matrix of the multiplication of some specific pairs of 3 x 3 matrices is 0.4 percent, which corresponds to an 8-bit accuracy.
Liu, H. K.↗
An efficient sparse matrix multiplication scheme for the CYBER 205 computer
This paper describes the development of an efficient algorithm for computing the product of a matrix and vector on a CYBER 205 vector computer. The desire to provide software which allows the user to choose between the often conflicting goals of minimizing central processing unit (CPU) time or storage requirements has led to a diagonal-based algorithm in which one of four types of storage is selected for each diagonal. The candidate storage types employed were chosen to be efficient on the CYBER 205 for diagonals which have nonzero structure which is dense, moderately sparse, very sparse and short, or very sparse and long; however, for many densities, no diagonal type is most efficient with respect to both resource requirements, and a trade-off must be made. For each diagonal, an initialization subroutine estimates the CPU time and storage required for each storage type based on results from previously performed numerical experimentation. These requirements are adjusted by weights provided by the user which reflect the relative importance the user places on the two resources. The adjusted resource requirements are then compared to select the most efficient storage and computational scheme.
Lambiotte, Jules J., Jr.↗
Efficient spares matrix multiplication scheme for the CYBER 203
This work has been directed toward the development of an efficient algorithm for performing this computation on the CYBER-203. The desire to provide software which gives the user the choice between the often conflicting goals of minimizing central processing (CPU) time or storage requirements has led to a diagonal-based algorithm in which one of three types of storage is selected for each diagonal. For each storage type, an initialization sub-routine estimates the CPU and storage requirements based upon results from previously performed numerical experimentation. These requirements are adjusted by weights provided by the user which reflect the relative importance the user places on the resources. The three storage types employed were chosen to be efficient on the CYBER-203 for diagonals which are sparse, moderately sparse, or dense; however, for many densities, no diagonal type is most efficient with respect to both resource requirements. The user-supplied weights dictate the choice.
Lambiotte, J. J., Jr.↗
Review of NASTRAN development relative to efficiency of execution
This paper reviews the development of NASTRAN relative to the efficiency of execution, with particular emphasis on those items which have changed significantly since the original release of NASTRAN. Features discussed include main and secondary storage utilization, matrix packing, matrix assembly, matrix multiplication, matrix decomposition and equation solution. Also a brief look into the future discusses the questions of faster arithmetic units and more effective storage utilization. In some cases the improvements in NASTRAN efficiency have resulted from taking advantage of hardware developments, while in other cases increased efficiency has resulted from improvements in the state of the art for data processing or matrix operations. The modular design of NASTRAN has made it possible to improve the efficiency in many parts of NASTRAN without changing the basic design of the program.
Mccormick, C. W.↗
Matrix-vector multiplication in thin photorefractive GaAs crystals
Optical matrix-vector multiplication using four-wave mixing in a thin photorefractive GaAs crystal is demonstrated. Using a thin wafer of GaAs offers the potential to integrate the encoding spatial light modulators directly on the wave-mixing medium.
Cheng, Li-Jen↗
Fast polar decomposition of an arbitrary matrix
The polar decomposition of an m x n matrix A of full rank, where m is greater than or equal to n, can be computed using a quadratically convergent algorithm. The algorithm is based on a Newton iteration involving a matrix inverse. With the use of a preliminary complete orthogonal decomposition the algorithm can be extended to arbitrary A. How to use the algorithm to compute the positive semi-definite square root of a Hermitian positive semi-definite matrix is described. A hybrid algorithm which adaptively switches from the matrix inversion based iteration to a matrix multiplication based iteration due to Kovarik, and to Bjorck and Bowie is formulated. The decision when to switch is made using a condition estimator. This matrix multiplication rich algorithm is shown to be more efficient on machines for which matrix multiplication can be executed 1.5 times faster than matrix inversion.
Higham, Nicholas J.↗
Parallel Gaussian elimination of a block tridiagonal matrix using multiple microcomputers
The solution of a block tridiagonal matrix using parallel processing is demonstrated. The multiprocessor system on which results were obtained and the software environment used to program that system are described. Theoretical partitioning and resource allocation for the Gaussian elimination method used to solve the matrix are discussed. The results obtained from running 1, 2 and 3 processor versions of the block tridiagonal solver are presented. The PASCAL source code for these solvers is given in the appendix, and may be transportable to other shared memory parallel processors provided that the synchronization outlines are reproduced on the target system.
Blech, Richard A.↗
Applications of multiple-constraint matrix updates to the optimal control of large structures
Low-authority control or vibration suppression in large, flexible space structures can be formulated as a linear feedback control problem requiring computation of displacement and velocity feedback gain matrices. To ensure stability in the uncontrolled modes, these gain matrices must be symmetric and positive definite. In this paper, efficient computation of symmetric, positive-definite feedback gain matrices is accomplished through the use of multiple-constraint matrix update techniques originally developed for structural identification applications. Two systems were used to illustrate the application: a simple spring-mass system and a planar truss. From these demonstrations, use of this multiple-constraint technique is seen to provide a straightforward approach for computing the low-authority gains.
Smith, S. W.↗
An explicit form of the Mie phase matrix for multiple scattering calculations in the I, Q, U, and V representation
An explicit expression is obtained for the phase matrix in the I, Q, U, and V Stokes vector representation for a system containing a polydispersion of spherical particles. All of the symmetry relations derived by Hovenier using general arguments are established explicitly. Convenient algorithms are given for the computation of the phase matrix for a spherical polydispersion. Since this theory is so vitally important in radiative transfer, many researchers will need to compute these functions for realistic aerosols distributions. Therefore, results are presented for a haze L distribution so that other researchers will have a way of checking their programs which compute these quantities.
Kattawar, G. W.↗
Matrix-vector multiplication using digital partitioning for more accurate optical computing
Digital partitioning offers a flexible means of increasing the accuracy of an optical matrix-vector processor. This algorithm can be implemented with the same architecture required for a purely analog processor, which gives optical matrix-vector processors the ability to perform high-accuracy calculations at speeds comparable with or greater than electronic computers as well as the ability to perform analog operations at a much greater speed. Digital partitioning is compared with digital multiplication by analog convolution, residue number systems, and redundant number representation in terms of the size and the speed required for an equivalent throughput as well as in terms of the hardware requirements. Digital partitioning and digital multiplication by analog convolution are found to be the most efficient alogrithms if coding time and hardware are considered, and the architecture for digital partitioning permits the use of analog computations to provide the greatest throughput for a single processor.
Gary, C. K.↗
Using a Cray Y-MP as an array processor for a RISC Workstation
As microprocessors increase in power, the economics of centralized computing has changed dramatically. At the beginning of the 1980's, mainframes and super computers were often considered to be cost-effective machines for scalar computing. Today, microprocessor-based RISC (reduced-instruction-set computer) systems have displaced many uses of mainframes and supercomputers. Supercomputers are still cost competitive when processing jobs that require both large memory size and high memory bandwidth. One such application is array processing. Certain numerical operations are appropriate to use in a Remote Procedure Call (RPC)-based environment. Matrix multiplication is an example of an operation that can have a sufficient number of arithmetic operations to amortize the cost of an RPC call. An experiment which demonstrates that matrix multiplication can be executed remotely on a large system to speed the execution over that experienced on a workstation is described.
Lamaster, Hugh↗