SummaryMotivated by the recently popular probabilistic methods for low‐rank approximations and randomized algorithms for the least squares problems, we develop randomized algorithms for the total least squares problem with a single right‐hand side. We present the Nyström method for the medium‐sized problems. For the large‐scale and ill‐conditioned cases, we introduce the randomized truncated total least squares with the known or estimated rank as the regularization parameter. We analyze the accuracy of the algorithm randomized truncated total least squares and perform numerical experiments to demonstrate the efficiency of our randomized algorithms. The randomized algorithms can greatly reduce the computational time and still maintain good accuracy with very high probability.

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The number of high-dimensional datasets recording multiple aspects of a single phenomenon is increasing in many areas of science, accompanied by a need for mathematical frameworks that can compare multiple large-scale matrices with different row dimensions. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. We mathematically define a higher-order GSVD (HO GSVD) for N≥2 matrices D(i)∈R(m(i) × n), each with full column rank. Each matrix is exactly factored as D(i)=U(i)Σ(i)V(T), where V, identical in all factorizations, is obtained from the eigensystem SV=VΛ of the arithmetic mean S of all pairwise quotients A(i)A(j)(-1) of the matrices A(i)=D(i)(T)D(i), i≠j. We prove that this decomposition extends to higher orders almost all of the mathematical properties of the GSVD. The matrix S is nondefective with V and Λ real. Its eigenvalues satisfy λ(k)≥1. Equality holds if and only if the corresponding eigenvector v(k) is a right basis vector of equal significance in all matrices D(i) and D(j), that is σ(i,k)/σ(j,k)=1 for all i and j, and the corresponding left basis vector u(i,k) is orthogonal to all other vectors in U(i) for all i. The eigenvalues λ(k)=1, therefore, define the "common HO GSVD subspace." We illustrate the HO GSVD with a comparison of genome-scale cell-cycle mRNA expression from S. pombe, S. cerevisiae and human. Unlike existing algorithms, a mapping among the genes of these disparate organisms is not required. We find that the approximately common HO GSVD subspace represents the cell-cycle mRNA expression oscillations, which are similar among the datasets. Simultaneous reconstruction in the common subspace, therefore, removes the experimental artifacts, which are dissimilar, from the datasets. In the simultaneous sequence-independent classification of the genes of the three organisms in this common subspace, genes of highly conserved sequences but significantly different cell-cycle peak times are correctly classified.

This introduction to computer-based problem-solving using the MATLAB environment is highly recommended for students wishing to learn the concepts and develop the programming skills that are fundamental to computational science and engineering (CSE). Through a teaching by examples approach, the authors pose strategically chosen problems to help first-time programmers learn these necessary concepts and skills. Each section formulates a problem and then introduces those new MATLAB language features that are necessary to solve it. This approach puts problem-solving and algorithmic thinking first and syntactical details second. Each solution is followed by a talking point that concerns some related, larger issue associated with CSE. Collectively, the worked examples, talking points, and 300+ homework problems build intuition for the process of discretization and an appreciation for dimension, inexactitude, visualization, randomness, and complexity. This sets the stage for further coursework in CSE areas. The interplay between programming and mathematics throughout the text reinforces the student s ability to reason numerically and geometrically. Audience: Undergraduate students whose mathematical maturity is at the level of Calculus I will find this book extremely useful, especially as preparation for further courses in computing and mathematics. It can also be used as a MATLAB reference at any level. Contents: Preface; MATLAB Glossary; Programming Topics; Software; Chapter 1: From Formula to Program; Chapter 2: Limits and Error; Chapter 3: Approximation with Fractions; Chapter 4: The Discrete versus the Continuous; Chapter 5: Abstraction; Chapter 6: Randomness; Chapter 7: The Second Dimension; Chapter 8: Reordering; Chapter 9: Search; Chapter 10: Points, Polygons, and Circles; Chapter 11: Text File Processing; Chapter 12: The Matrix: Part II; Chapter 13: Acoustic File Processing; Chapter 14: Divide and Conquer; Chapter 15: Optimization; Appendix A: Refined Graphics; Appendix B: Mathematical Facts; Appendix C: MATLAB, Java, and C; Appendix D: Exit Interview; Index

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others but that none are completely satisfactory.Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.

This companion article discusses portability and optimization issues of the GEMM-based level 3 BLAS model implementations and the performance evaluation benchmark. All software comes in all four data types (single- and double-precision, real and complex) and are designed to be easy to implement and use on different platforms. Each of the GEMM-based routines has a few machine-dependent parameters that specify internal block. sizes, cache characteristics, and branch points for alternative code sections. These parameters provide means for adjustment to the characteristics of a memory hierarchy.

Abstract

The GEMM-based level 3 BLAS model implementations, which are structured to e ectively reduce data tra c in a memory hierarchy, and the performance evaluation benchmark, which is a tool for evaluating and comparing di erent implementations of the level 3 BLAS with the GEMM-based model implementations are presented in [5]. Here, the installation and tuning of the Fortran 77 model implementations, as well as the use and installation of the performance evaluation benchmark are described. All software come in all four data precisions and are designed to be easy to implement and use on di erent platforms. Each of the GEMM-based routines has a few system dependent parameters that specify internal block sizes, cache characteristics, and intersection points for alternative code sections.

Multiple independent matrix problems of very small size appear in a variety of different fields. In this work, we study the implementation of elementary linear algebra subroutines so as to best use vectorizing compilers and vector hardware for multiple small problem instances. We cheek the performance improvement over the single-instance optimized codes on different vector supercomputers. We also describe how to automate the transformation of a singleinstance linear algebra solver into a multiple instance solver

Let A be an m-by-n matrix with m=m1m2 and n=n1n2. We consider the problem of finding (mathematical formula omitted) so that (mathematical formula omitted) is minimized. This problem can be solved by computing the largest singular value and associated singular vectors of a permuted version of A. If A is symmetric, definite, non-negative, or banded, then the minimizing B and C are similarly structured. The idea of using Kronecker product preconditioners is briefly discussed.

1. The Radix-2 Frameworks. Matrix Notation and Algorithms The FFT Idea The Cooley-Tukey Factorization Weight and Butterfly Computations Bit Reversal and Transposition The Cooley-Tukey Framework The Stockham Autosort Frameworks The Pease Framework Decimation in Frequency and Inverse FFTs 2. General Radix Frameworks. The General Radix Ideas Index Reversal and Transposition Mixed-Radix Factorizations Radix-4 and Radix-8 Frameworks The Split-Radix Frameworks 3. High Performance Frameworks. The Multiple DFT Problem Matrix Transposition The Large Single-Vector FFT Problem Multi-Dimensional FFT Problem Distributed Memory FFTs Shared Memory FFTs 4. Selected Topics. Prime Factor FFTs Convolution FFTs of Real Data Cosine and Sine Transforms Fast Poisson Solvers Bibliography Index.

This chapter is a three-level introduction to the field of matrix computations. The first section is very informal and is designed to acquaint the reader with the basic tools of the trade. The tools range from the algorithmic (What is a Householder matrix, and how they can be used to solve the least square problem?) to the analytic (What happens to the solution of a least square problem if we perturb the data?). In the second section, we discuss the matrix factorizations that figure heavily in numerical linear algebra. For each factorization, we survey algorithms, associated mathematical properties, and applications. Sprinkled throughout this section are special topics that illustrate the power of the factorization paradigm. In the final section, we use one factorization (Cholesky) to illustrate various aspects of high performance matrix computations. Successful computing nowadays requires the design of codes that pay careful attention to the flow of data during execution. Our goal is to make the reader more aware of these data movement concerns.

Computing the QR factorization of a matrix A ∈ R m×n involves finding an orthogonal matrix Q ∈ R m×n and an upper tringular matrix R ∈ R m×n such that A = QR. This factorization has a prominent role to play in numerical linear algebra especially because of its bearing on the least square problem. A detailed description of the QR factorization and the various ways that it can be computed may be found in Golub and Van Loan (1983).

1. A Subset of Fortran 77. Basics, Logical Operations, Loops, Arrays, Subprograms, Input/Output, Complex Arithmetic, Programming Tips, Fortran 77 Built-in Functions 2. The BLAS. Bookkeeping Operations, Vector Operations, Norm Computations, Givens Rotations, Double Precision and Complex Versions 3.LINPACK. Triangular Systems, General Systems, Symmetric Systems, Banded Systems, The QR Factorization, The Singular Value Decomposition, Double Precision and Complex Versions 4. MATLAB. Basics, Loops and Conditionals, Built-in Functions, Working with Submatrices, Functions, Factorizations, Miscellaneous.

ESPRIT is an interesting new method for solving the Direction-of-Arrival estimation problem. It involves some rather tricky matrix manipulations. We show how these calculations can be carried out using only unitary transformations of the data. No inverses or cross-products are required making the new method extremely robust.