Row Compression and Nested Product Decomposition of a Hierarchical Representation of a Quasiseparable Matrix

This research introduces a row compression and nested product decomposition of an nxn hierarchical representation of a rank structured matrix A, which extends the compression and nested product decomposition of a quasiseparable matrix. The hierarchical parameter extraction algorithm of a quasiseparable matrix is efficient, requiring only O(nlog(n))operations, and is proven backward stable. The row compression is comprised of a sequence of small Householder transformations that are formed from the low-rank, lower triangular, off-diagonal blocks of the hierarchical representation. The row compression forms a factorization of matrix A, where A = QC, Q is the product of the Householder transformations, and C preserves the low-rank structure in both the lower and upper triangular parts of matrix A. The nested product decomposition is accomplished by applying a sequence of orthogonal transformations to the low-rank, upper triangular, off-diagonal blocks of the compressed matrix C. Both the compression and decomposition algorithms are stable, and require O(nlog(n)) operations. At this point, the matrix-vector product and solver algorithms are the only ones fully proven to be backward stable for quasiseparable matrices. By combining the fast matrix-vector product and system solver, linear systems involving the hierarchical representation to nested product decomposition are directly solved with linear complexity and unconditional stability. Applications in image deblurring and compression, that capitalize on the concepts from the row compression and nested product decomposition algorithms, will be shown

Eigenvalue Methods for Interpolation Bases

This thesis investigates eigenvalue techniques for the location of roots of polynomials expressed in the Lagrange basis. Polynomial approximations to functions arise in almost all areas of computational mathematics, since polynomial expressions can be manipulated in ways that the original function cannot. Polynomials are most often expressed in the monomial basis; however, in many applications polynomials are constructed by interpolating data at a series ofpoints. The roots of such polynomial interpolants can be found by computing the eigenvalues of a generalized companion matrix pair constructed directly from the values of the interpolant. This affords the opportunity to work with polynomials expressed directly in the interpolation basis in which they were posed, avoiding the often ill-conditioned transformation between bases. Working within this framework, this thesis demonstrates that computing the roots of polynomials via these companion matrices is numerically stable, and the matrices involved can be reduced in such a way as to significantly lower the number of operations required to obtain the roots. Through examination of these various techniques, this thesis offers insight into the speed, stability, and accuracy of rootfinding algorithms for polynomials expressed in alternative bases

Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by exploiting the correlation between the computations with matrices and polynomials, extending the techniques of the matrix sign iteration, and exploiting the structure of the companion matrix of the input polynomial. We extend some of the proposed techniques to the approximation of the real eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm

A rational QZ method

We propose a rational QZ method for the solution of the dense, unsymmetric generalized eigenvalue problem. This generalization of the classical QZ method operates implicitly on a Hessenberg, Hessenberg pencil instead of on a Hessenberg, triangular pencil. Whereas the QZ method performs nested subspace iteration driven by a polynomial, the rational QZ method allows for nested subspace iteration driven by a rational function, this creates the additional freedom of selecting poles. In this article we study Hessenberg, Hessenberg pencils, link them to rational Krylov subspaces, propose a direct reduction method to such a pencil, and introduce the implicit rational QZ step. The link with rational Krylov subspaces allows us to prove essential uniqueness (implicit Q theorem) of the rational QZ iterates as well as convergence of the proposed method. In the proofs, we operate directly on the pencil instead of rephrasing it all in terms of a single matrix. Numerical experiments are included to illustrate competitiveness in terms of speed and accuracy with the classical approach. Two other types of experiments exemplify new possibilities. First we illustrate that good pole selection can be used to deflate the original problem during the reduction phase, and second we use the rational QZ method to implicitly filter a rational Krylov subspace in an iterative method

Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations

This thesis consists of two parts. In Part I, we describe an algorithm for approximating the Green\u27s function for elliptic problems with variable coefficients in arbitrary dimension. The basis for our approach is the separated representation, which appears as a way of approximating functions of many variables by sums of products of univariate functions. While the differential operator we wish to invert is typically ill-conditioned, its conditioning may be improved by first applying the Green\u27s function for the constant coefficient problem. This function may be computed either numerically or, in some case, analytically in a separated format. The variable coefficient Green\u27s function is then computed using a quadratically convergent iteration on the preconditioned operator, with sparsity maintained via representation in a wavelet basis. Of particular interest is that the method scales linearly in the number of dimensions, a feature that very desirable in high dimensional problems in which the curse of dimensionality must be reckoned with. As a corollary to this work, we described a randomized algorithm for maintaining low separation rank of the functions used in the construction of the Green\u27s function. For certain functions of practical interest, one can avoid the cost of using standard methods such as alternating least squares (ALS) to reduce the separation rank. Instead, terms from the separated representation may be selected using a randomized approach based on matrix skeletonization and the interpolative decomposition. The use of random projections can greatly reduce the cost of rank reduction, as well as calculation of the Frobenius norm and term-wise Gram matrices. In Part II of the thesis, we highlight three practical applications of sparse and separable approximations to the analysis of renewable energy data. In the first application, error estimates gleaned from repeated measurements are incorporated into sparse regression algorithms (LASSO and the Dantzig selector) to minimize the statistical uncertainty of the resulting model. Applied to real biomass data, this approach leads to sparser regression coefficients corresponding to improved accuracy as measured by k-fold cross validation error. In the second application, a regression model based on separated representations is fit to reliability data for cadmium telluride (CdTe) thin-film solar cells. The data is inherently multi-way, and our approach avoids artificial matricization that would typically be performed for use with standard regression algorithms. Two distinct modes of degradation, corresponding to short- and long-term decrease in cell efficiency, are identified. In the third application, some theoretical properties of a popular chemometrics algorithm called orthogonal projections to latent structures (O-PLS) are derived

Orthogonal similarity transformation of a symmetric matrix into a diagonal-plus-semiseparable one with free choice of the diagonal

In this paper we describe an orthogonal similarity transformation for transforming arbitrary symmetric matrices into a diagonal-plus-semiseparable matrix, where we can freely choose the diagonal. Very recently an algorithm was proposed for transforming arbitrary symmetric matrices into similar semiseparable ones. This reduction is strongly connected to the reduction to tridiagonal form. The class of semiseparable matrices can be considered as a subclass of the diagonalplus- semiseparable matrices. Therefore we can interpret the proposed algorithm here as an extension of the reduction to semiseparable form.status: publishe