Bidiagonal Decomposition Research Articles

Accurate bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices and applications

An algorithm for the computation of the bidiagonal decomposition of strictly totally positive h-Bernstein-Vandermonde matrices is presented. These matrices are collocation matrices of h-Bernstein bases (a generalization of the Bernstein basis) of the space of polynomials of degree less than or equal to n. The algorithm is fast (its computational complexity is O(n2)) and has high relative accuracy. The computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices: linear system solving, eigenvalue computation and inverse computation. Numerical experiments are also included.

Accurate eigenvalues and exact zero Jordan blocks of totally nonnegative matrices

We present a new algorithm which, given a bidiagonal decomposition of a totally nonnegative matrix, computes all its eigenvalues to high relative accuracy in floating point arithmetic in \(O(n^3)\) time. It also computes exactly the Jordan blocks corresponding to zero eigenvalues in up to \(O(n^4)\) time.

Accurate computation of the Moore–Penrose inverse of strictly totally positive matrices

The computation of the Moore–Penrose inverse of structured strictly totally positive matrices is addressed. Since these matrices are usually very ill-conditioned, standard algorithms fail to provide accurate results. An algorithm based on the Q R factorization and which takes advantage of the special structure and the totally positive character of these matrices is presented. The first stage of the algorithm consists of the accurate computation of the bidiagonal decomposition of the matrix. Numerical experiments illustrating the good behavior of our approach are included.

Accurate computations with collocation matrices of a general class of bases

SummaryIn this paper, we provide algorithms for computing the bidiagonal decomposition of the collocation matrices of a very general class of bases of interest in computer‐aided geometric design and approximation theory. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these matrices, such as the calculation of their inverses, their eigenvalues, or their singular values. Numerical experiments illustrate the results.

Accurate computations with Lupaş matrices

Lupaş q-analogues of the Bernstein functions play an important role in Approximation Theory and Computer Aided Geometric Design. Their collocation matrices are called Lupaş matrices. In this paper, we provide algorithms for computing the bidiagonal decomposition of these matrices and their inverses to high relative accuracy. It is also shown that these algorithms can be used to perform to high relative accuracy several algebraic calculations with these matrices, such as the calculation of their inverses, their eigenvalues or their singular values. Numerical experiments are included.

Accurate bidiagonal decomposition of totally positive Cauchy–Vandermonde matrices and applications

Cauchy–Vandermonde matrices play a fundamental role in rational interpolation theory and in other fields. When all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings, these matrices are totally positive. In this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy. These algorithms can be used to solve with high relative accuracy other algebraic problems, such as the computation of all singular values, all eigenvalues or the solution of certain linear systems. The error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed.

Accurate and fast computations with positive extended Schoenmakers–Coffey matrices

SummarySchoenmakers–Coffey matrices are correlation matrices with important financial applications. Several characterizations of positive extended Schoenmakers–Coffey matrices are presented. This paper provides an accurate and fast method to obtain the bidiagonal decomposition of the conversion of these matrices, which in turn can be used to compute with high relative accuracy the eigenvalues and inverses of positive extended Schoenmakers–Coffey matrices. Numerical examples are included.

Bidiagonal decomposition of rectangular totally positive Said–Ball–Vandermonde matrices: Error analysis, perturbation theory and applications

An algorithm for computing the bidiagonal decomposition of Said–Ball–Vandermonde matrices in the rectangular case is presented, which allows one to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is fast and possesses high relative accuracy. The error analysis of the algorithm is also addressed, along with the perturbation theory for the bidiagonal factorization of totally positive Said–Ball–Vandermonde matrices. Some numerical experiments showing the good behavior of the algorithm are included.

On the extension of some total positivity inequalities

Some inequalities recently obtained for totally positive matrices are extended to more general classes of matrices. Among these larger classes of matrices, sign consistent, sign regular and matrices with signed bidiagonal decomposition are considered. The class of matrices with a signed bidiagonal decomposition contains the nonsingular totally positive matrices and their inverses.

Divide and Conquer the CS Decomposition

We develop a divide-and-conquer algorithm for the bidiagonal CS decomposition (CSD). This complements an earlier algorithm based on simultaneous QR iteration. The new algorithm is designed to provide the efficiency gains of familiar divide-and-conquer algorithms on both serial and parallel architectures. The solution uses many components of existing algorithms, particularly the bidiagonal SVD algorithm of Gu and Eisenstat, but extra steps and reparameterizations are required to maintain orthogonality and consistent singular vectors, especially when the singular vectors are ill conditioned. The algorithm supports the stable computation of the generalized singular value decomposition (GSVD) in addition to the CSD.

Accurate computations with collocation matrices of rational bases

The rational Bernstein basis plays a crucial role in Computer Aided Geometric Design. The collocation matrices of this basis are called rational Bernstein–Vandermonde matrices. In this paper we provide algorithms for computing the bidiagonal decomposition of rational Bernstein–Vandermonde matrices and their inverses with high relative accuracy. Similar results are obtained for the collocation matrices of another important rational basis: the rational Said–Ball basis. It is also shown that these algorithms can be used to perform accurately some computations with these matrices, such us the calculation of their inverses, their eigenvalues or their singular values. Numerical experiments illustrate the results.

Accurate computations of matrices with bidiagonal decomposition using methods for totally positive matrices

SUMMARYA class of sign‐symmetric P‐matrices including all nonsingular totally positive matrices and their inverses as well as tridiagonal nonsingular H‐matrices is presented and analyzed. These matrices present a bidiagonal decomposition that can be used to obtain algorithms to compute with high relative accuracy their singular values, eigenvalues, inverses, or their LDU factorization. Copyright © 2012 John Wiley & Sons, Ltd.

Bidiagonal decompositions, minors and applications

Matrices, called ε-BD matrices, that have a bidiagonal decomposition satisfying some sign constraints are analyzed. The ε-BD matrices include all nonsingular totally positive matrices, as well as their matrices opposite in sign and their inverses. The signs of minors of ε-BD matrices are analyzed. The zero patterns of ε-BD matrices and their triangular factors are studied and applied to prove the backward stability of Gaussian elimination without pivoting for the associated linear systems.

Accurate and efficient LDU decomposition of diagonally dominant M-matrices

NA † Abstract. An efficient method for the computation to high relative accuracy of the LDU decomposition of an n × n row diagonally dominant M-matrix is presented, assuming that the off-diagonal entries and row sums are given. This method costs an additional O(n2) elementary operations over the cost of Gaussian elimination, and leads to a lower triangular, column diagonally dominant matrix and an upper triangular, row diagonally dominant matrix. Comparisons with other methods in the literature are commented and illustrated. 1. Introduction. Recent advances in Numerical Linear Algebra have shown that certain classes of matrices allow computation of certain matrix functions to high relative accuracy, independently of the size of the classical condition number. Some of these classes of matrices are defined by special sign or other structure and require knowledge of some natural parameters to high relative accuracy. In most of those cases, accurate spectral computation (eigenvalues, singular values) is assured once we have an accurate matrix factorization with a suitable pivoting. For instance, the bidiagonal decomposition in the case of totally nonnegative matrices (see also (5), (10)) or an LDU factorization after a symmetric pivoting in the case of diagonally dominant matrices (cf. (4), (13), (14)). Let us focus now on the problem considered in this paper. An algorithm published in (2) computes with high relative accuracy the LDU factorization of an n × n row diagonally dominant M-matrix, if the off-diagonal entries and the row sums are given. The trick is to modify Gaussian elimination to compute the off-diagonal entries and the row sums of each Schur complement without performing subtractions. In addition, symmetric complete pivoting was used in (4) in order to obtain well conditioned L and U factors (U is even row diagonally dominant). This factorization is a special case of a

Polynomial least squares fitting in the Bernstein basis

The problem of polynomial least squares fitting in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein–Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition of the coefficient matrix A. Starting from that bidiagonal decomposition, an algorithm for obtaining the QR decomposition of A is the applied. Finally, a triangular system is solved by using the bidiagonal decomposition of the R-factor of A. Some numerical experiments showing the behavior of this approach are included.

Accurate computations with Said–Ball–Vandermonde matrices

A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said–Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said–Ball–Vandermonde matrices is presented, which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.

Generalized Triangular Decomposition in Transform Coding

A general family of optimal transform coders (TCs) is introduced here based on the generalized triangular decomposition (GTD) developed by Jiang This family includes the Karhunen-Loeve transform (KLT) and the generalized version of the prediction-based lower triangular transform (PLT) introduced by Phoong and Lin as special cases. The coding gain of the entire family, with optimal bit allocation, is equal to that of the KLT and the PLT. Even though the original PLT introduced by Phoong is not applicable for vectors that are not blocked versions of scalar wide sense stationary processes, the GTD-based family includes members that are natural extensions of the PLT, and therefore also enjoy the so-called MINLAB structure of the PLT, which has the unit noise-gain property. Other special cases of the GTD-TC are the geometric mean decomposition (GMD) and the bidiagonal decomposition (BID) transform coders. The GMD-TC in particular has the property that the optimum bit allocation is a uniform allocation; this is because all its transform domain coefficients have the same variance, implying thereby that the dynamic ranges of the coefficients to be quantized are identical.

Bidiagonal decomposition of totally positive Bernstein‐Vandermonde matrices

AbstractThe class of Bernstein‐Vandermonde matrices (a generalization of Vandermonde matrices arising when the monomial basis is replaced by the Bernstein basis) is considered. A convenient ordering of their rows makes these matrices strictly totally positive. By using results related to total positivity and Neville elimination, an algorithm for computing the bidiagonal decomposition of a Bernstein‐Vandermonde matrix is constructed. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate. One of the applications of our algorithm is the design of fast and accurate algorithms for solving Lagrange interpolation problems when using the Bernstein basis, an approach useful for the field of Computer Aided Geometric Design since it avoids the stability problems involved with basis transformations between the Bernstein and the monomial bases. A different application consists of the use of the bidiagonal decomposition as an intermediate step of the computation of the eigenvalues and the singular value decomposition of a totally positive Bernstein‐Vandermonde matrix. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A fast and accurate algorithm for solving Bernstein–Vandermonde linear systems

A fast and accurate algorithm for solving a Bernstein–Vandermonde linear system is presented. The algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of Neville elimination. The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate.

The Accurate and Efficient Solution of a Totally Positive Generalized Vandermonde Linear System

Vandermonde, Cauchy, and Cauchy--Vandermonde totally positive linear systems can be solved extremely accurately in O(n2 time using Bjorck--Pereyra-type methods. We prove that Bjorck--Pereyra-type methods exist not only for the above linear systems but also for any totally positive linear system as long as the initial minors (i.e., contiguous minors that include the first row or column) can be computed accurately. Using this result we design a new O(n2 Bjorck--Pereyra-type method for solving generalized Vandermonde systems of equations by using a new algorithm for computing the Schur function. We present explicit formulas for the entries of the bidiagonal decomposition, the LDU decomposition, and the inverse of a totally positive generalized Vandermonde matrix, as well as algorithms for computing these entries to high relative accuracy.