294 research outputs found
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Accurate solution of structured least squares problems via rank-revealing decompositions
Least squares problems min(x) parallel to b - Ax parallel to(2) where the matrix A is an element of C-mXn (m >= n) has some particular structure arise frequently in applications. Polynomial data fitting is a well-known instance of problems that yield highly structured matrices, but many other examples exist. Very often, structured matrices have huge condition numbers kappa(2)(A) = parallel to A parallel to(2) parallel to A(dagger)parallel to(2) (A(dagger) is the Moore-Penrose pseudoinverse of A) and therefore standard algorithms fail to compute accurate minimum 2-norm solutions of least squares problems. In this work, we introduce a framework that allows us to compute minimum 2-norm solutions of many classes of structured least squares problems accurately, i.e., with errors parallel to(x) over cap (0) - x(0)parallel to(2)/parallel to x(0)parallel to(2) = O(u), where u is the unit roundoff, independently of the magnitude of kappa(2)(A) for most vectors b. The cost of these accurate computations is O(n(2)m) flops, i.e., roughly the same cost as standard algorithms for least squares problems. The approach in this work relies in computing first an accurate rank-revealing decomposition of A, an idea that has been widely used in recent decades to compute, for structured ill-conditioned matrices, singular value decompositions, eigenvalues, and eigenvectors in the Hermitian case and solutions of linear systems with high relative accuracy. In order to prove that accurate solutions are computed, a new multiplicative perturbation theory of the least squares problem is needed. The results presented in this paper are valid for both full rank and rank deficient problems and also in the case of underdetermined linear systems (m < n). Among other types of matrices, the new method applies to rectangular Cauchy, Vandermonde, and graded matrices, and detailed numerical tests for Cauchy matrices are presented.This work was supported by the Ministerio de EconomÃa y Competitividad of Spain through grants MTM-2009-09281, MTM-2012-32542 (Ceballos, Dopico, and Molera) and MTM2010-18057 (Castro-González).Publicad
The Power of Bidiagonal Matrices
Bidiagonal matrices are widespread in numerical linear algebra, not least
because of their use in the standard algorithm for computing the singular value
decomposition and their appearance as LU factors of tridiagonal matrices. We
show that bidiagonal matrices have a number of interesting properties that make
them powerful tools in a variety of problems, especially when they are
multiplied together. We show that the inverse of a product of bidiagonal
matrices is insensitive to small componentwise relative perturbations in the
factors if the factors or their inverses are nonnegative. We derive
componentwise rounding error bounds for the solution of a linear system , where or is a product of bidiagonal
matrices, showing that strong results are obtained when the are
nonnegative or have a checkerboard sign pattern. We show that given the \fact\
of an totally nonnegative matrix into the product of bidiagonal
matrices, can be computed in flops and that in
floating-point arithmetic the computed result has small relative error, no
matter how large is. We also show how factorizations
involving bidiagonal matrices of some special matrices, such as the Frank
matrix and the Kac--Murdock--Szeg\"o matrix, yield simple proofs of the total
nonnegativity and other properties of these matrices
Toward accurate polynomial evaluation in rounded arithmetic
Given a multivariate real (or complex) polynomial and a domain ,
we would like to decide whether an algorithm exists to evaluate
accurately for all using rounded real (or complex) arithmetic.
Here ``accurately'' means with relative error less than 1, i.e., with some
correct leading digits. The answer depends on the model of rounded arithmetic:
We assume that for any arithmetic operator , for example or , its computed value is , where is bounded by some constant where , but
is otherwise arbitrary. This model is the traditional one used to
analyze the accuracy of floating point algorithms.Our ultimate goal is to
establish a decision procedure that, for any and , either exhibits
an accurate algorithm or proves that none exists. In contrast to the case where
numbers are stored and manipulated as finite bit strings (e.g., as floating
point numbers or rational numbers) we show that some polynomials are
impossible to evaluate accurately. The existence of an accurate algorithm will
depend not just on and , but on which arithmetic operators and
which constants are are available and whether branching is permitted. Toward
this goal, we present necessary conditions on for it to be accurately
evaluable on open real or complex domains . We also give sufficient
conditions, and describe progress toward a complete decision procedure. We do
present a complete decision procedure for homogeneous polynomials with
integer coefficients, {\cal D} = \C^n, and using only the arithmetic
operations , and .Comment: 54 pages, 6 figures; refereed version; to appear in Foundations of
Computational Mathematics: Santander 2005, Cambridge University Press, March
200
Vandermonde Neural Operators
Fourier Neural Operators (FNOs) have emerged as very popular machine learning
architectures for learning operators, particularly those arising in PDEs.
However, as FNOs rely on the fast Fourier transform for computational
efficiency, the architecture can be limited to input data on equispaced
Cartesian grids. Here, we generalize FNOs to handle input data on
non-equispaced point distributions. Our proposed model, termed as Vandermonde
Neural Operator (VNO), utilizes Vandermonde-structured matrices to efficiently
compute forward and inverse Fourier transforms, even on arbitrarily distributed
points. We present numerical experiments to demonstrate that VNOs can be
significantly faster than FNOs, while retaining comparable accuracy, and
improve upon accuracy of comparable non-equispaced methods such as the Geo-FNO.Comment: 21 pages, 10 figure
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