Steingrimsson (2009) outlined Luce’s (2002, 2004) proposed psychophysical theory and tested, for brightness, behavioral properties that, separately, gave rise to two psychophysical functions, Ψ⊕ and Ψ◦p . The function Ψ⊕ maps pairs of physical intensities onto the positive real numbers and represents subjective summation, and the function Ψ◦p represents a form of ratio production. This article ...

We discuss the spectral subspace perturbation problem for a selfadjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an a priori sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, a priori, tanΘ Theorem. We also extend the Davis–Kahan ta...

The Davis–Kahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions. 10 It relies on an eigenvalue separation condition between certain relevant population and sample eigenvalues. We present a variant of this result that depends only on a population eigenvalue separation condition, ma...

Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large-scale linear systems of equations and eigenvalue problems as...

In this paper, we propose an implicitly restarted block Lanczos bidiagonalization (IRBLB) method for computing a few extreme or interior singular values and associated right and left singular vectors of a large matrix A. Our method combines the advantages of a block routine, implicit shifting, and the application of Leja points as shifts in the accelerating polynomial. The method neither requir...

We describe efficient implementations of the Subtractive Optimally Localized Averages (SOLA) mollifier method for solving linear inverse problems in, e.g., inverse helioseismology. We show that the SOLA method can be regarded as a constrained least squares problem, which can be solved by means of standard “building blocks” from numerical linear algebra. We compare the standard implementation of...

In this work we study the minimization of a linear functional defined on a set of approximate solutions of a discrete ill-posed problem. The primary application of interest is the computation of confidence intervals for components of the solution of such a problem. We exploit the technique introduced by Eldén in 1990, utilizing a parametric programming reformulation involving the solution of a ...

Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining a partial Lanczos bidiagonalization of the matrix of the given system of equations. This paper explores the possibility of instead computing a partial Arnoldi decomposition of the given matrix. Computed examples illustrate that this approach may require fewer matrix-vector product evaluation...

We present a fast 5D (frequency and 4 spatial axes) reconstruction method that uses Multichannel Singular Spectrum Analysis / Cazdow algorithm. Rather than embedding the 4D spatial volume in a Hankel matrix, we propose to embed the data into a block Toeplitz form. Rank reduction is carried out via Lanczos bidiagonalization with fast block Toeplitz matrix-times-vector multiplications via 4D Fast...

This paper presents a preconditioning method based on a recursive multilevel lowrank approximation approach. The basic idea is to recursively divide the problem into two and apply a low-rank approximation to a matrix obtained from the Sherman-Morrison formula. The low-rank approximation may be computed by the partial Singular Value Decomposition (SVD) or it can be approximated by the Lanczos bi...