Yves Nievergelt

Total least squares: state-of-the-art regression in numerical analysis

Total least squares regression (TLS) fits a line to data where errors may occur in both the dependent and independent variables. In higher dimensions, TLS fits a hyperplane to such data. The elementary algorithm presented here fits readily in a first course in numerical linear algebra.

Motor and Speech Disorders in Classic Galactosemia

Purpose To test the hypothesis that children with classic galactosemia and speech disorders are at risk for co-occurring strength and coordination disorders. Method This is a case-control study of 32 children (66% male) with galactosemia and neurologic speech disorders and 130 controls (50% male) ages 4-16 years. Speech was assessed using the Percentage of Consonants Correct (PCC) metric from responses to the Goldman-Fristoe Test of Articulation-2 and from a 5-min recorded speech sample, hand and tongue strength using the Iowa Oral Performance Instrument, and coordination using the Movement Assessment Battery for Children. The number of days on milk during the neonatal period was obtained by parent report. Analyses of covariance, distributions, and correlations were used to evaluate relationships among speech, strength, coordination, age, gender, and days on milk. Results Children with galactosemia had weaker hand and tongue strength and most (66%) had significant coordination disorders, primarily affecting balance and manual dexterity. Among children with galactosemia, children with more speech errors and classified as childhood apraxia of speech (n = 7) and ataxic dysarthria (n = 1), had poorer balance and manual dexterity, but not weaker hand or tongue strength, compared to the children with fewer speech errors. The number of days on milk during the neonatal period was associated with more speech errors in males but not in females. Conclusion Children with galactosemia have a high prevalence of co-occurring speech, coordination, and strength disorders, which may be evidence of a common underlying etiology, likely associated with diffuse cerebellar damage, rather than distinct disorders.

A tutorial history of least squares with applications to astronomy and geodesy

This article surveys the history, development, and applications of least squares, including ordinary, constrained, weighted, and total least squares. The presentation includes proofs of the basic theory, in particular, unitary factorizations and singular-value decompositions of matrices. Numerical examples with real data demonstrate how to set up and solve several types of problems of least squares. The bibliography lists comprehensive sources for more specialized aspects of least squares.

Scalar fused multiply-add instructions produce floating-point matrix arithmetic provably accurate to the penultimate digit

Combined with doubly compensated summation, scalar fused multiply-add instructions redefine the concept of floating-point arithmetic, because they allow for the computation of sums of real or complex matrix products accurate to the penultimate digit. Particular cases include complex arithmetic, dot products, cross products, residuals of linear systems, determinants of small matrices, discriminants of quadratic, cubic, or quartic equations, and polynomials.

A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres

SummaryThe algorithm proved here solves the problem of orthogonal distance regression for the maximum norm with hyperplanes and hyperspheres. For each finite set of points in a Euclidean space of any dimension, the algorithm determines – through finitely many arithmetic operations – all the hyperplanes and hyperspheres that minimize the maximum Euclidean distance measured perpendicularly from the data. The algorithm finds all the slabs (bounded by parallel hyperplanes) and all the spherical shells (bounded by concentric hyperspheres) that contain all the data and are “rigidly supported” by the data (for which there does not exist any other pair of parallel hypersurfaces of the same type that intersect the data at the same points.) The computational complexity of the algorithm increases as the number of data points raised to the dimension of the ambient space. The solutions are then the midrange hyperplanes in the thinnest slabs, and the midrange hyperspheres in the thinnest shells. Their sensitivity to perturbations of the data is of the order of a power of the reciprocal of the smallest angle between two median hyperplanes separating two pairs of data points. The methods of proof consist in showing that if a pair of parallel hyperplanes or hyperspheres is not rigidly supported but encompasses all the data, then there exists a projective shift of their common projective center producing a thinner slab or shell that still contains all the data.

The Concept of Elasticity in Economics

The following pages introduce the notion of elasticity and present its main mathematical and graphical properties, illustrating them on real economic cases, such as the consumption of marijuana and the demand for hospital beds.

Fitting helices to data by total least squares

Abstract Total least squares and the singular value decomposition provide a method to fit cylindrical helices to data, with applications in—but not restricted to—engineering and nuclear physics.

Computing circles and spheres of arithmitic least squares

Abstract A proof of the existence and uniqueness of L. Moura and R. Kitneys circle of least squares leads to estimates of the accuracy with which a computer can determine that circle. The result shows that the accuracy deteriorates as the correlation between the coordinates of the data points increases in magnitude. Yet a numerically more stable computation of eigenvectors yields the limiting straight line, which a further analysis reveals to be the line of total least squares. The same analysis also provides generalizations to fitting spheres in higher dimensions.

Aitken's and Steffensen's accelerations in several variables

SummaryAitkens acceleration of scalar sequences extends to sequences of vectors that behave asymptotically as iterations of a linear transformation. However, the minimal and characteristic polynomials of that transformation must coincide (but the initial sequence of vectors need not converge) for a numerically stable convergence of Aitkens acceleration to occur. Similar results hold for Steffensens acceleration of the iterations of a function of several variables. First, the iterated function need not be a contracting map in any neighbourhood of its fixed point. Instead, the second partial derivatives need only remain bounded in such a neighbourhood for Steffensens acceleration to converge quadratically, even if ordinary iterations diverge. Second, at the fixed point the minimal and characteristic polynomials of the Jacobian matrix must coincide to ensure a numerically stable convergence. By generalizing the work that Noda did on the subject between 1981 and 1986, the results presented here explain the numerical observations reported by Henrici in 1964 and 1982.

SCHMIDT-MIRSKY MATRIX APPROXIMATION WITH LINEARLY CONSTRAINED SINGULAR VALUES

Abstract Schmidt and Mirskys theorems identify the matrix with a specified rank that lies closest to another matrix, with distances measured by any matrix norm invariant under the unitary group. The more general result presented here identifies the nearest matrix whose singular values satisfy any specified set of linear constraints. A typical application consists in determining the oblate spheroid that most closely fits a specified ellipsoid.