On the augmented system approach to sparse least-squares problems

Summary We study the augmented system approach for the solution of sparse linear least-squares problems. It is well known that this method has better numerical properties than the method based on the normal equations. We use recent work by Arioli et al. (1988) to introduce error bounds and estimates for the components of the solution of the augmented system. In particular, we find that, using iterative refinement, we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute. The final error and all our error estimates are much better than the classical or Skeel's error analysis (1979) indicates. Moreover, we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the Björck scaling (1967) on these estimates. We illustrate this with runs both on large-scale practical problems and contrived examples, comparing the numerical behaviour of the augmented systems approach with a code using the normal equations. These experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach, it is comparable or better when the least-squares matrix has a full row, and is, in any case, much more stable and robust.This author was visiting Harwell and was funded by a grant from the Italian National Council of Research (CNR), Istituto di Elaborazione dell'Informazione-CNR, via S. Maria 46, I-56100 Pisa, ItalyThis author was visiting Harwell from Faculty of Mathematics and Computer Science of the University of Amsterdam     

An experiment with arithmetic precision in linear algebra computations

Several techniques for experimental determination of floating point precision in practical computations are examined, and applied to linear algebra algorithms. These techniques are simple enough to be directly applicable to existing production codes, requiring a very limited amount of software on many machines, and yet they yield interesting information on the numerical precision of a computation. Our choice of linear algebra algorithms includes a direct solver (namely the MA32 program from the Harwell Library) and several variants of preconditioned conjugate gradients (the methods DIAG, INV, MINV and POL of Reference 1). The results may be of interest as method selection criteria, and thus complement Mflop performance data available from several sources.     

Prevalence of voice disorders in African American and European American preschoolers

Several studies have reported prevalence rates for voice disorders in school-aged children. Less is known, however, about such prevalence in preschoolers, and whether racial, ethnic, or cultural diversity may influence it. The presence of voice disorders in a total of 2445 African-American and European-American preschool children, 1246 males and 1199 females, from 2 to 6 years of age is reported here. Presence of a voice disorder characterized by hoarseness was identified by a three-prong approach including teacher identification, investigator screening, and parent identification. Speech-language pathologists listened individually to each child's speech as they engaged each child in play-conversation activities. A voice disorder was identified on the basis of the judgment of two speech-language pathologists. Voice disorders characterized by hoarseness were identified in 95 children or 3.9% of the total sample by the investigators. Statistical analysis revealed no significant differences for age, gender, or race.     

“Dibutylmagnesium”, a convenient reagent for the synthesis of useful organic magnesium reagents MgA2 including cyclopentadienyls,aryloxides, and amides. Preparation of Zr(C5H5)Cl3. X-ray structure of [Mg{μ-N(SiMe)3C6H4N}(SiMe3)-o(OEt2)]2

n-Heptane-soluble “di-butylmagnesium” (I) (a commercially available material, prepared by addition of LiBu^s to MgBuⁿCl, and subsequent addition of ca. 5% MgOct₂ⁿ) has been found to be a useful starting material for obtaining numerous organic magnesium compounds. This is illustrated by its reaction with a number of protic compounds HA to give in good yields Mg(C₅H₅)₂, Mg(C₅H₄Me)₂, or the new compounds MgA₂: IV (A = C₅H₄SiMe₃), V [A = C₅H₃(SiMe₃)₂], VII (A = OC₆H₂Bu₂^t-2,6-Me-4), and X [A₂ = N(SiMe₃)C₆H₄N(SiMe₃)-o(OEt₂)]. The value of such compounds MgA₂ as mild ligand transfer reagents is illustrated by the synthesis of Zr(C₅H₃X₂)Cl₃ (X = H or SiMe₃). Compound X was isolated from OEt₂ solution as the crystalline dimer

with two o-$\text{N}$(SiMe₃)C₆H₄$\text{N}$(SiMe₃) ligands bridging two magnesium atoms and a terminal OEt₂ ligand completing a distorted tetrahedral environment around each Mg. Some key parameters are: MgN_t 1.997(7), MgN_b 2.083(8), MgO 2.041(7) Å; OMgN_t 112.1(3), OMgN_b 119.7(3), and N_tMgN_b 118.5(3)°.