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The algebraic matrix Riccati equation AX+XAT-XFX+C=0, where matrices A, B, C, F ∈ ℝn × n are given and a solution X ∈ ℝn × n is sought, plays a fundamental role in optimal control problems. Large-scale systems typically appear if the constraint is described by a partial differential equation (PDE). We provide a nonlinear multigrid algorithm that computes the solution X in a data-sparse, low-rank format and has a complexity of 𝒪(n), subject to the condition that F and C are of low rank and A is the finite element or finite difference discretization of an elliptic PDE. We indicate how to generalize the method to ℋ-matrices C, F and X that are only blockwise of low rank and thus allow a broader applicability with a complexity of 𝒪(nlog(n)p), p being a small constant. The method can also be applied to unstructured and dense matrices C and X in order to solve the Riccati equation in 𝒪(n2). Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the Sylvester equation AX-XB+C=0 where the matrix C∈ℂn×m is of low rank and the spectra of A∈ℂn×n and B∈ℂm×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ℇ∈(0,1) there exists a matrix &Xtilde; of rank k=O(log(1/ℇ)) such that ∥X-&Xtilde;∥2⩽ℇ∥X∥2. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62: 89–108). The blockwise rank of the approximation is again proportional to log(1/ℇ). Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Aim: Child growth is modulated by numerous factors and, particularly in infancy and early childhood, often tends to follow apparently irregular patterns, with many centiles crossed before the later growth channels are reached. The aim of this study was to visualize the diversity of individual growth. Design: The study investigated 333 girls and 329 boys without chronic illnesses from four paediatric practices in Kiel, Germany. The children were measured on natural, i.e., on various occasions, when they were presented to their doctors for preventive care examinations and for minor illnesses, at the age of 0.25 (range ± 0.08) y, 0.5 (range ± 0.16) y, 0.75 (range ± 0.16) y, 1.0 (range ± 0.25) y, and at the ages of 1.5, 2, 3, 4 and 5 (range ± 0.25) y. Each individual growth curve was converted into a series of height SDS (standard deviation scores) using one of the most reputable longitudinal German growth studies as background reference. Height SDS was then converted into residual height SDS (differences between height SDS of each measurement and average personal height SDS of the respective child). Cluster analysis was used to identify groups of children (clusters) with similarities in residual height SDS patterns (growth tracks). The clusters contained a minimum of at least 10 children. Single children or small sets of individuals below the minimum number were rejected from further analysis. Results: In males, 10 growth tracks were identified, each consisting of 11 to 52 boys. Growth in 111 boys was so heterogeneous that they could not be assigned to growth tracks. In females, 11 growth tracks were identified, each consisting of 12 to 48 girls; 112 girls could not be assigned. Approximately 7% of boys and 15% of girls showed evidence of a mild intermittent growth spurt at the end of infancy. Some growth tracks were almost horizontal, or showed declining residual height SDS up to the age of 3 and 4 y, with no evidence of growth spurts during early childhood. Others showed sharply declining growth in early infancy, or irregular patterns. Similar results were obtained when using cross-sectional standards as background reference. Conclusion: Cluster analysis provides evidence that the substantial diversity in infant and early child growth is limited to a small number of narrow but characteristic tracks of yet unknown biological significance. [ABSTRACT FROM AUTHOR]