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The symmetric Procrustes problem
No full textThe following "symmetric Procrustes" problem arises in the determination of the strain matrix of an elastic structure: find the symmetric matrix X which minimises the Frobenius (or Euclidean) norm of AX — B, where A and B are given rectangular matrices. We use the singular value decomposition to analyse the problem and to derive a stable method for its solution. A perturbation result is derived and used to assess the stability of methods based on solving normal equations. Some comparisons with the standard, unconstrained least squares problem are given
FORTRAN codes for estimating the one-norm of a real or complex matrix, with appliations to condition estimation
No full textFORTRAN 77 codes SONEST and CONEST are presented for estimating the 1-norm ( or the infinity-norm) of a real or complex matrix, respectively. The codes are of wide applicability in condition estimation since explicit access to the matrix, A, is not required; instead, matrix-vector products Ax and ATx are computed by the calling program via a reverse communication interface. The algorithms are based on a convex optimization method for estimating the 1-norm of a real matrix devised by Hager. We derive new results concerning the behavior of Hager's method, extend it to complex matrices, and make several algorithmic modifications in order to improve the reliability and efficiency
Computing a nearest symmetric positive semidefinite matrix
No full textThe nearest symmetric positive semidefinite matrix in the Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2, where H is the symmetric polar factor of B=(A + AT)/2. In the 2-norm a nearest symmetric positive semidefinite matrix, and its distance δ2(A) from A, are given by a computationally challenging formula due to Halmos. We show how the bisection method can be applied to this formula to compute upper and lower bounds for δ2(A) differing by no more than a given amount. A key ingredient is a stable and efficient test for positive definiteness, based on an attempted Choleski decomposition. For accurate computation of δ2(A) we formulate the problem as one of zero finding and apply a hybrid Newton-bisection algorithm. Some numerical difficulties are discussed and illustrated by example
Computing real square roots of a real matrix
No full textBjörck and Hammarling [1] describe a fast, stable Schur method for computing a square root X of a matrix A (X2 = A). We present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix. For a nonsingular real matrix A conditions are given for the existence of a real square root, and for the existence of a real square root which is a polynomial in A; the number of square roots of the latter type is determined. The conditioning of matrix square roots is investigated, and an algorithm is given for the computation of a well-conditioned square root
A Survey of Condition Number Estimation for Triangular Matrices
No full textWe survey and compare a wide variety of techniques for estimating the condition number
of a triangular matrix, and make recommendations concerning the use of the estimates in applications.
Each of the methods is shown to bound the condition number; the bounds can broadly be categorised as
upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms. For each
bound we examine by how much, at worst, it can overestimate or underestimate the condition number.
Numerical experiments are presented in order to illustrate and compare the practical performance of the
condition estimators
Numerical Linear Algebra in Statistical Computing
Get PDFSome of the factors to be considered when applying the
techniques of numerical linear algebra to statistical problems
are discussed with reference to three particular examples: the
use of the normal equations in regression problems; the use of
perturbation theory to assess the effects of errors in
regression matrices; and the phenomenon of benign degeneracy, in
which the numerical problem becomes more difficult even as the
associated statistical problem becomes easier
Fractals
No full textThe idea of a fractal as a mathematical object and as a model of natural phenomena is introduced by way of simple examples. The characterization of fractals both mathematically and experimentally is then discussed. Finally a model exemplifying the generation of fractal geometries in nature is discussed
Topological dimension and local coordinates from time series data
No full textA method for the estimation of the topological dimension of a manifold from time series data is presented. It is based on the approximation of the manifold near a point chi by its tangent space at chi. The dimension of the tangent space is estimated by constructing a maximal set of linearly independent vectors from the data near chi using the method of singular value decomposition. The method is used to analyse experimental data obtained from a nonlinear electronic oscillator in a chaotic state
Efficient Algorithms for Computing the Condition Number of a Tridiagonal Matrix
No full textLet A be a tridiagonal matrix of order n. We show that it is possible to compute
and hence condo (A), in O(n) operations. Several algorithms which perform this task are given and their
numerical properties are investigated.
If A is also positive definite then I[A-[[o can be computed as the norm of the solution to a positive
definite tridiagonal linear system whose coeffcient matrix is closely related to A. We show how this
computation can be carried out in parallel with the solution of a linear system Ax b. In particular we
describe some simple modifications to the LINPACK routine SPTSL which enable this routine to compute
condt (A), efficiently, in addition to solving Ax b
Computing the Polar Decomposition---with Applications
No full textA quadratically convergent Newton method for computing the polar decomposition of a
full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the
initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed
in practice.
To add to the known best approximation property of the unitary polar factor, the Hermitian polar
factor H of a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to A
and 1/2(A / H) is shown to be a best Hermitian positive semi-definite approximation to A. Perturbation
bounds for the polar factors are derived.
Applications of the polar decomposition to factor analysis, aerospace computations and optimisation
are outlined; and a new method is derived for computing the square root of a symmetric positive definite
matrix