Showing papers in "Numerische Mathematik in 1965"

Numerical methods for solving linear least squares problems

Abstract: A common problem in a Computer Laboratory is that of finding linear least squares solutions. These problems arise in a variety of areas and in a variety of contexts. Linear least squares problems are particularly difficult to solve because they frequently involve large quantities of data, and they are ill-conditioned by their very nature. In this paper, we shall consider stable numerical methods for handling these problems. Our basic tool is a matrix decomposition based on orthogonal Householder transformations.

Linear least squares solutions by householder transformations

Abstract: Let A be a given m×n real matrix with m≧n and of rank n and b a given vector. We wish to determine a vector x such that $$\parallel b - A\hat x\parallel = \min .$$ where ∥ … ∥ indicates the euclidean norm. Since the euclidean norm is unitarily invariant $$\parallel b - Ax\parallel = \parallel c - QAx\parallel $$ where c=Q b and Q T Q = I. We choose Q so that $$QA = R = {\left( {_{\dddot 0}^{\tilde R}} \right)_{\} (m - n) \times n}}$$ (1) and R is an upper triangular matrix. Clearly, $$\hat x = {\tilde R^{ - 1}}\tilde c$$ where c denotes the first n components of c.

Numerical calculation of elliptic integrals and elliptic functions

Abstract: This paper is a continuation of [2, 3]. It contains anALGOL program for the incomplete elliptic integral of the third kind based on a theory described in [4]. This program replaces the inadequate one based on the Gauβ-transformation which was published in [2]. In addition, anAlgol program for a general complete elliptic integral is presented.

Positive operators and an inertia theorem

Abstract: : A generalization of the theorem of Perron-Frobenius is used to prove an inertia theorem, which contains Lyapunov's theorem and Stein's theorem as special cases. It is pointed out that there is a close relation between the inertia theorem and known results on M-matrices. (Author)

Mildly nonlinear elliptic partial differential equations and their numerical solutoin. II

Abstract: : High speed digital computer methods are studied for obtaining solutions of difference equation analogues of mildly nonlinear elliptic boundary value problems. The problem is formulated in terms of finding a vector X which satisfies AX = - f(X) + Y, where A is an irreducible M matrix, Y is a given vector, and f is a given function. Two general iteration techniques and related convergence theorems are explored. The methods and ideas also extend to a large class of ordinary differential equation boundary value problems. (Author)

Symmetric decomposition of positive definite band matrices

Abstract: The method is based on the following theorem. If A is a positive definite matrix of band form such that $${a_{ij}} = 0{\rm{ (|}}i - j| >m{\rm{)}}$$ (1) then there exists a real non-singular lower triangular matrix L such that $$L{L^T} = A,{\rm{ where }}{l_{ij}} = 0{\rm{ (}}i - j >m{\rm{)}}{\rm{.}}$$ (2)

On Newton's method

Abstract: In this note we discuss Newton's method in a setting somewhat more restrictive than customary In this setting, however, we claim to have proved superlinear convergence of the Newton process without assuming twice differentiability or Lipschitz continuity of the first derivative of the operator A further feature is that the iteration to be discussed is not initially but is eventually the Newton process With this feature global rather than local convergence is achieved

Elimination with weighted row combinations for solving linear equations and least squares problems

Abstract: Let A be a matrix of n rows and m columns, m≦n. If and only if the columns are linearly independent, then for any vector b there exists a unique vector x minimizing the Euclidean norm of \(b - Ax,\parallel b - Ax\parallel = \mathop {\min }\limits_\xi \parallel b - A\xi \parallel .\).

An elementary proof of the hopf inequality for positive operators

Abstract: Recently, OSTROWSKI [41 has established the connection between a theory of BIRXHOFF [1] on positive linear transformations and a theory of HOl'F [3] on positive linear integral operators. A central result of these investigations is an inequality of HOPF, which OSTROWSKI could extend to the more general case of positive linear transformations studied by BIRKHOFF. This inequality deals with the oscillation of a pair of positive vectors and gives a bound for the decrease of the oscillation under a positive linear transformation. I t leads to theoretical as well as practical applications of some importance especially in the theory of positive matrices E4], e.g. estimates for the modulus of the second eigenvalue in terms of the Perron root. A simplified derivation may therefore be worthwhile.

A class of linear programming problems requiring a large number of iterations

Abstract: : A coordinate-free description of the simplex algorithm (for nondegenerate linear programming problems) is supplied, and is used to show that the number of iterations can be larger than was previously known. For 0 m n, there is constructed a nondegenerate linear programming problem whose bounded (n - m)-dimensional feasible region is defined by means of m linear equality constraints in n nonnegative variables, and in which, after starting from the worst choice of an initial feasible vertex, m(n - m - l) + 1 simplex iterations are required in order to reach the optimal vertex. It is conjectured that this is the maximum possible number of iterations (for arbitrary 0 m n), but the conjecture is proved only for n m + 4. (Author)

On a linear diophantine problem of Frobenius: an improved algorithm

Abstract: Since nn is in general considerably less than ha, and since 7 (B) is less than 7(A) by the same amount, the work necessary to compute F(B) rather than 7(A) is correspondingly reduced (see Section 5 of [6]). Furthermore, the reduction in the order of the matrix considerably lessens the storage requirements of the algorithm. This matrix, B, is defined in equation (3.t); the order, p~, of B is, in a sense to be discussed later in the paper, the least that can be obtained in general for the computation of g [Pl . . . . . p~] using these techniques. We begin by repeating some of the relevant definitions and lemmas required for this paper.

Quadrature methods based on the Euler-Maclaurin formula and on the Clenshaw-Curtis method of integration

Abstract: The Euler-Maclaurin formula [11 may be used to derive a wide range of quadrature formulas including the Newton-Cotes formulas [2]. Some of these are derived, including two which are more accurate than Simpson's rule and which share some of its advantages. These are especially useful in the automatic evaluation of integrals which are badly behaved at their end points. The Clenshaw-Curtis method [3] is also examined and it is shown to be considerably more accurate in practice than the equivalent trapezoidal rule even though the Clenshaw-Curtis method appears to converge to the trapezoidal rule as the number of abscissas approaches infinity [41. When the Clenshaw-Curtis formula is used in a way different from that put forward by the original authors it is suggested that it may have significant advantages over other methods of numerical integration in many problems.

A note on the numerical solution of a periodic parabolic problem

Abstract: A direct method is presented for the numerical determination of solutions of the heat conduction equation having periodT in time. A simple error analysis which improves previous results is given.