Showing papers on "Basis (linear algebra) published in 1978"
TL;DR: In this article, a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements.
Abstract: Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.
1,211 citations
TL;DR: In this article, the SCC-DV-Xα molecular orbital method was applied to metal clusters and the numerical basis functions were utilized in the present calculations, and it was proved that the self-consistent charge (SCC) approximation to the SCF method gives accurate orbital energies.
Abstract: Applications of the discrete variational (DV) Xα molecular orbital method based on the self-consistent Hartree-Fock-Slater model to metal clusters are presented. Numerical basis functions are utilized in the present calculations. Variations of orbital energies and populations with exchange scaling parameter α are investigated. It is proved that the self-consistent-charge (SCC) approximation to the SCF method gives accurate orbital energies. The numerical basis SCC-DV-Xα method is shown to be very efficient for studies of rather large metal clusters such as Ni 13 .
877 citations
TL;DR: A new derivation of a set, of complete invariants and a corresponding canonical form first given by Morse is provided, which yields relatively simple proofs and economical matrix algorithms.
Abstract: The class {∑} of all linear multivariable systems is partitioned into equivalence classes by the group consisting of all basis, all state feedback and all output injection transformations. This paper provides a new derivation of a set, of complete invariants and a corresponding canonical form first given by Morse. Strong reachability and strong observability concepts are the key tools. The method yields relatively simple proofs and economical matrix algorithms.
164 citations
01 Jun 1978
TL;DR: In this article, a quaternion is regarded as a four-parameter representation of a coordinate transformation matrix, where the four components of the quaternions are treated on an equal basis.
Abstract: A quaternion is regarded as a four-parameter representation of a coordinate transformation matrix, where the four components of the quaternion are treated on an equal basis. This leads to a unified, compact, and singularity-free approach to determining the quaternion when the matrix is given.
163 citations
162 citations
TL;DR: In this article, the friction between two juxtaposed leptodermous systems in relative motion arising from the exchange of particles between them was studied and a universal key function related to the flux between two parallel surfaces as a function of their separation was derived.
117 citations
82 citations
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TL;DR: In this paper, a common basis of convergence analysis is given for a large class of iterative procedures which we call general approximation methods, including the Remes algorithm, the nonlinear Chebyshev approximation, the classical Newton method along with its variants such as Newton's method for partially ordered spaces and for degenerate tangent spaces.
Abstract: In the present work a common basis of convergence analysis is given for a large class of iterative procedures which we call general approximation methods. The concept of strong uniqueness is seen to play a fundamental role. The broad range of applications of this proposed classification will be made clear by means of examples from various areas of numerical mathematics. Included in this classification are methods for solving systems of equations, the Remes algorithm, methods for nonlinear Chebyshev-approximation, the classical Newton method along with its variants such as Newton's method for partially ordered spaces and for degenerate tangent spaces. As an example of the latter the approximation with exponential sums having coalescing frequencies is discussed, that is the case where the tangent space is degenerate.
80 citations
TL;DR: In this article, the concept of universal basis sets for electronic structure calculation is illustrated by presenting results obtained when basis sets are transferred from one atom to another, and a single Slater-orbital basis set, consisting of nine 1s and six 2p functions, produces Hartree-Fock total energies and orbital energies.
Abstract: The concept of a universal basis set for electronic structure calculation is illustrated by presenting results obtained when basis sets are transferred from one atom to another. A single Slater-orbital basis set, consisting of nine 1s and six 2p functions, produces Hartree-Fock total energies and orbital energies in good agreement with the most accurate calculations of these energies obtained using different basis sets individually optimized for each atom. Transferability of integrals is a natural consequence of the use of the same basis set for each atom in a molecule.
72 citations
TL;DR: In this paper, the concept of universal basis sets for electronic structure calculations is explored by presenting energy results obtained when basis sets are transferred from one atom to another, and a single universal basis set is shown to give uniformly accurate descriptions of the matrix Hartree-Fock and correlation energies of the He, Be and Ne atoms.
Abstract: The concept of a ’’universal’’ basis set for electronic structure calculations is explored by presenting energy results obtained when basis sets are transferred from one atom to another. The calculations are performed using the diagrammatic techniques of many‐body perturbation theory. A single universal basis set is shown to give uniformly accurate descriptions of the matrix Hartree–Fock and correlation energies of the He, Be, and Ne atoms.
49 citations
01 May 1978
TL;DR: In this paper, the authors presented a method for matrix element generation in the form of a weighted sum of molecular integrals in which the weighting coefficients represent the integrated value of the wavefunctions over spin coordinates.
Abstract: Advanced techniques are developed to provide efficient economic treatment of the large scale eigenvalue problem posed when configuration interaction is carried out on SCF basis sets of moderate size. When the characteristic properties of the hamiltonian matrix are examined in light of the type of solution required, partitioning of the configuration space is shown to result in an expansion of the problem about a limited core of states, where the small but cumulative interactions of vast regions of the remaining space are reduced to the form of an effective potential. With proper selection of the core, the evaluation of this potential can be readily and accurately truncated to a level involving minimum expenditure in time and effort. In particular only diagonal elements and a strip of the full CI matrix are required to achieve an accuracy of 1 – 5 kcal/mole with complete treatment for configuration spaces of order tens of thousands. In addition, a close look at current theory on the generation of matrix elements between spin symmetry adapted configurations leads to simplified expressions where the matrix elements are derived in the form of a weighted sum of molecular integrals in which the weighting coefficients represent the integrated value of the wavefunctions over spin coordinates. For typical cases of low multiplicity and limited numbers of open shells the list of unique parameters needed to generate all weights are shown to be readily stored as a program library. Actual times for matrix element generation are believed to be an order of magnitude faster than current techniques. Practical demonstration of the accuracy and efficiency of the method is provided by calculations on formaldehyde, water, and ethylene.
TL;DR: In this paper, an algorithm is presented which greatly facilitates the complete exploitation of state feedback in the assignment of the entire closed-loop eigenstructure of controllable multi-input systems.
Abstract: In this paper an algorithm is presented which greatly facilitates the complete exploitation of state feedback in the assignment of the entire closed-loop eigenstructure of controllable multi-input systems. This algorithm is a generalization of the algorithm of MacLane and Birkhoff (1968) for the computation of a basis for the null space of a matrix and is ideally suited to digital computer implementation. The algorithm readily yields the vectors which are required (Porter and D'Azzo 1978) for the simultaneous assignment of Jordan canonical forms, eigenvectors, and generalized eigenvectors to the plant matrices of closed-loop controllable multivariable linear systems. The effectiveness of the algorithm is illustrated by assigning the entire closed-loop eigenstructure of a third-order two-input discrete-time system in such a way that the resulting closed-loop system exhibits time-optimal behaviour.
TL;DR: In this article, a criterion is proposed for determining which explicit Runge-Kutta formulas are the most promising as a basis for developing good library subroutines for solving nonstiff initial-value problems associated with ordinary differential equations.
Abstract: A criterion is proposed for determining which explicit Runge–Kutta formulas are the most promising as a basis for developing good library subroutines for solving nonstiff initial-value problems associated with ordinary differential equations. The criterion is based on a theoretical measure of the cost of solving classes of linear homogeneous problems with constant coefficients. It takes into account the trade-off between reliability and efficiency, as well as the user’s accuracy requirement. Results are presented for twenty-one formulas; one of the recently developed formula pairs due to Verner appears to be one of the most promising currently available. Moreover, our results support the thesis that the use of local extrapolation will usually improve the performance of a method. These conclusions are also supported by some empirical results collected over a relatively wide class of nonstiff problems. Finally, some suggestions are given on how to exploit these results, particularly if local extrapolation i...
01 Jan 1978
TL;DR: In this paper, the authors discuss techniques for implementing Lemke's algorithm for the linear complementarity problem in a numerically robust way as well as a method for recovering from loss of feasibility or singularity of the basis.
Abstract: This note discusses techniques for implementing Lemke's algorithm for the linear complementarity problem in a numerically robust way as well as a method for recovering from loss of feasibility or singularity of the basis. This recovery method is valid for both positive semi-definite M matrices and those with positive principal minors. It also allows a user to start from an advanced basis for such problems.
TL;DR: The degree of ill-conditioning, for a general inner-product space, in terms of the basis is characterized, and it is shown, for example, that the powers {1, z, z 2,…} are a universally bad choice of basis.
Abstract: It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.
01 Jan 1978
TL;DR: The vector-network method is a combination of vector dynamics and some concepts of graph theory which serves as the basis for a “self-formulating” computer program which can simulate the response of a dynamic system, given only the system description.
Abstract: This paper describes the “vector-network” method for creating mathematical models of dynamic mechanical systems. The vector-network method is a combination of vector dynamics and some concepts of graph theory; it serves as the basis for a “self-formulating” computer program which can simulate the response of a dynamic system, given only the system description. The vector-network method also permits us to observe a useful but little-known “principle of orthogonality” which is an extension of Tellegen’s theorem for electrical networks, discovered in 1952. Many basic dynamic concepts, such as the principle of virtual work and the instantaneous balance of power, are special cases of this principle.
TL;DR: In this article, the atom-density basis functions are split into inner and outer parts, and the atomic charges reflect polarity of the molecule reasonably well and are relatively independent of the orbital bases used in spanning the molecular wave function.
Abstract: Electron population analyses of several molecular one-electron density functions have been studied by least-squares projection methods into several atomic-density basis functions. All studies have been restricted to spherically symmetrical functions, which have been fitted to the atomic-density functions for the ground-state atoms hydrogen through neon. It is found that when the atom-density basis functions are split into inner (K shell) and outer (L shell) parts, then the atomic charges reflect polarity of the molecule reasonably well and, moreover, are relatively independent of the orbital bases used in spanning the molecular wave function. The standard density basis sets given here can be used for a similar electron population analysis of accurate X-ray diffraction data.
TL;DR: This paper considers sets ofinteger vectors containing the zero vector and closed under addition, the integral monoids, and provides conditions under which they contain a finite subset of integer vectors which generate the entire monoid as nonnegative integer combinations.
Abstract: We consider sets of integer vectors containing the zero vector and closed under addition, the integral monoids, and provide conditions under which they contain a finite subset of integer vectors which generate the entire monoid as nonnegative integer combinations. The paper concludes with some applications to the theory of integer programming.
TL;DR: The number of irreducible factors of a given monic polynomial over GF( q ) is equal to the dimension of the space of characteristic sequences associated with $f( x )$.
Abstract: The number of irreducible factors of a given monic polynomial $f( x )$ over $GF( q )$ is equal to the dimension of the space of characteristic sequences associated with $f( x )$. A basis for this space can be used to obtain the irreducible factors.
01 Jan 1978
TL;DR: This paper discusses persistance in staircase models and presents a considerable refinement of the above factorization algorithm, which has been implemented in an experimental code and use being made of LU and QR factorization and updating techniques for the solution of small sub-systems of equations.
Abstract: : Time-staged and multi-staged linear programs usually have a structure that is block triangular. Basic solutions to such problems typically have the property that similar type activities persist in the basis over several consecutive time-periods. When this occurs the basis is close to being square block triangular. In 1955 Dantzig suggested a way of factorizing the basis to take advantage of this property. This paper discusses persistance in staircase models and then presents a considerable refinement of the above factorization algorithm. The method has been implemented in an experimental code, with use being made of LU and QR factorization and updating techniques for the solution of small sub-systems of equations. An in-depth analysis is made of the work involved, and computational experience on several dynamic models is reported. (Author)
01 Jun 1978
TL;DR: In this paper, two non-Gaussian ergodic models are given, one of a genuinely probabilistic character similar to Lauritzen's model, and another based on a formal probability theory in rotation group space.
Abstract: : The paper deals with mathematical models suitable as a basis for the statistical treatment of collection. As a preparation, stochastic processes on the circle are discussed first; such processes are simple to understand and exhibit already essential features of the problem. Then the paper treats stochastic processes on the sphere, which may be suitable as statistical models for collocation. Lauritzen's theorem on the nonexistence of ergodic Gaussian stochastic process models for collocation is seen to be essentially dependent on the Gaussian character. Two non-Gaussian ergodic models are given, one of a genuinely probabilistic character similar to Lauritzen's model, and another based on a formal probability theory in rotation group space. This second model gives a statistical foundation of the usual homogeneous and isotropic covariance analysis of the anomalous gravity field; it also provides a basis for the study of the statistical distribution of quantities related to this field. This model allows a formal statistical treatment of the anomalous gravitational field which is independent of an interpretation of this field as some genuinely physical stochastic process and seems, therefore, to be preferable. (Author)
TL;DR: The comrade matrix as mentioned in this paper is a generalization of the companion matrix, and arises when a polynomial is expressed in terms of a basis set of orthogonal polynomials.
Abstract: The comrade matrix is a generalization of the companion matrix, and arises when a polynomial is expressed in terms of a basis set of orthogonal polynomials. The work begun in a previous paper is here continued for multivariable systems, and a number of generalizations of standard results are described. Topics covered include controllability, canonical forms, polynomial and state-space realizations and linear feedback. The flexibility offered by an arbitrary choice of basis promises to be useful for applications.
TL;DR: In this paper, the coreflective hull of metric spaces is shown to be productive if and only if there exists no uniformly sequential cardinal number in the metric space, and three basis examples of productive sub-classes are constructed (connected with products of discrete spaces, proximally fine spaces, and uniformly sequentially continuous mappings).
Abstract: Using the fact that each product of uniform quotient mappings is a quotient mapping, new conditions are given for the finite and countable productivity of a coreflective sub-class of uniform spaces. Three basis examples of productive coreflective sub-classes are constructed (connected with products of discrete spaces, proximally fine spaces, and uniformly sequentially continuous mappings) and the coreflective hull of metric spaces is shown to be productive if and only if there exists no uniformly sequential cardinal number.
TL;DR: In this paper, it was shown that there exists a binary subtreeS of S such that either all chains of S lie in S or no chain of S lies in S. As an application, they proved the following result on Banach spaces: if (xs)s is a bounded sequence of elements in a Banach spaceE, there is a subtreeS ofS such that for any chainβ ofS ∈ β the sequence (xss ∈β is either a weak Cauchy sequence or equivalent to the usuall 1 basis.
Abstract: LetS be the binary tree of all sequences of 0’s and 1’s. A chain ofS is any infinite linearly ordered subset. Letℋ be an analytic set of chains, we show that there exists a binary subtreeS’ ofS such that either all chains ofS’ lie inℋ or no chain ofS’ lies inℋ. As an application, we prove the following result on Banach spaces: If (xs)sɛs is a bounded sequence of elements in a Banach spaceE, there exists a subtreeS’ ofS such that for any chainβ ofS’ the sequence (xs)s ∈β is either a weak Cauchy sequence or equivalent to the usuall1 basis.
TL;DR: In this article, the authors define the order and symbol of a distribution on a manifold X and the space in which it lies, and prove the elementary properties of the symbol, and give some examples.
Abstract: A definition is given, for an arbitrary distribution g on a manifold X, of the order and symbol of g at a point (x, ?) of the cotangent bundle TX. If X = R', the order of g at (0, $) is the growth order as X co of the distributions gT(x) = e < g(x/ ); if the order is less than or equal to N, the N-symbol of g is the family gt modulo O (T N 1/2). It is shown that the order and symbol behave in a simple way when g is acted upon by a pseudo-differential operator. If g is a Fourier integral distribution, suitable identifications can be made so that the symbol defined here agrees with the bundle-valued symbol defined by Hormander. PREFACE Since the introduction of pseudo-differential operators, the analysis of distributions on a manifold X has involved the geometry of the cotangent bundle T*X. With the notion of wavefront set [6], one can localize a distribution at a nonzero cotangent vector ( to obtain its "microgerm", just as one localizes at x E X to obtain the ordinary germ. On the base space, X, one can go beyond the local level to the infinitesimal one; for a C function, the result is its "jet", which can be thought of as a function on the tangent space T,X. In this paper, we describe an analogous procedure for distributions: given a distribution g on X and a cotangent vector {, we construct a jet-like object called its symbol, which is a distribution on TX depending on certain parameters. (In [13], we show that the symbol may be thought of as an object on TCT*X, thus completing the analogy with jets.) In Chapter I, we define the symbol and the space in which it lies, we prove the elementary properties of the symbol, and we give some examples. Chapter II, written in collaboration with K. Sklower, establishes the relation between our symbol and the wavefront set. In Chapter III, we show that our symbol construction contains the one given by Hormander [6] for a very special class of distributions-the Fourier integral distributions. This last result was the basis for the whole paper. In lectures at the Nordic Received by the editors November 24, 1976. AMS (MOS) subject classifications (1970). Primary 46F10; Secondary 53C15, 58G15. C American Mathematical Society 1978
TL;DR: In this paper, it was shown that for every pair A, B in the Brunovski (1966) and Luenberger (1967) controllable canonical form, there corresponds a unique polynomial matrix X(8) which has a canonical structure.
Abstract: The problem of determining the structure of the basis matrices of all possible controllability subspaces of a controllable pair [A, [Btilde]] in the Brunovski (1966) and Luenberger (1967) controllable canonical form is considered. Departing from a characterization of the c.s.'s of [A, [Btilde]] given by Warren and Eckberg (1975) it is shown that to every pair A, B in the Brunovski (1966) and Luenberger (1967) controllable canonical form, there corresponds a unique polynomial matrix X(8) which has a canonical structure. Using the results on rational vector spaces obtained by Forney (1975) it is seen that this polynomial matrix qualifies as a minimal basis which uniquely identifies a rational vector space (s). A correspondence between the polynomial n-tuples x(8)∊(8) and the c.s.'s of [A, [Btilde]] loads to simple expressions that describe the structure of the bases of all c.s. of [A, [Btilde]] of all possible dimensions.
TL;DR: In this paper, the authors investigated the scattering problem for a system of differential equations on the interval [0,a], where A is a positive matrix-valued function which jumps to I for x > a, and the system of resonances is described, and an expression for the resonance states in terms of the Jost solution is given.
Abstract: The scattering problem is investigated for a system of differential equations on the interval [0,a], where A is a positive matrix-valued function which jumps to I for x > a. The scattering matrix for large spectral parameter is studied, the system of resonances is described, and an expression for the resonance states in terms of the Jost solution is given. A relation is established between the resonances and the poles of the analytic continuation of the Green function. It is proved that the syrtem of resonance states corresponding to complex zeros of the scattering matrix has serial structure; namely, it splits into n Carleson series. The completeness of the system of resonances is investigated, and it is established that this system forms a Riesz basis in the corresponding space with the energy metric.Bibliography: 27 titles.
TL;DR: In this article, an exact theory of the dynamics in simple classical liquids is presented and employed to give formal justification for approximations of the memory function of the phase-space correlation function derived previously by Sjogren and Sjolander.
01 Feb 1978
TL;DR: The problem of finding a minimal asymptotic basis for a set of natural numbers has been studied by Nathanson and Stohr as mentioned in this paper, who showed that there exists a set S which has infinitely many finite sums of distinct powers of 3, but no minimal base.
Abstract: The set A of natural numbers is an asymptotic basis for S if the sets S and 2A eventually coincide. An asymptotic basis A for S is minimal if no proper subset of A is a basis for S. Sets S are constructed which possess infinitely many asymptotic bases, but no minimal asymptotic basis. Let A and S be subsets of the natural numbers N = (0, 1, 2, 3 . . . . ), and let 2A denote the sum set {a + a'1 a, a' E A). Then A is an asymptotic basis (of order 2) for S if the sets 2A and S eventually coincide, that is, if the symmetric difference (S \ 2A) U (2A \ S) is finite. The asymptotic basis A for S is minimal if no proper subset of A is an asymptotic basis for S. Minimal asymptotic bases for the natural numbers have been investigated by Erdos, Hartter, Nathanson, and Stohr [1H6], [8], [9], [14]. The simplest example of a minimal asymptotic basis for N is the set consisting of all finite sums of distinct powers of 3 (Nathanson [8]). Of course, a set, such as f2'),'. ., which has no asymptotic basis certainly has no minimal asymptotic basis. Let us call two sets asymptotically equivalent if their symmetric difference is finite, and asymptotically inequivalent if their symmetric difference is infinite . The object of this note is to construct a set S which has infinitely many asymptotically inequivalent asymptotic bases, but no minimal asymptotic basis . Notation . Lower case letters denote natural numbers, and upper case letters denote sets of natural numbers . Let A + B = (a + bl a E A, b E B) . Let [a, b] denote the interval of integers n such that a 2 and mk+1 > 2mk +PkPk+3. Let Ck = (c E [mk, mk+1]Jc 0 (modpk)}, and let C = Uk 2 Ck = (c;}°o , where co < c, < c 2 < . . . . Then C is an asymptotic basis for N such that (I) lim sup s-m(c; + , c;) = oo, and (II) C \ F is an asymptotic basis for Nfor every finite set F. PROOF. Clearly, ci+ 1 c; --* oo as i -* oo . Received by the editors February 14, 1977 . AMS (MOS) subject classifications (1970) . Primary 1OL05, 101-10,, 10.199 .