The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

This paper concerns a harmonic projection method for computing an approximation to an eigenpair of a large matrix . Given a target point and a subspace that contains an approximation to , the harmonic projection method returns an approximation to . Three convergence results are established as the deviation of from approaches zero. First, the harmonic Ritz value converges to if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector converges to if the Rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue --in other words, the method can miss if it is very close to . To this end, we propose to compute the Rayleigh quotient of with respect to and take it as a new approximate eigenvalue. is shown to converge to once tends to , no matter how is close to . Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

     

Sequential and parallel synchronous alternating iterative methods

The so-called parallel multisplitting nonstationary iterative Model A was introduced by Bru, Elsner, and Neumann [Linear Algebra and its Applications 103:175-192 (1988)] for solving a nonsingular linear system using a weak nonnegative multisplitting of the first type. In this paper new results are introduced when is a monotone matrix using a weak nonnegative multisplitting of the second type and when is a symmetric positive definite matrix using a -regular multisplitting. Also, nonstationary alternating iterative methods are studied. Finally, combining Model A and alternating iterative methods, two new models of parallel multisplitting nonstationary iterations are introduced. When matrix is monotone and the multisplittings are weak nonnegative of the first or of the second type, both models lead to convergent schemes. Also, when matrix is symmetric positive definite and the multisplittings are -regular, the schemes are also convergent.

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

     

An analysis of the Rayleigh-Ritz method for approximating eigenspaces

This paper concerns the Rayleigh-Ritz method for computing an approximation to an eigenspace of a general matrix from a subspace that contains an approximation to . The method produces a pair that purports to approximate a pair , where is a basis for and . In this paper we consider the convergence of as the sine of the angle between and approaches zero. It is shown that under a natural hypothesis--called the uniform separation condition--the Ritz pairs converge to the eigenpair . When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that has distinct eigenvalues or is diagonalizable.

     

The hyperdeterminant and triangulations of the 4-cube

The hyperdeterminant of format is a polynomial of degree in unknowns which has terms. We compute the Newton polytope of this polynomial and the secondary polytope of the -cube. The regular triangulations of the -cube are classified into -equivalence classes, one for each vertex of the Newton polytope. The -cube has coarsest regular subdivisions, one for each facet of the secondary polytope, but only of them come from the hyperdeterminant.

     

Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

Minimality and other properties of the width- nonadjacent form

Let be an integer and let be the set of integers that includes zero and the odd integers with absolute value less than . Every integer can be represented as a finite sum of the form , with , such that of any consecutive 's at most one is nonzero. Such representations are called width- nonadjacent forms (-NAFs). When these representations use the digits and coincide with the well-known nonadjacent forms. Width- nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the -NAF. We show that -NAFs have a minimal number of nonzero digits and we also give a new characterization of the -NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on -NAFs and show that any base 2 representation of an integer, with digits in , that has a minimal number of nonzero digits is at most one digit longer than its binary representation.

     

Solving polynomials by radicals with roots of unity in minimum depth

Let be an algebraic number field. Let be a root of a polynomial which is solvable by radicals. Let be the splitting field of over . Let be a natural number divisible by the discriminant of the maximal abelian subextension of , as well as the exponent of , the Galois group of over . We show that an optimal nested radical with roots of unity for can be effectively constructed from the derived series of the solvable Galois group of over .

     

Upper bounds for the prime divisors of Wendt's determinant

Let be an even integer, . The resultant of the polynomials and is known as Wendt's determinant of order . We prove that among the prime divisors of only those which divide or can be larger than , where and is the th Lucas number, except when and . Using this estimate we derive criteria for the nonsolvability of Fermat's congruence.

     

Two contradictory conjectures concerning Carmichael numbers

Erdos conjectured that there are Carmichael numbers up to , whereas Shanks was skeptical as to whether one might even find an up to which there are more than Carmichael numbers. Alford, Granville and Pomerance showed that there are more than Carmichael numbers up to , and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdos must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data), and so we herein derive conjectures that are consistent with Shanks's observations, while fitting in with the viewpoint of Erdos and the results of Alford, Granville and Pomerance.

     

The Postage Stamp Problem: An algorithm to determine the

Given an integral ``stamp" basis with and a positive integer , we define the -range as

. For given and , the extremal basis has the largest possible extremal -range

We give an algorithm to determine the -range. We prove some properties of the -range formula, and we conjecture its form for the extremal -range. We consider parameter bases , where the basis elements are given functions of . For we conjecture the extremal parameter bases for .

     

Analysis of PSLQ, an integer relation finding algorithm

Let be either the real, complex, or quaternion number system and let be the corresponding integers. Let be a vector in . The vector has an integer relation if there exists a vector , , such that . In this paper we define the parameterized integer relation construction algorithm PSLQ, where the parameter can be freely chosen in a certain interval. Beginning with an arbitrary vector , iterations of PSLQ will produce lower bounds on the norm of any possible relation for . Thus PSLQ can be used to prove that there are no relations for of norm less than a given size. Let be the smallest norm of any relation for . For the real and complex case and each fixed parameter in a certain interval, we prove that PSLQ constructs a relation in less than iterations.

     

Linear algebra algorithms for divisors on an algebraic curve

We use an embedding of the symmetric th power of any algebraic curve of genus into a Grassmannian space to give algorithms for working with divisors on , using only linear algebra in vector spaces of dimension , and matrices of size . When the base field is finite, or if has a rational point over , these give algorithms for working on the Jacobian of that require field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess' 1999 Ph.D. thesis, which works with function fields as extensions of . However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of .

     

On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an -regular solution, a first-order error bound in the norm is shown and used to derive a second-order error bound in the norm. For the cubic Schrödinger equation with an -regular solution, first-order convergence in the norm is used to obtain second-order convergence in the norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and -conditional stability for error propagation, where for the Schrödinger-Poisson system and for the cubic Schrödinger equation.

     

Optimal a priori error estimates for the -version of the local discontinuous Galerkin method for convection-diffusion problems

We study the convergence properties of the -version of the local discontinuous Galerkin finite element method for convection-diffusion problems; we consider a model problem in a one-dimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh-width , in the polynomial degree , and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both and . The theoretical results are confirmed in a series of numerical examples.

     

Normal bases via general Gauss periods

Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive th root of unity, one obtains under certain conditions a normal basis for over , where is a prime and for some integer . We generalize this construction by allowing arbitrary integers with , and find in many cases smaller values of than is possible with the previously known approach.

     

Computing automorphisms of abelian number fields

Let be an abelian number field of degree . Most algorithms for computing the lattice of subfields of require the computation of all the conjugates of . This is usually achieved by factoring the minimal polynomial of over . In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of , which is based on -adic techniques. Given and a rational prime which does not divide the discriminant of , the algorithm computes the Frobenius automorphism of in time polynomial in the size of and in the size of . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of .

     

Bounds for computing the tame kernel

The tame kernel of the of a number field  is the kernel of some explicit map , where the product runs over all finite primes  of  and is the residue class field at . When is a set of primes of , containing the infinite ones, we can consider the -unit group  of . Then has a natural image in . The tame kernel is contained in this image if  contains all finite primes of  up to some bound. This is a theorem due to Bass and Tate. An explicit bound for imaginary quadratic fields was given by Browkin. In this article we give a bound, valid for any number field, that is smaller than Browkin's bound in the imaginary quadratic case and has better asymptotics. A simplified version of this bound says that we only have to include in  all primes with norm up to  , where  is the discriminant of . Using this bound, one can find explicit generators for the tame kernel, and a ``long enough' search would also yield all relations. Unfortunately, we have no explicit formula to describe what ``long enough' means. However, using theorems from Keune, we can show that the tame kernel is computable.

     

On iterates of Möbius transformations on fields

Let be a quadratic polynomial over a splitting field , and be the set of zeros of . We define an associative and commutative binary relation on so that every Möbius transformation with fixed point set is of the form ``plus' for some . This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Möbius transformation to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of are distinct and the iterates of converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.