David J. Jeffrey

• B.Sc.
• Professor (Full) at Western University

Listening to stories of sickness lies at the heart of the patient–doctor relationship. Storytelling serves as a powerful learning tool fostering empathy, attentive listening, clinical curiosity and reflection, key elements of narrative competence. The patient’s story, their history, forms the fundamental core of diagnosis, and their emotions are ce...

We derive and prove formulae for solving cubic equations, with an eye to providing computer implementations. We consider two families of solutions: the formulae of Tartaglia—Cardano and those of Vi`ete. We show that even if the coefficients in the cubic are purely real, and the cubic roots are all real, it is still the case that complex numbers app...

The Gram–Schmidt process is a standard topic in beginning linear algebra courses. The computations, while straightforward, often require working with square roots, which can be difficult for students. We demonstrate how rational orthogonal matrices can be used to design questions examining the Gram–Schmidt method which avoid the appearance of squar...

A Maple implementation of partitioned matrices is described. A recursive block data structure is used, with all operations preserving the block abstraction. These include constructor functions, ring operations such as addition and product, and inversion. The package is demonstrated by calculating the PLU factorization of a block matrix.

The infinite exponential tower is studied through the associated iteration c₁ = 0 and cₙ₊₁ = eᶜₙ λ, for complex λ. For a subset of λ values, the sequence displays stable 2-cycles, that is to say as n → ∞ we observe that the odd subsequence c₂ₙ₋₁ → A whereas the even subsequence c₂ₙ → B, with A ≠ B. Thus, A and B obey B=eᴬ λ and A = eᴮ λ. Numerical...

An exponential tower is obtained by repeatedly raising a number to itself, as in xˣ, x^{x^x}, and so on. When formulated as an iterative process, it may converge to a single value, diverge to infinity, cycle between multiple values, or wander forever. Representing this property graphically leads to a beautiful fractal object. In order to plot the f...

This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the “old fashioned way”. We discuss some technological tools such as Maple, Matlab, Python, and Jupyter Notebooks, and some choices of topics tha...

The Lambert W function is a multivalued function whose principal branch has been studied in detail. Non-principal branches, however, have been much less studied. Here, asymptotic series expansions for the non-principal branches are obtained, and their properties, including accuracy and convergence are studied. The expansions are investigated by map...

This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the ``old fashioned way''. We discuss some technological tools such as Maple, Matlab, Python, and Jupyter Notebooks, and some choices of topics t...

The plotting of Riemann surfaces by computational software is discussed. The link between the branches of a multi-valued function $g(z)$, defined on the range of $g(z)$, and a Riemann surface, defined on the domain of $g(z)$, is emphasized. The connection between the two is clarified by defining the \textit{charisma} of the argument $z$ to the func...

When evaluating or simplifying mathematical expressions, the question arises of how to handle inverse functions. The problem is that for a non-injective function f:D→R, the inverse is generally not a function R→D since there may be multiple pre-images for a given point. The majority of work in this area has fallen into two camps: either the inverse...

The recursively-constructed family of Mandelbrot matrices Mn for n = 1, 2, … have nonnegative entries (indeed just 0 and 1, so each Mn can be called a binary matrix) and have eigenvalues whose negatives −λ=c give periodic orbits under the Mandelbrot iteration, namely zk=zk−12+c with z0=0, and are thus contained in the Mandelbrot set. By the Perron–...

Subresultant chains over rings of multivariate polynomials are calculated using a speculative approach based on the Bézout matrix. Our experimental results yield significant speedup factors for the proposed approach against comparable methods. The determinant computations are based on fraction-free Gaussian elimination using various pivoting strate...

We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, whil...

We give here some problems and puzzles that need a combination of thought and computation to solve. Please submit your solutions to the journal at mapletransactions.org. Include the problem number with your solution.

We study the problem of xy=yx, first proposed by Daniel Bernoulli in 1728. We present Maple’s parametric solution and a solution using the Lambert W function. This leads us to consider an implementation in Maple of new simplifications of the Lambert W function. The method uses a mixture of exact and floating-point computation.

The recursively-constructed family of Mandelbrot matrices $M_n$ for $n=1$, $2$, $\ldots$ have nonnegative entries (indeed just $0$ and $1$, so each $M_n$ can be called a binary matrix) and have eigenvalues whose negatives $-\lambda = c$ give periodic orbits under the Mandelbrot iteration, namely $z_k = z_{k-1}^2+c$ with $z_0=0$, and are thus contai...

Calculus students are taught that an indefinite integral is defined only up to an additive constant, and as a consequence generations of students have assiduously added ‘+ C ’ to their calculus homework. Although ubiquitous, these constants rarely garner much attention, and typically loiter without intent around the ends of equations, feeling negle...

This paper offers what seems at first to be a minor technical correction to the current practice of computing indefinite integrals, and introduces the idea of a "Kahanian constant of integration". However, the total impact of this minor correction is potentially large because the current practice is taught early at the university level and to very...

We consider the problem of obtaining expressions for integrals that are continuous over the entire domain of integration where the true mathematical integral is continuous, which has been called the problem of obtaining integrals on domains of maximum extent (Jeffrey, 1993). We develop a method for correcting discontinuous integrals using an extens...

We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauss--Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, wh...

The comprehensive LU decomposition of a parametric matrix consists of a case analysis of the LU factors for each specialization of the parameters. Special cases can be discontinuous with respect to the parameters, the discontinuities being triggered by zero pivots encountered during factorization. For polynomial matrices, we describe an implementat...

This paper describes an application of Maple in the teaching of linear algebra. The topic is the construction of an orthogonal basis for a set of vectors or a matrix using Householder transformations. We present a method for generating matrices which, when subject to using Householder transformations, require only rational computations and give rat...

We present a poster and software demonstration regarding the calculation of indefinite integrals, or anti-derivatives, when parameters are present.

We begin with a discussion of general design decisions made in implementing the Lambert W function in Maple . Many of these decisions are not system-specific and apply to any implementation of W; also they touch some of the fundamental issues in computer-algebra systems. A specific topic is the choice of a branch structure for W, and a new approach...

We consider some integer sequences connected with combinatorial applications. Specifically, we consider Stirling partition and cycle numbers, associated Stirling partition and cycle numbers, and Eulerian numbers of the first and second kinds. We consider their evaluation in different contexts. One context is the calculation of a single value based...

We consider exact matrix decomposition by Gauss-Bareiss reduction. We investigate two aspects of the process: common row and column factors and the influence of pivoting strategies. We identify two types of common factors: systematic and statistical. Systematic factors depend on the process, while statistical factors depend on the specific data. We...

Nonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (AD...

We describe the development of a term-rewriting system for indenite integration; it is also called a rule-based evaluation system. The development is separated into modules, and we describe the module for a wide class of integrands containing the tangent function.

We revisit fraction-free Gaussian elimination as a method for finding exact solutions of linear systems over integral domains, specifically integers and univariate polynomials. We conducted several experiments regarding common folklore about these methods such as pivoting strategies. Moreover, we find that the classical algorithms produce a unsettl...

The Lambert W function possesses branches labelled by an index k. The value of W therefore depends upon the value of its argument z and the value of its branch index. Given two branches, labelled n and m, the branch difference is the difference between the two branches, when both are evaluated at the same argument z. It is shown that elementary inv...

We describe a simple package of Matlab programs which implements an extended-precision class in Matlab. We give some examples of how this class can be used to demonstrate the effects of rounding errors and truncation errors in scientific computing. The package is based on a representation called Double-Double, which represents each floating-point r...

An implementation (in Maple) of the multivalued elementary inverse functions is described. The new approach addresses the difference between the single-valued inverse function defined by computer systems and the multivalued function which represents the multiple solutions of the defining equation. The implementation takes an idea from complex analy...

We study discrete dynamical systems, or recurrence relations, of the general form [EQUATION] with explicitly known series coefficients αk and α1 ≠ 0. We associate with the discrete system an interpolating continuous system Y (t), such that Y (n) = yn. The asymptotic behaviour of yn can then be investigated through Y (t). The corresponding continuou...

We consider, from a symbolic point of view, a pair of definite integrals containing Lambert W, recently considered from a numerical point of view by Walter Gautschi. We transform the integrals to a shape that can be integrated in special cases by a computer-algebra system or by using tables of integrals, such as Prudnikov et al.

This paper summarizes research done in the Mechanical Pulping Program of the Network of Centres of Excellence for Wood Pulp. The fundamental mechanisms of wood refining were studied by applying cyclical forces to wood blocks. An image analysis technique was developed to estimate the degree of fibre separation, and it was shown that for large strain...

This paper summarizes research done in the Mechanical Pulping Program of the Network of Centres of Excellence for Wood Pulps to understand the mechanical properties of individual fibres and fibre development mechanisms. A theoretical model was developed to predict the force exerted on a single fibre within a floc caught between refiner bars. Two un...

We consider LU factoring of matrices in the context of exact and symbolic computation, as opposed to floating-point computation. Although initially developed for Gaussian elimination, fraction-free methods have been extended to LU factoring and related forms. We present surprising evidence that the rows and columns of the three matrices in the frac...

We introduce a Matlab mprec arbitrary precision library with applications to numerical analysis. For maximum efficiency arithmetic operators and algebraic functions are implemented in the mpreal class. The examples are chosen to reflect the diversity of types of problems for which multiple precision can play a useful role.

We describe the development of a term-rewriting system for indefinite integration; it is also called a rule-based evaluation system. The development is separated into modules, and we describe the module for a wide class of integrands containing the tangent function.

We study some series expansions for the Lambert $W$ function. We show that known asymptotic series converge in both real and complex domains. We establish the precise domains of convergence and other properties of the series, including asymptotic expressions for the expansion coefficients. We introduce an new invariant transformation of the series....

This article describes an efficient and robust algorithm and implementation for the evaluation of the Wright ω function in IEEE double precision arithmetic over the complex plane.

The Lambert W function has a number of integral expressions, including integrals of Bernstein, Thorin, Poisson, Stieltjes, Pick and Burniston–Siewert types. We give explicit integral expressions for W for each of these types. We also give integrals for a number of functions containing W.

We consider dominant three-, four- and five-loop contributions to λ, the quartic scalar coupling-constant's β-function in the Standard Model. We find that these terms accelerate the evolution of λ to nonperturbative values, thereby lowering the unification bound for which scalar-couplings are still perturbative. We also find that these higher order...

We show that many functions containing $W$ are Stieltjes functions. Explicit Stieltjes integrals are given for functions $1/W(z)$, $W(z)/z$, and others. We also prove a generalization of a conjecture of Jackson, Procacci & Sokal. Integral representations of $W$ and related functions are also given which are associated with the properties of their b...

Based on the homotopy analysis method (HAM), a general analytical approach for obtaining approximate series solutions to nonlinear two-point boundary value problems in finite domains is proposed. To demonstrate its effectiveness, this approach is applied to solve three nonlinear problems, and the analytical solutions obtained are more accurate than...

We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci and Sokal. In addition, we consider the relationship of functions of W with the class of completely monotonic functions and show...

The applications of the Lambert W function (also known as the W function) to D-dimensional Bose gases are presented. We introduce two sets of families of logarithmic transcendental equations that occur frequently in thermodynamics and statistical mechanics and present their solution in terms of the W function. The low temperature T behavior of free...

We consider a sequence of polynomials appearing in expressions for the derivatives of the Lambert W function. The coefficients of each polynomial are shown to form a positive sequence that is log-concave and unimodal. This property implies that the positive real branch of the Lambert W function is a Bernstein function. Comment: 6 pages

We consider a new invariant transformation of some previously known series for the Lambert W function. The transformations contain a parameter p which can be varied, while retaining the basic series structure. The parameter can be used to expand the domain of convergence of the series. The speed of convergence, that is the accuracy for a given numb...

The definition of the LU factoring of a matrix usually requires that the matrix be invertible. Current software systems have extended the definition to non-square and rank-deficient matrices, but each has chosen a different extension. Two new extensions, both of which could serve as useful standards, are proposed here: the first combines LU factori...

This paper describes continuing progress on the development of a repository of transformation rules relevant to indefinite integration. The methodology, however, is not restricted to integration. Several optimization goals are being pursued, including achieving the best form for the output, reducing the size of the repository while retaining its sc...

Based on a sequence of numerical computations, a conjecture is presented regarding the class of functions $H(x;a)=\exp(a)-(1+a/x)^x$, and the open problem of determining the values of $a$ for which the functions are completely monotonic with respect to $x$. The critical value of $a$ is determined here to sufficient accuracy to show that it is not a...

In this paper, the homotopy analysis method (HAM) is applied to solve a parameterized sixth order boundary value problem which, for large parameter values, cannot be solved by other analytical methods for finding approximate series solutions. Convergent series solutions are obtained, no matter how large the value of the parameter is.

In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by Ganji et al. [D.D. Ganji, H. Tari, M.B. Jooybari, Variational iteration method and homotopy perturbation method for no...

Based on the homotopy analysis method (HAM), an e-cient approach is proposed for obtaining approximate series solutions to fourth order two-point boundary value problems. We apply the approach to a linear problem which involves a parameter c and cannot be solved by other analytical methods for large values of c, and obtain convergent series solutio...

We present some applications of the Lambert W function (W function) to the formalism of quantum statistics (QS). We consider the problem of finding extrema in terms of energy for a general QS distribution, which involves the solution of a transcendental equation in terms of the W function. We then present some applications of this formula including...

A new algorithm for the automatic computation of the complete root classiflcation of a real parametric polynomial is described. The improvement lies in the fact that previous algorithms required 'revised sign lists' derived from the discriminant sequence. This is replaced here by the direct use of 'sign lists' derived from the discriminant sequence...

A new Maple package for solving parametric systems of polynomial equations and inequal-ities is described. The main idea for solving such a system is as follows. The parameter space R d is divided into two parts: the discriminant variety W and its complement R d \W . The dis-criminant variety is a generalization of the well-known discriminant of a...

A new procedure for finding exact travelling wave solutions to the modified Camassa–Holm and Degasperis–Procesi equations is proposed. It turns out that many new solutions are obtained. Furthermore, these solutions are in general forms, and many known solutions to these two equations are only special cases of them.

Taking the specific problem domain of indefinite integration, we describe the on-going development of a repository of mathematical knowledge based on transformation rules. It is important that the repository be not confused with a look-up table. The database of transformation rules is at present encoded in Mathematica, but this is only one convenie...

A multiple precision library for floating-point calculations to any number of digits has been implemented in Matlab. The library is based on the ARPREC library. One application is discussed in detail, namely the evalu-ation in the complex plane of special functions in regions of bad conditioning. Through the use of Matlab classes, all the basic ari...

This paper considers the evaluation of indefinite integrals by transformation rules. The transformation rules are contained in a repository of mathematical information, and encapsulate not only knowledge of standard transformations, but also – and more importantly – conditions under which a transformation should be applied. It is precisely the cond...

It is shown that the Lambert W function cannot be expressed in terms of the elementary, Liouvillian, functions. The proof is based on a theorem due to Rosenlicht. A related function, the Wright ω function, is similarly shown to be not Liouvillian.

We consider a monic polynomial of even degree with symbolic coefficients. We give a method for obtaining an expression in the coefficients (regarded as parameters) that is a lower bound on the value of the polynomial, or in other words a lower bound on the minimum of the polynomial. The main advantage of accepting a bound on the minimum, in contras...

Given a real parametric polynomial p(x) and an interval (a;b) ‰R, the Complete Root Classiflcation (CRC) of p(x) on (a;b) is a collection of all possible cases of its root classifl- cation on (a;b), together with the conditions its coe-cients must satisfy for each case. In this paper, a new algorithm is proposed for the automatic computation of the...

Various extensions of the tanh-function method and their implementations for finding explicit travelling wave solutions to nonlinear partial differential equations (PDEs) have been reported in the literature. However, some solutions are often missed by these packages. In this paper, a new algorithm and its implementation called TWS for solving sing...

Gaussian elimination and LU factoring have been greatly studied from the algorithmic point of view, but much less from the point view of the best output format. In this paper, we give new output formats for fraction free LU factoring and for QR factoring. The formats and the algorithms used to obtain them are valid for any matrix system in which th...

We study the solutions of the matrix equation $S\exp(S) = A$. Our motivation comes from the study of systems of delay differential equations $y'(t) = A y(t-1)$, which occur in some models of practical interest, especially in mathematical biology. This paper concentrates on the distinction between \emph{evaluating a matrix function} and \emph{solvin...

A study is made of the behaviour of cohesive sediment in turbulent flowfields, such as are found in strongly tidal river estuaries. A model is developed which incorporates the fact that cohesive sediments, usually clays, consist of particles which can flocculate because of the electrical charges on them. During the cycle of erosion and deposition t...

A combination of symbolic and numerical methods is used to extend the reach of the purely symbolic methods of physics. One particular physics problem is solved in detail, namely, a computation of the electric potential in the space between a sphere and a containing cylinder. The potential is represented as an infinite sum of multipoles, whose coeff...

The symbolic-numeric computing described here consists of an extensive symbolic pre-processing of systems of differential-algebraic equations (DAE), followed by the numerical integration of the system obtained. The application area is multibody dynamics. We deal symbolically with a DAE system using differentiation and elimination methods to find al...

A vector-product space is a component-free representation of the common three-dimensional Cartesian vector space. The components of the vectors are invisible and formally inaccessible, although expres- sions for the components could be constructed. Expressions that have been built from the scalar and vector products can be simplifled us- ing a rule...

A new Maple package RootFinding[Parametric]for solving para-metric systems of polynomial equations and inequalities is described. The main idea for solving such a system is as follows. The parameter space R d is divided into two parts: the discriminant variety W and its complement R d \W . The discriminant variety is a generalization of the well-kn...

We describe a method for managing large expressions in sym- bolic computations which combines a hierarchical representation with signature calculations. As a case study, the problem of factoring matri- ces with non-polynomial entries is studied. Gaussian Elimination is used. Results on the complexity of the approach together with benchmark cal- cul...

The Complete Root Classification for a univariate polynomial with symbolic coefficients is the collection of all the possible cases of its root classification, together with the conditions its coefficients should satisfy for each case. Here an algorithm is given for the automatic computation of the complete root classification of a polynomial with...

We consider algebraic numbers defined by univariate polynomials over the rationals. In the syntax of Maple, such numbers are expressed using the RootOf function. This paper defines a canonical form for RootOf with respect to affine transformations. The affine shifts of monic irreducible polynomials form a group, and the orbits of the polynomials ca...