Dual Numbers and Operational Umbral Methods
Nicolas Behr, Giuseppe Dattoli, Ambra Lattanzi, Silvia Licciardi
AArticle
Dual Numbers and Operational Umbral Methods
Nicolas Behr *, Giuseppe Dattoli , Ambra Lattanzi and Silvia Licciardi Université de Paris, Institut de Recherche en Informatique Fondamentale (IRIF), F-75013 Paris, France;[email protected] ENEA—Frascati Research Center, Via Enrico Fermi 45, 00044 Rome, Italy; [email protected] (G.D.),[email protected] (S.L.) H. Niewodnicza ´nski Institute of Nuclear Physics, Polish Academy of Science, Kraków, Poland;[email protected] * Correspondence: [email protected]
Abstract:
Dual numbers and their higher order version are important tools for numericalcomputations, and in particular for finite difference calculus. Based upon the relevant algebraic rulesand matrix realizations of dual numbers, we will present a novel point of view, embedding dualnumbers within a formalism reminiscent of operational umbral calculus.
Keywords: dual numbers, operational methods, umbral image techniques
1. Introduction
The dual numbers (DNs), introduced during the second half of the XIXth century [1–5], can beviewed as abstract entities much like the ordinary complex numbers, and are defined as z = x + (cid:101) y , ( x , y ) ∈ R (1)where the corresponding “imaginary” unit or dual number unit (DNU) (cid:101) is a nilpotent number, (cid:101) = (cid:101) (cid:54) =
0. (2)The dual numbers were originally introduced within the context of geometrical studies, and laterexploited to deal with problems in pure and applied mechanics [6,7]. For instance, it has beendemonstrated in [8–10] how to formulate the equations of rigid body motion in terms of just three“dual” equations instead of their six “real” counterparts (thereby realizing an equivalence betweenspherical and spatial kinematics). More recently, as further discussed in the present paper, theirimportance has been recognized in numerical analysis to reduce round-off errors [11]. We believe thatthe use of dual numbers in the applied sciences is not as widespread as it could be, and that manynew fields of research would benefit from their relevant introduction. An important domain in whichthey may bring significant novelties is that of the perturbative techniques in classical and quantummechanics.The main contribution of this paper consists in fixing the underlying algebraic rules of thedual numbers in the wider context of umbral and operational calculus. The paper is organized asfollows: Section 2 delivers a basic mathematical introduction to dual numbers. Section 3 is devotedto the description of the computational procedure based upon dual numbers and umbral calculus.In Section 4, we will provide insight into how this powerful method can be applied to deal withproblems arising in different contexts. For illustration, we will consider the Schrödinger and the heatequation, cornerstones in the respective fields of Physics. Section 5 provides a conclusion with furtherconsiderations for future works. a r X i v : . [ m a t h . G M ] M a y of 10
2. Higher order dual numbers
The DN algebraic rules [12,13], summarized below, are a straightforward consequence of theprevious identity (2) (with z = x + (cid:101) y and w = u + (cid:101) v ): Component-wise algebraic addition z + w = x + u + (cid:101) ( y + ν ) Product z · w = xu + (cid:101) ( x ν + yu ) Inverse z − = x (cid:16) − (cid:101) yx (cid:17) ( x (cid:54) = ) Power z n = x n (cid:16) + n (cid:101) yx (cid:17) ( n ∈ Z ≥ , x (cid:54) = ) (3)While the addition operation is entirely analogous to the component-wise addition operation ontwo-dimensional vectors, the last three operations (product, inverse and power) characterize thedistinguishing special algebraic properties of dual numbers (DNs). The multiplication is commutative,associative and distributive, thus the DNs form a two-dimensional associative and commutativealgebra over the real numbers.We will now extend this traditional dual number formalism as motivated by the following typeof problem. Consider the Taylor expansion up to some order k (denoted ≈ k ) of an at least k -foldcontinuously differentiable function f around a point x , f ( x + y ) ≈ k k ∑ m = y m m ! f ( m ) ( x ) . (4)Following the automatic differentiation paradigm [14–16], since in practice the function f will beimplemented in some algorithmic from, it may be advantageous to formulate truncations such as (4)in terms of generalized (or higher order) dual numbers . To this end, let us introduce the families of squarematrices ˆ (cid:101) k ± , ˆ1 k and ˆ0 k with entries (for i , j =
1, . . . , k ) (cid:0) ˆ (cid:101) k ± (cid:1) i , j : = δ j , i ± , (cid:0) ˆ1 k (cid:1) i , j : = δ i , j , (cid:0) ˆ0 k (cid:1) i , j : = δ i , j denotes the Kronecker symbol. It is straightforward to verify that for all k ≥ (cid:96) ≥ (cid:16) ˆ (cid:101) (cid:96) k ± (cid:17) i , j = δ j , i ± (cid:96) ⇒ (cid:0) ˆ (cid:101) k ± (cid:1) k = f ( x ) suitably with a component-wise actionon square matrices, we find (for k ≥ f (cid:0) x ˆ1 k + y ˆ (cid:101) k ± (cid:1) = k − ∑ m = m ! y m f ( m ) ( x ) ˆ (cid:101) mk ± . (7)For example, setting k =
2, reproduces the well-known dual number identity [16] f (cid:0) x ˆ1 + y ˆ (cid:101) + (cid:1) = f ( x ) ˆ1 + y f (cid:48) ( x ) ˆ (cid:101) + = (cid:32) f ( x ) y f (cid:48) ( x ) f ( x ) (cid:33) . (8) of 10 It may be verified that e.g. for the choice “+” in (5), the first row of the resulting matrices in (7) containsthe terms of the Taylor expansion up to order k −
1. More explicitly, introducing the auxiliary notationsfor the row vector (cid:104) k e | and the column vector | k (cid:105) of length k ≥ (cid:104) k e | : = (
1, 0, . . . , 0 ) , | k (cid:105) : = , (9)let us define the order k evaluation operation acting on some function F ( ˆ (cid:101) k + ) depending on a generalizeddual number as (cid:104) F ( ˆ (cid:101) k + ) (cid:105) k : = (cid:104) k e | F ( ˆ (cid:101) k + ) | k (cid:105) . (10)We thus find that (cid:10) f (cid:0) x ˆ1 k + y ˆ (cid:101) k + (cid:1)(cid:11) k = k − ∑ m = y m m ! f ( m ) ( x ) . (11)Recently, expansions such as (7) have received considerable interest in the field of numericalanalysis [17]. Referring to [16] for an overview, various alternative types of "numbers" have beenstudied for the purpose of finding optimized numerical schemes for computing k -th order derivativesof functions. For example, it has been demonstrated that the use of so-called hyper-dual numbers resultsin first and second derivative calculations that are exact, regardless of the step size [11].For later convenience, motivated by the identity (for k ≥ (cid:0) ˆ (cid:101) k + x (cid:1) = k − ∑ r = x r r ! ˆ (cid:101) rk + , (12)we may introduce the so-called truncated exponential polynomials [18] e n ( x ) defined through the series e n ( x ) : = n ∑ r = x r r ! , (13)which may be expressed in terms of generalized dual numbers as e n ( x ) e n − ( x ) ... e ( x ) : = exp (cid:0) ˆ (cid:101) n + + x (cid:1) | n + (cid:105) . (14)One may thus easily verify the property e (cid:48) n ( x ) = e n − ( x ) . (15)Having provided a matrix-based extension of ordinary to k -th order dual numbers of arbitraryorder k ≥
2, we will now proceed to develop a computational procedure embedding dual numberswith other techniques inspired by the operational umbral formalism.
3. Umbral-type methods and Dual Numbers
Starting from this section, we will employ the notational simplification of writing (cid:101) for the dualnumber unit (DNU) ˆ (cid:101) k ± of generalized dual numbers (cf. Eq. (5)), making the order k ≥ of 10 explicit only via the analogue of the notation (10), and masking the matrix nature of ˆ (cid:101) k ± . Thus forsome function F ≡ F ( (cid:101) ) , we write F (cid:32) k G : ⇔ G = (cid:16) F (cid:12)(cid:12) (cid:101) k + → (cid:17) (cid:12)(cid:12) (cid:101) → (16)for the truncation of F via setting (cid:101) k + = (cid:101) =
1. It is straightforward to verify that thisformal definition may be implemented in terms of the matrix representations introduced in Section 2via use of (10) as G = (cid:104) F ( ˆ (cid:101) k + + ) (cid:105) k + . (17)Consider then the dual complex parameter ˆ z ≡ ˆ z ( a , b ) : = a + (cid:101) b . (18)Following the principles of umbral calculus [19,20], we will treat the dual complex parameter ˆ z as anordinary algebraic quantity in calculations of integrals, derivatives and other operations, delaying theevaluation of ˆ z via performing the operation (cid:32) k to the very end of the computations. We will nowillustrate the computational benefits of this approach via a number of examples. We first consider a Gaussian-type function explicitly containing in its argument the dual complexparameter (18), whence the dual-shifted Gaussian function f ( x ) = e − α x + ˆ z ( a , b ) x . (19)Assuming for instance third order dual numbers (i.e. (cid:101) = f ( x ) (cid:32) e − α x + ax (cid:104) + bx + ( bx ) (cid:105) , (20)which is easily recognized as the product of a shifted Gaussian with a second degree polynomial.In full analogy to the umbral operational methods of [20], it is then straightforward to calculatethe following integral of the function f of (19) via the standard Gaussian integral formula (cid:90) + ∞ − ∞ f ( x ) dx = (cid:113) πα e ˆ z ( a , b ) α = (cid:113) πα e a α + ab α (cid:101) + b α (cid:101) . (21)The term on the right has in fact a definite meaning, since the use of the generating function of the twovariable Hermite polynomials [21] ∞ ∑ n = t n n ! H n ( x , y ) = e xt + yt (22a) H n ( x , y ) = e y ∂ x x n = n ! (cid:98) n (cid:99) ∑ r = x n − r y r ( n − r ) ! r ! (22b) Albeit the term umbral calculus has been introduced in the seminal papers by Roman and Rota [19], in the following wewill make reference to the formalism developed in [20] which enriches the original formalism with the wealth of techniquesderived from the operational calculus. of 10 permits to cast the r.h.s. of (21) into the form (cid:113) πα e a α + ab α (cid:101) + b α (cid:101) = (cid:113) πα e a α ∑ m ≥ (cid:101) m m ! H m (cid:16) ab α , b α (cid:17) (cid:32) k (cid:113) πα e a α H e k (cid:16) ab α , b α (cid:17) . (23)Here, H e k ( x , y ) denotes the Hermite-based truncated exponential polynomial [22–24] defined as H e k ( x , y ) : = k ∑ r = r ! H r ( x , y ) . (24) Let us consider as a further example g ( x ) : = e − ˆ z ( a , b ) x (25)and the following infinite integral (for Re ( a ) > (cid:90) + ∞ − ∞ g ( x ) dx = (cid:114) π ˆ z = (cid:114) π a + (cid:101) b (cid:32) k (cid:114) π a k ∑ r = (cid:18) − r (cid:19) (cid:18) ba (cid:19) r . (26)Here, by invoking the operation (cid:32) k , we obtain a finite series, thus obviating the need to impose anycondition on the relevant convergence range. The calculus of higher order dual numbers may be further refined via combining it with the wealthof techniques available from the theory of special functions and symbolic calculus as put forwardin [20,25–29]. Consider for illustration the following identity, known from the theory of two-variableHermite polynomials [30], ∂ nx e α x = H n ( α x , α ) e α x (27)which allows to simplify the task of calculating successive derivatives of the dual Gaussian introducedin (25), such as in the computation ∂ nx e − ˆ zx (27) = H n ( − zx , − ˆ z ) e − ˆ zx (22b) = n ! (cid:98) n (cid:99) ∑ r = ∑ s ≥ ( − ) n − r + s n − r x n − ( r − s ) ( n − r ) ! r ! s ! ˆ z n − r + s .Another interesting type of calculus concerns infinite integrals involving rational functions suchas Φ ( x ; a , b ) : = + ˆ zx (cid:32) k + ax k ∑ r = (cid:16) − bx + ax (cid:17) r . (28)For example, the infinite integral (cid:90) + ∞ − ∞ + ˆ zx dx = π √ ˆ z (29)may be easily transformed into truncated form in full analogy to the calculation summarized in (26). Referring to [31] for the precise technical details (compare also [30]), suffice it here to providethe following definition for the action of the formal integration operator ˆ I on the formal variable v (for α ∈ C ): ˆ I ( v α ) : = Γ ( α ) . (30) of 10 Then an interesting variant of the example presented in (28) may be obtained asˆ I (cid:20) (cid:90) + ∞ − ∞ v Φ ( x ; a , vb ) dx (cid:21) = ˆ I (cid:34) v π (cid:112) z ( a , v β ) (cid:35) (cid:32) k (cid:114) π a k ∑ r = Γ ( − r )( r ! ) (cid:18) ba (cid:19) r . (31)In summary, the combination of the concept of higher order dual numbers with techniques fromsymbolic and umbral-image type calculus appears to offer a large potential in view of novel toolsof computation. To corroborate this claim, we will now present some first high-level results in thisdirection.
4. Dual numbers and solution of heat- and Schrödinger-type equations
Before entering the main topic of this section, let us recall a few useful “operational rules”, startingwith the
Glaisher identity [32,33] e τ d dx e − α x = √ + τα e − α x + τα , (32)which can also be understood as the solution of the heat equation with a Gaussian as initial function.It will prove particularly useful in the following to note that according to the definition of the Hermitepolynomials H n ( x , y ) as given in (22b), an alternative interpretation of (32) is provided in terms ofthe double lacunary exponential generating function H ( λ ; x , y ) of the polynomials H n ( x , y ) , where weemploy notations as in [34] e τ d dx e − α x = ∑ n ≥ ( − α ) n n ! H n ( x , τ ) = H ( − α ; x , τ ) . (33)By specializing eq. (32) to α = ˆ z (with ˆ z = a + (cid:101) b the dual complex parameter (18)), we obtain theoperational identity e τ d dx e − ˆ zx = √ + τ ˆ z e − ˆ zx + τ ˆ z . (34)Via the simple factorizations 1 + z τ = γ ( a , τ ) γ (cid:16) b (cid:101)γ ( a , τ ) , τ (cid:17) ,ˆ z + z τ = a γ ( a , τ ) + b (cid:101) [ γ ( a , τ )] γ (cid:16) b (cid:101)γ ( a , τ ) , τ (cid:17) , γ ( c , τ ) = + c τ , (35)we may transform the identity (34) as e τ d dx e − ˆ zx = H (cid:16) − b (cid:101) [ γ ( a , τ )] ; x , τγ ( a , τ ) (cid:17) H ( − a ; x , τ ) . (36)By re-inserting the definition of the first double-lacunary EGF, using the Glaisher-identity (32) for thesecond one and finally truncating to order k , we eventually arrive at the compact result e τ d dx e − ˆ zx (cid:32) k e − α x γ ( a , τ ) (cid:112) γ ( a , τ ) k ∑ n = n ! (cid:16) b [ γ ( a , τ )] (cid:17) n H n ( x , τγ ( a , τ )) , γ ( a , τ ) = + a τ . (37)For example, by evaluating the above expression for second order dual numbers, one finds e τ d dx e − ˆ zx (cid:32) e − ax γ ( a , τ ) (cid:112) γ ( a , τ ) (cid:18) − b γ ( a , τ ) H ( x , τγ ( a , τ )) + b γ ( a , τ ) H ( x , τγ ( a , τ )) (cid:19) . (38) of 10 The above result may be interpreted as the solution of the heat-type equation ∂ τ F ( x , τ ) = ∂ x F ( x , τ ) , F ( x , 0 ) = e − ˆ zx . (39)An analogous problem has been addressed in [30] within the framework of a different method.The techniques we have envisaged may be further exploited to treat the paraxial propagation of theso-called flattened distributions , introduced in [35] to study the laser field evolution in optical cavitiesemploying super-Gaussian mirrors [36]. These cavities shape beams whose transverse distribution isnot reproduced by a simple Gaussian, but by a function exhibiting a quasi-constant flat-top , expressiblethrough a function of the type E ( x ; p ) : = e −| x | p , p ∈ Z > . (40)The paraxial propagation of these beams has less obvious properties than, say, Laguerre or HermiteGauss modes [36]. In order to overcome this drawback, Gori introduced the so-called flattenedbeams [35] which permit a fairly natural expansion in terms of Gauss Laguerre/Hermite modes, thusproviding a straightforward solution to the corresponding paraxial wave equation.Invoking our formalism as developed so far, we may approximate the aforementioned Gori beamsin the form E ( x ; p ) ≈ Y ( x ; α | m ) : = e − α x e m ( x ) . (41)Here, e m ( x ) denotes the truncated exponential polynomials introduced in (13), and both parameters α and m depend on p (see [30] for further details). Recalling from (18) the definition ˆ z ( a , b ) : = a + b (cid:101) ofthe dual complex parameter, the r.h.s. of (41) may be equivalently expressed as e − ˆ z ( α , − ) x (cid:32) m Y ( x ; α | m ) , (42)whence as an instance of a dual Gaussian as described in Section 3.2. The problem of the relevantpropagation can accordingly be reduced to that of an ordinary Gaussian mode, namely to the solutionof the Schrödinger type equation i ∂ τ Ψ ( x , τ ) = − ∂ x Ψ ( x , τ ) , Ψ ( x , 0 ) = Y ( x ; α | m ) . (43)Consequently, by invoking the operational identity (34), the paraxial evolution of a flattened beammay be expressed in the form Ψ ( x , τ ) = e i τ∂ x e − ˆ z ( α , − ) x = (cid:112) + i τ ˆ z ( α , − ) e − ˆ z ( α , − ) x + i τ ˆ z ( α , − ) , (44)which reproduces indeed the known solution of our problem (compare [30]).In a forthcoming paper we will discuss this specific application in further detail by applying themethod to the problem of designing super-Gaussian optical systems.
5. Weyl formula and modified Hermite polynomials
The wide flexibility of the method we propose is corroborated by the following further example,relevant to the use of operational ordering tools. Let us consider an evolution equation of the form ∂ τ F ( x , τ ) = [ γ∂ x − ˆ zx ] F ( x , τ ) , F ( x , 0 ) = f ( x ) . (45) of 10 The relevant procedure for combining differential calculus with the umbral formalism is describedin [37]. Following this approach, the solution of (45) can be expressed as F ( x , τ ) = e τ ( γ∂ x − ˆ zx ) f ( x ) . (46)In order to evaluate the solution (46) explicitly, we need to suitably “factorize” the exponential operator.This so-called disentanglement operation may be implemented via the Weyl formula [38] e ˆ X + ˆ Y = e − [ ˆ X , ˆ Y ] e ˆ X e ˆ Y , (47)which is applicable whenever the identities [ ˆ X , [ ˆ X , ˆ Y ]] = [[ ˆ X , ˆ Y ] , ˆ Y ]] = F ( x , τ ) = e − τ γ ˆ z e − ˆ zx τ f ( x + γτ ) . (48)Thus the solution at any desired truncation order k may be obtained by invoking the dual numberevaluation operation (cid:32) k of (16).As already mentioned above, the Weyl formula applies in the example presented becausethe algebraic structure of the argument of the exponential in (46) satisfies a special property: thecommutators of the associated generators reduce to a constant after the first commutation bracket. Amore interesting extension is given by the case in which the generators are embedded into a solvableLie algebra. In this case, the combined use of the dual number formalism and of the Wei-Normanordering method [39] leads to new and interesting results. They deserve a separate treatment that willbe reported in a forthcoming paper.As a final example, we define modified Hermite polynomials H n ( x , ˆ z ) , whence ordinary two-variableHermite polynomials H n ( x , y ) as introduced in (22b) evaluated at y = ˆ z , with ˆ z ≡ ˆ z ( a , b ) the dualcomplex parameter of (18), H n ( x , ˆ z ) = e ˆ z ∂ x x n . (49)It is straightforward to verify that these modified polynomials inherit all the relevant properties fromthe polynomials H n ( x , y ) , such as the recurrences ∂ x H n ( x , ˆ z ) = nH n − ( x , ˆ z ) , H n + ( x , ˆ z ) = xH n ( x , ˆ z ) + z ∂ x H n ( x , ˆ z ) , (50)and we find that they satisfy the second order differential equation2 ˆ z ∂ x H n ( x , ˆ z ) + x ∂ x H n ( x , ˆ z ) = nH n ( x , ˆ z ) . (51)The explicit form of these truncated polynomials is easily obtained. For example, by using third orderdual numbers, which implies e ˆ z ∂ x (cid:32) e a ∂ x (cid:16) + b ∂ x + b ∂ x (cid:17) , (52)we find the explicit formula H n ( x , ˆ z ) (cid:32) H n ( x , a ) + b ∂ a H n ( x , a ) + b ∂ a H n ( x , a ) , (53)where we have invoked the well-known identity ∂ x H n ( x , y ) = ∂ y H n ( x , y ) . (54) of 10
6. Final Comments
The method we have outlined in this paper offers many computational advantages to treatproblems where truncated expansions (not necessarily of Taylor type) of functions are involved.At its core, the umbral formalism and the notion of higher order dual numbers allow to delay theexplicit expansions to later stages in a given calculation, thus opening the possibility to exploitnumerous efficient computation strategies from the theory of operational calculus and special functions.The technique we have introduced in this paper is amenable for new applications in variousdifferent fields. We have presented herein the solution of parabolic equations in transport problems,and within such a context a fairly important example has been provided by treating the propagationof flattened beams [30,35] in optics. For brevity, we have just outlined the procedure in terms ofa 1-dimensional computation. The relevant extension to the 3-dimensional case does not requireany particular conceptual effort, but only a consistent numerical implementation. In a forthcominginvestigation, we will further extend the method and study its potential for treating perturbativeproblems in classical and quantum mechanics.
Author Contributions: conceptualization, G.D.; methodology, N.B., G.D.; validation, N.B., G.D., S.L.; formalanalysis, N.B., G.D., A.L., S.L.; writing – original draft preparation, N.B., G.D., A.L.; writing – review and editing,N.B., A.L., S.L.
Funding:
The work of N.B. is supported by funding from the European Union’s Horizon 2020 research andinnovation programme under the Marie Skłodowska-Curie grant agreement No 753750. A.L. was supported bythe NCN research project OPUS 12 no. UMO-2016/23/B/ST3/01714 and by the NAWA project: Program im.Iwanowskiej PPN/IWA/2018/1/00098. S.L. was supported by a
Enea-Research Center Individual Fellowship . Acknowledgments:
N.B. would like to thank the LPTMC (Paris 06) and ENEA Frascati for warm hospitality.
Conflicts of Interest:
The authors declare no conflicts of interest. The founding sponsors had no role in the designof the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in thedecision to publish the results.
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