Methods for the computation of slowly convergent series and finite sums based on Gauss-Christoffel quadratures

Transcription

1 ISSN: c 24 Universidad de Jaén Web site: jja.ujaen.es Jaen J. Approx. 6() (24), Jaen Journal on Approximation Methods for the computation of slowly convergent series and finite sums based on Gauss-Christoffel quadratures Gradimir V. Milovanović Abstract In this paper we give an account on summation/integration methods for the computation of slowly convergent series and finite sums. The methods are based on Gauss-Christoffel quadrature rules with respect to some nonclassical weight functions over R or R +. For constructing such rules we use the recent progress in symbolic computation and variable-precision arithmetic, as well as our Mathematica package OrthogonalPolynomials [4, 26]. Together with our own results, we are also taking into consideration the use of other known results, especially the classical summation formulae of Euler-Maclaurin and Abel-Plana, in order to apply them afterwards in the computational techniques that we have developed recently. We present the Laplace transform method [5] and the contour integration method [2, 23], and give several numerical examples in order to illustrate the efficiency of different summation/integration methods. Keywords: Summation, Gaussian quadratures, weight function, three-term recurrence relation, convergence, Laplace transform, contour integration. MSC: Primary 65B; Secondary 4A25, 4A55, 65B5, 65D3, 65D32, 3E2, 33C9. Communicated by F. Marcellán Received August 7, 23 Accepted January 7, 24 This work was supported in part by the Serbian Ministry of Education, Science and Technological Development. 37

2 38 Gradimir V. Milovanović. Introduction For slowly convergent series and finite sums, which appear in many problems in mathematics, physics and other sciences, there are several numerical methods based on linear and nonlinear transformations. In general, starting from the sequence of partial sums of the series, these transformations give other sequences with a faster convergence to the sum of the series. There is a rich literature on this subject (cf. references in Mastroianni and Milovanović [9]). In this paper we give an account on some summation processes for series (n = ) and finite sums, n (±) k f(k), (.) with a given function z f(z) with certain properties with respect to the variable z, based on ideas related to Gauss-Christoffel quadratures. In a general case, the function f can depend on other parameters, e.g., f(z;x,...), so that these summation processes can be applied also to some classes of functional series, not only to numerical series. The basic idea in our methods is to transform the sum (.) to an integral with respect to some measure dµ on R (or R + ), and then to approximate this integral by a finite quadrature sum, n (±) k f(k) = R g(t)dµ(t) N A ν g(x ν ), (.2) where the function g is connected with f in some way, and the weights A ν A ν (n) and abscissae x ν x (n) ν, ν =,...,N, are chosen in such a way as to approximate closely the sum (.) for a large class of functions with a relatively small number N n. In our approach we take the Gaussian quadrature sums as the sum on the right-hand side in (.2). Together with our own results, we also consider some other known results, in particular the classical summation formulae of Euler-Maclaurin and Abel-Plana, in order to combine them with the computational techniques that we have developed recently. For the construction of Gaussian quadratures with respect to nonstandard measures dµ on R or R + we use the recent progress in symbolic computation and variable-precision arithmetic and our Mathematica package OrthogonalPolynomials [4, 26]. The approach enables us to overcome the numerical instability in generating coefficients of the three-term recurrence relation for the corresponding orthogonal polynomials with respect to the measure dµ. The paper is organized as follows. Section 2 is devoted to the classical summation formulae of Euler-Maclaurin and Abel-Plana. Recurrence coefficients for some orthogonal polynomial systems on ν=

3 Methods for the computation of slowly convergent series and finite sums 39 R and R + are studied in Section 3. For some of them, these coefficients can be expressed in an explicit form. In Sections 4 and 5 we present the Laplace transform method and the contour integration method, respectively. Finally, in order to illustrate the efficiency of different summation/integration methods we give a few numerical examples in Section Summation formulae of Euler-Maclaurin and Abel-Plana The first result on summation/integration methods was the well-known Euler-Maclaurin formula, discovered by Leonard Euler (77 783) in 732 in connection with the so-called Basel problem (or in modern terminology, for the Riemann zeta function, with determining ζ(2)), n f(k) = k=m n m f(x)dx+ 2 (f(m)+f(n))+ r j= B [ ] 2j f (2j ) (n) f (2j ) (m) +E r (f), (2.) (2j)! which holds for any m N, n,r N, m < n, and f C 2r [m,n], where B 2j are the Bernoulli numbers. This formula was also found independently by Colin Maclaurin ( ) in 738, who used (2.) for calculating integrals. Bernoulli numbers are defined by the generating function t e t = j= B j t j. j! For example, B =, B = /2, B 2 = /6, B 3 =, B 4 = /3, B 5 =, B 6 = /42, etc. The error term E r (f) in (2.) can be expressed in different forms (cf. [24]); e.g., if f C 2r+2 [m,n] then there exists η (m,n) such that E r (f) = (n m) B 2r+2 (2r+2)! f(2r+2) (η). (2.2) A history of this formula was given by Barnes [2], and some details can be found in [28,, 2, 3, 3, 24]. Alternatively, the Euler-Maclaurin summation formula (2.) is related to the so-called composite trapezoidal rule, n T m,n f := f(k) = n 2 f(m)+ f(k)+ 2 f(n). k=m k=m+

4 4 Gradimir V. Milovanović Namely, the error of this rule can be expressed in terms of the Euler-Maclaurin formula (2.), T m,n f n m f(x)dx = r j= B [ ] 2j f (2j ) (n) f (2j ) (m) +Er T (f). (2.3) (2j)! In other words, it means that the trapezoial sum T m,n f can be improved by values of derivatives at the end points of the interval of integration, through corrections terms in the form presented on the right hand side in (2.3). The Euler-Maclaurin summation formula is a standard formula and it is implemented in the package Mathematica as the function NSum with option Method -> Integrate. There is also another summation formula, the Abel-Plana formula, which is not so well known like the Euler-Maclaurin formula. In 82 Giovanni (Antonio Amedeo) Plana (78 864) obtained the summation formula k= f(k) f(x)dx = 2 f()+i f(iy) f( iy) e 2πy dy, (2.4) which holds for analytic functions, f in Ω = { z C : Rz }, that satisfy the conditions: lim y + e 2πy f(x±iy) =, uniformly in x on every finite interval, and such that f(x+iy) f(x iy) e 2πy dy exists for every x and tends to zero when x +. This formula was also proved by Niels Henrik Abel (82 829) in 823. In addition, Abel also proved an interesting alternating series version, under the same conditions, namely, k= For the finite sum S n,m f = ( ) k f(k) = 2 f()+i n ( ) k f(k), it becomes k=m S n,m f = 2[ ( ) m f(m)+( ) n f(n+) ] + f(iy) f( iy) dy. (2.5) 2sinhπy [ ψn+ (y) ψ m (y) ] w A (y)dy, (2.6)

5 Methods for the computation of slowly convergent series and finite sums 4 where the Abel weight on R, ω A (x), and the function ψ m (y) are given by w A x (x) = 2sinhπx and ψ m(y) = ( ) mf(m+iy) f(m iy). (2.7) 2iy Let m,n N, m < n, and C(ε) be a closed rectangular contour with vertices at m ±ib, n ± ib, b > (see Figure ), and with semicircular indentations of radius ε around m and n. Let f be an analytic function in the strip Ω m,n = { z C : m Rz n } and suppose that for every m x n, lim y + e 2πy f(x±iy) = uniformly in x, and that f(x+iy) f(x iy) e 2πy dy exists. The integral C(ε) f(z) e i2πz dz, with ε and b +, leads to the Abel-Plana formula in the form where T m,n f n m f(x)dx = φ m (y) = f(m+iy) f(m iy) 2iy ( φn (y) φ m (y) ) w P (y)dy, (2.8) and w P (y) = y e 2πy. (2.9) Practically, the Abel-Plana formula (2.8) gives the error of the composite trapezoidal formula (like the Euler-Maclaurin formula). In order to find the moments of the Plana weight function x w P (x) on R, we note first that if k is odd, the moments are zero, i.e., µ k (w P ) = x k w P (x)dx = x k x dx =. e 2πx For even k, we have R µ k (w P ) = 2 R x k+ e 2πx dx = 2 + (2π) k+2 t k+ e t dt,

6 42 Gradimir V. Milovanović Figure : Rectangular contour C(ε) which can be exactly expressed in terms of the Riemann zeta function ζ(s), µ k (w P ) = 2(k +)!ζ(k +2) (2π) k+2 = ( ) k/2b k+2 k +2, because the number k +2 is even. Thus, in terms of Bernoulli numbers, the moments are, k is odd, µ k (w P ) = ( ) k/2b k+2 k +2, k is even. (2.) Now, by the Taylor expansion for φ m (y) (and φ n (y)) on the right-hand side in (2.8), φ m (y) = f(m+iy) f(m iy) 2iy = j= ( ) j y 2j 2 f (2j ) (m), (2j )! and using the moments (2.), the Abel-Plana formula (2.8) reduces to the Euler-Maclaurin formula, T m,n f n m f(x)dx = j= ( ) j ( ) (2j )! µ 2j 2(w P ) f (2j ) (n) f (2j ) (m)

7 Methods for the computation of slowly convergent series and finite sums 43 = j= B ( ) 2j f (2j ) (n) f (2j ) (m), (2j)! because of µ 2j 2 (w P ) = ( ) j B 2j /(2j) (for more details see Dahlquist [5, 6, 7]). A similar summation formula is the so-called midpoint summation formula. It can be obtained by combining two Plana formulas for f(z /2) and f((z +m )/2). Namely, i.e., ( z +m ) ( T m,2n m+2 f T m,n+ f z ) = 2 2 n f(k) k=m n+/2 m /2 f(x)dx = where the midpoint weight function is given by w M (x) = w P (x) w P (2x) = n f(k), k=m [ φm /2 (y) φ n+/2 (y) ] w M (y)dy, x e 2πx + and φ m /2 and φ n+/2 are defined in (2.9), taking m := m /2 and m := n+/2, respectively. There are also several other summation formulas. For example, the Lindelöf formula for alternating series is ( ) k f(k) = ( ) m dy f(m /2+iy) 2coshπy, k=m where the Lindelöf weight function is given by w L (x) = 2coshπy = e πx +e πx. All these weights are even functions on R. As we mentioned in Section, a summation problem can be also transformed into the integration with respect to a weight function defined on the half line R +. In the sequel we present some of the most important weight functions on R and R +, including their moments. The recursion coefficients β k for the weights on R are also presented.

8 44 Gradimir V. Milovanović 3. Recurrence coefficients for some orthogonal polynomial systems on R and R + We denote the space of all algebraic polynomials defined on R by P, and by P N P the space of polynomials of degree at most N (N N). Suppose that for a given weight function w all moments µ k = R xk w(x)dx, k, exist, are finite and µ >. Then, for each N N, there exists the N-point Gauss-Christoffel quadrature rule N f(x)w(x)dx = A (N) ν f(x (N) ν )+R N (f), (3.) R ν= which is exact for all polynomials of degree 2N (f P 2N ). The Gauss-Christoffel quadrature formula (3.) can be characterized as an interpolatory formula for which its node polynomial π N (x) = N ( (N)) ν= x x ν is orthogonal to PN with respect to the inner product defined by (p,q) = p(x)q(x)w(x)dx (p, q P). R Because of the property (xp,q) = (p,xq), these (monic) orthogonal polynomials π k satisfy the fundamental three term recurrence relation π k+ (x) = (x α k )π k (x) β k π k (x), k =,,..., (3.2) withπ (x) = andπ (x) =, where{α k } = {α k (w)} and{β k } = {β k (w)} aresequencesofrecursion coefficients which depend on the weight w. The coefficient β may be arbitrary, but is conveniently defined by β = µ = R w(x)dx. For even weights on R, the coefficients α k are zero, so that the recurrence relation (3.2) becomes π k+ (x) = xπ k (x) β k π k (x), k =,,... (3.3) The quadrature nodes x (N) ν, ν =,...,N, are eigenvalues of the Jacobi matrix α β β α β2. J n (w) = β2 α.. 2, βn βn α N

9 Methods for the computation of slowly convergent series and finite sums 45 and the first components of the corresponding normalized eigenvectors v ν = [v ν,... v ν,n ] T (with vν T v ν = ) give the Christoffel numbers, A (N) ν = λ N,ν = β vν, 2, ν =,...,N. Unfortunately, the recursion coefficients are known explicitly only for some narrow classes of orthogonal polynomials, as e.g. for the classical orthogonal polynomials (Jacobi, the generalized Laguerre, and Hermite polynomials). However, for a large class of the so-called strongly non-classical polynomials these coefficients can be constructed numerically. Basic procedures for generating these coefficients are the method of (modified) moments, the discretized Stieltjes Gautschi procedure and the Lanczos algorithm, and they play a central role in the so-called constructive theory of orthogonal polynomials, which was developed by Walter Gautschi in the eighties on the last century. In [9] he starts with an arbitrary positive measure dµ(t), which is given explicitly, or implicitly via moment information, and considers the basic computational problem: for a given measure dµ and for a given n N, generate the first n coefficients α k (dµ) and β k (dµ) for k =,,...,n. The problem is very sensitive with respect to small perturbations in the data. The basic references are [9,, 9, 25]. Recent progress in symbolic computation and variable-precision arithmetic now makes possible to generate the recurrence coefficients α k and β k directly by using the original Chebyshev method of moments in sufficiently high precision. The corresponding software for such a purpose, as well as many other calculations with orthogonal polynomials and different quadrature rules, is now available: Gautschi s package SOPQ in Matlab, and our Mathematica package OrthogonalPolynomials (see [4] and [26]). These packages are downloadable from the web sites archives/22/wxg/codes/ and gvm/, respectively. Thus, all that is required is a procedure for the symbolic calculation of moments or their calculation in variableprecision arithmetic. 3.. The weight functions on R In this part we give an account of some of the most important (even) weight functions on R, including their moments and the coefficients β k in the three-term recurrence relation (3.3) for the corresponding orthogonal polynomials. For a given sequence of moments (mom), our Mathematica package OrthogonalPolynomials enables us to get recurrence coefficients {al,be} in a symbolic form, {al,be}=achebyshevalgorithm[mom, Algorithm -> Symbolic]; Graphics of these weight functions are displayed in figures 2 and 3.

10 46 Gradimir V. Milovanović Sonin-Markov weight: w SM (x) = x β exp( x 2 ), β > The corresponding orthogonal polynomials {π k } + k= are known as the Sonin-Markov or generalized Hermite polynomials. They can be expressed in terms of the generalized (monic) Laguerre polynomials, π 2k (x) = ˆL α k(x 2 ), π 2k+ (x) = xˆl α+ k (x 2 ) (α = (β )/2). Using [9, p. 2, theorems 2.2. and 2.2.2], we have the following equations for the coefficients β k in (3.3), and where α (ν) k polynomials ˆL α k β 2k +β 2k+ = α () k = 2k +α+, β 2k β 2k = β () k = k(k +α) β 2k+ +β 2k+2 = α (2) k = 2k +α+2, β 2k β 2k+ = β (2) k = k(k +α+), and β (ν) k, ν =,2, are the recurrence coefficients for the generalized monic Laguerre, respectively. From the previous equations we get and ˆL α+ k β k = k +β, k odd, 2 k 2, k even, and β = µ = R wsm (x)dx = Γ((β +)/2). In the special case β =, this weight function reduces to the well-known Hermite weight w H (x) = exp( x 2 ). This is the most popular (classical) weight function on the real line. The recurrence coefficients of the classical Hermite polynomials orthogonal with respect to this weight function on R are β = π, β k = k 2, k =,2,... Remark 3.. There are several generalizations of the Sonin-Markov weight function. For example, the Freud weight is defined by w F (x) = exp( x α ), α, or, in general, w(x) = exp( 2Q(x)), where Q is a function (e.g., a polynomial in a special case) with some properties.

11 Methods for the computation of slowly convergent series and finite sums wh Hermite. wa Abel wl Lindelof.2.5 wlog logistic Figure 2: Weight functions on R: Cases 4 Abel weight: w A (x) = x/(2sinh(πx)) The moments for this weight are, k odd, µ k = ( 2 k+2 ) ( ) k/2 B k+2, k even k +2

12 48 Gradimir V. Milovanović.5.4. wp Plana.3 wm midpoint w7.5 w Figure 3: Weight functions on R: Cases 5 8 and the coefficients in the three-term recurrence relation (3.3) for the corresponding orthogonal polynomials are known explicitly (see [9, p. 59]), β = µ = 4, β k = k(k +), k =,2,... 4

13 Methods for the computation of slowly convergent series and finite sums 49 Lindelöf weight: w L (x) = /(2cosh(πx)) The moments for this weight are {, k odd, µ k = 2(4π) k k! [ ζ ( ( k +, 4) ζ k +, 3 4)], k even, where ζ(s, a) is the generalized Riemann zeta function, defined by ζ(s,a) = ν= (ν +a) s. The recurrence coefficients are known explicitly (see [9, p. 59]), β = µ = 2, β k = k2 4, k =,2,... Logistic weight: w log (x) = e πx /(+e πx ) 2 = /(2cosh(πx/2)) 2 = [w L (x/2)] 2 The moments for this weight are µ k =, k odd, 2 π ( )k/2 ( 2 k ) B k, k even and the coefficients in the three-term recurrence relation (3.3) for the corresponding orthogonal polynomials are also known explicitly (see [9, p. 59]), β = µ = π, β k = Plana weight: w P (x) = x /(e 2πx ) k4 4k 2, k =,2,... The moments for this weight function are given in (2.), and our package OrthogonalPolynomials gives the sequence of recurrence coefficients {β k } k in the rational form: { 2,, 79 2, , , , , ,

14 5 Gradimir V. Milovanović , , , ,... }. Midpoint weight: w M (x) = x /(e 2πx +) The moments for this weight are given by, k is odd, µ k = ( ) k/2 ( 2 (k+) ) B k+2, k is even. k +2 The sequence {β k } k in rational form is { 24, 7 4, , , , , , , ,... The weight w 7 (x) = x2 e πx ( e πx ) 2 = ( x 2 sinh(πx/2) ) 2 = 4 [wa (x/2)] 2 In this case the moments are, k is odd, µ k = ( ) k/2 2 k+2b k+2, k is even. π For the corresponding sequence {β k } k we obtain { } 2 3π, 4 5, 72 35, 8 2, 2 33, 26 43, , , , 33 33, 484 6, , , , , ,.... }.

15 Methods for the computation of slowly convergent series and finite sums 5 After some experiments, we conjectured and proved that β = µ = 2 3π, β k = k(k +)2 (k +2) (2k +)(2k +3), k N. The weight w 8 (x) = x 2 eπx/2 +e πx/2 ( (e πx/2 e πx/2 ) = 2cosh(πx/2) x 2 2 sinh(πx/2) In this case the moments are, k is odd, µ k = 2 k+3 π (2k+2 ) B k+2, k is even. Also, in this case we proved that { β = µ = 4 (k +) 2 π, β, k is odd, k = k(k +2), k is even. ) Weight functions on R + In this part we mention four important weight functions defined on R +. Bose-Einstein and Fermi-Dirac weights: ε(t) = t e t and ϕ(t) = e t +, respectively These functions and the corresponding quadratures with respect to them are widely used in solid state physics. The moments can be exactly calculated in terms of Riemann zeta function, and µ k (ε) = µ k (ϕ) = t k+ e t dt = (k +)!ζ(k +2), k N, t k e t + dt = { log2, k =, ( 2 k )k!ζ(k +), k >.

16 52 Gradimir V. Milovanović Hyperbolic weights on R + Consider the hyperbolic weights w (t) = cosh 2 and w 2 (t) = sinht t cosh 2 t. (3.4) The moments for these weights can be expressed in the form:, k =, + µ () k = t k log2, k =, w (t)dt = 2 k 4 k k!ζ(k), k 2 and, k =, + ( π ) k Ek µ (2) k = t k w 2 (t)dt= k, k(odd), 2 2k [ ψ (k ) 4 k ( 4 ) ψ(k ) ( 3 4 )], k(even) 2, respectively, where E k are Euler s numbers and ψ(z) is the so-called digamma function, i.e., the logarithmic derivative of the gamma function, ψ(z) = Γ (z)/γ(z). In the cases studied in this subsection, the recurrence coefficients α k and β k in (3.2) are not rational numbers, so that the option Algorithm -> Symbolic in their construction (by the procedure achebyshevalgorithm in our Mathematica package OrthogonalPolynomials) cannot be applied. Therefore, in these cases, we use the variable-precision arithmetic to overcome the numerical instability in the numerical construction of α k and β k, by setting WorkingPrecisionto be sufficiently large. For example, if we want to construct the first 5 recursion coefficients, with WorkingPrecision->8, we only need to execute the following commands: << OrthogonalPolynomials mombe=table[(k+)!zeta[k+2],{k,,99}]; {al,be}=achebyshevalgorithm[mombe, WorkingPrecision->8]; Taking larger WorkingPrecision, for example, {al,be}=achebyshevalgorithm[mombe, WorkingPrecision->]; N[Max[Abs[al/al-],Abs[be/be-]],3] we get the maximal relative error in the previous recurrence coefficients {al,be} to be.6 43.

17 Methods for the computation of slowly convergent series and finite sums Laplace transform method Consider the two series T = a k and S = ( ) k a k. (4.) Suppose that the general term of the series in (4.) is expressible in terms of the Laplace transform, or its derivative, of a known function. Let Then for a k = f(k), we have i.e., T = f(s) = e kt g(t)dt = T = Similarly, for alternating series, we have Also, if a k = f (k), we can get and S = f (k) = ( ) k f (k) = f(k) = ( ) k f(k) = e st g(t)dt, Rs. ( + e )g(t)dt kt = g(t) t te kt g(t)dt = e t g(t)dt, e t t e t dt. (4.2) ( g(t)) e t dt. (4.3) + ( ) k te kt g(t)dt = t ( g(t)) e t dt (tg(t)) e t + dt. Thus, the summation of series is now transformed to an equivalent integration problem.

18 54 Gradimir V. Milovanović The first idea for the numerical integration of these integrals is the application of the Gauss- Laguerre quadrature, but its convergence can be very slow because of the presence of poles on the imaginary axis at the points ±2πi, ±4πi,... (in the case of integrals for T) and ±πi, ±3πi,... (in the case of integrals for S). Another approach for the integration over (, + ), h(t)w(t)dt = N A ν h(x ν )+R N (h), (4.4) ν= with respect to the weight functions w(t) = ε(t) (Bose-Einstein weight) and w(t) = ϕ(t) (Fermi-Dirac weight), was developed by Gautschi and Milovanović [5] (see also [] and [23]). 5. Contour integration method An alternativesummation/integrationprocedurefortheseries(4.)with a k = f(k), when the function f is analytic in the region { z C Rz α, m < α < m } (5.) was considered in [2] (see also [22] and [23]). This method requires the indefinite integral F of f chosen so as to satisfy certain decay properties. Here we consider a more general version of this method which can be applied also to finite sums. In fact, we consider the series T m,n = n f(k) and S m,n = k=m n ( ) k f(k), (5.2) k=m with an analytic (holomorphic) function f in { G = z C : α Rz β, Iz δ }, π where m,n Z, m < n +, and m < α < m, n < β < n+, δ >, Γ = G. By contour integration of a product of functions z f(z)g(z) over the rectangle Γ in the complex plane, where g(z) = π/tanπz and g(z) = π/sinπz, we are able to reduce the summation of the series T m,n and S m,n to a problem of numerical integration with Gaussian quadrature rules on (,+ ), respectively, with respect to the first and second hyperbolic weight functions of (3.4). The moments of these weights are given in subsection 3.2.

19 Methods for the computation of slowly convergent series and finite sums 55 For a holomorphic function f in G, by Cauchy s residue theorem, we obtain T m,n = π f(z) 2πi tanπz dz and S m,n = π f(z) 2πi sinπz dz. Γ Γ After integration by parts, these formulas reduce to T m,n = ( π ) 2F(z)dz and Sm,n = 2πi sinπz 2πi where F is an integral of f. Assume now the following conditions for the function F (cf. [8, p. 57]): (C) F is a holomorphic function in the region (5.), (C2) lim t + e c t F (x+it/π) =, uniformly for x α, (C3) lim + x + e c t F (x+it/π) dt =, where c = 2 or c =, when we consider T m,n or S n,m, respectively. Γ Γ ( π ) 2cosπzF(z)dz, (5.3) sinπz Setting α = m /2, β = n+/2 and letting δ +, we can prove that the integrals in (5.3) over Γ reduce to integrals along the lines z = α+iy and z = β+iy ( < y < + ). Namely, under condition (C2), the integrals on the lines z = x±i(δ/π), α x β, tend to zero when δ +. In the case n =, the condition (C3) implies that the integrals over the line z = β+iy, when δ + and β +, also tend to zero. Thus, T m,n = Φ m,n (t)w (t)dt and S m,n = Ψ m,n (t)w 2 (t)dt, (5.4) where w and w 2 are hyperbolic weight functions defined in (3.4) and the functions t Φ m,n (t) and t Ψ m,n (t) can be expressed in terms of the real and imaginary parts of the function F(z). Formulas (5.4) suggest to apply Gaussian quadrature to the integrals on the right, using the weight functions w and w 2, respectively, i.e., g(t)w s (t)dt = N A N ν,sg(τν,s)+r N N,s (g) (s =,2), (5.5) ν=

20 56 Gradimir V. Milovanović where A N ν,s and τn ν,s, ν =,...,N (s =,2), are parameters of these quadratures and the remainders R N,s (g) = (s =,2) for all g P 2N. The required recursion coefficients for the corresponding orthogonal polynomials(in the numerical construction of Gaussian quadratures(5.5)) can be computed using the moments which are given in subsection 3.2 (last case). In this way we obtain the following statements (cf. [2, 23]): Theorem 5.. Let F be an integral of f such that conditions (C) (C3) are satisfied with c = 2. If A N ν, and τn ν,, ν =,...,N, are the weights and nodes of the N-point Gaussian quadrature rule (5.5) with respect to the weight function w (t) = /cosh 2 t on R +, then T m,n = n f(k) = k=m N ν= ( τ N ) A N ν, ν,φ m,n +R N, (Φ m,n ), π where ( Φ m,n (t) = Φ m 2, t ) ( Φ n+ π 2, t ) π and Φ(x,y) is defined by Φ(x,y) = [F(x+iy)+F(x iy)] = RF(x+iy). 2 Theorem 5.2. Let F be an integral of f such that conditions (C) (C3) are satisfied with c =. If A N ν,2 and τn ν,2, ν =,...,N, are the weights and nodes of the N-point Gaussian quadrature rule (5.5) with respect to the weight function w 2 (t) = sinht/cosh 2 t on R +, then S m,n = n N ( τ N ) ( ) k f(k) = A N ν,2 ν,2ψ m,n +R N,2 (Ψ m,n ), π k=m ν= where ( Ψ m,n (t) = ( ) m Ψ m 2, t ) ( +( ) n Ψ n+ π 2, t ) π and Ψ(x,y) is defined by Ψ(x,y) = [F(x+iy) F(x iy)] = IF(x+iy). 2i

21 Methods for the computation of slowly convergent series and finite sums 57 At the end of this section we mention an integral representation of the well-known (generalized) Mathieu series S m (r) = 2n (n 2 +r 2 ) m+ and Sm (r) = ( ) n 2n (n 2 +r 2 ) m+, n n obtained by applying the previous transformation [27]: Theorem 5.3. For each m N and r > we have S m (r) = π m [m/2] j= ( ) j ( m 2j) (r 2 x ) m 2jx 2j [ (x2 ) r m w (πx)dx 4 +r 2] and S m (r) = π m [(m )/2] j= ( ) m (r ) m 2j x ( ) j 2 x 2 + 2j+ 2j + 4 [ (x2 ) r m w 2 (πx)dx, 4 +r 2] where w and w 2 are the hyperbolic weights given in (3.4). By means of these newly established integral forms of generalized Mathieu series, we have also obtained a new integralexpressionfor the Bessel function of the first kind of half integer order J m /2, solving a related Fredholm integral equation of the first kind with nondegenerate kernel (see [27]). The series S (r) was introduced and studied for the first time by Émile Leonard Mathieu (835 89) in his book devoted to the elasticity of solid bodies [2]. 6. Numerical examples Example 6.. Consider two simple series T = (k +) 2 = π2 + 6 and S = ( ) k (k +) 2 = π2 2.

22 58 Gradimir V. Milovanović Summation of the series T (a) Application of the Plana formula We first apply the formula (2.8), with n =, to the series T, using m as an integer parameter. In that case we have T = f(k) = = m f(k) 2 f(m)+t m, m f(k) 2 f(m)+ = U m (f) where φ m (y) and w P (x) are defined in (2.9), U m (f) = R m φ m (y)w P (x)dy, m f(k) 2 f(m)+ f(x)dx f(x)dx φ m (y)w P (x)dy R and f(x) = /(x+) 2. For the integrali(φ m ) = R φ m(y)w P (x)dy, we construct the Gaussian quadratureswith the Plana weight w P (x), Q N (φ m ) = N A (N) ν φ m (x (N) ν ), for N = 5(5)5 (that is to say for N from 5 to 5 with ν= step 5, we use this notation later with the same sense) and Precision->5, using our Mathematica package: << orthogonalpolynomials moments=table[if[oddq[k],,(-)^(k/2)bernoullib[k+2]/(k+2)], {k,,99}]; {al,be}=achebyshevalgorithm[moments, Algorithm -> Symbolic]; pq[n_]:=agaussiannodesweights[n,al,be,workingprecision->55, Precision->5]; nw=table[pq[n], {n,5,5,5}]; The corresponding nodes and weights are the elements of the list nw in the Mathematica code. For example, for N = 5 the nodes are nw[[3]][[]] and the weights are nw[[3]][[2]] (the lengths of these lists are 5). m

23 Methods for the computation of slowly convergent series and finite sums 59 N m = m = 2 m = 3 m = 4 m = 5 m = BEQ 5 2.( 6) 2.4( 8) 8.( ) 5.4( ) 5.7( 2).( 5) 3.( 4) 2.7( 8) 4.4( ) 2.5( 3 3.4( 5) 8.7( 7).5( 8).( 8) 5.7( 9) 7.( 3).( 5 4.5( 8) 3.6( 2) 3.4( ) 3.2( 3) 2 2.2( ) 3.3( 4).9( 7) 3.( 2) 9.5( 23).9( 2) 8.( 8) ( ) 2.9( 5) 7.5( 9) 5.3( 22) 7.9( 25) 3.( 4).8( 22) 3.2( ) 3.9( 6) 5.( 2).8( 23).4( 26) 2.( 5) 3.9( 27) ( 2) 6.9( 7) 5.( 2).( 24) 4.7( 28) 6.8( 7) 8.2( 32) 4.4( 2).5( 7) 6.9( 22) 8.4( 26) 2.3( 29) 2.6( 8).7( 36) ( 3) 4.( 8).( 22) 9.( 27).6( 3) 8.6( 9) 3.3( 4) 5 2.6( 3).2( 8) 2.3( 23).2( 27).4( 3).2( 9) 6.5( 46) Table : Applications of Plana formula and Laplace transform method to the series T. Relative errors in the Gaussian approximation (numbers in parentheses indicate decimal exponents). The relative errors in the Gaussian approximations T T (N,m) = U m (f) Q N (φ m ), r N,m = T (N,m) T T = I(φ m ) Q N (φ m ) T, are given in Table for N = 5(5)5 and m = ()5. The rapid speed of convergence of the Gaussian rule (with the Plana weight) as m increases is due to the poles ±i(m + ) (second order!) of the function 2(m+) y φ m (y) = [(m+) 2 +y 2 ] 2 moving away from the real line. This influence of the poles to the convergence can be also reduced if the integration is performed only over the semiaxis R +. For example, if we apply the original Plana formula (2.4), where the integration is over R + (practically, with respect to the Bose-Einstein function), i.e., T = f(k) = f(x)dx 2 f() 2π 2 ( t ) t φ 2π e t dt, the relative errors in the corresponding Gaussian approximation are presented in the penultimate column of the same table (with m = ).

24 6 Gradimir V. Milovanović (b) Application of the Laplace transform method In the last column of Table (BEQ) we give the relative errorsin N-point Gaussian approximation (w.r.t. the Bose-Einsten weight) in the case of the Laplace transform method. In that case, for f(s) = (s+) 2 we have g(t) = te t, so that, according to (4.2) and (4.4), T = (k +) 2 = e t t N e t dt A ν exp( x ν ), where x ν and A ν, ν =,...,N, are nodes and weights in (4.4) when w(t) = ε(t) = t/(e t ). It is interesting to note that the previous trick of moving m with the Laplace transform method does not lead to acceleration of convergence. For example, in this case we have that T = m + (k +) 2 + m (k +m) 2 = ν= (k +) 2 + e mt t e t dt. The convergence of the corresponding quadrature process (as m increases) slows down considerably (see [2]). The reason for this is the behavior of the function t e mt, which tends to a discontinuous function when m +. On the other hand, the function is entire, which explains the ultimately much better results for m =. (c) Application of the contour integration method We put T = m + (k +) 2 + k=m m (k +) 2 = (k +) 2 +T m,, where the first sum on the right-hand side for m = is empty. We apply now the Gaussian rule with respect to the hyperbolic weight w (t) = /cosh 2 t to the second sum, T m,, T m, = ( w (t)φ m 2, t ) dt Q N, (Φ m, ) = π N ν= ( A N ν,φ m 2, τn ν, π where f(z) = /(z +) 2, F(z) = /(z +) (the integration constant being zero on account of the condition (C3)), and Φ(x,y) = { } 2 [F(x+iy)+F(x iy)] = R x+ = z + (x+) 2 +y 2. Notice that Φ m, (t) = Φ(m /2,t/π) because lim Φ(n+/2,t/π) =. n + ),

25 Methods for the computation of slowly convergent series and finite sums 6 N m = m = 2 m = 3 m = 4 m = 5 5.3( 7) 5.4( 9).9( ) 8.6( 2) 3.7( 3).( ).( 3).7( 6) 7.9( 8) 2.( 9) 5.( 3) 3.8( 7) 3.7( 2).( 22) 3.8( 25) 2.4( 5) 4.( 2).2( 24).9( 27) 2.3( 29) ( 8).( 22) 2.( 27) 2.6( 3) 2.5( 33) 3 2.6( 9).4( 25).( 3) 2.( 33) 8.( 37) ( 2) 3.2( 27) 2.3( 32) 2.3( 36) 2.5( 4) 4.3( 22) 3.6( 3).5( 34) 3.8( 39).6( 43) ( 24) 4.( 3) 6.5( 38) 4.2( 42) 3.4( 46) 5.9( 25) 6.5( 33) 6.6( 39) 2.8( 44).2( 48) Table 2: Application of the contour integration method to the series T. Relative errors in the Gaussian approximation. The relative errors in the Gaussian approximations, T T (N,m) = m (k +) 2 +Q N,(Φ m, ), canbeexpressedintheformr N,m = (T (N,m) T)/T andtheyarepresentedin Table2forN = 5(5)5 and m = ()5. As we can see, the convergence is very fast! Summation of the series S According to (2.6) (for n = ), the series S can be expressed in the form S = = m ( ) k f(k) = ( ) k f(k)+ m ( ) k f(k)+ 2 ( )m f(m) R ( ) k f(k) ψ m (y)w A (y)dy, where the function y ψ m (y) and the Abel weight w A (y) are defined in (2.7). The recurrence coefficients for the orthogonal polynomials in this case are known explicitly and given in subsection

26 62 Gradimir V. Milovanović 3. (Abel weight), so that we can calculate the parameters of the corresponding Gaussian formula in the following way: << orthogonalpolynomials al = Table[, {k,,49}]; be = Join[{/4}, Table[k(k+)/4, {k,,49}]]; pq[n_]:=agaussiannodesweights[n,al,be,workingprecision->55, Precision->5]; nw=table[pq[n], {n,5,5,5}]; Relative errors in Gaussian approximations for N = 5(5)5 and m =, m = 2, and m = 3, are presented in Table 3. Applying the Gaussian quadrature rule with respect to the Fermi-Dirac weight function ϕ(t) = /(e t +) to the integral on the right side in (4.3), we obtain approximationswith relative errors given also in Table 3 (FDQ-column). N Gauss-Abel formula (2.6) Eq. (4.3) Theorem 5.2 m = m = 2 m = 3 FDQ m = m = 2 m = 3 m = 4 m = 5 5.9( 3) 5.7( 5) 3.4( 6).2( 3).( 4).9( 6) 2.2( 7).5( 8) 4.5( ) 2.3( 4) 2.3( 6) 4.7( 8) 6.5( 8) 3.5( 7).9( 9) 2.3( 2).( 2).( 4) 5 6.( 5) 2.9( 7) 3.( 9) 2.( 2) 5.( 9) 8.( 3).9( 5) 3.2( 6) 9.2( 8) 2 2.2( 5) 6.5( 8) 3.9( ) 5.5( 7) 8.9( ) 6.6( 4).( 6) 6.2( 9) 7.6( 2) 25.( 5).9( 8) 7.7( ).3( 2) 3.4( 2) 6.2( 6) 6.8( 9) 2.5( 2).3( 23) 3 5.3( 6) 7.2( 9) 2.( ) 3.( 26) 2.4( 3) 2.4( 8) 9.7( 2).2( 24).3( 26) 35 3.( 6) 3.( 9) 6.5( 2) 6.4( 3) 2.( 4).( 9) 4.3( 23).3( 25) 2.( 28) 4.9( 6).5( 9) 2.4( 2).3( 35).5( 5) 5.6( 2).3( 24).( 27) 9.8( 3) 45.2( 6) 7.6( ) 9.9( 3) 2.7( 4) 4.2( 7) 8.3( 23) 6.7( 26) 3.8( 3) 2.( 32) 5 8.2( 7) 4.2( ) 4.5( 3) 5.4( 45).( 7).7( 23) 2.6( 27) 4.( 3) 8.9( 35) Table 3: Application of summation/integration methods to the series S. Relative errors in Gaussian approximation. Finally, applying the contour integration method to the series S (see Theorem 5.2), for N = 5(5)5 nodes and m = ()5, we get the Gaussian approximations, S S (N,m) = m ( ) k (k +) 2 +Q N,2(Ψ m, ),

27 Methods for the computation of slowly convergent series and finite sums 63 where Q N,2 (Ψ m, ) = N ν= ( A N ν,2ψ m 2, τn ν,2 π The relative errors, r N,m = (S (N,m) S)/S, are presented in the last five columns in Table 3. As we can see the Laplace transform method is most efficient in this case. Example 6.2. We consider now the series T (a) = k(k +a) (a ). T () appears in a study of spirals and defines the well-known Theodorus constant (see Davis [8]). The series is slowly convergent. The first terms of T () give the result T () = (only 3-digit accuracy). In 99 Gauutschi [] calculated T (a) by using the method of the Laplace transform for different values of a ( 32). As a increases, the convergence of the Gauss quadrature formula slows down considerably. For example, when a = 8, the corresponding quadrature with N = 4 nodes gives a result with relative error In a special case for a =, Gautschi [4] has recently proved that T () = 2 π D( t) t w(t)dt, x where D is Dawson s integral D(x) = e x2 e t2 dt and w is the corresponding weight function given by w(t) = t /2 ε(t) = t/(e t ). In the construction of Gaussian quadratures with respect to this weight, the moments are µ k = ). t k+/2 (k e t dt = Γ + 3 ) ( ζ k + 3 ), k =,,..., 2 2 where the gamma function and the Riemann zeta function are computable by variable-precision calculation. Using the Chebyshev algorithm with sufficiently high precision, Gautschi [4] has obtained Gaussian quadratures and applied them to the summation of this series for N = 5()75.

28 64 Gradimir V. Milovanović N Q (N) () Table 4: Gaussian approximation of the sum T (). Now, we directly apply the method of contour integration to T m (a), where T (a) = m +T m (a), T m (a) = k(k +a) k=m k(k +a). Then we use the Gaussian quadrature formula with respect to the weight function w (t) = /cosh 2 t on R + to calculate T m (a). In order to construct Gaussian rules for N, we need the recursion coefficients α k and β k for k N = 99, i.e., the moments for k 2N = 99. Taking the WorkingPrecisionto be 6, we obtain the first hundred recursion coefficients α k and β k, with relative errors less than For this series we have f(z) = and F(z) = 2 z (arctan z(z +a) a a π ), 2 where the integration constant is taken so that F( ) =. Thus, we obtain m T (a) Q (N) m (a) = + k(k +a) N ν= ( A N ν,φ m 2, τn ν, π where Φ(x,y) is defined in Theorem 5.. The Gaussian approximation Q (N) m (a) for a =, m =, and N = 5()55 are presented in Table 4. Boxed digits are in error.as we can see the method is very efficient. ),

29 Methods for the computation of slowly convergent series and finite sums m m 2 m 3 m 4 m 5 m m 2 Figure 4: Relative errors in the Gaussian approximation T () Q (N) m () for N = 5(5) and some selected values of m. Relative errorsof Q (N) m () for N = 5(5) and different values of m ( 2) are presented in Figure 4 in a log-scale. Numerical results also show that the convergence is slightly faster if the parameter a is larger. For example, if a =, then taking m = 2 and N = 5(5)25, the corresponding relative errors in the Gaussian approximations are 2.32( 2),.6( 33), 6.( 43),.8( 5),.89( 59), respectively. References [] Apostol T. M. (999) An elementary view of Euler s summation formula, Amer. Math. Monthly 6, [2] Barnes E. W. (95) The Maclaurin sum-formula, Proc. Lond. Math. Soc. 2-3(), [3] Butzer P. L., P. J. S. G. Ferreira, G. Schmeisser and R. L. Stens (2) The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections

30 66 Gradimir V. Milovanović with the approximate sampling formula of signal analysis, Result. Math. 59(3-4), [4] Cvetković A. S. and G. V. Milovanović (24) The Mathematica Package OrthogonalPolynomials, Facta Univ. Ser. Math. Inform. 9, [5] Dahlquist G. (997) On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules, I, BIT 37(2), [6] Dahlquist G. (997) On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules, II, BIT 37, [7] Dahlquist G. (999) On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules, III, BIT 39, [8] Davis P. J. (993) Spirals. From Theodorus to Chaos. With contributions by Walter Gautschi and Arieh Iserles, A.K. Peters, Wellesley, MA. [9] Gautschi W. (982) On generating orthogonal polynomials, SIAM J. Sci. Stat. Comput. 3, [] Gautschi W. (99) A class of slowly convergent series and their summation by Gaussian quadrature, Math. Comp. 57(95) [] Gautschi W. (24) Orthogonal Polynomials. Computation and Approximation, Oxford University Press, Oxford. [2] Gautschi W. (27) Leonhard Eulers Umgang mit langsam konvergenten Reihen, Elem. Math. 62, [3] Gautschi W. (28) Leonhard Euler: his life, the man, and his works, SIAM Rev. 5, 3 33.

31 Methods for the computation of slowly convergent series and finite sums 67 [4] Gautschi W. (2) The spiral of Theodorus, numerical analysis, and special functions, J. Comput. Appl. Math. 235(4), [5] Gautschi W. and G. V. Milovanović (985) Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series, Math. Comp. 44, [6] Golub G. and J. H. Welsch (969) Calculation of Gauss quadrature rules, Math. Comp. 23, [7] Henrici P. (997) Applied and Computational Complex Analysis. Volume 2. Special functions Integral transforms Asymptotics Continued Fractions, Pure and Applied Mathematics, John Wiley & Sons Inc., New York - London - Sydney. [8] Lindelöf E. (95) Le Calcul des Résidus et ses Applications á la théorie des fonctions, Gauthier Villars, Paris. [9] Mastroianni G. and G. V. Milovanović (28) Interpolation Processes. Basic Theory and Applications, Springer Monographs in Mathematics, Springer Verlag, Berlin. [2] Mathieu É. L. (89) Traité de Physique Mathematique VI VII: Theory de l Elasticité des Corps Solides, Gauthier Villars, Paris. [2] Milovanović G. V. (994) Summation of series and Gaussian quadratures, in: Approximation and Computation: a Festschrift in honor of Walter Gautschi, R.V.M. Zahar (ed.), Birkhäuser, Basel Boston Berlin, pp , [22] Milovanović G. V. (995) Summation of series and Gaussian quadratures, II, Numer. Algorithms (-2), [23] Milovanović G. V. (2) Summation processes and Gaussian quadratures, Sarajevo J. Math. 7(2), 85 2.

32 68 Gradimir V. Milovanović [24] Milovanović G. V. (23) Families of Euler-Maclaurin formulae for composite Gauss-Legendre and Lobatto quadratures, Bull. Cl. Sci. Math. Nat. Sci. Math. 38 (23), [25] Milovanović G. V. (24) Orthogonal polynomials on the real line, in: Walter Gautschi: Selected Works and Commentaries, Volume 2, C. Brezinski and A. Sameh (eds.), Birkhäuser, Basel, ch.. [26] Milovanović G. V. and A. S. Cvetković (22) Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balk., New Ser. 26, [27] Milovanović G. V. and T. K. Pogány (23) New integral forms of generalized Mathieu series and related applications, Appl. Anal. Discrete Math. 7, [28] Mills S. (985) The independent derivations by Leonhard Euler and Colin Maclaurin of the Euler-Maclaurin summation formula, Arch. Hist. Exact Sci. 33, 3. Gradimir V. Milovanović, Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Beograd, Serbia. gvm@mi.sanu.ac.rs