Numerische Mathematik

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1 Numer. Math. (2000) 86: Digital Object Identifier (DOI) /s Numerische Mathematik Quadrature rules for rational functions Walter Gautschi 1, Laura Gori 2, M. Laura Lo Cascio 2 1 Department of Computer Sciences, Purdue University, West Lafayette, IN , USA 2 Dipartimento MeMoMat, Università di oma La Sapienza, Via A. Scarpa 16, oma, Italy eceived June 21, 1999 / evised version received September 14, 1999 / Published online June 21, 2000 c Springer-Verlag 2000 Summary. It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turán, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. Mathematics Subject Classification (1991): 65D32 0. Introduction The idea of constructing quadrature rules that are exact for rational functions with prescribed poles, rather than for polynomials, has received some attention in recent years; see, e.g., [9], [10], [11], [2], [4]. These rational quadrature rules have proven to be quite effective if the poles are chosen so as to simulate the poles present in the integrand; see [3] for an application to integrals occurring in solid state physics. The work so far has been exclusively centered around quadrature rules of Gaussian or Newton-Cotes type. Here we construct rational versions of other important quadrature rules, specifically the Gauss-Kronrod and the Gauss-Turán rule, and Cauchy principal value quadrature rules. It is found that the accuracy is enhanced similarly as has been observed for Gauss-type quadrature rules. 1. ational Gauss-Kronrod quadrature Let dλ(t) be a positive measure on the real line and {τν G } n the nodes of the n-point Gaussian quadrature rule for dλ. The Gauss-Kronrod rule is Correspondence to: W. Gautschi

2 618 W. Gautschi et al. a formula of the type (1.1) g(t)dλ(t) = n n+1 λ K ν g(τν G )+ λ K µ g(τµ K )+n K (g), which is exact for all polynomials of degree 3n +1. Here, in analogy to [2], [4], we are interested in making (1.1) exact on the space of dimension 3n +2, (1.2) S 3n+2 = Q m P 3n+1 m, 0 m 3n +2, where (1.3) Q m = span{g : g(t) =(1+ζ µ t) r,r=1, 2,...,r µ, M µ =1, 2,...,M; r µ = m}, and P k is the space of polynomials of degree k (with Q 0 =, P 1 = the empty sets). The ζ µ are real or complex numbers satisfying (1.4) ζ µ 0, 1+ζ µ t 0on supp(dλ), µ =1, 2,...,M. Exactly as in [2], one proves the following theorem. Theorem 1.1. Let (1.5) ω m (t) = M (1 + ζ µ t) rµ (a polynomial of exact degree m). Assume that the measure dλ/ω m admits a (2n +1)-point (ordinary) Gauss-Kronrod formula (1.6) p(t) dλ(t) ω m (t) = n n+1 wν K p(t G ν )+ wµ K p(t K µ ), all p P 3n+1, having nodes t G ν and t K µ t G ν contained in the support of dλ. Define (1.7) τν G = t G ν,τµ K = t K µ ; λ K ν = wν K ω m (t G ν ), λ K µ = wµ K ω m (t K µ ). Then the formula (1.1) is exact on the space S 3n+2 defined in (1.2) and (1.3). ecall that the t G ν in (1.6) are the zeros of ˆπ n ( )=π n ( ; dλ/ω m ),the polynomial of degree n orthogonal with respect to the measure dλ/ω m, and t K µ are the zeros of the polynomial πn+1 of degree n +1orthogonal to all lower-degree polynomials relative to the (oscillating) measure ˆπ n dλ/ω m.

3 Quadrature rules for rational functions 619 We present numerical results analogous to those in Examples 4.1 and 4.2 of [4]. To compute the Gauss-Kronrod formula (1.6), we first use a fortran implementation of an algorithm due to Laurie [8] to generate the required Jacobi-Kronrod matrix (cf. [5]). As input to this routine, one needs sufficiently many recursion coefficients for the orthogonal polynomials ˆπ k. These are obtained by a double-precision version of the routine abmod of [4]. Finally, a double-precision version of the procedure gqrat of [4] is used to compute the rational Gauss formulae when applicable (i.e., when m 2n; cf. [4]), and the desired Gauss-Kronrod formulae. Example 1.1. I 1 (ω) = 1 1 πt/ω sin(πt/ω) dt, ω > 1. Here, dλ(t) =dt on [ 1, 1], and the integrand has simple real poles at ω, ω, 2ω, 2ω,.... This suggests taking r µ =1,M = m in (1.3), and (1.8) ζ µ = ( 1)µ, µ =1, 2,...,m. ω µ+1 2 In view of symmetry, we choose m even, specifically m =2 3n+2 2, m = 2n, m =2 n+1 2, m =2, and, for comparison, m =0. esults obtained (in IEEE double-precision arithmetic) by means of the rational Gauss and the rational Gauss-Kronrod formulae for dλ(t) =dt on [ 1, 1], along withthe respective relative errors, are shown in Table 1.1 for ω =2, ω =1.5, and ω =1.1. The true values of I 1 (ω) needed to compute errors are obtained by Maple to 20 decimals. Incidentally, it was noted by Maple that for ω =2the value is expressible in terms of the Catalan constant C as I 1 (2)=8C/π. It can be seen that for each fixed n the larger m>0, i.e., the more poles of the integrand are taken into account, the smaller the error in both quadrature rules. (There are a few exceptions when the error is near the machine precision.) Compared to ordinary Gauss and Gauss-Kronrod rules (m =0), the improvement is rather spectacular, even if only one pair of poles (m =2) is incorporated. Example 1.2. I 2 (ω) = 1 t 1/2 Γ (1+t) 0 t+ω dt, ω > 0. Again, all poles of the integrand are real and simple, now located at 1, 2, 3,... and at ω. Furthermore, dλ(t) =t 1/2 dt on [0, 1]. We may therefore put r µ =1,M = m in (1.3) and (1.9) ζ µ = 1 µ, µ =1, 2,...,m 1; ζ m = 1 ω. The corresponding rational Gauss and Gauss-Kronrod quadratures are applied withthe same values of m as in Example 1.1, and for ω =2, ω =1, and ω =.5. The true values of I 2 (ω) are again obtained by Maple to 20 decimals.

4 620 W. Gautschi et al. Table 1.1. Numerical results for Example 1.1, ω = 2, 1.5, and 1.1 omega = 2.0 n m rat G rat GK err ratg err ratgk D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-15 omega = 1.5 n m rat G rat GK err ratg err ratgk D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-15

5 Quadrature rules for rational functions 621 Table 1.1. (continued) D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-15 omega = 1.1 n m rat G rat GK err ratg err ratgk D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-07

6 622 W. Gautschi et al. Table 1.2. Numerical results for Example 1.2, ω =0.5 n m rat G rat GK err ratg err ratgk D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-14 The results are similar to those in Example 1.1, except that convergence is generally faster for bothquadrature rules and, curiously, smaller values of m>0 give slightly better results than larger values. We show in Table 1.2 as representative only the results for ω = ational Gauss-Turán quadrature With dλ(t) the measure introduced in 1, we now wishto construct a quadrature formula of the type (2.1) which is exact on g(t)dλ(t) = n 2s λ (σ) ν g (σ) (τ ν )+n T (g) σ=0 (2.2) S 2(s+1)n = Q m P 2(s+1)n m 1, 0 m 2(s +1)n, with Q m, P k as defined in (1.3) (1.4). If m =0, this is the classical Gauss- Turán formula, exact for all polynomials of degree 2(s +1)n 1.

7 Quadrature rules for rational functions 623 Theorem 2.1. Let ω m be defined as in Theorem 1.1. Assume that the measure dλ/ω m admits an (ordinary) Gauss-Turán formula (2.3) p(t) dλ(t) ω m (t) = n 2s w ν (σ)t p (σ) (t T ν ), all p P 2(s+1)n 1, σ=0 having nodes t T ν contained in the support of dλ. Define (2.4) 2s ( ρ ) λ (σ) ν = w ν (ρ)t ω m (ρ σ) (t T ν ), ν =1, 2,...,n, σ =0, 1,...,2s, σ ρ=σ (2.5) τ ν = t T ν, ν =1, 2,...,n. Then the formula (2.1) is exact on the space S 2(s+1)n defined in (2.2) Proof. To prove exactness on S 2(s+1)n, let g be an arbitrary element of this space. Then either g Q m or g P 2(s+1)n m 1. In either case ω m g P 2(s+1)n 1. Indeed, if g Q m, say g(t) =(1+ζ µ t) r, then ω m g P m r, and m 2(s +1)n, r 1 yields the assertion. In the other case, the assertion is trivial. Consequently, by (2.3), g(t)dλ(t) = ω m (t)g(t) dλ(t) n 2s ω m (t) = w ν (ρ)t (ω m g) (ρ) (t T ν ). ρ=0 Applying Leibniz s rule of differentiation followed by an interchange of summation, we obtain for the inner sum 2s w ν (ρ)t ρ=0 2s ρ σ=0 2s = g (σ) (t T ν ) σ=0 2s = σ=0 ( ρ σ ) ω (ρ σ) m (t T ν )g (σ) (t T ν ) ρ=σ λ (σ) ν g (σ) (τ ν ), w (ρ)t ν ( ρ σ ) ω (ρ σ) m (t T ν ) the last expression by definition of λ (σ) ν and τ ν in (2.4) and (2.5), respectively. Summing over ν yields (2.1) withzero remainder term. ecall that the t T ν in (2.3) are the zeros of π n,s thenth-degree s- orthogonal polynomial relative to the measure dλ/ω m. This polynomial, and its zeros, are computed by a procedure described in [6, 2], which requires the solution of a system of 2n nonlinear equations for certain implicitly

8 624 W. Gautschi et al. defined recursion coefficients. For n =1, 2, 3,... we sequentially call on the MINPACK procedure hybrd1 to solve this system, using as initial approximation the solution of the preceding system (if n>1), suitably extended. The weights w ν (σ)t are then computed following the procedure in [6, 3] (where dλ has to be replaced by dλ/ω m ). In our numerical examples all poles are simple and real, so that ω m has the form m ω m (t) = (1 + ζ µ t). Moreover, 1+ζ µ t>0on the support of dλ. Computing the rational Gauss- Turán formula (2.1), (2.4) requires successive derivatives of ω m (cf. (2.4)). For simplicity, we do the computations only for s =1and s =2, so that at most four derivatives of ω m are needed. To compute them, let m ( ) k ζµ s k (t) =, k =1, 2, 3,..., 1+ζ µ t and note that From ω m ω m = s k (t) = ks k+1(t). m ζ µ 1+ζ µ t = s 1(t), that is, ω m = s 1 ω m, one obtains by repeated differentiation ω m = s 1 ω m + s 1 ω m = s 2 ω m + s 1 s 1 ω m =( s 2 + s 2 1 )ω m, ω m =( s 2 +2s 1s 1 )ω m +( s 2 + s 2 1 )s 1ω m =(2s 3 2s 1 s 2 )ω m +( s 1 s 2 + s 3 1 )ω m =(2s 3 3s 1 s 2 + s 3 1 )ω m, ω m =(2s 3 3s 1 s 2 3s 1 s 2 +3s2 1 s 1 )ω m +(2s 3 3s 1 s 2 + s 3 1 )s 1ω m =( 6s 4 +3s s 1s 3 3s 2 1 s 2)ω m +(2s 1 s 3 3s 2 1 s 2 + s 4 1 )ω m =( 6s 4 +8s 1 s 3 +3s 2 2 6s2 1 s 2 + s 4 1 )ω m, etc. The first example is meant to test our computer programming. ( 1+ ) 5 dt. Example 2.1. I = 1 1 The exact value of the integral is (2.6) I = 2 + log ζ µ 1+ζ µt 5 1+ζ µ 1 ζ µ,

9 Quadrature rules for rational functions 625 Table 2.1. Test results for Example 2.1 s n m rat GTerr D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-15 and the rational Gauss-Turán formula with dλ(t) =dt on [ 1, 1] and Q m as defined in (1.3), where r µ = 1,M = m, should yield exact results whenever 5 m<2(s +1)n. This has been confirmed numerically for the ζ µ in (1.8), ω =1.1, m =2, m =2 (n +1)/2, m =2n, with s either 1 or 2, and is illustrated in the top half of Table 2.1 for m =2n. If m =2(s+1)n, the polynomial part in (2.2) is empty, and hence the constant 1 will in general not be integrated exactly. This is illustrated in the bottom half of Table 2.1, where the errors are definitely not zero at the beginning, although they approach zero rather rapidly as n increases, more so for s =2 than for s =1. If the constant 1 in the integrand of I is removed and, accordingly, also the 2 in (2.6), then one gets exact results also in the limit case m =2(s +1)n as soon as m>5. For purposes of comparison, we show in Table 2.2 the results of ordinary Gauss-Turán quadrature (m =0). Example 2.2. The integral I 1 (ω) of Example 1.1.

10 626 W. Gautschi et al. Table 2.2. Ordinary Gauss-Turán quadrature for Example 2.1 s n m Gauss-Turan err D D D D D D D D D D D D D D D D D D D D-14 In order to apply Gauss-Turán quadrature to this integral, we must compute successive derivatives (the first four if 1 s 2) of the function f(x) = x sin x. An elementary computation (or Maple) yields f (x) sin x =1 x cot x, f (x) sin x = x 2 cot x +2x cot 2 x, f (x) sin x =3 5x cot x + 6 cot 2 x 6x cot 3 x, f (x) sin x =5x 20 cot x 24 cot 3 x +28x cot 2 x +24x cot 4 x, the limiting values at x =0being respectively 0, 1 3, 0, To avoid cancellation effects near the origin x =0, we compute these derivatives in quadruple precision, before rounding them to double precision. The results of the rational Gauss-Turán quadratures are shown in Table 2.3 for ω =1.1, with m =2(s +1)n, m =2n, m =2 (n +1)/2, m =2, and with m =0for conparison. For the same s and n, the accuracy can be seen to be markedly better for larger values of m, and significantly so when compared with m =0. 3. ational Gauss formulae for Cauchy principal value integrals We consider now Gaussian quadrature rules for Cauchy principal value integrals g(t)dλ(t)/(t x), where x is contained in the interior of the

11 Quadrature rules for rational functions 627 Table 2.3. Numerical results for Example 2.2, ω =1.1 s n m rat GTerr D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-15

12 628 W. Gautschi et al. Table 2.3. (continued) D D D D D D D D D D D D D D D D D D D D-04 support of dλ. The elevated degree of exactness characteristic for Gausstype quadrature formulae is achievable only if x is taken to be one of the quadrature nodes. Letting (3.1) λ 0 (x) = dλ(t) n t x λ ν τ ν x, the formula can be written in the form (cf. [1, 3.2.1]) (3.2) g(t) n t x dλ(t) = λ ν τ ν x g(τ ν)+λ 0 (x)g(x)+n C (g). We are interested in making this exact in spaces containing both polynomials and rational functions. An appropriate space for (3.2) is (3.3) S 2n+1 = Q m P 2n m, 0 m 2n, with Q m, P k as defined in (1.3) (1.4). Theorem 3.1. Let ω m be defined as in Theorem 1.1. Assume that the measure dλ/ω m admits an (ordinary) n-point Gaussian quadrature formula (3.4) p(t) dλ(t) n ω m (t) = wν G p(t G ν ), all p P 2n 1 having nodes t G ν x contained in the support of dλ. Define (3.5) τ ν = t G ν, λ ν = w G ν ω m (t G ν ). Then the formula (3.2) is exact on the space S 2n+1 defined in (3.3). Proof. Ifg S 2n+1, then (3.6) ω m (t) g(t) g(x) t x P 2n 1.

13 Quadrature rules for rational functions 629 Indeed, if g P 2n m, this is obvious. If g Q m, say g(t) =(1+ζ µ t) r, 1 r r µ, then ( ) g(t) g(x) = (1 + ζ µt) r (1 + ζ µ x) r 1 1+ζµt r =(1+ζ µ t) r 1+ζ µx, t x t x t x where the ratio on the right is a polynomial (in t) of degree r 1. Therefore, the function (of t) on the left of (3.6) is a polynomial in P m r+r 1 = P m 1 P 2n 1, since m 2n. We now use the familiar decompositon g(t) t x dλ(t) = g(t) g(x) dλ(t) ω m (t) t x ω m (t) + g(x) dλ(t) t x. Because of (3.6), if g S 2n+1, we can apply (3.4) to the first integral on the right, giving g(t) n t x dλ(t) = wν G ω m (t G ν ) g(tg ν ) g(x) t G + g(x) dλ(t) ν x t x n g(τ ν ) g(x) = λ ν + g(x) dλ(t) τ ν x t x by virtue of (3.5). Thus, g(t) t x dλ(t) = (3.7) n λ ν τ ν x g(τ ν) { +g(x) dλ(t) n t x λ ν τ ν x whichproves (3.2) withzero remainder. emark 3.1. If m n there is an alternative expression for λ 0 (x) in (3.1), which is in terms of the orthogonal polynomials p n ( )=p n ( ; dλ) relative to the original measure dλ. Indeed, if we define (3.8) q n (x) = p n(t) t x dλ(t), then (3.1 ) λ 0 (x) = q n(x) n p n (x) This follows readily from the identity p n (x) dλ(t) t x = λ ν τ ν x p n(τ ν ) ω m (t) p n(x) p n (t) t x. dλ(t) ω m (t) + q n(x), },

14 630 W. Gautschi et al. if we apply (3.4) to the integral on the right. This is legitimate if m+n 1 2n 1, i.e., if m n. For m =0we have ω m (t) =1, and the orthogonal polynomials π n and p n coincide. In particular, the nodes τ ν are the zeros of p n, and (3.1 ) simplifies to the (well-known) formula (3.1 ) λ 0 (x) = q n(x) (m =0). p n (x) The p k (x), q k (x) are solutions of the basic three-term recurrence relation for the measure dλ, (3.9) y k+1 =(x a k )y k b k y k 1, k =0, 1, 2,..., withinitial conditions p 1 (x) =0,p 0 (x) =1 and q 1 = 1, q 0 (x) = dλ(t) t x. (The coefficient b 0 in (3.9) must be defined by b 0 = dλ(t).) Serious numerical problems arise in the use of (3.2) when x is close to one of the quadrature nodes in (3.5), say τ µ. Indeed, as x approaches τ µ, one term in the quadrature sum tends to + and another to, even though there is a finite limit. To cope with this difficulty, one might want to write the quadrature sum in (3.7) as (3.7 ) λ ν τ ν x g(τ ν)+g(x) dλ(t) ν µ t x λ ν τ ν x + λ µ ν µ and evaluate the last term with special care. Our first example is an adaptation of Example Example 3.1. I(ω, x) = g(τ µ ) g(x) τ µ x πt/ω dt,ω>1, 1 <x<1. 1 sin(πt/ω) t x πt/ω sin(πt/ω) and dλ(t) =dt. The space Q m is constructed Here, g(t) = as described in Example 1.1. esults of the application of (3.2) to evaluate I(ω, x) are shown in Table 3.1 for ω =1.1, x =.95, and m =2n, m =2 n/2, m =2. As expected, the more poles of g are incorporated, the more accurate the results. If m = 0 (i.e., the poles are ignored), it takes a 38-point quadrature rule (3.2) to achieve results close to machine (double) precision. The (relative) errors are computed by comparison with quadruple-precision results. Example 3.2. Generalized Bose-Einstein integral G k (η, θ) = tk θt e η+t, η > 0, θ

15 Quadrature rules for rational functions 631 Table 3.1. Numerical results for Example 3.1, ω =1.1, x =.95 n m integral err D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D-15 For η < 0 and k = 1 2, 3 2, 5 2, these integrals are widely used in solid state physics. Positive values of η, however, do not seem to have physical meaning. Nevertheless, from a computational point of view, consideration of these (now Cauchy principal value) integrals may be of interest. The role of x in (3.2) is now played by η, which becomes more evident if the integral is written in the form G k (η, θ) = 0 t θt t η t k 1 e t dt. e η e t t η

16 632 W. Gautschi et al. Table 3.2. The Hilbert transform (3.8) for x =1 α λ 0(1,α) 1/ / / Table 3.3. Numerical results for Example 3.2, k = 1 2, η =1, θ =10 4, m =2n n m G err D D D D D D D D D D D D D D D D-12 Thus, g(t) = t θt, d(η, t) = e η e t d(η, t) t η and dλ(t) =t k 1 e t dt on [0, ] is a generalized Laguerre measure. Some care is required in the evaluation of d(η, t) for t close (or equal) to η. We suggest Taylor expansion d(η, t) =e η ν=0 ( 1) ν (ν + 1)! (t η)ν, say for t η 1, and direct evaluation otherwise. The biggest challenge in applying the method of Theorem 3.1 to this example is the computation of the Hilbert transform (3.8) λ 0 (x,α)= 0 tα e t t x dt, α = k 1, x>0. We refer, however, to [7] for a detailed discussion of computing this transform (as well as the one for the Hermite measure). For the three values k = 1 2, 3 2, 5 2 of interest here, taking, for example, x =1, one finds the values in Table 3.2.

17 Quadrature rules for rational functions 633 The desired integral is now easily computed, using the software in [4] to generate the weights λ ν and nodes τ ν. esults obtained in IEEE double precision are shown in Table 3.3 for k = 1 2, η =1, θ =10 4, and m =2n. We also experimented with other choices of m, specifically with m = 2 n/2, m =2, and m =0. Interestingly, the first of these led to somewhat faster convergence than m =2n, but the other two choices yielded distinctly slower convergence, requiring values of n =11and n =23, respectively, to achieve the same accuracy. esults for k = 3 2 and k = 5 2 are similar. eferences 1. Gautschi, W. (1981) A survey of Gauss-Christoffel quadrature formulae. In E.B. Christoffel: The influence of his work on mathematics and the physical sciences, P.L. Butzer, F. Fehér, (eds.) Birkhäuser, Basel, pp Gautschi, W. (1993) Gauss-type quadrature rules for rational functions. In Numerical integration IV, H. Brass and G. Hämmerlin,(eds.) International Series of Numerical Mathematics, vol Birkhäuser, Basel, pp Gautschi, W. (1993) On the computation of generalized Fermi-Dirac and Bose-Einstein integrals, Comput. Phys. Comm. 74, Gautschi, W. (1999) Algorithm 793: GQAT Gauss quadrature for rational functions. ACM Trans. Math. Software 25, Gautschi, W. (1999) Orthogonal polynomials and quadrature. Electr. Trans. Numer. Math. 9, Gautschi, W., Milovanović, G.V. (1997) s-orthogonality and construction of Gauss- Turán-type quadrature formulae. J. Comput. Appl. Math. 86, Gautschi, W., Waldvogel, J.. Computing the Hilbert transform of the generalized Laguerre and Hermite weight functions (submitted for publication) 8. Laurie, D.P. (1997) Calculation of Gauss-Kronrod quadrature rules. Math. Comp. 66, López Lagomasino, G. and Illán, J. (1984) A note on generalized quadrature formulas of Gauss-Jacobi type. In Constructive theory of functions. Publ. House Bulgarian Acad. Sci., Sofia, pp López Lagomasino, G. and Illán Gonzales, J. (1987) Sobre los métodos interpolatorios de integración numérica y su conexión con la aproximación racional. ev. Ciencias Matém. 8, (2), Van Assche, W., Vanherwegen, I. (1993) Quadrature formulas based on rational interpolation. Math. Comp. 61,