TECHNOMETRICS@, VOL. 21, NO. 3, AUGUST 1979 A Modified 3-Spline Method for Evaluating the Euler Digamma Function F. T. Lindstrom Department of Statistics Oregon State University Corvallis, OR 97331 The modified k splines are introduced. A simple linear model results from applying a variational principle to a modified 3-spline approximation of #’ on [1,2]. An approximation of I) is obtained analytically by integration. The coefficients needed to use this approximate method are tabulated. KEY WORDS compartment I .V. drug distribution compartment models with random metabolism parameters (Lind- Splines Modified splines Strom and Birkes, [7]). Digamma function Assuming an iid sample of size n, (tl, C,), (tzCz), Gamma function * * * (t,,, C,), with tk being the kth time from adminis- Log gamma function tration of the drug (at time t = 0) and with the initial drug concentration being C, then the density of C is assumed to be fc(c) defined in equation (I). INTRODUCTION By applying the usual likelihood procedures we The Euler digamma function and its derivative (the obtain the MLE’s i and x of r and X respectively trigamma function) are often found in applications satisfying the following equations work where sampling is assumed to be done with gamma distributed data (Chapman [3]; Harter and ai = eh;’ 7 O<a<l, (2) Moore [6]; Neyman and Scott [8]; Stacy and Mihram [ll]; and Choi and Wette [4]). These works, and others the author has found, all rely on the digamma and trigamma function to be approximated via the well-known asymptotic expression (Abramowitz and Stegun [ 11, page 258 formula 6.3.18) or by inter- - -r 4 &-!L (4) polation in the existing National Bureau of Standards i tables. As was the case in the two fine papers by Chapman where the jointly minimal sufficient statistics d, and d, [3] and Choi and Wette [4], a colleague and I had a are defined as need to estimate the population parameters, via max- imum likelihood methods, in the apparently little (5) studied log gamma distribution fe(c>= ~(fIog(~))“(~)““(~) (1) , for and of course q(r) is the Euler digamma function 0 < c I co, O<t<w. This distribution arises in a natural way in simple one Hence, even though we are sampling from a log gamma distributed population, we still have a need to Received March 1978; revised August 1978 solve for i in the transcendental expression (2). With 308 F. T. LINDSTROM rC, appearing explicitly in equation (2) we therefore data analysis. We use splines here simply to give a have a need to evaluate it for any r on (1, a). close I, function approximation over 1 < r I 2. We now present a numerical scheme for evaluating Since both economy of computer time and the $(r)on 1 < r < 03. accuracy of the polynomial approximation technique should be balanced, it was determined early on in this work that to achieve the same level of accuracy using MODIFIED SPLINE METHOD the classical B splines (smooth cubits), requires about In Chapman’s work [3] an equation, analogous three times as much computation as the modified 3- with our equation (2), was obtained for estimating r. spline technique using a reasonable number of uni- He used a table (Pairman [9]) and an inverse inter- formly spaced knots [lo]. The variational method polation method to find i. Choi and Wette [4] used a (least squares) of obtaining “model parameters” has Newton-Raphson method on their transcendental ex- the useful property of distributing the approximation pression, analogous to our equation (2), to find i. line more uniformly among the data points while the They needed the trigamma function as well for this purely interpolatory constraint problem may allow technique. An important point caught by Choi and “large” between point deviations, i.e. “the classical Wette [4] but evidentally overlooked by Chapman [3] polynomial wiggle” as it is known in numerical anal- is that the di- and trigamma functions satisfy the ysis. It was therefore decided to use the variational following recurrence relations method here rather than the purely interpolatory ap- proach. rCi(r + 1) = 4Yr) + +, l<r<a, (8) Let the set of knot points {T,Jk,ln on [l, 21 be given. Suppose that the value of the function to be fit, and f(r) say, is known at m + 1 points, t,, j = 0, 1,2 . * . m, t,, = 1 and t, = 2. It is customary to choose the knots $‘(r + 1) = +‘(r) - f , l<r<a. (9) Tk not equal to the “observations points” tl, although there is no mathematical reason (such as singularity) With these two recurrence relations and our mod- that they cannot be. The modified 3 splines, consid- ern high-speed computers we now have the capabili- ered here in detail, serve to illustrate the approxima- ties of easily computing # and #’ with any argument tion procedure. They are C[l, 21 and are defined as such that 1 < r < 03. An important detail apparently follows: overlooked by Choi and Wette [4] is that when using PI(t) = a0 + a,t + (ret2 + ast3, 1 = to I t I T,, equations (8) and (9) one must always know a priori or find via some other means $ and $’ with argu- pdt) = pdt) + 4 - Tl)‘, 7’1 5 t 5 Ts, ments lying on the closed interval [ 1, 21. For example, ti(4.6) is readily computed to be rc/(4.6) = #(3.6) + l/3.6 pdt) = pl-t(t) + w+z(t - T,-I)‘, T,-, I t g T,, = $(2.6) + l/2.6 + l/3.6 = q(1.6) + l/1.6 + l/2.6 + l/3.6 = #(1.6) + 1.2874 Pa+l(t) = p,(t) + a n+s(t - T,,)‘, T,, I t I t, = 2, which demonstrates the need to know #(1.6) so that (11) ti(4.6) may be computed. Our method supplies this where instead of the usual purely interpolatory con- “missing” piece very nicely. straints, it is assumed that the variational condition To be sure, for arguments r 2 10.0, the well-known asymptotic form K+4 ,. cm,)- W,)Y 1 W) - log(r) - - (10) 2r holds. (Abramowitz and Stegun [l], page 258) serves as an Before proceeding, it should be pointed out that approximation to G(r) and similarly for $‘(r) by dif- the low order 3 splines have been chosen because they ferentiating in (10) with respect to r. (1) tend to be less sensitive to errors in the data and (2) tend to give a curvature sign and magnitude more consistent with a slowly varying function such as the Modijied k splines $ function. Ahlberg, et al. [2] give the essential mathematical Carrying out the indicated minimization in equa- details of the family of k splines. Svante Wold [ 121 tion (12) yields the usual linear model design matrix has an excellent paper detailing the use of splines in notation (Draper and Smith [5]) TECHNOMETRICS@, VOL. 21, NO. 3, AUGUST 1979 MODIFIED 3-SPLINE METHOD FOR EVALUATING THE EULER DIGAMMA FUNCTION 309 Required continuity conditions at Knot points f(t) t t t t t nlfl tn n2+1 n n +l nlll L b bd LL n pi n T T t Tl T2 T3 Tk+l n-l n m t--t FIGURE1. Sketch of typical spline interpolation. (X’X)/3 = X'Y (13) ary to choose m considerably larger than n + 4. This we will do shortly. The knot points are placed accord- where ing to the “thumb rules” stated by Wold [12]. x = (Xl, x2, x3, . * Xn+,), X is m X (n + 4) and the columns ofX are TABLE I-$’ data from Abramowitz and Stegun [I]. x, = (1, 1, I, e-*,1) r 9’ (r) x2 = (11,L c3 . * * I fm)' x3 = (f1zt*2t32, * * *, tm2) 1.00 1.6449341 x, = (tl3tz3t32, * * f, tm3)' 1.05 1.5323573 1.10 1.4332992 1.15 1.3455594 x n,,, = (O,O,O * * - 0, (tn*+1 - TlJ3, * * - (&I - T,J3)’ 1.20 1.2673772 1.30 1.1342534 x n2+,= (0, o,o . * - 0, * . *o, (GzZ+l- TZY, 1.35 1.0771937 * - * (t, - T*)3) 1.40 1.0253566 1.45 0.9780788 1.55 0.8950541 x n+, = (0,O * * * 0, (fn,+l - Tn)” 1.60 0.8584319 . (t, - T,)‘) 1.65 0.8245905 and 1.70 0.7932328 1.80 0.7369741 1.85 0.7116545 Figure 1 shows a sketch of a typical spline approxi- 1.90 0.6879721 mation set up for either interpolatory or variational constraints. It should be remembered that for X’X to 1.95 0.6657762 be nonsingular we must have m > n + 4. It is custom- 2.00 0.6449341 TECHNOMETRICS@, VOL. 21, NO. 3, AUGUST 1979 310 F. T. LINDSTROM TABLE 2-Knot points and a values for #’ data on [I, 21 for the modified 3-spline approximation. 4(t) = -Y + s,’ V(5) dt, lI?<2. (15) Since y = 0.5772157 ao = 9.1153865 T1 = 1.25 a1 = -14.7073784 PI(f) = cl0 + a,t + a# + a3t3, 1 <tl T,, T2 = 1.50 u2 = 9.3903813 T3 = 1.75 c13 = -2.1535377 P&J = ~4) + a,0 - Tl13, T, < t I T,, "4 = 1.3036181 (16) “(‘) L p3(f) = p&) + a,(t - Tz)‘, a5 = 0.4655709 Tz I t I T3, = 0.1870607 "6 p,(t) = pa(t) + 4 - TsY, T3I tI2, we have ModiJed 3-spline approximations to $’ p&) = -y + cl& - 1) + CYl(P- 1)/2 The $’ data listed in Table 1 were taken from + a2(t3 - 1)/3 + cYs(t’ - 1)/4, Abramowitz and Stegun, [ 11.