Analytical Approximations for the Principal Branch of the Lambert W Function Roy M. Howard School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO Box U1987, Perth, 6845, Australia ABSTRACT. A geometric based approach for specifying approximations to the Lambert W function, which can achieve any set relative error bound over the interval , is detailed. Approximations that can achieve arbitrarily high accuracy for the interval , based on a two point spline approximation, are specified. Iterative methods can be used to improve the accuracy of the approximations. Applications include, first, analytical expressions, with set relative error bounds, for the Lambert W function over the interval . Second, approximations, with an arbitrarily low relative error, for upper and lower bounds for the Lambert W function. Third, analytical expressions for the evaluation of and the integral of , for , without knowledge of . Fourth, a direct approach for evaluating the Lambert W function to achieve a prior set error constraint. 1. INTRODUCTION The Lambert W function is associated with Lambert [19] and is a multivalued complex function with the single valued function, associated with the branch, being denoted . The principle branch of the Lambert W function, , is the function defined by the inverse of for the case of , i.e. (1) The Lambert W function does not have an explicit analytical form but is of importance consistent with increasing applications as detailed in the literature, e.g. [7], [3], [25], [9], [6], [14], [26], [20], [18], [4], and [11]. Generalizations of the Lambert W function are also of interest, e.g. [8] and [22]. Efficient numerical methods for computing values of the Lambert W function have long been known, e.g. [10], and with an advance detailed by Fukushima [12]. The focus of this paper is on the principle branch and the real case which continues to receive research interest, e.g. [17]. The graph of the Lambert W function, for this case, is shown in Figure 1 and, for notational simplicity, is denoted in this paper. Existing analytical approximations for the principle branch and real case of the Lambert W func- tion, e.g. [5], [3] and [17], are, in general, custom and cannot be directly generalized to obtain approximations of arbitrarily high accuracy. This paper provides a geometrical basis for defining such approximations. 0   1 e– 0  0  W y  W y  y 0   W y  kth Wk W0 y f x  xe x = = Re x  1– x W0 y  f 1– y .= = W Correspondence: r.howard@curtin.edu.au https://adac.ee/ https://doi.org/10.28924/ada/ma.2.14 The advances in this paper are twofold. First, a systematic geometric approach for defining approximations to the Lambert W function, over the interval , with quad- ratic convergence. Convergence is proved. The approximations can be used to specify, with an arbitrarily low relative error, upper and lower bounds for the Lambert W function. Second, a systematic method for combining series expansions at , and the origin, to define arbitrarily accurate approximations for the interval . Applications of the approximations include analytical expressions, with set relative error bounds, for the Lambert W function over the interval , evaluation of and the integral of , for , without knowledge of , and a direct approach for evaluating the Lambert W function to achieve a prior defined error. A review of published approximations for the Lambert W function is provided in sec- tion 2. The proposed geometric approach for establishing approximations to the Lambert W function is detailed in section 3. Convergence is discussed in section 4 and in section 5 convergent two point spline based approximations, for the interval , are detailed. The use of iterative methods, to improve the accuracy of approximations, is dis- cussed in Section 6. Applications are detailed in section 7 and conclusions are stated in section 8. 1.1. Notation and Properties. The notation of is used which is consistent with the principle value of the Lambert W being the inverse function of , , . Relevant properties of the Lambert W function are detailed in Appendix A. For an arbitrary function , defined over the interval , an approximating function has a relative error, at a point , defined according to . The rel- ative error bound for the approximating function, over the interval , is defined according to (2) The notation is used. Mathematica has been used to facilitate analysis and to obtain numerical results. In general, relative error results, associated with approximations to the Lambert W function, have been obtained by sampling specified intervals, in either a linear or a logarithmic manner, as appropriate, with points. FIGURE 1. Graph of and its inverse which is the Lambert W function for the principle branch and the real case. f x  xe x = W y  x y f x  xe x = 1– 1 e–  1 e– 1–  0   1 e– 1 e– 0  0  W y  W y  y 0   W y  1 e– 0  x W y = y f x  xe x = = x 1– y 1 e– f    fA x1 re x1  1 fA x1  f x1 –=    reB max re x1  : x1    .= f k  x  x k k d d f x = 1000 https://doi.org/10.28924/ada/ma.2.14 2. PUBLISHED APPROXIMATIONS A Taylor series expansion, at the origin, for the Lambert W function is well known, e.g. [17], eqn. 2, and yields the following approximation which has a limited region of conver- gence: (3) The relationship implies (positive sign for ; negative sign for ) and, hence: (4) Such approximations suggest the more general approximation for the Lambert W function of (5) Fritsch et al., [10], eqn. 6, utilizes an initial approximation of which is suitable for . 2.1. Published Approximations. The following is an overview of indicative published approximations for the Lambert W function. Additional useful references include [26] and [14]. Boyd [5], eqn. 5-7, proposed the approximation (6) which is valid for . Whilst the approximation is sharp at , it is not zero at the origin and, thus, is not sharp at this point. It has a relative error bound for the inter- val of . Barry et al. [3], eqn. 12, proposed the approximation (7) for . This approximation was modified, [3], eqn. 15, to be valid for according to (8) T y  y y 2 – 3y 3 2 -------- 8y 4 3 --------– 125y 5 24 -------------- 54y 6 5 ----------- 16807y 7 720 --------------------+– 16384y 8 315 --------------------– + + += y 1 e ---. y xe x = x x ln+ y ln= x y 0 x y 0 x y y 1« y ln y 1.»    x W y  1 y+ ln = y 1– e ------. W y  y ln y e WBd y  1 2 1 ey+  10 ln 10 ln ln– ------------------------------------------------- 11 ey+ ln 11 ey+ ln ln–tanh+– = 1 1 10 ------ 1 ey+ ln 7 5 ---– 3– 40 ------ 1 ey+ ln 7 5 ---– 2 exp+ y 1 e– y 1 e–= 1   0.0499 WB1 y  6 5 --- y 12 5 ------ y 1 12y 5+ ln -----------------------------------ln ---------------------------------------------------------ln= y 0 y 1 e– WB2 y  1 +  6 5 --- y 12 5 ------ y 1 12y 5+ ln -----------------------------------ln ---------------------------------------------------------ln  2y 1 2y+ ln -------------------------ln –=  0.4586887.= https://doi.org/10.28924/ada/ma.2.14 The relative error bound, for , is . Iacono and Boyd, [17], eqn. 17, proposed the following approximation which is valid for (9) This approximation yields a relative error bound of for . An improved approximation, [17], eqn. 19, 20, is (10) which yields a relative error bound for of (the bound occurs at of the order of ) for the case of optimally chosen as . The approximation is sharp at . 2.1.1. Padè Approximations. Padè approximates for the Lambert W function for the interval have been proposed, e.g. [21], eqn. 34: (11) Higher order Padè approximates are detailed in Fukushima [13]. 2.1.2. Comparison of Approximations. The relative errors in the above specified approximations are shown in Figure 2 and Figure 3. 2.2. Classic Iterative Approximations. The classical iterative approximation for the Lambert W function, e.g. [17], eqn. 8-10, is based on the fundamental relationship , , and an initial approximation as specified in y 0 1.96 10 3– y 1– e   WI1 y  1 y 1 1 y+ ln 2 ----------------------+ --------------------------------+ .ln= 3.53 10 2– y 0   WI2 y  1– a 1 b 1 ey++ 1 c 1 1 ey++ ln+ ---------------------------------------------------ln += c e 1 a 1– 2 a– 1 2 e1 a ln– ----------------------------------------= b 2 a ------- c+ = y 0 4.53 10 3– y 10 12 a a 2.036= y 1 e–= y 1– e 1  WL y  1 123y 40 ------------ 21y 2 10 -----------+ + 1 143y 40 ------------ 713y 2 240 --------------+ + ------------------------------------------ 1 y+ .ln= FIGURE 2. Graph of the relative errors, over the interval , in published approximations to .1– e 1  W y  y re y  WI1 y  WI2 y  WB2 y  WBd y  WL y  WL y  WB2 y  WI1 y  x y ln x ln–= x y 0 x W0 y  1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 Equation 5. A first order approximation arises by substitution of this approximation into the expression to yield (12) Iteration yields (13) In general: (14) 2.2.1. Comparison of Approximations. The relative error in the iterative approximations, of orders zero to three, are detailed in Figure 4 and Figure 5. The relative errors decrease, for large values of , at an increasing rate as the order of approximation is increased. The approximations are poor for which is consistent with the assumptions made in the iteration. 2.2.2. Alternative Iterative Approximations. An alternative iterative approach is to utilize the relationship and solve for the error given an initial approximation FIGURE 3. Graph of the relative errors in published approximations to .W y  y re y  WI1 y  WI2 y  WBd y  WL y  WB2 y  WL y  x ln W1 y  y ln 1 y+ ln ln– y 1 y+ ln ---------------------- .ln= = W2 y  y ln y ln 1 y+ ln ln–ln– y y 1 y+ ln ----------------------ln ---------------------------------ln .= = Wi y  y ln Wi 1– y  ln– = i 1 2 3     W0 y  1 y+ .ln= y y 10 FIGURE 4. Graph of the relative errors, over the interval , in iterative approximations to . The relative errors in the approximations for and are high. 1– e 1  W y  W2 W3 y WI3 y  re y  W0 y  W1 y  W2 y  x y ln x ln–= https://doi.org/10.28924/ada/ma.2.14 of . For fixed, consider an initial approximation of , with an error of , which implies and, thus, e.g. [17], eqn. 11: (15) For the case where the error is small and , a first order Taylor series for the logarithm function yields (16) and the first order approximation (17) The general iteration formula, e.g. [17], eqn. 12 (18) then follows. Higher order iteration, based on a higher order approximation for , is detailed in [10]. With a starting value, utilized in Fritsch et al. [10], of (suitable for ) it follows that a first order approximation is (19) The relative error in this approximation is shown in Figure 5 and the relative error bound for the interval is . A first order iteration, based on Equation 18, with , yields the approximation [17], eqn. 18: FIGURE 5. Graph of the relative errors in iterative approximations to .W y  y W0 y  W2 y  W3 y  W1 y  WI3 y  re y  WF y  WF y  x0 y x0 0 y x0 0+  x0 0+ exp= x0 0+ y ln x0 0+  ln– y x0 -----ln 1 0 x0 -----+ .ln–= = 0 x0 1« x0 0+ y x0 -----ln 0 x0 -----–  0 x0 1 x0+ -------------- y x0 -----ln x0–  x1 x0 0+ x0 1 x0+ -------------- 1 y x0 -----ln+ .= xi 1+ xi 1 xi+ ------------- 1 y xi ----ln+ = 1 0 x0+ ln x0 y ln= y e WF y  y ln 1 y ln+ ---------------------- 1 y y ln -------------ln+ .= e   1.47 10 2– x0 y  1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln= https://doi.org/10.28924/ada/ma.2.14 (20) The relative error in this approximation is shown in Figure 4 and Figure 5. The relative error bound, over the interval , is . 2.3. Approximations via Newton-Raphson Iteration. Consistent with the illustration shown in Figure 6, a direct Newton-Raphson method for solving for , in the equation , is: (21) A first iteration, based on a known approximating function for , is, e.g. [17], eqn. 15: (22) Halley’s method can similarly be utilized, e.g. [26]. With an initial value of , the first and second order approximations, respectively, are: (23) (24) 2.3.1. Results. Graphs of the variation of the relative error, with iteration level, are shown in Figure 7 and Figure 8 for the case of an initial approximation of . WI3 y  1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln 1 1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln+ --------------------------------------------------------------------- 1 y 1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln ------------------------------------------------------------ln+ .= 0   2.16 10 4– x y xe x = xi xi 1– xi 1– e xi 1– y– e xi 1– xi 1– e xi 1–+ -----------------------------------------– xi 1– xi 1– ye xi 1–– – 1 xi 1–+ ----------------------------------.–= = g W W y  g y  g y  g y  exp y– g y  exp g y  g y  exp+ --------------------------------------------------------------------– g y  g y  y g– y  exp– 1 g y + ------------------------------------------------.–= FIGURE 6. Newton-Raphson iteration for determining an approximation to the solution, denoted , of for fixed and based on an initial value of . xo y x x exp= y x0 x xo h x0  x0x1 h x  xe x y–= x2 h x1  x0 1 y+ ln= WN1 y  1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------– = WN2 y  1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------– –= 1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------– y 1 y+ ------------ 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------exp– 1 1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------–+ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- . x0 1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 Note, for higher order iteration, the relative error increases, from an increasingly low level, as increases. 3. GEOMETRIC BASIS FOR ITERATIVE APPROXIMATIONS TO LAMBERT W FUNCTION 3.1. Geometric Basis. To establish a systematic, geometrically based, approach for establishing approximations to the Lambert W function of arbitrarily high accuracy, consider a set value of . The Lambert W function associated with , denoted , is such that and is the point defined by the intersection of the two curves and as illustrated in Figure 9 for the case of . The geometry associated with this intersection of the two curves is the basis for an initial approximation and for the iterative approximations detailed in the following theorem. Theorem 3.1. Iterative Approximations for Lambert W Function. For fixed, , approximations to the Lambert W function can be iteratively defined according to (25) y FIGURE 7. Graph of the relative errors, in the zero to fourth order iterative approximations to , based on Newton-Raphson iteration with an initial approximation of . W y  WN0 y  1 y+ ln= y WN1 y  re y  WN2 y  WN4 y  WN3 y  WN0 y  FIGURE 8. Graph of the relative errors, in the zero to fifth order iterative approximations to , based on Newton-Raphson iteration with an initial approximation of . W y  WN0 y  1 y+ ln= y WN0 y  re y  WN3 y  WN4 y  WN5 y  y y xo xo ye xo– = x ye x– y 0 y y 1 e– WLi WLi 1– 1 WUi 1– +  1 WLi 1– + --------------------------------------------= i 1 2    WL0 y= WUi y WLi ---------ln = i 1 2    WU0 0.= https://doi.org/10.28924/ada/ma.2.14 Proof. Consider the geometry detailed in Figure 9, for the case of , and an initial approximation to of which is based the intersection of the first order Taylor series for at the origin, i.e. , and . An associated approximation, denoted , arises from the intersection of the level defined by and the curve . The solution is (26) Second order approximations follow from the intersection of with a first order Taylor series of , based on the point , i.e. , to yield (27) and by the intersection of this level with the curve to yield (28) The general iterative form: (29) then follows. Whilst the geometry, and the defined approximations, are clearly defined for the case of , the approximations are also valid for as simulation results, shown in Figure 10, demonstrate. 3.1.1. Explicit Approximations. Approximations to the Lambert W function, of orders one to five, are: (30) y 0 xo WL1 y 1 y+ = ye x– y yx– x WU1 WL1 ye x– FIGURE 9. Illustration of the geometry underpinning an iterative relationship to find an approximation to which is the solution of for fixed, .xo x y x– exp= y y 0 WL1 WL2 WL2 x WL1 xo y x WU1 ye x– xo WU2 y yx– WL1 x WU1 – WL1 – WU1 y WL1 ---------ln 1 y+ .ln= = x ye x– WU1 WL1 x WU1 – WL1 – WL2 WL1 1 WU1 +  1 WL1 + -----------------------------------= ye x– WU2 y WL2 ---------ln 1 2y+ 1 1 y+ ln+ ------------------------------- .ln= = WLi WLi 1– 1 WUi 1– +  1 WLi 1– + --------------------------------------------= WUi y WLi ---------ln = i 2 3    y 0 y 1 e– 0  WL1 y  y 1 y+ ------------= WU1 y  1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 (31) (32) (33) (34) (35) (36) where (37) WL2 y  y 1 1 y+ ln+  1 2y+ ---------------------------------------= WU2 y  1 2y+ 1 1 y+ ln+ -------------------------------ln= WL3 y  y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 3y y 1 y+ ln+ + --------------------------------------------------------------------------------------------------= WU3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln= WL4 y  = y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ 1 4y y 1 2y+ 1 1 y+ ln+ -------------------------------ln y 1 y+ ln 2 1 2y+ 1 1 y+ ln+ -------------------------------ln++ + + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- WU4 y  = 1 4y y 1 2y+ 1 1 y+ ln+ -------------------------------ln y 1 y+ ln 2 1 2y+ 1 1 y+ ln+ -------------------------------ln++ + + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ln WL5 y  yn5 y  d5 y  ----------------= WU5 y  d5 y  n5 y  -------------ln= n5 y  1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ = 1 1 4y y+ + 1 2y+ 1 1 y+ ln+ -------------------------------ln y 1 y+ ln 2 1 2y+ 1 1 y+ ln+ -------------------------------ln++ 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ln+ https://doi.org/10.28924/ada/ma.2.14 (38) 3.1.2. Notes. The approximation is the approximation stated in Equation 5 and is the basis for the Newton-Raphson iteration leading to Equation 23 and Equation 24. Consider the iteration formula specified in Equation 18 ([17], eqn. 12). With a starting value of , it follows that a first order iteration leads to the approximation (39) which is as specified by Equation 31. 3.1.3. Results. The relative errors associated with the approximations, specified in Theorem 3.1, are shown in Figure 10 and Figure 11, whilst the relative error bounds, for the interval , are detailed in Table 1. Note the quadratic convergence. TABLE 1. Relative error bounds, over the interval , for the iterative approximations to the Lambert W function defined in Theorem 3.1. Iteration order: i Relative error bound for Relative error bound for 1 increasing 0.381 2 increasing 0.0569 3 4 5 6 d5 y  1 5y y 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ + += y 1 2y+ 1 1 y+ ln+ ------------------------------- 2 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ln + y 1 y+  3 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ + 1 2y+ 1 1 y+ ln+ ------------------------------- 2 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ln ln WU1 y  1 y+ ln= x0 y 1 y+ = W1 y  y 1 2y+ --------------- 1 1 y+ ln+  = WL2 y  0   0   WLi WUi 8.32 10 3– 1.33 10 3– 3.88 10 6– 7.23 10 7– 1.08 10 12– 2.15 10 13– 9.33 10 26– 1.90 10 26– https://doi.org/10.28924/ada/ma.2.14 3.2. Alternative Iterative Formulas. The approximations and for the Lambert W function, as specified in Theorem 3.1, can be also be defined via iterative formulas based on numerator and denominator expressions. Theorem 3.2. Alternative Iterative Formulas for Lambert W. The approximations and , , for the Lambert W function can be specified according to (40) where (41) Proof. The proof is detailed in Appendix B. FIGURE 10. Graph of the relative errors in the iterative approximations to the Lambert W function defined in Theorem 3.1. y WL1 y  re y  WL2 y  WU1 y  WU2 y  WL3 y  WU3 y  WU4 y  FIGURE 11. Graph of the relative errors in the iterative approximations to the Lambert W function defined in Theorem 3.1. y WL1 y re y  WU1 y  WL2 y  WU2 y  WL3 y  WU3 y  WL4 y  WU4 y  WLi WUi WLi WUi i 1 2    WLi y  ni y  di y  ------------= WUi y  di y  ln ni y  y ------------ln– = i 1 2    ni y  ni 1– y  1 di 1– y  ln ni 1– y  y -------------------ln–+ = n0 y  y= d0 y  1= di y  ni 1– y  di 1– y .+= https://doi.org/10.28924/ada/ma.2.14 3.2.1. Explicit Formulas. Explicit formulas, for the first to fourth order approximations, are: (42) (43) (44) (45) (46) 3.3. Alternative Geometrical Approach. An alternative approach that leads to the approximations , , , as specified in Theorem 3.1, is to utilize the transformation , , in the relationship which implies . The geometry underpinning the iteration that leads to the approximations is illustrated in Figure 12. Theorem 3.3. Alternative Geometrical Iteration. For fixed, , iterative approximations for the Lambert W function can be defined according to (47) and it is the case that as specified in Theorem 3.1. Proof. Consider the case of fixed and the geometry illustrated in Figure 12 which is based on affine approximations to find, iteratively, the solution to . The ini- n1 y  y= d1 y  1 y+ = n2 y  y 1 1 y+ ln+ = d2 y  1 2y+ = n3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ = d3 y  1 3y y 1 y+ ln+ + = n4 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ = 1 1 3y y 1 y+ ln+ + ln 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ln–+ , d4 y  1 3y y 1 y+ ln+ + y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ .+= WU1 WU2  x z ln= x 0 z 1 y xe x = y z z ln= FIGURE 12. Illustration of the geometry underpinning the iterative relationship to find approximations to which is the solution of for fixed.zo z z ln y= y 1 z y zo z z ln z2z1 z 1– z1 z1 ln z z1–  z1 ln 1+ + y2 y1 z1 1 y+= y y 0 zi zi 1– zi 1– zi 1– ln y– zi 1– ln 1+ -----------------------------------------– = z1 1 y+ = xi zi ln = xi WUi y = y zo y z z ln= https://doi.org/10.28924/ada/ma.2.14 tial value is established by the intersection of a first order Taylor series for at the point , which is , and the level . The solution is . A first order Taylor series for at this point is and the intersection of this approximation with the level is the point defined according to (48) Iteration in this manner leads to the stated general iteration formulas. 3.3.1. Explicit Formulas. Approximations, of orders one to three, are: (49) (50) (51) 3.4. Iterative Algorithm for (-1/e,0]. For the case of , the geometric approach, illustrated in Figure 13, can be utilized to establish an algorithm for determining approximations to the Lambert W function. Theorem 3.4. Iterative Algorithm for (-1/e,0]. An iterative algorithm for defining approximations to , for , is: (52) Proof. Consider the illustration shown in Figure 13. With an initial approximation for of , a first order Taylor series for , based on the point , with , is . The intersection of this Taylor series with , at the point , leads to (53) The value of associated with is . A first order Taylor series approxima- tion for at the point is and the intersection of this approxima- tion with , at the point , leads to z z ln z 1= z 1– y z1 1 y+= z z ln z1 z1 ln z z1–  z1 ln 1+ + y z2 z2 z1 z1 z1 ln y– z1 ln 1+ -----------------------------.–= z1 y  1 y+ = x1 WU1 y  1 y+ ln = = z2 y  1 2y+ 1 1 y+ ln+ -------------------------------= x2 WU2 y  1 2y+ 1 1 y+ ln+ -------------------------------ln = = z3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------= x3 WU3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ ----------------------------------------------------------------------------------------------- .ln= = y 1– e 0  x W y = y 1 e– 0  xi yi 1– 1 xi 1–+  1 yi 1–+ -------------------------------------= x0 0= y0 y= yi ye xi– .= xo x0 0= ye x– x0 y0  y0 y= y0 1 x x0– –  x x1 x1 y0 1 x0+  1 y0+ -------------------------.= ye x– x1 y1 ye x1– = ye x– x1 y1  y1 1 x x1– –  x x2 https://doi.org/10.28924/ada/ma.2.14 (54) The value of associated with this value is . Iteration in this manner leads to the general formula as stated in the theorem. 3.4.1. Explicit Approximations. Approximations, for orders one to four, are: (55) (56) 3.4.2. Results. The relative error in the iterative approximations specified in Theorem 3.4 are shown in Figure 14. For orders two, and higher, the approximation are more accurate than the approximations detailed in Theorem 3.1. 3.5. Improved Approximations for [0,∞). Improved approximations, for the interval , can be established by using the iteration formula specified in Theorem 3.3 and by using FIGURE 13. Illustration of the geometry underpinning the iterative relationship to find approximations to , the solution of , for the case of fixed and . xo x ye x– = y y 1 e– 0  x1 x y x x2 y1 ye x– 1= xo y1 1 x x1– –  ye x– y2 ye x– 2= y 1 x x0– –  x0 x2 y1 1 x1+  1 y1+ -------------------------.= ye x– y2 ye x2– = WE1 y  y 1 y+ ------------= WE2 y  y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------= WE3 y  y y 2 3y+  1 y+ ey 1 y+  + 1 y+  y e y 1 y+  +  y y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+ ----------------------------------------------------------------------------------------------------------------------------------------= 0   (57)WE4 y  y y 2 3 4y+  2y 1 y+ ey 1 y+  y 1 y+  y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+ + + 1 y+  y 1 3y e y 1 y+  + +  1 y+  y e y 1 y+  +  ----------------------------------------------------exp 1 y+  y e y 1 y+  +  y y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+  y y y 2 3y+  1 y+ ey 1 y+  +  1 y+  y e y 1 y+  +  y y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+ ----------------------------------------------------------------------------------------------------------------------------------------exp+ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------= https://doi.org/10.28924/ada/ma.2.14 an initial approximation of rather than . The resulting approximations, of orders one to four, are: (58) (59) (60) (61) The improvement in the relative error bounds for the interval , for values of close to optimum, are detailed in Table 2. For the case of a third order approximation, and for , the maximum relative error is . Thus, the approximation (62) FIGURE 14. Graph of the relative error in approximations to , for the interval , based on the iterative algorithms specified in Theorem 3.4 and Theorem 3.1. W y  1 e– 0  y WE2 y  re y  WE4 y  WE1 y  WL1 y  WE3 y  WL2 y  WU1 y  WU2 y  WL3 y  WU3 y  WU4 y  WL4 y  z1 y  1 ky+= z1 y  1 y+= z1 y  1 ky+ = Wk1 y  1 ky+ ln = z2 y  1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------= Wk2 y  1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln = z3 y  1 2 k+ y y 1 ky+ ln+ + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ -----------------------------------------------------------------------------------------------------= Wk3 y  z3 y  ln = z4 y  = 1 3 k+ y y 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln y 1 ky+ ln 2 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln++ + + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ 1 1 2 k+ y y 1 ky+ ln+ + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ -----------------------------------------------------------------------------------------------------ln+ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Wk4 y  z4 y  .ln= 0   k k 4465 10000 ---------------= 1.023 10 4– Wk3 y  1 2 k+ y y 1 ky+ ln+ + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ -----------------------------------------------------------------------------------------------------ln = k 4465 10000 ---------------= https://doi.org/10.28924/ada/ma.2.14 1 0.381 0.570 0.518 0.465 represents a good compromise between accuracy and complexity for the interval with a relative error bound of close to . 3.6. Higher Accuracy via Iterative Quadratic Approximations. The iterative approximation detailed in Theorem 3.1 can be improved upon by utilizing quadratic, rather than affine, approximations as illustrated in Figure 15. Theorem 3.5. Iterative Quadratic Approximations for Lambert W Function. An iterative formula, based on quadratic approximations, for the Lambert W function, and valid for , is (63) Proof. The proof is detailed in Appendix C. 3.6.1. Explicit Approximations. Approximations, of orders one to three, are: (64) TABLE 2. Relative error bounds, over the interval , in the approximations , , to the Lambert W function. Iteration order: i Relative error bound: k = 1 Relative error bound: k = 0.4 Relative error bound: k = 0.45 Relative error bound: k = 0.5 2 0.0569 0.0335 0.0250 0.0184 3 4 5 6 0   10 4– 0   Wki y  i 1 2  6    1.33 10 3– 1.87 10 4– 1.05 10 4– 1.50 10 4– 7.23 10 7– 6.46 10 9– 4.96 10 9– 9.97 10 9– 2.15 10 13– 7.98 10 18– 1.11 10 17– 4.43 10 17– 1.90 10 26– 1.24 10 35– 5.53 10 35– 8.78 10 34– FIGURE 15. Illustration of the geometry underpinning an iterative relationship, based on quadratic approximations, to find an approximation to which is the solution of for fixed, . xo x y x– exp= y y 0 WL1 WL2 WL2 x WL1 xo x WU1 ye x– xo WU2 y 1 x– x 2 2 -----+ WL1 x WU1 – WL1 x WU1 – 2 2 --------------------------WL1 +– y y 1 e– WLi 1 WLi 1– -------------- 1 WLi 1– 1 WUi 1– +  1 WLi 1– 2 – 2WLi 1– 1 WUi 1– + +–+ = WUi y WLi ---------ln = WL1 y 1 y+ ------------= WU1 1 y+ .ln= WL1 y  y 1 y+ ------------= WU1 y  1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 (65) (66) (67) 3.6.2. Results. The relative errors in the approximations specified in Theorem 3.5 are shown in Figure 16 and Figure 17. The relative error bounds, for the interval , are detailed in Table 3 and the convergence is cubic in nature. The approximation has a relative error bound for the interval of . 4. CONVERGENCE Consider the case of , , and the error definitions (68) associated with the upper and lower approximations detailed in Theorem 3.1 and as illustrated in Figure 18. The following results hold: TABLE 3. Relative error bounds, over the interval , for the approximations to the Lambert W function detailed in Theorem 3.5. Iteration order i Maximum relative error in Maximum relative error in 1 increasing 0.381 2 increasing 0.0122 3 4 5 WL2 y  1 y --- 1 2y y 1 y+ ln 1 4y 2y 2 2y 1 y+  1 y+ ln+ + +–+ += WU2 y  y 2 1 2y y 1 y+ ln 1 4y 2y 2 2y 1 y+  1 y+ ln+ + +–+ + -----------------------------------------------------------------------------------------------------------------------------------------------ln= WL3 y  1 1 2y y 1 y+ ln q2 y –+ + ------------------------------------------------------------------ = y 1 2y y 1 y+ ln q2 y –+ + 1 y 2 1 2y y 1 y+ ln q2 y –+ + ------------------------------------------------------------------ln+ –+ y 2 1 2y y 1 y+ ln q2 y –+ + 2– + 2y 1 2y y 1 y+ ln q2 y –+ + 1 y 2 1 2y y 1 y+ ln q2 y –+ + ------------------------------------------------------------------ln+ q2 y  1 4y 2y 2 2y 1 y+  1 y+ ln+ + += WU3 y  y WL3 i  ---------------- .ln= 0   WU3 0   1.26 10 6– 0   WLi WUi 1.02 10 5– 1.26 10 6– 1.66 10 17– 2.04 10 18– 7.88 10 53– 9.63 10 54– y 0 xo W y = Li xo WLi – = Ui WUi xo– = ith https://doi.org/10.28924/ada/ma.2.14 Theorem 4.1. Convergence of Iterative Algorithm. For the case of fixed and , the errors associated with the lower approximations , , detailed in Theorem 3.1, are (69) FIGURE 16. Graph of the relative errors in approximations to as specified in Theorem 3.5.W y  y WL2 y  re y  WL1 y  WU1 y  WU2 y  WU3 y  FIGURE 17. Graph of the relative errors in approximations to as specified in Theorem 3.5. W y  y WL1 y re y  WL2 y  WU1 y  WU2 y  WL3 y  WU3 y  FIGURE 18. Error definitions associated with the upper and lower approximations to . ith xo W y = WLi WLi 1+ x xo y x WUi ye x– xo WUi 1+ UiLi y xo W y = WLi i 1 2    Li Li 1– WLi 1– WUi 1– WLi 1– –  1 WLi 1– + -------------------------------------------------------– Li 1– WLi 1– Li 1– Ui 1– +  1 WLi 1– + ---------------------------------------------------–= = L1 xo y 1 y+ ------------– .= https://doi.org/10.28924/ada/ma.2.14 The following results hold: First, the sequence is a monotonically increasing sequence, i.e. . Second, the errors , , define a monotonically decreasing sequence, i.e. . Third, convergence is guaranteed, i.e. , and . Proof. The proof of these results is detailed in Appendix D. 5. SPLINE BASED APPROXIMATION FOR [-1/e,0] The approximations, detailed above in Theorem 3.1 and Theorem 3.4, are sharp at the origin but not at the point . It is useful to have approximations that are sharp at both points and which converge throughout the interval , e. g. [3], eqn. 7 and [17], eqn. 20. The latter approximation is sharp at but not at the origin. One approach, with potential, is to utilize the two point spline approximation for a function as specified by Howard, [16], eqn. 40. For an interval , the order approximation can be written in the form (see Appendix E) (70) where (71) 5.1. Spline Based Approximations for [-1/e,0]. The spline approximation detailed in Equation 70 has the potential to provide approximations for the Lambert W function in the interval with exact values at the end points of this interval. However, an initial problem is that the derivatives of the Lambert W function are undefined at the point . This problem can be overcome by utilizing two suitable transformations. Lemma 1. Transformations. With , , the first transformation (72) with , and , , yields (73) (74) WLi WLi WLi 1–  Li i 1 2    Li 1+ Li Lii  lim 0= WLii  lim xo= WUii  lim xo= 1 e– 1– e 0  1– e f    nth fn x   x– n 1+ an r x – r r 0= n  x – n 1+ bn r  x– r r 0= n + = x   , an r 1  – n 1+ ---------------------------- f r u–    r u– ! ----------------------- n u+ ! u!n! ------------------- u 0= r  1  – u --------------------  = bn r 1  – n 1+ ---------------------------- 1– r u– f r u–    r u– ! -------------------------------------------- n u+ ! u!n! ------------------- u 0= r  1  – u --------------------. = 1 e– 0  1 e– f x  xe x = x 1– y1 g1 x1  1 e --- f x1 1– + == x x1 1–= x1 0 y y1 1 e ---–= y 1– e ------ y1 0 g1 x1  1 e --- 1 x1 1– e x1+  = x1 0  W y  f 1– y  g1 1– y 1 e ---+ 1– = y 1– e ------.= https://doi.org/10.28924/ada/ma.2.14 With the second transformation of , it follows that (75) (76) Proof. The proofs for these results are detailed in Appendix F. 5.1.1. Graphs and Values. Consistent with Equation 76, the Lambert W function is defined in terms of for . Relevant values associated with the points and are specified in Table 4. The graph of , and its inverse , are shown in Figure 19. 5.1.2. Spline Based Approximations. Consistent with Equation 76, and the values tabulated in Table 4, an approximation for the Lambert , over the interval , requires an approximation to the inverse of , , i.e. , , to be determined. A spline approximation for , of order , requires derivatives, of orders zero to , at the points and to be determined. Such values can be determined from the derivatives of at the points and as detailed in Appendix G. The following approximations result. Theorem 5.1. Spline Based Approximations for the Lambert W Function. The order spline based approximation for the Lambert W function, based on the transformation and the points and , is (77) TABLE 4. Values associated with the points and . g x1  g1 x1 = g x1  1 e --- 1 x1 1– e x1+  = x1 0  W y  g 1– y 1 e ---+ 1– = y 1 e ---– . g 1– y 1 e– 0  y 1 e–= y 0= g x1  g 1– y2  FIGURE 19. Graph of and its inverse . y2 g x1 = g 1– y2  x1 y2 g 1– y2  g x1  1 e y 1 e–= y 0= y y1 y 1 e+= x W y = x1 x 1+= y2 g x1 = 1– e 0 1– 0 0 0 1 e 0 1 1 e W 1 e– 0  g x1  x1 0 1  g 1– y2  y2 0 1 e  g 1– n n 0 1 e g 0 1 kth g 1 e–  0 W k y  1– 2e y 1 e ---+ 1 1 y 1 e ---+ 2 y 1 e ---+ 3 y 1 e ---+ 3 2 + + + + 2k y 1 e ---+ k ++ = https://doi.org/10.28924/ada/ma.2.14 for and for appropriately defined constants. Proof. The proof is detailed in Appendix G. 5.1.3. Explicit Approximations. Explicit approximations, of orders one to four, are: (78) (79) (80) (81) k 1 2    W 1 y  1– 2e y 1 e ---+ 1 2 e 1 1 2e ---------- 3 2 2 ----------–+ y 1 e ---+– e 1 2– 2 e -------+ y 1 e ---+++= W 2 y  1– 2e y 1 e ---+ 1 2e 3 ---------- y 1 e ---+– 6e 2 1–  7 2 -------– 2 2 e ----------– y 1 e ---+ –+ e 3 2 17 2 ------- 8– 6 2e– 4 2 e ----------– y 1 e ---+ 3 2 + e 2 10 2 3 ------------- 3– 5e 2 -------– 2 2– y 1 e ---+ 2 += W 3 y  1– 2e y 1 e ---+ 1 2e 3 ---------- y 1 e ---+– 11e 36 --------- y 1 e ---+ –+ e 3 2 191 9 --------- 125 3 2 ----------– 25 e 2 ------------- 8 2 e ---------- 6 2 e 3 2 ----------+ + + y 1 e ---+ 3 2 + e 2 281 6 --------- 146 2 3 ----------------– 32 2e 22 2 18 2 e -------------+ + + y 1 e ---+ 2 – e 5 2 335 9 --------- 40 2– 55e 3 2 2 ----------------- 20 2 e 18 2 e -------------+ + + y 1 e ---+ 5 2 + e 3 371 36 --------- 34 2 3 -------------– 8 2e 2 6 2e 6 2+ + + y 1 e ---+ 3 += W 4 y  1– += 2e y 1 e ---+ 1 2e 3 ---------- y 1 e ---+– 11e 36 --------- y 1 e ---+ 43e 3 2 135 2 ----------------- y 1 e ---+ 3 2 –+ + e 2 4075 27 2 ------------- 895 12 ---------– 91e 2 ---------– 61 2 -------– 27 2 e -------------– 64 2 3e 2 -------------– y 1 e ---+ 2 – e 5 2 6658 2 27 ------------------- 2126 9 ------------– 315e 3 2 2 --------------------– 110 2e– 102 2 e ----------------– 256 2 3e 3 2 ----------------– y 1 e ---+ 5 2 + e 3 8467 2 27 ------------------- 1175 4 ------------– 207 2e 2 – 149 2e– 144 2– 128 2 e ----------------– y 1 e ---+ 3 – e 7 2 4904 2 27 ------------------- 502 3 ---------– 245e 5 2 2 --------------------– 90 2e 3 2 – 90 2e– 256 2 3 e ----------------– y 1 e ---+ 7 2 + e 4 10843 135 2 ---------------- 1315 36 ------------– 55e 3 2 -----------– 41e 2 2 -----------– 21 2e– 64 2 3 -------------– y 1 e ---+ 4  https://doi.org/10.28924/ada/ma.2.14 5.1.4. Results. The relative errors in the spline based approximations to the Lambert W function, as specified in Theorem 5.1, of orders one to five, are shown in Figure 20. The relative error bounds over the interval , for first to fifth order approximations, respectively, are: , , , and . By construction, the approximations are sharp at the points and . The results shown in Figure 20 indicate that the sequence of approximations have good convergence to the Lambert W function over the interval and modest convergence over the interval . 6. IMPROVED APPROXIMATIONS VIA ITERATION Iteration is, potentially, effective in improving the accuracy of an initial approximation. One potential approach is to utilize the iteration potential in the fundamental relation- ship for the Lambert W function as specified by Equation 116. An alternative approach is to utilized the Newton-Raphson method. These two approaches are detailed below. 6.1. Inherent Iteration. The basis for an iterative relationship for the Lambert W function is Equation 116, i.e. (82) It then follows that an approximation, , for , can potentially be improved upon according to (83) (84) 1 e– 0  1.27 10 2– 9.68 10 4– 8.69 10 5– 8.46 10 6– 8.65 10 7– 1 e– 0 1 e– 0  0 1  FIGURE 20. Graph of the relative errors in the spline based approximations, of orders one to five, to the Lambert W function as specified in Theorem 5.1. y order 1 order 5 re y  1 e– W y  y ln W y  ln– .= W0 y  W y  W1 y  y ln W0 y  ln–= W2 y  y ln y ln W0 y  ln–ln–= W21 y  y ln y ln W1 y  ln–ln–= https://doi.org/10.28924/ada/ma.2.14 (85) (86) etc. The number of possible permutations for approximations to the Lambert W function is clearly large as the iteration order increases. Representative relative error bounds, for the interval , are tabulated in Table 5 for the base approximations of , and as specified in Theorem 3.1. Graphs of the relative errors, based on the approximations and , are shown, respectively, in Figure 21 and Figure 22. W3 y  y ln y ln y ln W0 y  ln–ln–ln–= W31 y  y ln y ln y ln W1 y  ln–ln–ln–= W32 y  y ln y ln y ln W2 y  ln–ln–ln–= W3 21 y  y ln y ln y ln W21 y  ln–ln–ln–= W4 y  y ln y ln y ln y ln W0 y  ln–ln–ln–ln–= 0  W0 y  WU3 y = W0 y  WU4 y = W0 y  WU5 y = W0 y  WU4 y = W0 y  WU5 y = FIGURE 21. Graph of the relative errors in approximations to , based on (Equation 35). W y  W0 y  WU4 y = y W0 W1 W3 21 W32 W4 W2 W3 W31 W21 re y  FIGURE 22. Graph of the relative errors in approximations to , based on (Equation 36). W y  W0 y  WU5 y = y W0 W1 W3 21 W32 W2 W3 W31 W21 re y  W4 https://doi.org/10.28924/ada/ma.2.14 6.1.1. Explicit Approximations. The approximation , based on , is (87) and has a maximum relative error bound, over the interval , of which is a factor of lower that the original approximation whose relative error bound is . The approximation , based on , is (88) where is specified by Equation 35. The maximum relative error bound, over the interval , is which is a factor of lower that the original approximation whose relative error bound is . The approximation , based on , is TABLE 5. Relative error bounds, for the interval , based on iteration and for the specified base approximations of (Equation 33), (Equation 35) and (Equation 36). Iteration Form 0  WU3 y  WU4 y  WU5 y  W0 y  WU3 y = W0 y  WU4 y = W0 y  WU5 y = W0 1.33 10 3– 7.23 10 7– 2.15 10 13– W1 4.08 10 4– 1.84 10 7– 4.95 10 14– W2 1.97 10 4– 6.03 10 8– 1.31 10 14– W21 1.40 10 4– 2.47 10 8– 3.95 10 15– W3 1.40 10 4– 2.47 10 8– 3.95 10 15– W31 1.46 10 4– 1.23 10 8– 1.34 10 15– W32 2.33 10 4– 7.43 10 9– 5.04 10 16– W3 21 6.65 10 4– 5.36 10 9– 2.11 10 16– W4 1.46 10 4– 1.23 10 8– 1.34 10 15– W3 W0 y  WU3 y = W3 3 y  y ln y ln y ln WU3 y  ln–ln–ln–= y ln y ln y ln 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------lnln–ln–ln–= 0  1.40 10 4– 9.5 WU3 y  1.33 10 3– W2 W0 y  WU4 y = W2 4 y  y ln y ln WU4 y  ln–ln– = WU4 y  0  6.03 10 8– 12 WU4 y  7.23 10 7– W3 W0 y  WU4 y = https://doi.org/10.28924/ada/ma.2.14 (89) and has a maximum relative error bound, over the interval , of which is a factor of lower that the original approximation whose relative error bound is . 6.2. Newton-Raphson Iteration. Consistent with Equation 22, the approximations stated in Theorem 3.1, Theorem 3.2 and Theorem 3.5 can be utilized as the basis for iterative approximations based on the Newton-Raphson method. Results are detailed in Table 6. As an example, the approximation arising from a first order iteration of (Equation 33), has a maximum relative error bound, over the interval , of which is a factor of lower that the original approximation whose relative error bound is . The approximation is: (90) TABLE 6. Relative error bounds, for the interval , based on Newton-Raphson iteration and for the specified base approximations, , of , , and . Iteration Order (Equation 31) (Equation 33) (Equation 35) (Equation 36) W3 4 y  y ln y ln y ln WU4 y  ln–ln–ln– = 0  2.47 10 8– 29.3 WU4 y  7.23 10 7– 0  W0 y  WU2 y  WU3 y  WU4 y  WU5 y  W0 y  WU2 y = W0 y  WU3 y = W0 y  WU4 y = W0 y  WU5 y = 0 5.69 10 2– 1.33 10 3– 7.23 10 7– 2.15 10 13– 1 8.28 10 3– 5.12 10 6– 1.49 10 12– 1.30 10 25– 2 3.48 10 4– 9.61 10 11– 6.98 10 24– 5.02 10 50– 3 9.95 10 7– 3.91 10 20– 1.62 10 46– 7.67 10 99– 4 8.21 10 12– 7.08 10 39– 9.04 10 92– 1.81 10 196– 5 5.60 10 22– 2.43 10 76– 2.85 10 182– 1.02 10 391– WU3 y  0  5.12 10 6– 260 1.33 10 3– W1 3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln –= 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 3y y 1 y+ ln+ +  --------------------------------------------------------------------------------------------------– 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- https://doi.org/10.28924/ada/ma.2.14 7. APPLICATIONS 7.1. Approximation with Fixed Relative Error Bound. There are many applications where the Lambert W function is used to model the physical nature/characteristics of an entity which are positive in nature. In many of these cases the parameter values are not known with high accuracy. For such cases, highly accurate computation of the Lambert W function is not required and a fixed approximation, with a set relative error bound, is useful rather than relying on, for example, iterative approximations where the relative error achieved depends on the initial approximate used. The relatively simple approximation for the Lambert W function, as specified by Equation 33, i.e. (91) with a relative error bound of over the interval , is likely to be useful. One application, for example, is in the evaluation of the collector current in a common emitter circuit, e.g. [2], eqn. 21. For the interval , the approximation detailed in Equation 79, is of modest complexity, is sharp at the points and and has a relative error bound of . For highly accurate approximations over the interval , an explicit analytical approx- imation, with a relative error bound of , can be specified by utilizing the fifth order iterative approximation, denoted , arising from the iteration specified by Equation 47 and with , (see Table 2). An alternative analytical expression, with a relative error bound of over the interval , is (92) where and are defined, respectively, in Equation 37 and Equation 38. This expres- sion arises from a first iteration of the Newton-Raphson method (Equation 22) utilizing the fifth order approximation specified in Equation 36. The relative error bound is specified in Table 6. 7.2. Upper/Lower Bounds for Lambert W. There is interest in upper/lower bounds for the Lambert W function, e.g. [15] and [24]. Alzahrani and Salem, [1], detail bounds for . The following bounds were proposed by Hoorfar (see, [17], eqn. 21) (93) For the interval , the relative error bound associated with the lower bounded func- tion is ; the relative error bound for the upper bounded function is . WU3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln = 1.33 10 3– 0  1 e– 0  W 2 y  1 e– 0 9.68 10 4– 0  7.98 10 18– Wk5 z1 1 ky+= k 0.4= 1.30 10 25– 0  W1 5 y  d5 y  n5 y  -------------ln d5 y  n5 y  -------------ln yn5 y  d5 y  ----------------– 1 d5 y  n5 y  -------------ln+ -----------------------------------------------– = n5 d5 WU5 y  W 1– y y ln -------------ln y ln ln 2 y ln -----------------------+ W y  y y ln -------------ln e e 1– ----------- y ln ln y ln -----------------------.+  e   0.0568 0.207 https://doi.org/10.28924/ada/ma.2.14 The bounds proposed in [17], eqn. 25, 27, have modest relative errors for and are: (94) The relative error bounds in the lower and upper bounded functions, over the interval , respectively, are and . The relative errors decrease for . By construction, , , as defined in Theorem 3.1, are a sequence of increasingly accurate lower bounds for . Similarly, , , is a sequence of increasingly accurate upper bounds for . For example: (95) with relative error bounds for the interval of, respectively and (see Table 1). Higher order approximations lead to lower relative bounds and these can be made arbitrarily small. The relative error bounds associated with over the interval are, respectively, and . The relative error bounds associated with over the interval are, respectively, and . One potential application for the upper bound is a bound for the prime counting func- tion, e.g. [27]. 7.3. Spline Approximations Based on Upper/Lower Bounds. Consider the upper, , and lower, , bounded functions for the Lambert W function as illustrated in Figure 23 and as defined in Theorem 3.1. For fixed at , a spline approximation, as specified by Equation 70 and based on the points , and , , can readily be determined. From such an approximation, an approximation to can then be specified. Theorem 7.1. Spline Approximations Based on Upper/Lower Bounds. Consider the lower and upper bounded approximations, and , defined in Theorem 3.1. The zero y e y y ln -------------ln y y ln -------------ln 1 y y ln -------------ln+ ---------------------------------– 1 y ln ln y ln -----------------------–ln W y   y y ln -------------ln 1 y ln ln y ln -----------------------– 1 1 y ln ln y ln -----------------------–ln 1 y y ln -------------ln+ -------------------------------------------– .ln– e  5.96 10 3– 4.10 10 3– y 10» WLi y  i 1 2    W y  WUi y  i 1 2    W y  WL3 y  y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 3y y 1 y+ ln+ + --------------------------------------------------------------------------------------------------= W y   WU3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ ----------------------------------------------------------------------------------------------- ln= y 0  0  8.32 10 3– 1.33 10 3– WL4 y  W y  WU4 y   0  3.88 10 6– 7.23 10 7– WL5 y  W y  WU5 y   0  1.08 10 12– 2.15 10 13– ith WUi WLi y yo uo uo exp uo  uo WLi yo = vo vo exp vo  vo WUi yo = xo W yo = ith WLi WUi https://doi.org/10.28924/ada/ma.2.14 order spline approximation for the Lambert W function, based on the approximations and , is (96) The order spline approximation for the Lambert W function, based on the approx- imations and , is (97) where , and is defined by Equation 120. Proof. The proof is detailed in Appendix H. 7.3.1. Results. Results are detailed in Table 7 and clearly show the high accuracy of the approximations. For example, the zero order spline approximations, as specified by Equation 96, yields relative error bounds, for the interval , of , and , respectively, based on third, fourth and fifth order approximations for the FIGURE 23. Illustration of upper and lower bounded approximations to the Lambert W function and the points , , which are the basis for spline based approximations. uo uo exp uo  vo vo exp vo  uo uo exp uo WLi yo = y xo W yo = WLi y  W y  x spline approx. based on WUi y  vo WUi yo = vo vo expyo ith WLi WUi W0 i y  WLi y WUi y  WUi y  exp WLi y  exp– y WUi y  WLi y – + WUi y  WUi y  exp WLi y  WLi y  exp– -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------.= nth ith WLi WUi Wn i y  vo vo exp y– n 1+ vo vo exp uo uo exp– n 1+ ------------------------------------------------------------------------ = y uo uo exp– r n r+ !uo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------- + pr u– uo e r u– uo– r u– ! 1 uo+ 2 r u–  1– ------------------------------------------------------------- n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  + y uo uo exp– n 1+ vo vo exp uo uo exp– n 1+ ------------------------------------------------------------------------  vo vo exp y– r n r+ !vo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------- + 1– r u– pr u– vo e r u– vo– r u– ! 1 vo+ 2 r u–  1– ------------------------------------------------------------------ n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  uo WLi y = vo WUi y = pk 0   3.84 10 5– 8.56 10 12– 6.80 10 25– https://doi.org/10.28924/ada/ma.2.14 Lambert W function detailed in Equation 3.1 Such convergence is approximately quadratic. 7.3.2. Application. The Omega constant, defined as , can be evaluated by using Equation 96 with relative errors, respectively, of , , and for the case of upper and lower bounded approximations of orders two to five, i.e. , , and . 7.4. Asymptotic Approximations. As is evident in the results shown in Figure 11, apart from the results for and , the relative errors in the approximations defined in Theorem 3.1, for a set order, decrease as their argument increases, i.e. for a set order, the approximations asymptotically approach the Lambert W function as their arguments become unbounded. Thus: (98) 7.4.1. Lambert W Function and Prime Counting Function. The prime number theorem states that the relative error between the prime counting function and decreases to zero as , i.e. (99) TABLE 7. Relative error bounds, over the interval , for spline approximations to the Lambert W function based on upper and lower bounded functions. Upper/lower bounded functions Spline order Approximation Relative error bound , 1 increasing re as (Equation 31) 2 increasing re as , 0 (Equation 32, Equation 33) 1 2 3 4 , 0 (Equation 34, Equation 35) 1 2 3 4 , 0 (Equation 36) 1 W 1  1.94 10 5– 4.71 10 11– 2.76 10 22– 9.48 10 45– W0 2 1  W0 3 1  W0 4 1  W0 5 1  0   WL2 y  WU2 y  W1 2 y  W2 2 y  WL3 y  WU3 y  W0 3 3.84 10 5– W1 3 1.92 10 8– W2 3 1.46 10 11– W3 3 1.31 10 14– W4 3 1.27 10 17– WL4 y  WU4 y  W0 4 8.56 10 12– W1 4 5.60 10 22– W2 4 5.05 10 32– W3 4 5.18 10 42– W4 4 5.68 10 52– WL5 y  WU5 y  W0 5 6.80 10 25– W1 5 3.01 10 48– WL1 y  WL2 y  WLi y  W y  i 3    WUi y  W y  i 1 2 3    .  y  y y ln y   y  y y .ln https://doi.org/10.28924/ada/ma.2.14 As (Equation 115), an equivalent statement for the prime number theo- rem is (100) With the manipulation of (101) and with for large, it follows that (102) which has been proposed by Visser [27] and briefly discussed by Iacono and Boyd, [17], section 4.4. Visser has proved that is an upper bound for the prime counting func- tion whilst is a lower bound for large. The magnitude of the relative error in the approximation of is lower than the magnitude of the relative error in the approximation for greater than around with the relative error decreasing as increases. However, the magnitude of the relative error in the approxi- mation is of the order of for . 7.5. Floor and Integral of Floor of Lambert W. It is possible to specify approximations for the Lambert W function, with a set accuracy bound, if an explicit expression for the floor of the Lambert function can be specified. The graph of is shown in Figure 24. Theorem 7.2. Floor of Lambert W. The floor of the Lambert W function can be ascertained, without knowledge of the function itself, according to (103) where, is fixed, is the unit step function and is an upper bound for the Lambert W function as specified in Theorem 3.1. W z z ln  z ln=  y  y W y y ln . W y y ln  y ln xe x ln x x ln+ W y  W y  ln+ = = = = W y  W y  ln» y  y  y W y   y W y  y y ln y  y  y W y   y  y W y y ln  y 5000 y  y  y W y  0.05 y 10 9 = W y  FIGURE 24. Graph of and . W y  W y  y W y  W y  2e 2e 4e 4 3e 3 W y  u y ke k –  k 1=   u y ke k –  k 1= 1 WUi y +  = = y 0  i 1 2    u WUi y  https://doi.org/10.28924/ada/ma.2.14 Proof. This result follows from the fact that and , , fixed, is an upper bound for , i.e. . It then follows that the upper limit of the summation can be specified as . 7.5.1. Notes. The simplest upper bound, specified in Theorem 3.1, for the Lambert W function is and, thus: (104) The graphs of , for , are shown in Figure 25 and it follows that the use of as the upper limit for the summation results in an increasing small number of additional zero terms in the summation defining as increases. The use of results, depending on the value of , in an additional zero term in the summation defining . 7.5.2. Integral of Floor of Lambert W. Using the result for the floor of the Lambert W function, as specified in Theorem 7.2, it is possible to explicitly specify the integral of the floor of the Lambert W function. Theorem 7.3. Integral of Floor of Lambert W. The integral of the floor of the Lambert W function can be explicitly specified according to (105) where is specified in Theorem 7.2. Proof. The required result follows, consistent with the graph of shown in Figure 24, according to W ke k  k= WUi y  i 1 2    i W y  W y  WUi y  1 WUi y + WU1 y  1 y+ ln= W y  u y ke k –  k 1= 1 1 y+ ln+  = y 0 . 1 WUi y  W y –+ i 1 2 3   1 WU1 y + 1 1 y+ ln+= W y  y 1 WU2 y + y W y  FIGURE 25. Graph of for . 1 WUi y  W y –+ i 1 2 3  y 1 WUi y  W y –+ i 1= i 2= i 3= W   d 0 y  e e 1– 2 ------------------- 1– e W y  1 W y  2 W y  2 –+ + + W y  1– W y + e1 W y + W y  2 e W y  1– +  += W y  y W y  e W y  –  W y  W y  https://doi.org/10.28924/ada/ma.2.14 (106) where the following result has been used: (107) 7.6. Set Accuracy Approximation for Lambert W. Consider a set accuracy limit of required for the evaluation of . This can be achieved by a step approximation, with a resolution of , and such an approximation is: (108) where , , fixed, is a set function defined in Theorem 3.1. As an example, the error in the approximation to , with a resolution of , is shown in Figure 26. 7.6.1. Computationally Efficient Implementation. The direct approximation detailed in Equation 108, for a set error level of , requires, approximately, a summation of terms. The approach detailed below requires, approximately, the summation of terms which represents a significant reduction for modest to large. For example, for and , the direct approach requires a summation of approximately terms whilst the approach detailed below requires close to terms. W   d 0 y  k k 1+ ek 1+ ke k –  k 0= max 0 W y  1–   W y  y W y  e W y  – += e e 1– 2 ------------------- 1– e W y  1 W y  2 W y  2 –+ + + W y  1– W y + e1 W y + W y  2 e W y  1– +  += W y  y W y  e W y  –  k k 1+ ek 1+ ke k –  k 1= n 1–  e e 1– 2 ------------------- 1– e n 1 n 2n 2 –+  ne 1 n+ 1– n+  n 2 e 1– n+ + + + .=  W y   W y   u y kek–  k 1= 1 1  --- WUi y +  = y 0 . WUi i 1 2    i W y   1 10= FIGURE 26. Graph in the error in for the case of .W y   0.1= y W y  W y –  10 q– = 10 q W y  W y  11q 1–+ q W y  10= q 6= 10 7 75 https://doi.org/10.28924/ada/ma.2.14 Theorem 7.4. Direct Evaluation of Lambert W with Specified Resolution. The Lambert W function can be evaluated, with a maximum error of , according to (109) where is defined in Theorem 7.2 and is the digit in the decimal expansion of : (110) Proof. The floor of the Lambert W function has been defined in Theorem 7.2. Consider the illustration shown in Figure 27. With (111) and with being the digit to the right of the decimal point, it follows that (112) Iteration with finer resolution, and from the point defined by , yields etc.  10 q– = W y  W y  d1 y   dq y + + + = W y  10 q dq y  qth W y  d1 y  1 10 ------ u y W y  k 10 ------+ W y  k 10 ------+exp– k 1= 10 =  d2 y  1 100 --------- u y W y  d1 y  k 100 ---------+ + W y  d1 y  k 100 ---------+ +exp– k 1= 10  =  dq y  1 10 q -------- u y W y  di y  i 1= q 1–  k 10 q --------+ + W y  di y  i 1= q 1–  k 10 q --------+ +exp– k 1= 10  .= W y  W y  d1 y  d2 y  + + += dq y  qth d1 y  1 10 ------ u y W y  k 10 ------+ W y  k 10 ------+exp– . k 1= 10 = W y  d1 y + d2 y  FIGURE 27. Illustration of demarcation that underpins determination of , to a set resolution of , between and . W y   ke k k 1+ ek 1+ y k + k 1+ k + ek + k ke k W y  k 2+ k 2+ ek 2+ k 1+ ek 1+ k 1 –+ ek 1 –+ k 1 –+ https://doi.org/10.28924/ada/ma.2.14 The number of terms in the summation defined by Equation 109 comprises approxi- mately terms for the evaluation of , plus terms for the summations comprising and terms for the summation of , , . 7.6.2. Note. Theorem 7.4 defines a series for the Lambert W function, which, by construction is convergent, i.e. (113) and is such that (114) 8. CONCLUSION A geometric based approach for iteratively specifying approximations to the Lambert W function, which can achieve any set relative error bound over the interval , was detailed. The approximations are also valid for the interval but are not sharp at the point . Convergence was proved. For the interval , arbitrarily accurate approximations, based on a two point spline approximation, were specified. Iteration, either by using the iteration structure inherent in the definition of the Lambert W func- tion, or via the Newton-Raphson method, leads to significantly improved approximations albeit with increasing complex functional forms. Applications of the approximations were detailed and include, first, analytical expres- sions for the Lambert W function that achieve set relative error bounds over the interval . Second, based on the geometry inherent in the approximations, upper and lower bounds for the Lambert W function that can be made arbitrary accurate. Third, higher accuracy spline based approximations for the Lambert W function based on the defined upper and lower bounded functions. Fourth, analytical expressions for the evaluation of , and the integral of , without knowledge of for . Finally, a direct approach for evaluating the Lambert W function to achieve a prior defined error. Acknowledgement: The author is pleased to acknowledge the support of Prof. A. Zoubir, SPG, Technische Universität Darmstadt, Darmstadt, Germany, who hosted a visit where the research, underpinning this paper, was completed. APPENDIX A. PROPERTIES OF LAMBERT W FUNCTION The following are useful properties of the Lambert W function: (115) (116) The latter formula underpins the iteration: W y  W y  10q d1 y  d2 y   dq y   q 1– d1 y  d2 y +  d1 y  d2 y   dq y + + + W y  W y  d1 y  d2 y  + + += y 0  W y  W y  d1 y   dq y + + +– 10 q– . 0   1 e– 0  1 e– 1 e– 0  0  W y  W y  W y  y 0   W z z ln  z ln = z 0 W y  y W y  ------------ln y ln W y  ln– = = y 0. https://doi.org/10.28924/ada/ma.2.14 (117) To prove that , consider which implies and, thus, . The relationship follows from the definitions , which implies and, thus, . A.1. Differentiation. The derivatives of the Lambert W function are defined according to (118) (119) where the second inequalities follow from the relationship and the polyno- mial is defined according to (120) The coefficients in this expression are defined according to (https://oeis.org/A042977; [7], eqn 3.4; [23], p. 1370): (121) Explicit expressions are: (122) APPENDIX B. PROOF of Theorem 3.2 The iteration formula, as specified in Theorem 3.1, yields the first order approxima- tions as stated in Equation 30: (123) W y  y ln y ln W y  ln– ln– = W y  y ln y ln y ln W y  ln– ln– ln– =  W z z ln  z ln= f x  xe x = f z ln  z z ln= z ln f 1– z z ln  W z z ln = = W y  y ln W y  ln–= y xe x = x W y = y ln x x ln+= x y ln W y  ln–= W 1  y  e W y – 1 W y + ---------------------- W y  y 1 W y +  ------------------------------= = W k  y  W k y pk W y   y k 1 W y +  2k 1– -------------------------------------------- pk W y  e kW– y  1 W y + 2k 1– ------------------------------------------= = y W y eW y  = pk pk r  ck 0 ck 1 r ck 2 r 2  ck k 1– r k 1– .+ + + += cn k 0 k 0 1– n 1+ n n 1– k 0= n 1– cn 1– k 1–– 3 n 1–  k 1+ – cn 1– k– k 1+ cn 1– k 1++ 1 k n 3–  n 1– cn 1– k 1–– 3 n 1–  k 1+ – cn 1– k– k n 2–= n 1– cn 1– k 1–– n 1– != k n 1–=         = p1 r  1= p2 r  2 r+ – = p3 r  9 8r 2r 2 + + = p4 r  64 79r 36r 2 6r 3 + + + – = p5 r  625 974r 622r 2 192r 3 24r 4 + + + + = p6 r  7776 14543r 11758r 2 5126r 3 1200r 4 120r 5 + + + + + .–= WL1 y  y 1 y+ ------------ n1 y  d1 y  -------------= = WU1 y  1 y+ ln 1 ln– d1 y  ln n1 y  y -------------ln– = = https://doi.org/10.28924/ada/ma.2.14 where and . The second order approximations, as specified by Equation 31, can be written in the form (124) where (125) The third order approximations, as specified by Equation 32 and Equation 33, can be written in the form: (126) (127) where (128) Thus, iteration yields the general formulas: (129) where (130) APPENDIX C. PROOF of Theorem 3.5 Consider the geometry illustrated in Figure 15 and an initial approximation for , based on the intersection of the second order Taylor series for at the origin, i.e. n1 y  y= d1 y  1 y+= WL2 y  y 1 1 y+ ln+  1 2y+ --------------------------------------- n2 y  d2 y  -------------= = WU2 y  1 2y+ ln 1 1 y+ ln+ ln– d2 y  ln n2 y  y -------------ln– = = n2 y  y 1 1 y+ ln+  n1 y  1 d1 y  ln+ = = d2 y  1 2y+ n1 y  d1 y .+= = WL3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ 1 3y y 1 y+ ln+ + ------------------------------------------------------------------------------------------------------------------------------------ n3 y  d3 y  -------------= = WU3 y  1 3y y 1 y+ ln+ + ln 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ln–= d3 y  ln n3 y  y -------------ln– = n3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+= n2 y  1 d2 y  ln n2 y  y -------------ln–+ = d3 y  1 3y y 1 y+ ln+ + n2 y  d2 y .+= = WLi ni y  di y  ------------= WUi di y  ln ni y  y ------------ln– = ni y  ni 1– y  1 di 1– y  ln ni 1– y  y -------------------ln–+ = n0 y  y= d0 y  1=  di y  ni 1– y  di 1– y .+= xo ye x– https://doi.org/10.28924/ada/ma.2.14 , and which is . The problem with such an approximation is that it only yields a real solution for . A practical approach is to utilize an affine approximation at the origin of lead- ing to the first approximation of . To establish a further approximation, consider the point where the level intersects which is (131) A second level approximation follows by finding the intersection of a second order Taylor series at the point , which is (132) with to yield (133) and (134) Iteration in this manner leads to the general iteration formulas: (135) Simulation results (see Figure 16) indicate that the approximations also have good con- vergence for the interval but the approximations are not sharp at . APPENDIX D. PROOF of Theorem 4.1 Consider the case of fixed, , and the illustration shown in Figure 18. By con- struction, , and for . Further, (see Equation 25) (136) and it follows that , , is a monotonically increasing sequence. Using Equation 136 it follows that y 1 x– x 2 2+  x WL1 1 y --- 1 y 1 2y y 2 –++= 1 2– y 1 2+  y 1 x–  WL1 y 1 y+ = WL1 ye x– WU1 y WL1 ---------ln 1 y+ .ln= = WU1 WL1 x WU1 – WL1 – x WU1 – 2WL1 2 -------------------------------------+  x WL2 1 WL1 --------- 1 WL1 1 WU1 +  1 WL1 2 – 2WL1 1 WU1 + +–+= WU2 y WL2 --------- .ln= WLi 1 WLi 1– -------------- 1 WLi 1– 1 WUi 1– +  1 WLi 1– 2 – 2WLi 1– 1 WUi 1– + +–+ = WUi y WLi ---------ln .= 1 e– 0  y 1 e–= y y 0 Li 0 Ui 0 WUi WLi  i 1 2    WLi 1+ WLi 1 WUi + 1 WLi + ------------------- = WLi i 1 2    https://doi.org/10.28924/ada/ma.2.14 (137) As , and , it follows that (138) where . Thus, as monotonically increases with , it follows that mono- tonically decreases with , i.e. . It then follows that (139) Hence, convergence is guaranteed as . Thus: and . The result implies that and, thus, . APPENDIX E. ALTERNATIVE FORM FOR SPLINE APPROXIMATION The general form for a order, two point, spline approximation for a function , over the interval , has been detailed in Howard, [16], eqn. 40. The assumption is that the function is at least order differentiable over the interval . The approximation, denoted , can be written in the modified form: (140) In this equation, the double summation can be rewritten by utilizing the transformations and , , . The possible values of are detailed in Table 8 and for fixed, the valid values for are from the set . TABLE 8. Valid values of for , . i k 0 1 2 3 ... n-2 n-1 n 0 0 1 2 3 n-2 n-1 n 1 1 2 3 4 n-1 n 2 2 3 4 5 n 3 3 4 5 6 Li Li 1+ – xo WLi –  xo WLi 1 WUi +  1 WLi + ---------------------------------–– WLi WUi WLi –  1 WLi + ---------------------------------------.= = WUi WLi – Li Ui += Ui 0 Li 1+ Li WLi 1 WLi + ------------------– Li Ui +  1 WLi 1 WLi + ------------------– Li riLi= = ri 1 1 WLi + ------------------= WLi i ri i 0 ri 1+ r i 1  0 Li 1+ L1 rk k 1= i  L1 r1 i .   0 r1 1 1 y 1 y+ + --------------------------------= 1  Lii  lim 0= WLii  lim xo= Li 1+ Li WLi 1 WLi + ------------------– Li Ui + = Uii  lim 0= WUii  lim xo= nth f    f nth    fn fn x   x– n 1+  – n 1+ ---------------------------- f k    k! ---------------- n i+ ! i!n! ----------------- i 0= n k–  x – k i+  – i -------------------------- k 0= n  += x – n 1+  – n 1+ ---------------------------- 1– kf k    k! ------------------------------ n i+ ! i!n! ----------------- i 0= n k–   x– k i+  – i ------------------------- . k 0= n  r i k+= u i= k 0 1  n    i 0 1  n k–    r i k+= r i 0 1  r    r i k+= k 0 1  n    i 0 1  n k–    https://doi.org/10.28924/ada/ma.2.14 With , and , , Equation 140 can be written as (141) Thus: (142) where (143) APPENDIX F. PROOF of Lemma 1 First, the definitions of , and with , imply: (144) It then follows that which implies . With and , the required result of then follows. Second, the transformation of , , results in (145) As it then follows that and the final result follows: (146) n-2 n-2 n-1 n n-1 n-1 n n n TABLE 8. Valid values of for , . i k 0 1 2 3 ... n-2 n-1 n r i k+= k 0 1  n    i 0 1  n k–     r 0 1  n    u 0 1  r    i u= k r u–= fn x   x– n 1+  – n 1+ ---------------------------- x – r f r u–    r u– ! ----------------------- n u+ ! u!n! ------------------- u 0= r  1  – u -------------------- r 0= n  += x – n 1+  – n 1+ ----------------------------  x– r 1– r u– f r u–    r u– ! -------------------------------------------- n u+ ! u!n! ------------------- u 0= r  1  – u -------------------- . r 0= n  fn x   x– n 1+ an r x – r r 0= n  x – n 1+ bn r  x– r r 0= n += an r 1  – n 1+ ---------------------------- f r u–    r u– ! ----------------------- n u+ ! u!n! ------------------- u 0= r  1  – u -------------------- = bn r 1  – n 1+ ---------------------------- 1– r u– f r u–    r u– ! -------------------------------------------- n u+ ! u!n! ------------------- u 0= r  1  – u --------------------. = f g1 x x1 1–= x 1– y1 g1 x1  1 e --- x1 1– e x1 1– + = x1 0.= y1 1 e– f x1 1– = f 1– y1 1 e–  x1 1–= x1 g1 1– y1 = y y1 1 e–= f 1– y  g1 1– y 1 e+  1–= y2 g x1  g1 x1  y1= = = x1 0 g 1– y2  x1 g 1– y1 .= = x1 g1 1– y1 = g1 1– y1  g 1– y1 = f 1– y  g1 1– y 1 e ---+ 1– g 1– y 1 e ---+ 1.–= = https://doi.org/10.28924/ada/ma.2.14 APPENDIX G. PROOF of Theorem 5.1 With denoting the differentiation operator, the following well known results apply for an arbitrary function : (147) (148) (149) etc. Consider , as defined by Equation 75, over the interval . A spline approxima- tion (see Equation 70) for , of order , requires derivatives, of orders zero to at the points and , to be determined. Using the above formulas, such values can be determined from the derivatives of at the points and and values of these deriva- tives are tabulated in Table 9. To determine the derivative values at zero, the standard Taylor series expansion for the exponential function can be used to yield the alternative form for of (150) Using Equation 70, and the derivative values given in Table 9, the spline approximations for , based on the points and and for orders one to four, are: (151) (152) D f D f 1– z   1 f 1  x  ---------------- x f 1– z = = D 2  f 1– z   f 2  x – f 1  x   3 ----------------------- x f 1– z = = D 3  f 1– z   f 3  x – f 1  x   4 ----------------------- 3 f 2  x   2 f 1  x   5 --------------------------+ x f 1– z = = D 4  f 1– z   f 4  x – f 1  x   5 ----------------------- 10f 3  x f 2  x  f 1  x   6 -------------------------------------- 15 f 2  x   3 f 1  x   7 -----------------------------–+ x f 1– z = = g 0 1  g 1– n n 0 1 e g 0 1 g g x1  x1 2e ---------- 1 2 i 1+ x1 i i 2+ ! --------------------- i 1=  + .= g 1– y2  0 1 e g1 1– y2  2ey2 1 2 ey2 1 1 2e ---------- 3 2 2 ----------–+– ey2 2 1 2– 2 e -------++= g2 1– y2  2ey2 1 2ey2 3 ---------------– 6e 2 1–  7 2 -------– 2 2 e ----------– y2 2 –+ e 3 2 17 2 ------- 8– 6 2e– 4 2 e ----------– y2 3 e 2 10 2 3 ------------- 3– 5e 2 -------– 2 2– y2 4 + = https://doi.org/10.28924/ada/ma.2.14 (153) (154) In general: (155) TABLE 9. Values of the derivatives of at the points zero and one. order 0 0 1 2 3 4 5 g x1  g i  0  g i  1  1 e ------ 1 2e ---------- e 2 ------ 2 3 e ---------- e 1 e 4 ---– 5 12 2e ---------------- 3 e 2 ---------- 1 e– e 2 4 -----+ 11 45 2e ---------------- 2 e 1 3e– 9e 2 4 -------- 15e 3 32 -----------–+ 59 432 2e ------------------- 5 e 2 ---------- 1 8e– 27e 2 2 ----------- 15e 3 2 -----------– 21e 4 16 -----------+ + g3 1– y2  2ey2 1 2ey2 3 ---------------– 11ey2 2 36 -------------- e 3 2 191 9 --------- 125 3 2 ----------– 25 e 2 ------------- 8 2 e ---------- 6 2 e 3 2 ----------+ + + y2 3 +–+ e 2 281 6 --------- 146 2 3 ----------------– 32 2e 22 2 18 2 e -------------+ + + y2 4 – e 5 2 335 9 --------- 40 2– 55e 3 2 2 ----------------- 20 2 e 18 2 e -------------+ + + y2 5 + e 3 371 36 --------- 34 2 3 -------------– 8 2e 2 6 2e 6 2+ + + y2 6 = g4 1– y2  2ey2 1 2ey2 3 ---------------– 11ey2 2 36 -------------- 43e 3 2 y2 3 135 2 ----------------------–+ + e 2 4075 27 2 ------------- 895 12 ---------– 91e 2 ---------– 61 2 -------– 27 2 e -------------– 64 2 3e 2 -------------– y2 4 – e 5 2 6658 2 27 ------------------- 2126 9 ------------– 315e 3 2 2 --------------------– 110 2e– 102 2 e ----------------– 256 2 3e 3 2 ----------------– y2 5 + e 3 8467 2 27 ------------------- 1175 4 ------------– 207 2e 2 – 149 2e– 144 2– 128 2 e ----------------– y2 6 – e 7 2 4904 2 27 ------------------- 502 3 ---------– 245e 5 2 2 --------------------– 90 2e 3 2 – 90 2e– 256 2 3 e ----------------– y2 7 + e 4 10843 135 2 ---------------- 1315 36 ------------– 55e 3 2 -----------– 41e 2 2 -----------– 21 2e– 64 2 3 -------------– y2 8 = gk 1– y2  2ey2 1 1y2 2y2 2 3y2 3 4y2 4  2ky2 2k + + + + + +  https://doi.org/10.28924/ada/ma.2.14 by Equation 70 with , based on the points and for appropriately defined constants and, thus: (156) APPENDIX H. PROOF of Theorem 7.1 A zero order spline approximation is simply an affine approximation between the two specified points. Consistent with Figure 23, the zero order spline approximation to , denoted , is an affine approximation between the points and . Thus: (157) With the approximation it follows that (158) Simplification yields (159) Substitution of and yields the required result. H.1. General Result. The general result arises from the spline approximation, specified where , , and with (160) Here is defined by Equation 120. The order spline approximation, for , is W y  g 1– y 1 e ---+ 1–= 2e y 1 e ---+ 1 1 y 1 e ---+ 2 y 1 e ---+ 3 y 1 e ---+ 3 2  2k y 1 e ---+ k + + + + + 1.– W y  f0 uo uo exp uo  vo vo exp vo  f0 y  uo y uo uo exp–  vo uo– vo vo exp uo uo exp– --------------------------------------------------------- y uo uo exp vo vo exp .+= xo W yo = f0 yo  xo uo yo uo uo exp–  vo uo– vo vo exp uo uo exp– ---------------------------------------------------------.+ xo uovo vo exp uo exp–  yo vo uo– + vo vo exp uo uo exp– ------------------------------------------------------------------------------------------------. uo WLi yo = vo WUi yo = f y  W y = uo uo exp uo  vo vo exp vo  uo WLi yo = vo WUi yo = W 1  y  e W y – 1 W y + ----------------------= W k  y  pk W y  e kW– y  1 W y + 2k 1– ------------------------------------------= k 1 2   . pk nth y uo uo exp vo vo exp  https://doi.org/10.28924/ada/ma.2.14 (161) The results , imply that (162) assuming . Hence: (163) The required result follows: the approximation for , denoted , arises for the case of . Thus, . fn y  vo vo exp y– n 1+ vo vo exp uo uo exp– n 1+ ------------------------------------------------------------------------ = y uo uo exp– r W r u–  uo uo exp  r u– ! ------------------------------------------------- n u+ ! u!n! ------------------- u 0= r  1 vo vo exp uo uo exp– u ----------------------------------------------------------------- r 0= n  + y uo uo exp– n 1+ vo vo exp uo uo exp– n 1+ ------------------------------------------------------------------------  vo vo exp y– r 1– r u– W r u–  vo vo exp  r u– ! --------------------------------------------------------------------- n u+ ! u!n! ------------------- u 0= r  1 vo vo exp uo uo exp– u ----------------------------------------------------------------- r 0= n  W uo uo exp  uo= W vo vo exp  vo= W r u–  uo uo exp  pr u– uo e r u– uo– 1 uo+ 2 r u–  1– ----------------------------------------------= W r u–  vo vo exp  pr u– vo e r u– vo– 1 vo+ 2 r u–  1– ----------------------------------------------= r u fn y  vo vo exp y– n 1+ vo vo exp uo uo exp– n 1+ ------------------------------------------------------------------------ = y uo uo exp– r n r+ !uo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------- + pr u– uo e r u– uo– r u– ! 1 uo+ 2 r u–  1– ------------------------------------------------------------- n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  + y uo uo exp– n 1+ vo vo exp uo uo exp– n 1+ ------------------------------------------------------------------------  vo vo exp y– r n r+ !vo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------- + 1– r u– pr u– vo e r u– vo– r u– ! 1 vo+ 2 r u–  1– ------------------------------------------------------------------ n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  W yo  Wn i yo  y yo= Wn i yo  fn yo = https://doi.org/10.28924/ada/ma.2.14 https://doi.org/10.28924/ada/ma.2.14 https://doi.org/10.1080/10652469.2018.1528247 https://doi.org/10.1109/81.895330 https://doi.org/10.1016/S0378-4754(00)00172-5 https://doi.org/10.1007/s10910-018-0932-3 https://doi.org/10.1007/s10910-018-0932-3 https://doi.org/10.1016/S0893-9659(98)00097-4 https://doi.org/10.1007/BF02124750 https://doi.org/10.1155/2021/6695559 https://doi.org/10.1017/S0022377805003788 https://doi.org/10.30538/oms2021.0149 https://doi.org/10.1016/j.cam.2012.11.021 https://scholar.google.com/citations?user=invpkckAAAAJ&hl=en&oi=sra https://doi.org/10.28924/ada/ma.2.14 https://doi.org/10.1016/j.bej.2012.01.010 https://doi.org/10.3390/mca24020035 https://doi.org/10.1007/s10444-017-9530-3 https://doi.org/10.1016/j.physleta.2015.12.004 https://doi.org/10.1111/2041-210X.12568 https://doi.org/10.1088/0143-0807/36/3/035030 http://arxiv.org/abs/1408.3999 https://doi.org/10.1093/imamat/hxu057 https://doi.org/10.1139/p00-065 https://doi.org/10.1016/j.cpc.2012.07.008 https://doi.org/10.3390/math6040056