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\ o£f<l Report No. 39^ 
 
 JVLaA^U 
 
 COO-lU69-0l62 
 
 HIGH ORDER STIFFLY STABLE METHODS FOR 
 ORDINARY DIFFERENTIAL EQUATIONS 
 
 by 
 
 April 1970 
 
 M. K. Jain 
 V. K. Srivastava 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
Report No. 39^ 
 
 HIGH ORDER STIFFLY STABLE METHODS FOR 
 ORDINARY DIFFERENTIAL EQUATIONS* 
 
 by 
 
 M. K. Jain 
 V. K. Srivastava 
 
 April 1970 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 This report is supported in part by contract U. S. AEC AT(ll-l)lU69 and 
 in part by the National Science Foundation under grant N3F-GJ-217, and 
 by the Department of Computer Science, University of Illinois, Urbana- 
 Ch amp ai gn . 
 
ACKNOWLEDGMENT 
 
 We wish to thank Professor C. W. Gear for his helpful 
 suggestions. Thanks are also extended to Miss Barbara Hurdle for 
 typing the final manuscript. 
 
ABSTRACT 
 
 Ordinary differential equations with greatly different time 
 constants arise in a wide variety of important physical problems. The 
 predictor-corrector method of solution is examined here. 
 
 The existence of stiffly stable methods of order as high as 
 eight has been known for some time. In this report, we have examined 
 a class of methods for finding methods of order greater than eight. 
 We find that the choice of 
 
 a(C) = 5 k ~ r (S-c) r , r = 0, 1, 2,... k and -1 <_ c < 1 
 
 leads to the methods of order k as high as eleven for some c and r. 
 The multistep methods are stiffly stable for k <_9 if r = 3, for 
 k <: 10 if r = U, and for k <_ 11 if r ■ 6. 
 
 We found no twelfth order and higher stiffly stable methods. 
 However, if we select 
 
 (P r ) o(0 = C k_r U r - c r ) or 
 
 k-r „ _r-i l 
 
 (q ) o(?) = c~ L z r A c 
 
 i=0 
 
 where r = 0, 1, 2,... k and -1 < c < 1. 
 
 We obtain stiffly stable methods of P type if k <_ 6 and of 
 
 Q type if k <_ 7- 
 
TABLE OF CONTENTS 
 
 Page 
 ABSTRACT 
 
 1. INTRODUCTION 1 
 
 2. HIGH ORDER STIFFLY STABLE METHODS k 
 
 3. CONCLUSIONS 8 
 
 LIST OF REFERENCES 9 
 
 APPENDIX 1 10 
 
 APPENDIX II 20 
 
 APPENDIX III ; 35 
 
LIST OF FIGURES 
 
 Figure Page 
 
 1 Stability and Accuracy Region 3 
 
 2 Locus of p(0/o(£) £ = e lQ , e[0, 2tt) ill 
 
 3 Seventh Order Formulas k2 
 k Eighth Order Formulas ^3 
 
 5 Ninth Order Formulas hk 
 
 6 Tenth Order Formulas ^5 
 
 7 Eleventh Order Formulas k6 
 
 8 Fifth Order Formula of Class II 1+7 
 
1 . INTRODUCTION 
 
 The mathematical analysis of the problems encountered in 
 reactor calculations, circuit analysis and chemical kinetics often 
 leads to stiff systems of ordinary differential equations, i.e. systems 
 with widely separated time constants. The standard methods for the 
 numerical integration of the ordinary differential equations are not well 
 suited to the stiff differential equations for reasons of stability, the 
 time steps have to be adopted to the fast decaying components while one 
 is interested in the slowly decaying ones. 
 
 A number of authors have proposed special algorithms to 
 overcome this difficulty (Lawson [l], Richard, Lanning and Torrey [2], 
 and Liniger and Willoughby [3]). However, there has remained still a 
 strong desire to develop methods of Runge-Kutta or multistep type with 
 improved stability and accuracy properties so that the above difficulty 
 is diminished. The object of this investigation is to develop high 
 order stiffly stable multistep methods. A linear k-step method for the 
 numerical solution of 
 
 y' = f(t, y), y(t Q ) = n 
 
 can be written in the form 
 
 (1) k k 
 
 cu y , = £ a v , . + h E 8. y' . 
 
 °n+l . . i °n+l-i . . i J n+l-i 
 
 i=l i=0 
 
 with a ^ and |a | + |g | ^ or symbolically, 
 
 (2) e (E) W* ■ h 0(E > ';» 
 
 where E is the translation operator (Ey = y ) and p and a are 
 polynomials defined by 
 
 te\ r k r 1 ^ 1 r k " 2 
 
 p(C) = a C - a C - « 2 £ -...- a fc 
 
 (3) a(c) = e c k + e 1 c 1 " 1 + 3 2 ? k " 2 +...+ e k 
 
Therefore, this formula can only be used if one knows the 
 values of the solution at k successive points. These k values will 
 he assumed to he given. Further, it can be assumed without loss of 
 generality that the polynomials p(0 and a(£) have no common factors 
 since in general case, (l) can be reduced to an equation of lower order. 
 
 Definition 1.1. The formula (1) will be said to be of order p >_ 
 if it fulfills the p + 1 conditions 
 
 (U) k 
 
 a = I a. 
 ° 1=1 X 
 
 k k 
 
 a = Z (l-i) S a. + s Z (l-i) S 6. , s = 1, 2,... p 
 
 i=l x i=0 X 
 
 Thus the method is of order p if for any y e a ^ and for some 
 
 nonzero o ,. 
 p+1 
 
 ( 5 ) k k 
 
 a y = I a. y ., . + h Z 3. y'.+c, h P+1 y P T^ 
 n+1 . . i J n+l-i . _ i J n+l-i p+1 J (t) 
 
 1=1 1=0 
 
 ♦ o( h P +2 ) 
 
 where y " is the (p+l)-st derivative of y evaluated for some t 
 between t 7 , and t . The last two terms represent the truncation 
 error. 
 
 Multistep methods were first investigated by Dahlquist [h] 
 who defined the following: 
 
 Definition 1.2. A multistep method is called A-stable if all solutions 
 of (1) tend to zero, as n -»■ °°, when it is applied to the differential 
 equation of the form y ' = Xy and X is a (complex) constant with negative 
 real parts. 
 
 He then proved that the order p of an A-stable linear 
 multistep method cannot exceed two. The order two method with the 
 smallest error term is the trapezoidal rule. Widlund [5] has shown the 
 
existence of methods of orders three and four which are stable in the 
 wedge |arg(hX) — tt J < a of the negative half plane for any 
 
 a < p-. Norsett [6] has extended these results to the methods of orders 
 
 five and six. Gear [7,8] has used the conditions which are necessary 
 for stiff differential equations. The requirements are shown in 
 Figure 1. 
 
 STABLE 
 
 hX = h - — plane 
 
 3y * 
 
 Figure 1 
 Stability and Accuracy Region 
 
 Thus the numerical methods suitable for stiff differential equations 
 depend on the parameters D, 0, a and on the definition of accuracy. 
 Gear has termed such methods as stiffly stable methods and has obtained 
 methods of order as high as six for suitable parameters D, 0, and a. 
 Dill [9] has used computer graphic techniques for finding methods of 
 orders seven and eight. We shall show here that the methods of order 
 as high as eleven can be obtained for suitable parameters. 
 
2. HIGH ORDER STIFFLY STABLE METHODS 
 
 The discretization error of the multistep method is defined 
 
 as the difference between the value y calculated from (l) and the 
 
 exact solution y(t ). 
 n 
 
 Define 
 
 e - y - y(t ) 
 
 n J n n 
 
 Then the error e obeys the difference equation 
 
 (6) 
 
 (a. - hAgJe = Z (a. + hAB. )e ., . - T 
 
 n+1 . , i l n+l-i n 
 
 1=1 
 
 for the equation y' = Ay. 
 Assuming that 
 
 we get from (6) the well known characteristic equation for £, 
 
 ( 7) k 
 
 (a - hX6 n )C n+1 - £ (a. + hAB.)C n+1-1 + T (y, t , h) = 
 . , i i n n 
 
 i=l 
 
 where T is the truncation error for the exact solution. For convergence, 
 
 n 
 
 it is necessary to bound the solution of the imhomogeneous difference 
 equation (7) (see, Henrici [10]). It depends on the stability of the 
 corresponding homogeneous equation obtained from (7) with T set to zero. 
 This difference equation is stable if and only if all the roots of the 
 polynomial equation 
 
 k 
 
 (a - hAB )£ n+1 - S (a. + hA6.)C X ■ 
 . , i i 
 1=1 
 
 or symbolically 
 
 (8) p(£) - hA o(5) = 
 
lie in or on the unit circle and those lying on the unit circle have 
 multiplicity of one. It should he noted that it is an asymptotic 
 condition only and is concerned with the convergence of y to y(t) 
 with h = (t - t )/n as n •*■ °° or hA -*■ 0. If stiff equations are to he 
 integrated with large values of h, then large values of hX must not 
 make (8) unstable. Letting hX ■> °°, the roots of (8) approach those of 
 o(C) = 0. This implies that the polynomial o(^) must not have roots 
 outside the unit circle and those roots £. for which |£.| =1 are 
 simple. Further, a(C) is of degree k and has no common root with p(C). 
 The stiff stability can he investigated as follows. 
 
 We want hX values such that (8) has roots inside the unit 
 circle or on the unit circle and simple. The region is hounded hy the 
 locus of p(£)/a(£) in the hX-plane for £ = e , e[0 s 2TT]„ This locus 
 can be plotted in hX plane and is of the type as shown in Figure 2. 
 At hX = infinity, the method is stable if a(0 is stable, so that any 
 region connected to that point will be stable by continuity argument. If 
 we prescribe k, then for stiffly stable methods the polynomial cU) 
 should be such that its roots lie within the unit circle or on the unit 
 circle and simple. Another restriction on aU) is that it has no 
 common factor with pU). Since p(c) will always have a root at +1, 
 a(0 must not have a root at +1. The locus of p ( £) /o( O for C = e , 
 £[0,2tt] should be as shown in Figure 2. 
 
 Selecting Coefficients of o(E,) 
 
 We shall consider the stiffly stable methods for which k = p. 
 
 A k-step stiffly stable method of order p (=k) requires the solution 
 
 of p + 1 equations in 2k + 1 unknowns (B can be taken as l) . The 
 
 k + 1 unknowns a., the coefficients of the polynomial p(?) can be 
 l ' 
 
 determined in terms of k unknowns B. , the coefficients of the 
 polynomial c(c)- The expressions for a. in terms of B. as obtained 
 on solving the equations (k) are given in Appendix 1. 
 
Besides the above metnioned restrictions on a(^), these 
 
 arbitrary coefficients 8. can be chosen to 
 
 1 
 
 1. control numerical stability, 
 
 2. minimize computational efforts, and 
 
 3. minimize the truncation error. 
 
 Gear has chosen 6. = 0, i = 1, 2,... k, i.e. o(£) = £ and obtained 
 stiffly stable methods for k <_ 6. k >_ 7 does not give a stable method. 
 
 These methods satisfy (l) and (2). Dill has prescribed 6=1 and 
 
 k— 1 
 8 = -.99, i.e. a (0 = 5 (£ - 0.99) and obtained a seventh order 
 
 formula. For eighth order formula he has selected 
 
 _k 2 2 
 a(S) = £T~ U - 1.8C + .81). These formulas also satisfy the 
 
 criterion (2) . 
 
 We have investigated a class of methods which will satisfy 
 the essential property (l). Some of our formulas will also satisfy 
 (2) and (3). 
 
 Class I 
 
 Choose 
 
 (9) c(0 = £ k " r U - c) r , r = 0, 1, 2,... k 
 
 where 
 
 We find 
 
 -1 < c < 1, 
 
 order of stable 
 formula 
 
 
 
 k<6 k<7 k<8 k<9 k<10 k<ll 
 
 There are no twelfth order and higher stiffly stable methods. Appendix 2 
 contains some of the special case formulas. The values of the parameters 
 D, c, max|^| (£. being the roots of p(C) = 0) and c (the coefficients 
 
 of the truncation error) are also given. The section of the locus 
 p(C)/o(£) for seventh to eleventh order formulas for different values of 
 r are shown in Figures 3 to 7, respectively. 
 
Class II 
 
 Here we take o(^) of the following form 
 
 o(0 = C k_r (C 
 
 c r ) 
 
 <v 
 
 i=0 
 k and -1 < c < 1. 
 
 whe re r = , 1 , 2 , . 
 
 The above P and Q type formulas for r = reduce to the 
 r r 
 
 case considered by Gear and are stiffly stable for k <_ 6. 
 
 By prescribing the values of c consistent with the stability 
 requirement, a locus was obtained which indicated the existence of 
 suitable parameters D and for a stiffly stable method. These results 
 are given in the table below. 
 
 r 
 
 1 2 3 k 5 6 
 
 P 
 r 
 
 k l6" k^T k^6 k<_7 k<_6 k^6 k<_6 
 
 Q 
 r 
 
 k<6 k<7 k<7 k<_6 k<_6 k^6 k<_6 
 
 r > 6 does not indicate the existence of stiffly stable methods. 
 Appendix 3 shows the coefficients for various order formulas of P 
 and Q type. The values of the parameters c, D, max|c| and c are 
 
 also tabulated. The section of the locus of p(£)/o(£) for fifth order 
 method is shown in Figure 8. 
 
3. CONCLUSIONS 
 
 The stiffly stable methods of order as high as eight are 
 already known. The aim of the present investigation has been to develop 
 methods of order higher than eight. 
 
 We have investigated the following classes of methods: 
 
 (1) oU) = K k " r (5 - c) r 
 
 (2) aU) = ? k " r U r - c r ) 
 
 (3) k-r r r i i 
 
 0(5) = r r S ? c 1 
 
 i=0 
 
 where r= 0, 1, 2,... k and -1 _f_c < 1. 
 
 We find that the choice (l) leads to the methods of order 
 as high as eleven. The methods are stiffly stable for k <_ 9 if 
 r = 3, for k <_ 10 if r = U, and for k <_ 11 if r = 6. Further, we 
 found no twelfth order and higher methods. The stiffly stable methods 
 of the types (2) and (3) do not exist for k _> 8. A comparitive study 
 of the methods developed in this report is under investigation. 
 
REFERENCES 
 
 [l] Lawson, J. D. "Generalized Runge-Kutta Processes for Stable Systems 
 with Large Lips chit z Constants," SIAM k_ t 3 (1967), pp. 372-380. 
 
 [2] Richard, P. I., Lanning, W. D. , and Torrey, M. D. "Numerical 
 Integration of Large, Highly-Damped, Nonlinear Systems," 
 SIAM Review 7, 3 (1965), pp. 376-380. 
 
 [3] Liniger, W. and Willoughby, R. A. "Efficient Numerical Integration 
 of Stiff Systems of Ordinary Differential Equations," IBM 
 Research Report RC 1970, December 1967. 
 
 [h] Dahlquist , G. "A Special Stability Criterion for Linear Multistep 
 Methods," BIT 3 (1963) , pp. 22-^3. 
 
 Widlund, 0. B. "A Note on Unconditionally Stable Linear Multistep 
 Methods," BIT 7 (1967), pp. 65-70. 
 
 [6] Norsett, S. P. "A Criterion for A(a )-Stability of Linear Multistep 
 Methods," BIT 9 (1969), PP 259-263. 
 
 [7] Gear, C. W. "Numerical Integration of Stiff Ordinary Differential 
 Equations," University of Illinois, Department of Computer 
 Science Report No. 221, January 1967. 
 
 [8] Gear, C. W. "The Automatic Integration of Stiff Ordinary Differential 
 Equations," Proceedings IFIP Congress, Edinburgh, August 1968. 
 
 [9] Dill, C. "A Computer Graphic Technique for Finding Numerical Methods 
 for Ordinary Differential Equations," University of Illinois, 
 Department of Computer Science Report No. 295, January 1969. 
 
 [10] Henrici, P. "Discrete Variable Methods in Ordinary Differential 
 Equations," John Wiley and Sons, Inc., New York, 1962 . 
 
10 
 
 APPENDIX I 
 
A. Third Order Formula: 
 
 a v = Z a v + h £ B v ' 
 y n+l . . i+1 °n-i . \ i y n+l-i 
 i=0 i=0 
 
 (1.1) 
 
 — — 
 
 
 a o 
 
 
 a 
 1 
 
 
 
 = 
 
 a 2 
 
 
 a 3 
 
 
 
 
 11 2 -1 
 
 18 -3 -6 
 
 -9 6 3 -18 
 
 2-1 2 11 
 
 and 
 
 \ - -(3S -B 1+ 6 2 -3B 3 )^ y<*> 
 
B. Fourth Order Formula: 
 
 n+l . n l+l n-i . n i n+l-i 
 
 i=0 i=0 
 
 (1.2) 
 
 12 
 
 a o 
 
 
 a l 
 
 
 a 2 
 
 = 
 
 a 3 
 
 
 a u 
 
 
 
 
 25 
 
 3-1 1-3 
 
 U8 -10 -8 6 -16 
 
 = -36 18 -18 36 
 
 16 -6 8 10 -U8 
 
 -3 1-1 3 2! 
 
 and 
 
 h 5 (5) 
 
 T n = -(12B - 3B 1+ 2B 2 - 3B 3 t 12 B,,) gj-^ 
 
C. Fifth Order Formula: 
 
 ''n+l . A l+l rf n-i . ~ l n+l-i 
 
 i=0 i=0 
 
 (1.3) 
 
 6o 
 
 — — 
 
 
 a o 
 
 
 a l 
 
 
 a2 
 
 
 
 = 
 
 a 3 
 
 
 % 
 
 
 a 5 
 
 i 
 
 
 1 
 
 137 12 -3 
 
 -3 12 
 
 300 -65 -30 15 -20 75 
 
 -300 120 -20 -60 60 -200 
 
 200 -60 60 20 -120 300 
 
 -75 20 -15 30 65 -300 
 
 12 
 
 2 -3 12 137 
 
 
 — — 
 
 
 B o 
 
 
 h 
 
 
 8 2 
 
 
 6 3 
 
 
 h 
 
 
 6 5 
 
 
 _ _ 
 
 and 
 
 T n = -(10B - 2 gl + S 2 
 
 * 2t k- m S ) hri 6] 
 
D. Sixth Order Formula: 
 
 n+l . „ i+l n-1 . - i "n+l-i 
 
 i=0 i=0 
 
 (l.iO 
 
 6o 
 
 *~ " 
 
 
 a o 
 
 
 a l 
 
 
 a 2 
 
 
 a 3 
 
 = 
 
 % 
 
 
 a 5 
 
 
 a 6 
 
 
 
 
 lUT 10 -2 1-1 2 -10 
 
 360 -77 -2h 9 -8 15 -72 
 
 -U50 150 -35 -U5 30 -50 225 
 
 1+00 -100 80 -80 100 -Uoo 
 
 -225 50 -30 U5 35 -150 U50 
 
 72 -15 
 
 _9 2U 77 -360 
 
 -10 
 
 2-1 1-2 10 1U7 
 
 and 
 
 T n = -(60B - 103 i + U3 2 - 3g 3 + ht h - 103 5 + 606 6 ) |2b-y^ T) 
 
E. Seventh Order Formula: 
 
 7 
 
 n+l . n l+l "'n-i . _ l n+l-i 
 
 i=0 i=0 
 
 (1.5) 
 
 1+20 
 
 
 
 a o 
 
 
 a l 
 
 
 a 2 
 
 
 a 3 
 
 
 
 = 
 
 a u 
 
 
 a 5 
 
 
 a 6 
 
 
 a 
 
 7 
 
 
 _ 
 
 
 1089 60 -10 
 
 -10 
 
 60 
 
 29I4O -609 -lUo 42 -28 35 -84 490 
 
 -4410 1260 -329 -252 126 l4o 315 1764 
 
 4900 -1050 700 -105 -420 350 -700 36T5 
 
 -3675 700 -350 420 105 -700 1050 -1+900 
 
 176U -315 i4o -126 252 329 -1260 41+10 
 
 -1+90 
 
 81+ -35 28 -1+2 ll+O 609 -29I+O 
 
 60 -10 
 
 1+ -3 
 
 1+ -10 
 
 60 1089 
 
 2 
 
 and 
 
 o 
 
 T n - -(105B - 15B 1 + 5B 2 - 3B 3 + 3B,, - 5B ? * 15B 6 - 105B 7 ) gjg- yf> 
 
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 1 
 
 CO 
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 OO 
 
 1 
 
 LA 
 
 o 
 
 H 
 
 H 
 
 
 -=t 
 
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 o 
 
 o 
 
 O 
 
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 o 
 
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 LA 
 
 
 -3- 
 
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 ^t 
 
 OJ 
 
 o 
 
 CO 
 
 VO 
 
 vo 
 
 | 
 
 
 
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 1 
 
 CO 
 
 LA 
 
 OJ 
 
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 VO 
 
 H 
 
 OO 
 
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 o 
 
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 OJ 
 
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 •H 
 
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 o 
 
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 rH 
 
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 H 
 
 1 
 
 
 OJ 
 
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 cn 
 
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 CM 
 
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 o 
 
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 o 
 
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 CM 
 
 
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 itn 
 
 o 
 
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 CM 
 
 
 
 
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 1 
 
 vo 
 
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 1 
 
 
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 o 
 
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 o 
 
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+ 
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 o 
 
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 rH 
 
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 o 
 
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 on 
 
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 m 
 
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 o 
 
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 no 
 
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 LfN 
 
 1 
 
 
 
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 i 
 
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 1 
 
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 on 
 
 
 
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 o 
 
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 OJ 
 
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 on 
 
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 o 
 
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 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
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 on 
 
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 o 
 
 o 
 
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 o 
 
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 1 
 
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 CO 
 
 en 
 
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 r-i 
 
 
 
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 on 
 
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 r-i 
 
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 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
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 o 
 
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 o 
 
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 o 
 
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 o 
 
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 o 
 
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 on 
 
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 H 
 
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 1 
 
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 c— 
 
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 1 
 
 on 
 
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 o 
 
 o 
 
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 on 
 
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 H 
 
 1 
 
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 on 
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 1 
 
 
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 en 
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 on 
 
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 o 
 
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 a 
 
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 o 
 
 on 
 
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 on 
 
 NO 
 
 NO 
 
 H 
 
 NO 
 
 NO 
 
 NO 
 
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 LfN 
 
 CM 
 
 ITS 
 
 ao 
 
 H 
 
 H 
 
 NO 
 
 OJ 
 
 CM 
 
 CO 
 
 ON 
 
 NO 
 
 
 1 
 
 LfN 
 
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 on 
 
 on 
 
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 en 
 
 on 
 
 en 
 
 | 
 
 
 
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 en 
 
 i 
 
 co 
 
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 1 
 
 t- 
 
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 1 
 
 en 
 
 H 
 1 
 
 
 
 
 o 
 
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 o 
 
 o 
 
 o 
 
 o 
 
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 o 
 
 o 
 
 LTN 
 
 o 
 
 OJ 
 
 CM 
 
 t— 
 
 o 
 
 o 
 
 o 
 
 NO 
 
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 o 
 
 o 
 
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 LTN 
 
 LfN 
 
 J- 
 
 NO 
 
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 OJ 
 
 O 
 
 CO 
 
 NO 
 
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 en 
 
 o 
 
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 on 
 
 CO 
 
 t— 
 
 t— 
 
 H 
 
 OJ 
 
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 On 
 
 t- 
 
 on 
 
 1 
 
 
 LfN 
 
 on 
 
 o 
 
 t— 
 
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 on 
 
 en 
 
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 H 
 
 
 
 
 i 
 
 H 
 
 CM 
 
 1 
 
 OJ 
 
 CM 
 1 
 
 CM 
 
 H 
 I 
 
 
 1 
 
 
 
 rH 
 
 o 
 
 o 
 
 o 
 
 o 
 
 CO 
 
 o 
 
 o 
 
 LfN 
 
 O 
 
 CM 
 
 o 
 
 H 
 
 CM 
 
 o 
 
 o 
 
 o 
 
 OJ 
 
 -=r 
 
 o 
 
 OJ 
 
 o 
 
 ON 
 
 OJ 
 
 Ir- 
 
 ON 
 
 en 
 
 NO 
 
 ON 
 
 on 
 
 -3- 
 
 OO 
 
 t— 
 
 J- 
 
 ,=1- 
 
 LTN 
 
 on 
 
 -J" 
 
 OJ 
 
 -=t 
 
 NO 
 
 H 
 
 _=f 
 
 NO 
 
 H 
 
 ON 
 
 o 
 
 OJ 
 
 CO 
 
 O 
 
 NO 
 
 OJ 
 
 CO 
 
 NO 
 
 en 
 
 o 
 
 t— 
 
 NO 
 
 on 
 
 
 1 
 
 on 
 
 i 
 
 LTN 
 
 H 
 
 OJ 
 OJ 
 
 1 
 
 LTN 
 OJ 
 
 H 
 CM 
 
 on 
 H 
 
 LfN 
 
 1 
 
 rH 
 
 i 
 
 i 
 
 1 
 
 
 
 
 
 
 II 
 
 
 
 
 o 
 
 1 
 
 H 
 
 o 
 
 H 
 
 CM 
 
 en 
 
 -4- 
 
 LTN 
 
 NO 
 
 t— 
 
 OO 
 
 ON 
 
 H 
 
 rH 
 
 1 S 
 
 3 
 
 s 
 
 a 
 
 3 
 
 3 
 
 s 
 
 a 
 
 a 
 
 3 
 
 3 
 
 8 1 
 
 r^ 
 
 OJ 
 H 
 
 o 
 
 OJ 
 
 o 
 
 OJ 
 
 t- 
 
 OJ 
 
 CQ 
 
 o 
 
 on 
 
 OJ 
 
 o 
 
 H 
 CQ 
 O 
 
 rH 
 OJ 
 
 ON 
 CQ 
 CM 
 
 I 
 
 CO 
 CQ 
 
 -4" 
 
 H 
 
 CQ 
 
 NO 
 
 CQ 
 LfN 
 
 LfN 
 CQ 
 LfN 
 
 CQ 
 
 on 
 
 CM 
 CQ 
 OJ 
 
 ca 
 o 
 H 
 
 OJ 
 
 I 
 o 
 
 CQ 
 O 
 H 
 
 on 
 
 OJ 
 
 i 
 
 EH 
 
 ON 
 
 •n 
 
20 
 
 APPENDIX II 
 
Third Order Formulas 
 
 a - 
 
 Table 1 
 3 Coefficients 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 °0 
 
 11 
 
 h9 
 
 257 
 
 15 1+8 
 
 a l 
 
 18 
 
 99 
 
 61+2 
 
 1+131 
 
 a 2 
 
 -9 
 
 -63 
 
 -531 
 
 -3672 
 
 "3 
 
 2 
 
 13 
 
 lk6 
 
 1089 
 
 6 
 
 6 
 
 30 
 
 200 
 
 2000 
 
 6 1 
 
 
 -18 
 
 -280 
 
 -1+200 
 
 6 2 
 
 
 
 98 
 
 29 U0 
 
 6 3 
 
 
 
 
 -686 
 
 Table 2 
 Values of the Parameters 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 C 
 
 
 
 0.6 
 
 0.7 
 
 0.7 
 
 D 
 
 -0.083 
 
 -0.011+ 
 
 -0.003 
 
 -0.003 
 
 max| c| , C¥l 
 
 0.1+26 
 
 0.515 
 
 0.751+ 
 
 0.839 
 
 C . + l 
 
 -0.1361+ 
 
 -0.1837 
 
 -0.3807 
 
 -0.8551 
 
B. Fourth Order Formulas 
 
 Table 1 
 a - 3 Coefficients 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 a 
 
 
 25 
 
 91 
 
 489 
 
 38849 
 
 5177 
 
 \ 
 
 48 
 
 222 
 
 472 
 
 139934 
 
 1902 4 
 
 a 
 2 
 
 -36 
 
 -198 
 
 -1620 
 
 -188922 
 
 -26244 
 
 a 
 3 
 
 16 
 
 82 
 
 768 
 
 113330 
 
 l6ll2 
 
 \ 
 
 -3 
 
 -15 
 
 -131 
 
 -25493 
 
 -3715 
 
 6 
 
 12 
 
 48 
 
 300 
 
 32000 
 
 7500 
 
 6 1 
 
 
 -36 
 
 -480 
 
 -8l600 
 
 -24000 
 
 6 2 
 
 
 
 192 
 
 69360 
 
 28800 
 
 S 3 
 
 
 
 
 -19652 
 
 -15360 
 
 B U 
 
 
 
 
 
 3072 
 
 Table 2 
 Values of the Parameters 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 C 
 
 
 
 0.75 
 
 0.8 
 
 0.8 
 
 0.8 
 
 D 
 
 -0.667 
 
 -0.164 
 
 -0.018 
 
 -0.007 
 
 -0.011 
 
 max| £ | 4 £#1 
 
 O.56I 
 
 0.752 
 
 0.821 
 
 0.858 
 
 0.896 
 
 c D+ i 
 
 -0.096 
 
 -0.1253 
 
 -0.1849 
 
 -0.35^6 
 
 -0.9740 
 
Fifth Order Formulas 
 
 a - 
 
 Table 1 
 3 Coefficients 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 1+ 
 
 5 
 
 a o 
 
 137 
 
 512 
 
 1877 
 
 166528 
 
 22157 
 
 154637 
 
 a i 
 
 300 
 
 1395 
 
 6090 
 
 622875 
 
 92700 
 
 705200 
 
 a 2 
 
 -300 
 
 -1560 
 
 -7860 
 
 -925500 
 
 -155460 
 
 -1288700 
 
 a 3 
 
 200 
 
 980 
 
 5180 
 
 684500 
 
 130760 
 
 1179 800 
 
 \ 
 
 -75 
 
 -360 
 
 -1815 
 
 -252750 
 
 -55215 
 
 -541175 
 
 a 
 5 
 
 12 
 
 57 
 
 282 
 
 37425 
 
 9372 
 
 99152 
 
 6 o 
 
 6o 
 
 240 
 
 960 
 
 96000 
 
 15360 
 
 187500 
 
 B l 
 
 
 -180 
 
 -l44o 
 
 -216000 
 
 -46080 
 
 -750000 
 
 B 2 
 
 
 
 5 1+0 
 
 162000 
 
 51840 
 
 1200000 
 
 6 3 
 
 
 
 
 -40500 
 
 -25920 
 
 -960000 
 
 \ 
 
 
 
 
 
 486o 
 
 384000 
 
 s 
 
 
 
 
 
 
 6l44o 
 
 Table 2 
 Values of Parameters 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 C 
 
 
 
 0.75 
 
 0.75 
 
 0.75 
 
 0.75 
 
 0.8 
 
 D 
 
 -2.327 
 
 -0.905 
 
 -0.272 
 
 -O.067 
 
 -0.033 
 
 -0.29 
 
 max|c| , £t*1 
 
 0.709 
 
 0.749 
 
 0.755 
 
 0.793 
 
 0.832 
 
 0.895 
 
 c D+ i 
 
 -0.0730 
 
 -0.0898 
 
 -0.1156 
 
 -0.1596 
 
 -0.2507 
 
 -0.7455 
 
Sixth Order Formulas 
 
 
 
 a 
 
 Table 1 
 - 3 Coefficients 
 
 
 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 1+ 
 
 5 
 
 6 
 
 a o 
 
 ll+7 
 
 281+ 
 
 209^ 
 
 7725 
 
 65539 
 
 269927 
 
 0.7519*+ 
 
 a 
 1 
 
 360 
 
 797 
 
 7392 
 
 31293 
 
 307832 
 
 13969^0 
 
 1+. 205^8 
 
 a 
 2 
 
 -1+50 
 
 -1050 
 
 -11115 
 
 -52965 
 
 -599970 
 
 -3018050 
 
 -9.81212 
 
 "3 
 
 1+00 
 
 900 
 
 9520 
 
 1+861+0 
 
 621520 
 
 31+85600 
 
 12.22517 
 
 a k 
 
 -225 
 
 -250 
 
 -5070 
 
 -26055 
 
 -361265 
 
 -2270525 
 
 -8.57888 
 
 a 
 5 
 
 72 
 
 159 
 
 15 81+ 
 
 7875 
 
 111861+ 
 
 -711252 
 
 3.215 
 
 a 6 
 
 -10 
 
 -22 
 
 -217 
 
 -1063 
 
 -11+1+1+2 
 
 -115290 
 
 -0.50269 
 
 e o 
 
 60 
 
 120 
 
 960 
 
 381+0 
 
 37500 
 
 187500 
 
 1 
 
 6 1 
 
 
 -60 
 
 -lUUO 
 
 -86 1+0 
 
 -120000 
 
 -750000 
 
 -5.1 
 
 B 2 
 
 
 
 5^0 
 
 61+80 
 
 1U 1+000 
 
 1200000 
 
 10.8375 
 
 3 3 
 
 
 
 
 -1620 
 
 -76800 
 
 -960000 
 
 -12.2825 
 
 6 1+ 
 
 
 
 
 
 15360 
 
 381+000 
 
 7.83009 
 
 6 5 
 
 
 
 
 
 
 -6lUU0 
 
 -2.66223 
 
 ^ 
 
 
 
 
 
 
 
 0.37715 
 
 Table 2 
 Values of the Parameters 
 
 r 
 
 
 
 1 
 
 2 
 
 3 
 
 1+ 
 
 5 
 
 6 
 
 C 
 
 
 
 0.5 
 
 0.75 
 
 0.75 
 
 0.8 
 
 0.8 
 
 O.85 
 
 D 
 
 -6.071 
 
 -3.607 
 
 -1.191 
 
 -0 . 1+09 
 
 -0.078 
 
 -0.039 
 
 -0.051 
 
 max| £ , 5^1 
 
 0.863 
 
 0.789 
 
 0.760 
 
 0.785 
 
 0.837 
 
 0.896 
 
 0.930 
 
 Vi 
 
 -0.0583 
 
 -0.0651+ 
 
 -0.081+3 
 
 -0.1071 
 
 -O.1569 
 
 -0.2521 
 
 -O.8605 
 
vo 
 
 e 
 u 
 o 
 
 P-H 
 
 u 
 
 ■s 
 
 o 
 
 -P 
 C! 
 <D 
 !> 
 <D 
 CQ 
 
 
 in 
 
 
 •P 
 
 
 C 
 
 
 a.' 
 
 
 ■H 
 
 H 
 
 CJ 
 
 
 ■H 
 
 (1) 
 
 Cm 
 
 H 
 
 «(H 
 
 fl 
 
 a; 
 
 Ed 
 
 
 
 EH 
 
 o 
 
 
 en 
 
 
 a 
 
 CO 
 
 LfN 
 
 OJ 
 
 H 
 
 on 
 vo 
 t— 
 vo 
 
 CM 
 CM 
 VO 
 O 
 O 
 OJ 
 H 
 CO 
 
 H 
 
 -=!• 
 CO 
 
 on 
 
 VO 
 
 vo 
 H 
 C\J 
 
 on 
 ltn 
 
 On 
 
 LfN 
 LTN 
 
 on 
 
 ON 
 
 H 
 
 o 
 
 OJ 
 
 CM 
 
 on 
 o 
 vo 
 
 o 
 
 CM 
 I 
 
 on 
 
 _=f 
 
 H 
 
 C— 
 
 on 
 
 on 
 
 CM 
 
 O 
 CM 
 t— 
 -=f 
 ON 
 OJ 
 On 
 
 m 
 
 o\ 
 
 i 
 
 on 
 
 CM 
 
 o 
 
 ON 
 
 vo 
 
 ON 
 CM 
 
 H 
 CM 
 VO 
 O 
 
 CM 
 
 on 
 
 o 
 
 LTN 
 LfN 
 
 CM 
 O 
 
 CO 
 
 H 
 
 LTN 
 CO 
 
 on 
 
 LTN 
 
 on 
 vo 
 
 CM 
 
 i 
 
 CO 
 ON 
 
 m 
 o 
 vo 
 
 vo 
 
 OJ 
 
 I 
 
 o 
 
 LTN 
 CO 
 
 On 
 CO 
 VO 
 
 on 
 o 
 
 ON 
 
 I 
 
 On 
 
 LTN 
 O 
 
 o 
 
 CM 
 VO 
 
 on 
 
 en 
 H 
 
 ON 
 
 o 
 H 
 
 ON 
 
 H 
 
 on 
 
 OJ 
 
 on 
 
 r-t 
 H 
 H 
 OJ 
 
 i 
 
 H 
 VO 
 
 on 
 I 
 
 o 
 
 CO 
 
 o 
 
 On 
 
 _=r 
 
 LT\ 
 CM 
 H 
 H 
 I 
 
 LTN 
 LA 
 H 
 
 -=r 
 
 LTN 
 
 on 
 o 
 On 
 
 LfN 
 
 o 
 
 CM 
 
 on 
 
 H 
 
 en 
 H 
 
 LTN 
 CM 
 
 LTN 
 I 
 
 OJ 
 
 on 
 
 I 
 
 LTN 
 CM 
 H 
 
 CO 
 
 vo 
 o 
 
 OJ 
 
 VO 
 
 VO 
 
 LfN 
 O 
 
 3 3 
 
 c— 
 
 -Si- 
 nn 
 
 CO 
 
 LfN 
 
 o 
 
 OJ 
 
 on 
 vo 
 
 LTN 
 VO 
 t- 
 
 -=r 
 
 H 
 
 i 
 
 vo 
 On 
 -=r 
 VO 
 
 t— 
 
 I 
 
 O 
 
 ■a 
 
 vo 
 o 
 
 H 
 I 
 
 CM 
 CM 
 
 H 
 
 CO 
 
 o 
 
 on 
 o 
 
 00 
 O 
 
 ■d 
 
 H 
 OJ 
 -Sf 
 O 
 LT\ 
 
 o 
 
 J- 
 on 
 
 CO 
 
 On 
 
 LTN 
 
 On 
 o 
 vo 
 
 H 
 
 o 
 
 ON 
 CO 
 LfN 
 
 o 
 
 LTN 
 CO 
 
 LfN 
 -=J- 
 
 OJ 
 
 m 
 
 ON 
 
 on 
 H 
 
 CO 
 
 on 
 on 
 
 o 
 i 
 
 i 
 
 -=f 
 
 on 
 
 vo 
 
 OJ 
 
 o 
 
 LfN 
 
 vo 
 
 vo 
 
 vo 
 
 LTN 
 
 _=r 
 
 on 
 
 OJ 
 
 o 
 
 
 
 LfN 
 
 t— 
 
 vo 
 
 -=t 
 
 CO 
 
 On 
 
 CO 
 
 vo 
 
 O 
 
 o 
 
 o 
 
 o 
 
 vo 
 
 
 
 CO 
 
 -4- 
 
 o 
 
 vo 
 
 c— 
 
 H 
 
 H 
 
 c— 
 
 on 
 
 H 
 
 CM 
 OJ 
 
 On 
 
 3 
 
 ■a 
 
 r-j 
 
 X 
 
 3 
 
 o 
 
 
 
 VO 
 
 LTN 
 
 CO 
 
 o 
 
 o 
 
 o 
 
 LfN 
 
 -=r 
 
 VO 
 
 j- 
 
 -3- 
 
 OJ 
 
 _=r 
 
 
 
 CM 
 
 ON 
 
 H 
 
 OJ 
 
 on 
 
 t— 
 
 CM 
 
 rH 
 
 
 VO 
 
 vo 
 
 on 
 
 H 
 
 
 
 H 
 
 LTN 
 
 OJ 
 
 H 
 
 1 
 
 H 
 
 o 
 
 H 
 
 1 
 
 -=f 
 
 H 
 1 
 
 
 
 M 
 
 H 
 
 CM 
 
 CO 
 
 1 
 
 
 
 on 
 
 CO 
 
 J" 
 
 J- 
 
 LfN 
 
 O 
 
 CO 
 
 -3- 
 
 ON 
 
 -3- 
 
 on 
 
 OJ 
 
 o 
 
 
 
 
 CM 
 
 ON 
 
 ON 
 
 H 
 
 VO 
 
 H 
 
 Lf\ 
 
 o 
 
 O 
 
 o 
 
 O 
 
 VO 
 
 
 
 
 ON 
 
 VO 
 
 H 
 
 o 
 
 LfN 
 
 LfN 
 
 ^t 
 
 o\ 
 
 3 
 
 H 
 
 ■d 
 
 o 
 
 
 
 
 VO 
 
 CO 
 
 on 
 
 o 
 
 on 
 
 J" 
 
 t— 
 
 t- 
 
 X 
 
 -3" 
 
 
 
 
 J- 
 
 ON 
 
 LfN 
 
 C-- 
 
 o 
 
 t— 
 
 OJ 
 
 CO 
 
 OJ 
 
 CM 
 
 j- 
 
 J- 
 
 
 
 
 On 
 
 ON 
 
 on 
 
 -3- 
 
 c— 
 
 i 
 
 H 
 
 CO 
 
 CO 
 
 LfN 
 1 
 
 VO 
 CM 
 
 
 
 -=r 
 
 CO 
 
 CO 
 
 1 
 
 H 
 
 vo 
 
 H 
 I 
 
 
 
 CM 
 
 o 
 
 o 
 
 H 
 
 o 
 
 o 
 
 O 
 
 LTN 
 
 VO 
 
 on 
 
 OJ 
 
 o 
 
 
 
 
 
 H 
 
 OJ 
 
 O 
 
 o 
 
 LfN 
 
 VO 
 
 ON 
 
 C— 
 
 O 
 
 o 
 
 CO 
 
 
 
 
 
 t— 
 
 o 
 
 CO 
 
 on 
 
 H 
 
 O 
 
 LfN 
 
 CO 
 
 ■d 
 
 ■d 
 
 On 
 
 
 
 
 
 CM 
 
 ON 
 
 -3" 
 
 -=r 
 
 on 
 
 on 
 
 LfN 
 
 ON 
 
 o 
 
 
 
 
 
 O 
 
 VO 
 
 C— 
 
 CM 
 
 On 
 
 on 
 
 -=t 
 
 CM 
 
 CO 
 
 -3- 
 
 t— 
 
 
 
 
 
 J- 
 
 -3- 
 H 
 
 OJ 
 1 
 
 VO 
 OJ 
 
 CO 
 
 H 
 
 1 
 
 CO 
 
 CM 
 1 
 
 
 VO 
 H 
 
 co 
 H 
 OJ 
 
 
 
 
 
 H 
 
 o 
 
 t— 
 
 o 
 
 o 
 
 o 
 
 LfN 
 
 OJ 
 
 O 
 
 CM 
 
 o 
 
 
 
 
 
 
 o 
 
 H 
 
 CO 
 
 LfN 
 
 o 
 
 t- 
 
 ON 
 
 on 
 
 o 
 
 vo 
 
 
 
 
 
 
 o 
 
 C— 
 
 LTN 
 
 VO 
 
 vo 
 
 on 
 
 CO 
 
 on 
 
 3 
 
 -3- 
 
 
 
 
 
 
 H 
 
 VO 
 
 -=f 
 
 H 
 
 CM 
 
 ON 
 
 O 
 
 H 
 
 LfN 
 
 
 
 
 
 
 OJ 
 
 vo 
 
 o 
 
 H 
 
 1 
 
 H 
 H 
 
 CO 
 
 1 
 
 (T) 
 
 H 
 
 1 
 
 
 CO 
 
 1 
 
 
 
 
 
 
 o 
 
 H 
 
 CM 
 
 on 
 
 -3- 
 
 LTN 
 
 vo 
 
 t— 
 
 o 
 
 H 
 
 OJ 
 
 on 
 
 -^ LfN 
 
 vo t— 
 
 !h 
 
 a 
 
 3 
 
 a 
 
 a 
 
 a 
 
 a 
 
 a 
 
 a 
 
 CQ 
 
 ca 
 
 CQ 
 
 CQ 
 
 CQ CQ 
 
 CQ CQ 
 
 w 
 
o 
 u 
 
 o 
 
 -P 
 CO 
 
 w 
 
 -p 
 
 OJ cd 
 
 0) 
 rH 0) 
 
 -9 ^ 
 
 ctJ -p 
 Eh 
 
 O 
 
 en 
 
 H 
 cd 
 
 > 
 
 en 
 
 
 
 H 
 
 O 
 
 H 
 
 
 LT\ 
 
 OJ 
 
 CO 
 
 V£> 
 
 
 c— 
 
 H 
 
 CO 
 
 00 
 
 
 • 
 
 • 
 
 • 
 
 * 
 
 
 o 
 
 O 
 1 
 
 O 
 
 o 
 
 1 
 
 \o 
 
 
 V£) 
 
 VD 
 
 CO 
 
 
 CM 
 
 _3" 
 
 t~ 
 
 t— 
 
 
 t- 
 
 H 
 
 00 
 
 H 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 o 
 
 O 
 
 o 
 
 o 
 I 
 
 \r\ 
 
 
 
 
 r-\ 
 
 
 
 LP> 
 
 LT\ 
 
 H 
 
 
 
 H 
 
 o 
 
 CM 
 
 
 t- 
 
 on 
 
 CO 
 
 H 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 o 
 
 o 
 i 
 
 o 
 
 o 
 
 -=r 
 
 
 
 
 LA 
 
 
 
 H 
 
 o\ 
 
 on 
 
 
 
 vo 
 
 CO 
 
 ON 
 
 
 c— 
 
 C— 
 
 [— 
 
 o 
 
 
 • 
 
 • 
 
 * 
 
 • 
 
 
 o 
 
 o 
 1 
 
 o 
 
 o 
 
 1 
 
 oo 
 
 
 
 
 UA 
 
 
 
 H 
 
 CO 
 
 vo 
 
 
 
 CO 
 
 H 
 
 t~ 
 
 
 c~- 
 
 t— 
 
 t- 
 
 o 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 o 
 
 H 
 1 
 
 o 
 
 o 
 
 1 
 
 CM 
 
 
 
 
 ON 
 
 
 
 CM 
 
 o\ 
 
 OJ 
 
 
 u~\ 
 
 J- 
 
 H 
 
 \o 
 
 
 \D 
 
 H 
 
 CO 
 
 o 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 o 
 
 -3- 
 1 
 
 o 
 
 o 
 1 
 
 H 
 
 
 
 
 ^o 
 
 
 
 LTN 
 
 CO 
 
 -=»■ 
 
 
 LT\ 
 
 _=r 
 
 H 
 
 U~\ 
 
 
 ^D 
 
 t— 
 
 o\ 
 
 o 
 
 
 i 
 
 • 
 
 * 
 
 • 
 
 
 O 
 
 1 
 
 o 
 
 o 
 1 
 
 Sh 
 
 o 
 
 P 
 
 H 
 IK 
 
 H 
 
 + 
 ft 
 
 o 
 
 I 
 
oo 
 
 C0 VO 
 -P 
 
 <U 
 
 •H 
 H O 
 
 •H 
 
 H «H 
 
 XJ <L> 
 
 03 O 
 
 Eh O u~\ 
 
 cq 
 
 
 
 o 
 ft 
 
 ■H 
 
 VO 
 
 oo 
 
 o 
 
 c— 
 
 -=f 
 
 oo 
 
 o 
 
 C— 
 
 _3" 
 
 On 
 
 H 
 
 o 
 
 -=r 
 
 in 
 
 -=r 
 
 ON 
 
 CO 
 
 LTN 
 
 VO 
 
 -=t 
 
 VO 
 
 D— 
 
 vo 
 
 OJ 
 
 t- 
 
 ON 
 
 VO 
 
 H 
 
 VO 
 
 oo 
 
 CO 
 
 i/n 
 
 -d- 
 
 ON 
 
 l/N 
 
 _3" 
 
 VO 
 
 VO 
 
 o 
 
 o 
 
 rH 
 
 c— 
 
 OJ 
 
 O 
 
 -=f 
 
 H 
 
 O 
 
 H 
 
 OJ 
 
 LTN 
 
 t- 
 
 vo 
 
 .Si" 
 
 o 
 
 
 H 
 
 00 
 
 1 
 
 Lf> 
 
 LTN 
 1 
 
 00 
 
 H 
 1 
 
 
 1 
 
 J- 
 
 t— 
 
 l/N 
 
 OO 
 
 O 
 
 l/N 
 
 D— 
 
 00 
 
 H 
 
 J- 
 
 On 
 
 oo 
 
 00 
 
 CO 
 
 00 
 
 oo 
 
 t— 
 
 j- 
 
 O 
 
 CO 
 
 oo 
 
 -d- 
 
 CO 
 
 OO 
 
 L/N 
 
 -=J- 
 
 oo 
 
 VO 
 
 o 
 
 OJ 
 
 CO 
 
 o 
 
 -=f 
 
 c— 
 
 On 
 
 VO 
 
 00 
 
 vo 
 
 o 
 
 o 
 
 vo 
 
 CO 
 
 o 
 
 00 
 
 00 
 
 H 
 
 H 
 
 OJ 
 
 H 
 
 H 
 
 00 
 
 -=f 
 
 CO 
 
 o 
 
 
 H 
 
 oo 
 
 l/n 
 
 LJA 
 
 1 
 
 oo 
 
 H 
 1 
 
 
 1 
 
 OJ 
 
 OJ 
 
 OJ 
 
 H 
 
 CO 
 
 OJ 
 
 -=f 
 
 CO 
 
 t— 
 
 CO 
 
 o 
 
 H 
 
 H 
 
 tf\ 
 
 o 
 
 -=t 
 
 LTN 
 
 OJ 
 
 >H 
 
 l/N 
 
 VO 
 
 00 
 
 -=t- 
 
 o 
 
 -=t 
 
 00 
 
 l/N 
 
 O 
 
 ON 
 
 t- 
 
 vo 
 
 H 
 
 l/N 
 
 H 
 
 OJ 
 
 0J 
 
 ON 
 
 CO 
 
 vo 
 
 l/N 
 
 OJ 
 
 CO 
 
 00 
 
 H 
 
 oo 
 
 H 
 
 H 
 
 OJ 
 
 H 
 
 H 
 
 OJ 
 
 00 
 
 00 
 
 o 
 
 
 H 
 
 00 
 
 1 
 
 i/n 
 
 l/N 
 
 1 
 
 oo 
 
 H 
 1 
 
 
 1 
 
 H 
 
 OJ 
 
 CO 
 
 H 
 
 -=r 
 
 o 
 
 OO 
 
 OJ 
 
 c— 
 
 vn 
 
 J- 
 
 t- 
 
 -J/ 
 
 o 
 
 vo 
 
 _=r 
 
 l/N 
 
 o 
 
 J - 
 
 tr— 
 
 l/N 
 
 CO 
 
 t— 
 
 t- 
 
 00 
 
 ON 
 
 OJ 
 
 VO 
 
 CO 
 
 l/N 
 
 vo 
 
 L/N 
 
 H 
 
 On 
 
 LTN 
 
 vo 
 
 o 
 
 o 
 
 O 
 
 ON 
 
 H 
 
 i/n 
 
 -=f 
 
 -^ 
 
 OJ 
 
 OJ 
 
 OJ 
 
 H 
 
 LTN 
 
 00 
 
 vo 
 
 o 
 
 OJ 
 
 o 
 
 
 H 
 
 oo 
 1 
 
 -=f 
 
 -3- 
 1 
 
 OJ 
 
 H 
 
 1 
 
 
 1 
 
 H 
 
 H 
 
 
 
 oo 
 
 CO 
 
 
 oo 
 
 vo 
 
 H 
 
 
 OJ 
 
 c— 
 
 t- 
 
 t— 
 
 CO 
 
 CO 
 
 vo 
 
 
 On 
 
 OJ 
 
 L/N 
 
 o 
 
 OJ 
 
 -=f 
 
 o 
 
 VO 
 
 vo 
 
 -=f 
 
 O 
 
 vo 
 
 oo 
 
 CO 
 
 -5f 
 
 oo 
 
 LTN 
 
 CO 
 
 H 
 
 i/> 
 
 l/N 
 
 J- 
 
 o 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 « 
 
 « 
 
 • 
 
 l/N 
 1 
 
 OJ 
 
 H 
 
 vo 
 H 
 
 1 
 
 H 
 
 C— 
 1 
 
 OJ 
 
 o 
 
 1 
 
 o 
 
 
 
 
 ON 
 
 VO 
 
 H 
 
 H 
 
 
 
 ON 
 
 t- 
 
 CO 
 
 OJ 
 
 OJ 
 
 vo 
 
 
 
 vo 
 
 vo 
 
 OJ 
 
 L/N 
 
 00 
 
 o 
 
 
 ON 
 
 OJ 
 
 OJ 
 
 LTN 
 
 oo 
 
 00 
 
 vo 
 
 
 VO 
 
 -=}■ 
 
 LTN 
 
 o 
 
 CO 
 
 vo 
 
 o 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 -=f 
 
 ON 
 
 o 
 
 r-\ 
 1 
 
 c— 
 
 OJ 
 
 1 
 
 o 
 
 o 
 
 1 
 
 
 
 
 VO 
 
 l/N 
 
 t- 
 
 OJ 
 
 
 
 
 l/N 
 
 vo 
 
 ON 
 
 o 
 
 _=r 
 
 
 
 
 00 
 
 o 
 
 OJ 
 
 -3- 
 
 CO 
 
 
 
 OJ 
 
 ON 
 
 oo 
 
 t— 
 
 OJ 
 
 o 
 
 
 
 vo 
 
 CO 
 
 H 
 
 OJ 
 
 vo 
 
 OJ 
 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 
 i 
 
 CO 
 
 ON 
 OJ 
 
 LTN 
 
 CO 
 
 -3" 
 
 H 
 
 1 
 
 CO 
 
 vo 
 
 D— 
 OJ 
 
 o 
 
 
 
 
 -3- 
 
 H 
 
 O 
 
 00 
 
 
 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 
 
 1 
 
 VO 
 
 L/N 
 
 1 
 
 OJ 
 
 o 
 1 
 
 
 
 
 J- 
 
 t~ 
 
 CO 
 
 OJ 
 
 CO 
 
 o 
 
 OJ 
 
 CO 
 
 OJ 
 
 OO 
 
 OO 
 
 -d- 
 
 oo 
 
 OJ 
 
 o 
 
 
 
 o 
 
 o 
 
 ON 
 
 o 
 
 o 
 
 oo 
 
 o 
 
 On 
 
 oo 
 
 o 
 
 O 
 
 o 
 
 o 
 
 J- 
 
 
 
 oo 
 
 o 
 
 LTN 
 
 -3- 
 
 CO 
 oo 
 
 l/N 
 
 H 
 
 vo 
 
 00 
 
 OJ 
 
 0J 
 
 H 
 
 OJ 
 OJ 
 
 fl 
 
 ^ 
 
 fl 
 
 H 
 X 
 
 o 
 
 L/N 
 
 
 
 t~— 
 
 H 
 
 VO 
 
 OJ 
 
 UN 
 
 t— 
 
 L/N 
 
 ^1- 
 
 r-\ 
 
 l/N 
 
 CO 
 
 vo 
 
 OJ 
 
 H 
 
 
 
 H 
 
 OJ 
 
 LTN 
 
 o 
 
 ON 
 
 OO 
 
 00 
 
 O 
 
 H 
 
 OJ 
 
 vo 
 
 H 
 
 LTN 
 
 OJ 
 
 
 
 H 
 
 vo 
 
 -Si- 
 nn 
 
 o 
 
 OJ 
 
 H 
 
 1 
 
 o 
 
 H 
 
 _=f 
 
 H 
 
 1 
 
 L/N 
 
 H 
 1 
 
 o 
 
 OJ 
 
 t- 
 o 
 
 H 
 1 
 
 
 
 00 
 
 l/N 
 
 o 
 
 O 
 
 -sj- 
 
 o 
 
 o 
 
 o 
 
 O 
 
 On 
 
 -=r 
 
 OO 
 
 OJ. 
 
 o 
 
 
 
 
 OO 
 
 OJ 
 
 VO 
 
 H 
 
 L/N 
 
 VO 
 
 OJ 
 
 H 
 
 oo 
 
 o 
 
 o 
 
 o 
 
 OJ 
 
 
 
 
 OO 
 
 t— 
 
 oo 
 
 CO 
 
 CO 
 
 ON 
 
 00 
 
 -St 
 
 vo 
 
 L/\ 
 
 CO 
 LTN 
 
 _sj- 
 
 LTN 
 
 
 ■d 
 
 3 
 
 ■d 
 
 On 
 
 VO 
 
 
 
 
 00 
 
 ON 
 
 OJ 
 
 o 
 
 -=t 
 
 OJ 
 
 CO 
 
 H 
 
 J- 
 
 OJ 
 
 vo 
 
 vo 
 
 OJ 
 
 
 
 
 CO 
 
 LTN 
 
 CO 
 
 -sj- 
 
 On 
 
 o 
 
 vo 
 
 LTN 
 
 OJ 
 
 t— 
 
 00 
 
 LfA 
 
 H 
 
 
 
 
 l/N 
 
 VO 
 
 l/N 
 
 vo 
 
 o 
 
 c— 
 
 t— 
 
 H 
 
 H 
 
 vo 
 
 H 
 
 vo 
 
 J- 
 
 
 
 
 H 
 
 t— 
 
 vo 
 H 
 
 1 
 
 H 
 
 OJ 
 
 ON 
 
 H 
 
 1 
 
 H 
 
 H 
 
 -sl- 
 1 
 
 H 
 
 1 
 
 
 H 
 1 
 
 L/N 
 
 -3- 
 
 1 
 
 
 
 OJ 
 
 L/N 
 
 o 
 
 CO 
 
 o 
 
 o 
 
 o 
 
 o 
 
 oo 
 
 L/N 
 
 oo 
 
 OJ 
 
 o 
 
 
 
 
 
 CO 
 
 o 
 
 oo 
 
 0O 
 
 LTN 
 
 vo 
 
 LTN 
 
 oo 
 
 O 
 
 o 
 
 o 
 
 -=J- 
 
 
 
 
 
 H 
 
 CO 
 
 00 
 
 CO 
 
 ON 
 
 OJ 
 
 oo 
 
 o 
 vo 
 
 o 
 
 CO 
 St 
 
 O 
 
 l/N 
 
 VO 
 00 
 
 ■a 
 
 3 
 
 o 
 
 CO 
 
 
 
 
 
 o 
 
 ON 
 
 vo 
 
 oo 
 
 ON 
 
 LTN 
 
 H 
 
 OJ 
 
 H 
 
 -H/ 
 
 CvJ 
 
 VO 
 
 
 
 
 
 OJ 
 
 CO 
 
 r— 
 
 H 
 
 1 
 
 OJ 
 OJ 
 
 ON 
 
 H 
 
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 LTN 
 
 
 On 
 
 On 
 
 
 LT\ 
 
 CM 
 
 CO 
 
 o 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 O 
 
 H 
 
 o 
 
 o 
 1 
 
 o\ 
 
 
 
 H 
 
 LP 
 
 -3- 
 CO 
 
 
 VO 
 
 CO 
 
 CO 
 
 o 
 
 
 O 
 
 H 
 1 
 
 o 
 
 o 
 1 
 
 0O 
 
 
 
 CO 
 
 vo 
 t— 
 
 
 CM 
 
 
 CO 
 
 t— 
 
 
 VO 
 
 CO 
 
 CO 
 
 o 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 O 
 
 H 
 1 
 
 o 
 
 o 
 1 
 
 t- 
 
 
 
 
 t- 
 
 
 
 CM 
 
 -d- 
 
 on 
 
 
 
 t- 
 
 CO 
 
 VO 
 
 
 vo 
 
 LTN 
 
 CO 
 
 o 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 
 o 
 
 CO 
 
 1 
 
 o 
 
 o 
 1 
 
 vo 
 
 
 
 
 H 
 
 
 
 CO 
 
 UA 
 
 t— 
 
 
 Lf\ 
 
 -=r 
 
 o\ 
 
 vo 
 
 
 t- 
 
 On 
 
 CO 
 
 o 
 
 
 « 
 
 • 
 
 • 
 
 * 
 
 
 o 
 
 CM 
 1 
 
 o 
 
 o 
 1 
 
 l/N 
 
 
 
 
 -3- 
 
 
 
 H 
 
 H 
 
 On 
 
 
 CO 
 
 O 
 
 VO 
 
 UA 
 
 
 • 
 
 
 CO 
 
 o 
 
 
 o 
 
 l/\ 
 1 
 
 o 
 
 o 
 1 
 
 -3- 
 
 
 
 
 o 
 
 
 
 CM 
 
 -H/ 
 
 LPv 
 
 
 CM 
 
 -=f 
 
 CO 
 
 LTA 
 
 
 o\ 
 
 -H/ 
 
 On 
 
 o 
 
 
 • 
 
 « 
 
 • 
 
 • 
 
 
 o 
 
 t- 
 
 o 
 
 o 
 1 
 
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 o 
 
 Q 
 
 H 
 
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 + 
 ft 
 o 
 
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 K 
 
 
 « 
 

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 C 
 
 
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 rH 
 
 
 ~ 
 
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 r- 
 
 u 
 
 r-H 
 
 t- 
 
 t- 
 
 H 
 
 o 
 
 vo 
 
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 O 
 
 t— 
 
 rH 
 
 vo 
 
 H 
 
 rH 
 
 
 
 
 
 
 
 l/N 
 
 CM 
 
 on 
 
 CM 
 _3- 
 
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 i-l 
 
 t— 
 
 -=r 
 
 o 
 
 on 
 
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 t- 
 
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 r-\ 
 
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 r-l 
 
 H 
 
 on 
 
 
 
 
 
 
 LTN 
 
 t— 
 
 H 
 
 L/N 
 
 r— 
 
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 VO 
 
 t— 
 
 l/N 
 
 l/N 
 
 O 
 
 t— 
 
 VO 
 
 O 
 
 L/N 
 
 CM 
 
 on 
 
 l/N 
 
 
 
 
 l/N 
 
 UN 
 
 t— 
 
 00 
 
 CO 
 
 -3- 
 
 t^ 
 
 o 
 
 o 
 
 
 8 
 
 on 
 
 -=t 
 
 i/n 
 
 on 
 
 &! 
 
 CM 
 
 t— 
 
 CO 
 
 CO 
 
 CM 
 
 o 
 
 
 
 l/N 
 
 CM 
 
 CM 
 
 on 
 
 r-i 
 
 L^ 
 
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 r-t 
 
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 t— 
 
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 CM 
 
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 l/N 
 
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 l/N 
 
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 H 
 
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 t— 
 
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 vo 
 
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 l/N 
 
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 L/N 
 
 on 
 
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 CM 
 
 o 
 
 o 
 
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 rH 
 
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 rH 
 
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 on 
 
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 on 
 
 on 
 
 l/N 
 
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 on 
 
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 rH 
 
 
 
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 on 
 
 i/n 
 
 l/N 
 
 t— 
 
 CO 
 
 on 
 
 CO 
 
 VO 
 
 L/N 
 
 t- 
 
 
 
 CM 
 
 l/N 
 
 VD 
 
 O 
 
 VO 
 
 VO 
 
 l/N 
 
 o\ 
 
 CM 
 
 
 
 i/n 
 
 i/n 
 
 l/N 
 
 t— 
 
 t— 
 
 CO 
 
 rH 
 
 on 
 
 l/N 
 
 on 
 
 CM 
 
 Os 
 
 
 
 CM 
 
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 l/N 
 
 r— 
 
 o 
 
 o 
 
 CM 
 
 on ■ 
 
 o 
 
 
 
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 CO 
 
 -=t 
 
 vo 
 
 CO 
 
 -3 
 
 ON 
 
 l/N 
 
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 c- 
 
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 rH 
 
 
 ^f 
 
 H 
 
 CO 
 
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 l/N 
 
 CM 
 
 VO 
 
 on 
 
 o 
 
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 CM 
 
 t- 
 
 on 
 
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 rH 
 
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 on 
 
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 rH 
 
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 on 
 
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 t— 
 
 H 
 
 l/N 
 
 H 
 
 o 
 
 o 
 
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 r-l 
 
 VO 
 
 1 
 
 on 
 
 H 
 
 rH 
 CM 
 
 1 
 
 on 
 
 CM 
 
 CO 
 
 r-t 
 1 
 
 rH 
 H 
 
 -3" 
 1 
 
 rH 
 
 1 
 
 
 
 1 
 
 H 
 
 H 
 
 H 
 
 rH 
 1 
 
 
 1 
 
 
 i 
 
 
 
 on 
 
 i/n 
 
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 l/N 
 
 CO 
 
 l/N 
 
 co 
 
 on 
 
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 co 
 
 
 t- 
 
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 vo 
 
 on 
 
 
 
 
 rH 
 
 CM 
 
 VO 
 
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 on 
 
 -3" 
 
 CO 
 
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 CM 
 
 vo 
 
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 -3 
 
 J- 
 
 t— 
 
 H 
 
 t- 
 
 CO 
 
 CM 
 
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 CM 
 
 l/N 
 
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 t— 
 
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 CM 
 
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 t— 
 
 o 
 
 VO 
 
 
 
 O 
 
 ON 
 
 co 
 
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 c— 
 
 -3- 
 
 L/N 
 
 t— 
 
 
 
 
 00 
 
 t- 
 
 00 
 
 VO 
 
 0\ 
 
 CM 
 
 ON 
 
 O 
 
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 ON 
 
 CO 
 
 
 CM 
 
 rH 
 
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 l/N 
 
 t- 
 
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 H 
 
 o 
 
 
 
 
 on 
 
 00 
 
 l/N 
 
 co 
 
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 l/N 
 
 ON 
 
 on 
 
 t- 
 
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 CM 
 
 rH 
 
 
 CM 
 
 H 
 
 on 
 
 CM 
 
 CM 
 
 H 
 
 t- 
 
 H 
 
 o 
 
 
 
 
 CM 
 
 t— 
 
 CM 
 
 l/N 
 
 on 
 
 H 
 
 t— 
 
 L/N 
 
 J- 
 
 CM 
 
 CM 
 
 O 
 
 rH 
 
 l/N 
 
 rH 
 
 vo 
 
 -3" 
 
 CO 
 
 on 
 
 O 
 
 O 
 
 o 
 
 
 
 
 
 rH 
 
 NO 
 
 on 
 
 o 
 
 CM 
 
 D- 
 
 O 
 
 -3- 
 
 H 
 
 1 
 
 
 
 1 
 
 H 
 
 r-i 
 
 1 
 
 
 
 
 1 
 
 
 
 
 
 
 1 
 
 .H 
 
 CM 
 
 1 
 
 CM 
 
 rH 
 
 1 
 
 H 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 00 
 
 o 
 
 i/n 
 
 on 
 
 -3- 
 
 on 
 
 VO 
 
 t— 
 
 O 
 
 ON 
 
 ON 
 
 -3- 
 
 VO 
 
 
 
 
 
 -3- 
 
 on 
 
 on 
 
 CO 
 
 VO 
 
 
 
 
 
 on 
 
 o\ 
 
 i/n 
 
 CM 
 
 vo 
 
 O 
 
 -3" 
 
 CO 
 
 O 
 
 -3- 
 
 on 
 
 t— 
 
 
 
 
 
 -3" 
 
 vo 
 
 t— 
 
 rH 
 
 CO 
 
 
 
 
 
 i/n 
 
 H 
 
 CO 
 
 00 
 
 CM 
 
 l/N 
 
 l/N 
 
 CO 
 
 ON 
 
 CM 
 
 CO 
 
 CM 
 
 
 
 
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 l^v 
 
 t— 
 
 rH 
 
 CM 
 
 rH 
 
 
 
 
 
 o\ 
 
 VO 
 
 rH 
 
 t— 
 
 t— 
 
 r-< 
 
 C— 
 
 LTN 
 
 ro 
 
 -3- 
 
 on 
 
 CO 
 
 
 
 on 
 
 t— 
 
 ON 
 
 ON 
 
 H 
 
 ON 
 
 on 
 
 
 
 
 
 on 
 
 ON 
 
 t— 
 
 on 
 
 L/N 
 
 co 
 
 H 
 
 t- 
 
 VO 
 
 VO 
 
 CM 
 
 H 
 
 
 CM 
 
 CO 
 
 on 
 
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 rH 
 
 on 
 
 o 
 
 
 
 
 
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 t— 
 
 rH 
 
 CM 
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 1 
 
 l/N 
 
 on 
 H 
 
 o 
 
 CM 
 1 
 
 t— 
 H 
 
 CM 
 
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 t— 
 
 rH 
 
 1 
 
 CM 
 O 
 
 rH 
 
 on 
 i 
 
 CM 
 
 CM 
 1 
 
 O 
 
 H 
 
 l/N 
 
 1 
 
 rH 
 
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 l/N 
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 1 
 
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 1 
 
 CM 
 
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 on 
 
 rH 
 
 l/N 
 
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 co 
 
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 vo 
 
 
 
 H 
 
 co 
 
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 t— 
 
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 vo 
 
 H 
 
 H 
 
 VO 
 
 vo 
 
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 rH 
 
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 CM 
 
 
 
 vo 
 
 vo 
 
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 CO 
 
 VO 
 
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 CM 
 
 CM 
 
 c— 
 
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 co 
 
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 t— 
 
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 t— 
 
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 C— 
 
 
 r— 
 
 CO 
 
 CM 
 
 ON 
 
 CO 
 
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 o 
 
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 on 
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 ■-{ 
 
 1 
 
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 on 
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 on 
 
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 on 
 
 c— 
 
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 CM 
 
 m 
 
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 l/N 
 
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 vo 
 
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 vo 
 
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 vo 
 
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 l/N 
 
 1 
 
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 rH 
 
 t— 
 rH 
 
 1 
 
 CO 
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 l-\ 
 
 ON 
 1 
 
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 on 
 
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 ^-1 
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 on 
 
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 on 
 
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 o 
 
 a 
 
 8 
 
 3 
 
 3 
 
 3 
 
 8 
 
 8 
 
 3 
 
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 en 
 
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 CO. 
 
 CQ 
 
 CQ 
 
 CQ 
 
 CQ 
 
 CQ 
 
 CQ 
 
 CQ 
 
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to 
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 0) 
 
 -p 
 
 cm 
 
 cu 
 
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 -p 
 
 «h 
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 W 
 
 3 
 
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 cd 
 > 
 
 ON 
 
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 ir\ 
 
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 ON 
 OO 
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 o 
 
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 ON 
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 On 
 
 H 
 
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 1 
 
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 LTN 
 'NO 
 
 O 
 
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 o 
 
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 cci 
 
 u 
 
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 Sh 
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 o 
 
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 t— -=y 
 
 vo 
 
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 o 
 
 CM 
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 I 
 
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 CO 
 
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 CM 
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 On 
 
 o 
 
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 On 
 
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 ft 
 
35 
 
 APPENDIX III 
 
A. Third Order Formulas 
 
 Polynomial 
 
 
 
 
 
 Table 1 
 
 
 
 
 a - 8 
 
 Coefficients 
 
 1 
 
 P 2 
 
 Q 2 
 
 P 3 
 
 S 
 
 
 
 185 
 
 16? 
 
 1321 
 
 1126 
 
 1 
 
 3^2 
 
 29U 
 
 2007 
 
 1962 
 
 2 
 
 -171 
 
 -165 
 
 -639 
 
 -95^ 
 
 3 
 
 lli 
 
 38 
 
 -hi 
 
 118 
 
 
 
 96 
 
 96 
 
 750 
 
 750 
 
 1 
 
 
 
 -2k 
 
 
 
 -U50 
 
 2 
 
 -5U 
 
 6 
 
 
 
 270 
 
 3 
 
 
 
 -162 
 
 -162 
 
 Table 2 
 Values of the Parameters 
 
 Polynomial 
 
 P 2 
 
 Q 
 H 2 
 
 P 3 
 
 Q 
 3 
 
 C 
 
 -0.75 
 
 -0.25 
 
 0.6 
 
 -0.6 
 
 D 
 
 -0.012 
 
 -0.06 
 
 -O.05U 
 
 -o.oi+o 
 
 max| £ | , £#1 
 
 0.7^7 
 
 0.1+77 
 
 0.6128 
 
 0.5529 
 
 C P + 1 
 
 -0.105U 
 
 -0.1587 
 
 -O.1726 
 
 -0.2558 
 
B. Fourth Order Formulas 
 
 a - 
 
 Table 1 
 (3 Coefficients 
 
 Polynomial 
 
 P 2 
 
 ^2 
 
 P 3 
 
 Q 
 ^3 
 
 % 
 
 a o 
 
 101 
 
 93 
 
 1063 
 
 185 
 
 13897 
 
 a l 
 
 200 
 
 204 
 
 212.8 
 
 402 
 
 29844 
 
 °2 
 
 -144 
 
 -180 
 
 -188U 
 
 -342 
 
 -23904 
 
 a 3 
 
 56 
 
 84 
 
 880 
 
 158 
 
 8812 
 
 % 
 
 -11 
 
 -15 
 
 -61 
 
 -33 
 
 -855 
 
 B 
 
 48 
 
 48 
 
 500 
 
 96 
 
 7500 
 
 6 1 
 
 
 
 -24 
 
 
 
 -48 
 
 -4500 
 
 8 2 
 
 -12 
 
 12 
 
 
 
 2k 
 
 2700 
 
 B 3 
 
 
 
 256 
 
 -12 
 
 -1620 
 
 "l, 
 
 
 
 
 
 972 
 
 Table 2 
 Values of the Parameters 
 
 Polynomial 
 
 P 2 
 
 Q 
 
 ^2 
 
 P 3 
 
 Q 3 
 
 Q 4 
 
 C 
 
 -0.5 
 
 -0.5 
 
 -0.8 
 
 -0.5 
 
 -0.6 
 
 D 
 
 -0.557 
 
 -0.4l4 
 
 -0.462 
 
 -0.472 
 
 -0.363 
 
 max] C| , C^l 
 
 0.527 
 
 0.662 
 
 0.833 
 
 0.578 
 
 0.6545 
 
 Vi 
 
 -0.0911 
 
 -0.1204 
 
 -0.0820 
 
 -0.1243 
 
 -0.1504 
 
Fifth Order Formulas 
 
 
 
 
 Table 
 
 1 
 
 
 
 
 
 
 a - 
 
 3 Coefficients 
 
 
 
 
 Polynomial 
 
 P 2 
 
 ^2 
 
 P 3 
 
 S 
 
 p u 
 
 % 
 
 s 
 
 a o 
 
 551 
 
 3137 
 
 138U58 
 
 208 
 
 U39 
 
 2077 
 
 U1U2 
 
 a i 
 
 1230 
 
 8320 
 
 310935 
 
 517 
 
 96U 
 
 5150 
 
 10225 
 
 a 
 2 
 
 -1180 
 
 -10220 
 
 -3U3740 
 
 -572 
 
 -972 
 
 -5660 
 
 -11120 
 
 a 
 3 
 
 7^0 
 
 7160 
 
 21)4580 
 
 388 
 
 66U 
 
 3760 
 
 7220 
 
 \ 
 
 -285 
 
 -2515 
 
 -53130 
 
 -1U8 
 
 -253 
 
 -1U15 
 
 -2530 
 
 a 5 
 
 U6 
 
 392 
 
 9813 
 
 23 
 
 36 
 
 2 1+2 
 
 3^7 
 
 e o 
 
 2H0 
 
 1500 
 
 60000 
 
 96 
 
 192 
 
 960 
 
 1920 
 
 6 1 
 
 
 
 -1200 
 
 
 
 -1+8 
 
 
 
 -U80 
 
 -960 
 
 3 2 
 
 -6o 
 
 960 
 
 
 
 2h 
 
 
 
 2 1+0 
 
 1+80 
 
 3 3 
 
 
 
 ^37^0 
 
 -12 
 
 
 
 -120 
 
 -2U0 
 
 \ 
 
 
 
 
 
 -12 
 
 60 
 
 120 
 
 S 
 
 
 
 
 
 
 
 -60 
 
 Polynomial 
 
 C 
 
 D 
 
 max|^| , &1 
 
 Table 2 
 Values of the Parameters 
 
 Q, 
 
 Q, 
 
 -0.5 -0.8 -0.9 -0.5 
 
 -2.325 -1.077 -1.101 -1.571 
 
 0.637 0.892 0.9^9 0.728 
 
 -0.0708 -0.0975 -0.0670 -0.0875 
 
 -0.5 
 
 -0.5 
 
 -0.5 
 
 -2.250 
 
 -1.553 
 
 _l.i4.U9 
 
 0.7U1 
 
 0.6875 
 
 0.722 
 
 -0.0720 
 
 -0.0886 
 
 -0.0913 
 
Sixth Order Formulas 
 
 a - B 
 
 Table 1 
 Coefficients 
 
 Polynomial 
 
 Q 2 
 
 P 3 
 
 s 
 
 ? u 
 
 Q 
 
 ^5 
 
 a o 
 
 3UU3 
 
 18439 
 
 5655 
 
 1^721+01 
 
 4520 
 
 °i 
 
 10156 
 
 45576 
 
 15655 
 
 3619208 
 
 12493 
 
 a 2 
 
 -1U810 
 
 -59130 
 
 -21125 
 
 -4572030 
 
 -16790 
 
 a 3 
 
 13280 
 
 50000 
 
 3760 
 
 4192080 
 
 14780 
 
 a 4 
 
 -7105 
 
 -252U5 
 
 -10525 
 
 -2334035 
 
 -8200 
 
 a 5 
 
 22208 
 
 8424 
 
 3306 
 
 662376 
 
 2615 
 
 a 6 
 
 -306 
 
 -1186 
 
 -455 
 
 -95198 
 
 -378 
 
 e o 
 
 1500 
 
 7500 
 
 2400 
 
 6xl0 5 
 
 1920 
 
 *1 
 
 -1200 
 
 
 
 -1200 
 
 
 
 -960 
 
 3 2 
 
 960 
 
 
 
 600 
 
 
 
 480 
 
 3 3 
 
 
 3840 
 
 -300 
 
 
 
 -240 
 
 3 4 
 
 
 
 
 -i44o6o 
 
 120 
 
 3 5 
 
 
 
 
 
 -60 
 
 Table 2 
 Values of the Parameters 
 
 Polynomial 
 
 Q 2 
 
 P 3 
 
 Q 
 3 
 
 P 4 
 
 Q 
 ^5 
 
 C 
 
 -0.8 
 
 -0.8 
 
 -0.5 
 
 -0.7 
 
 -0.5 
 
 D 
 
 -2.069 
 
 -4.452 
 
 -3.808 
 
 -4.694 
 
 -3.944 
 
 max| E,\ , £^1 
 
 0.934 
 
 0.868 
 
 O.861 
 
 0.933 
 
 0.839 
 
 Vi 
 
 -0.0732 
 
 -O.0566 
 
 -0.0671 
 
 -0.0573 
 
 -0.0677 
 
E. Seventh Order Formulas 
 
 Table 1 
 a - 3 Coefficients 
 
 Polynomial 
 
 Q 
 
 ^2 
 
 P 3 
 
 a 
 
 
 412310 
 
 136U0068U 
 
 a 
 1 
 
 13^2600 
 
 370394682 
 
 a 2 
 
 -2287481 
 
 -568618092 
 
 a 3 
 
 2519300 
 
 605263295 
 
 a k 
 
 -1809150 
 
 -430428180 
 
 a 5 
 
 853160 
 
 211815954 
 
 a 6 
 
 -234675 
 
 -59320212 
 
 a T 
 
 28556 
 
 7293237 
 
 e o 
 
 168000 
 
 525xl0 5 
 
 B l 
 
 -1U2800 
 
 
 
 e 2 
 
 121380 
 
 
 
 3 3 
 
 
 28946820 
 
 Table 2 
 Values of the Parameters 
 
 Polynomial 
 
 Q 2 
 
 P 3 
 
 C 
 
 -0.85 
 
 -0.82 
 
 D 
 
 -4.530 
 
 -12.022 
 
 max|c| , £7^1 
 
 0.992 
 
 0.927 
 
 Vi 
 
 -0.0589 
 
 -0.0474 
 
UO 
 
 LIST OF FIGURES 
 
Ul 
 
 15.00 
 
 o 
 
 Figure 2 
 
 i0 
 
 Locus of p(?)/a(C) C = e ,0 e[0, 2tt ] 
 
U2 
 
 r=T r .y 
 
 3 , 50 
 
 5.50 
 
 Figure 3 
 Seventh Order Formulas 
 
U3 
 
 o 
 
 : — I 
 
 -0750 1.50 
 
 3.50 
 
 
 5 
 
 50 
 
 Figure 
 
 k 
 
 
 
 Eighth Order 
 
 Formul 
 
 as 
 
 7 . 50 
 
hk 
 
 r= 
 
 r=9- 
 
 Figure 5 
 Ninth Order Formulas 
 
h5 
 
 
 r=9 
 
 1 . SO 
 
 3.50 
 
 5.5G 
 
 Figure 6 
 Tenth Order Formulas 
 
k6 
 
 r=10 
 
 -0750 
 
 Fi gure 7 
 Eleventh Order Formulas 
 
hi 
 
 Fi gure 8 
 Fifth Order Formula of Class II 
 
Form AEC-427 
 
 (6/68) 
 
 AECM 3201 
 
 U.S. ATOMIC ENERGY COMMISSION 
 
 UNIVERSITY-TYPE CONTRACTOR'S RECOMMENDATION FOR 
 
 DISPOSITION OF SCIENTIFIC AND TECHNICAL DOCUMENT 
 
 ( S»# Instructions on Riwrm Sid* ) 
 
 1. AEC REPORT NO. 
 
 COO-1U69-0162 
 
 2. TITLE 
 
 HIGH ORDER STIFFLY STABLE METHODS FOR 
 ORDINARY DIFFERENTIAL EQUATIONS 
 
 3. TYPE OF DOCUMENT (Check one): 
 
 PCI a. Scientific and technical report 
 
 I I b. Conference paper not to be published in a journal: 
 
 Title of conference 
 
 Date of conference 
 
 Exact location of conference 
 
 Sponsoring organization 
 
 □ c. Other (Specify) 
 
 4. RECOMMENDED ANNOUNCEMENT AND DISTRIBUTION (Check one): 
 
 EJ a- AEC's normal announcement and distribution procedures may be followed. 
 ~2 b. Make available only within AEC and to AEC contractors and other U.S. Government agencies and their contractors. 
 I | c. Make no announcement or distrubution. 
 
 5. REASON FOR RECOMMENDED RESTRICTIONS: 
 
 6. SUBMITTED BY: NAME AND POSITION (Please print or type) 
 
 C. W. Gear, Professor 
 
 and Principle Investigator 
 
 Organization 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 Signature 
 
 JKm^Y" 
 
 Date 
 
 April 1970 
 
 FOR AEC USE ONLY 
 
 7. AEC CONTRACT ADMINISTRATOR'S COMMENTS, IF ANY. ON ABOVE ANNOUNCEMENT AND DISTRIBUTION 
 RECOMMENDATION: 
 
 8. PATENT CLEARANCE: 
 
 Q a. AEC patent clearance has been granted by responsible AEC patent group. 
 LJ b. Report has been sent to responsible AEC patent group for clearance. 
 I I c. Patent clearance not required. 
 
JUN 12 
 
 1978 
 
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