1improved.mac is from the book "Computer Algebra in Applied 2Mathematics: An introduction to MACSYMA", by Richard H Rand, Pitman 3(1984). The version here was adapted from newimprv.bk1 by David 4Billinghurst. 5 6For given values of the parameters delta and e, either all the 7solutions are bounded (the equation is stable) or there exist 8unbounded solutions (the equation is unstable). The regions of 9stability are separated from thos of instability by "transition 10curves". 11 12This program computes the transition curves of Mathieu's equation 13using a method due to Levy and Keller (1963) which uses Fourier series 14to solve the perturbation equations. It is an improved version of 15recursiv.mac, as it stores intermediate results of the recursive 16functions A() and D() in arrays B[] and E[], rather than recalculating 17them each call. Some indirection is required, and the command 18REMARRAY is the only way to delete the values of arrays B and E, and 19would also delete any associated functions. 20 21The run below, using maxima-5.9.0cvs, reproduces the results on pages 22120-121 and page 140 of the book. 23 24(C1) load("./improved.mac"); 25(D1) ./improved.mac 26(C2) tc(); 27ENTER TRANSITION CURVE NUMBER N 280; 29ENTER DEGREE OF TRUNCATION 3010; 31 10 8 6 4 2 32 123707 e 68687 e 29 e 7 e e 33delta= - ---------- + -------- - ----- + ---- - -- 34 409600 294912 144 32 2 35 36(D2) FALSE 37(C3) tc(); 38ENTER TRANSITION CURVE NUMBER N 391; 40ENTER DEGREE OF TRUNCATION 4110; 42 10 9 8 7 6 5 4 3 43 114299 e 12121 e 83 e 55 e 49 e 11 e e e 44delta= - ---------- + --------- - ------ - ------ + ----- - ----- - --- + -- 45 6370099200 117964800 552960 294912 36864 4608 384 32 46 47 2 48 e e 1 49 - -- - - + - 50 8 2 4 51 52 10 9 8 7 6 5 4 3 53 114299 e 12121 e 83 e 55 e 49 e 11 e e e 54delta= - ---------- - --------- - ------ + ------ + ----- + ----- - --- - -- 55 6370099200 117964800 552960 294912 36864 4608 384 32 56 57 2 58 e e 1 59 - -- + - + - 60 8 2 4 61 62(D3) 63(C4) tc(); 64ENTER TRANSITION CURVE NUMBER N 652; 66ENTER DEGREE OF TRUNCATION 6710; 68 10 8 6 4 2 69 4363384401463 e 1669068401 e 1002401 e 763 e 5 e 70delta= ----------------- - ------------- + ---------- - ------ + ---- + 1 71 14447384985600 7166361600 4976640 3456 12 72 73 10 8 6 4 2 74 2499767 e 21391 e 289 e 5 e e 75delta= - -------------- + ---------- - ------- + ---- - -- + 1 76 14447384985600 7166361600 4976640 3456 12 77 78(D4) 79(C5) tc(); 80ENTER TRANSITION CURVE NUMBER N 810; 82ENTER DEGREE OF TRUNCATION 8320; 84 20 18 85 4011632808829219892175301 e 63642189915976296887 e 86delta= ----------------------------- - ------------------------ 87 1789497024366772224000000 44737425609169305600 88 89 16 14 12 10 90 7534554811777337 e 286241141477 e 8022167579 e 123707 e 91 + -------------------- - ---------------- + -------------- - ---------- 92 8182428094955520 468202291200 19110297600 409600 93 94 8 6 4 2 95 68687 e 29 e 7 e e 96 + -------- - ----- + ---- - -- 97 294912 144 32 2 98 99(D5) FALSE 100 101Reference: 102 103Levy, D.M. and Keller, J.B. "Instability Intervals of Hill's 104Equation", Comm. Pure Appl. Math. 16:469-476 (1963) 105 106Local Variables: *** 107mode: Text *** 108End: ***