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: ***