| 1 | ###
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| 2 | Legendre function
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| 3 |
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| 4 | Example
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| 5 |
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| 6 | legendre(x,3,0)
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| 7 |
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| 8 | Result
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| 9 |
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| 10 | 5 3 3
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| 11 | --- x - --- x
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| 12 | 2 2
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| 13 |
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| 14 | The computation uses the following recurrence relation.
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| 15 |
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| 16 | P(x,0) = 1
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| 17 |
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| 18 | P(x,1) = x
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| 19 |
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| 20 | n*P(x,n) = (2*(n-1)+1)*x*P(x,n-1) - (n-1)*P(x,n-2)
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| 21 |
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| 22 | In the "for" loop we have i = n-1 so the recurrence relation becomes
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| 23 |
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| 24 | (i+1)*P(x,n) = (2*i+1)*x*P(x,n-1) - i*P(x,n-2)
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| 25 |
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| 26 | For m > 0
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| 27 |
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| 28 | P(x,n,m) = (-1)^m * (1-x^2)^(m/2) * d^m/dx^m P(x,n)
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| 29 | ###
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| 30 |
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| 31 |
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| 32 |
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| 33 | Eval_legendre = ->
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| 34 | # 1st arg
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| 35 |
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| 36 | push(cadr(p1))
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| 37 | Eval()
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| 38 |
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| 39 | # 2nd arg
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| 40 |
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| 41 | push(caddr(p1))
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| 42 | Eval()
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| 43 |
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| 44 | # 3rd arg (optional)
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| 45 |
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| 46 | push(cadddr(p1))
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| 47 | Eval()
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| 48 |
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| 49 | p2 = pop()
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| 50 | if (p2 == symbol(NIL))
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| 51 | push_integer(0)
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| 52 | else
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| 53 | push(p2)
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| 54 |
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| 55 | legendre()
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| 56 |
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| 57 | #define X p1
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| 58 | #define N p2
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| 59 | #define M p3
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| 60 | #define Y p4
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| 61 | #define Y0 p5
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| 62 | #define Y1 p6
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| 63 |
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| 64 |
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| 65 | legendre = ->
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| 66 | save()
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| 67 | __legendre()
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| 68 | restore()
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| 69 |
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| 70 | __legendre = ->
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| 71 | m = 0
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| 72 | n = 0
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| 73 |
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| 74 | p3 = pop()
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| 75 | p2 = pop()
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| 76 | p1 = pop()
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| 77 |
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| 78 | push(p2)
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| 79 | n = pop_integer()
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| 80 |
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| 81 | push(p3)
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| 82 | m = pop_integer()
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| 83 |
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| 84 | if (n < 0 || isNaN(n) || m < 0 || isNaN(m))
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| 85 | push_symbol(LEGENDRE)
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| 86 | push(p1)
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| 87 | push(p2)
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| 88 | push(p3)
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| 89 | list(4)
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| 90 | return
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| 91 |
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| 92 | if (issymbol(p1))
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| 93 | __legendre2(n, m)
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| 94 | else
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| 95 | p4 = p1; # do this when X is an expr
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| 96 | p1 = symbol(SECRETX)
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| 97 | __legendre2(n, m)
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| 98 | p1 = p4
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| 99 | push(symbol(SECRETX))
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| 100 | push(p1)
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| 101 | subst()
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| 102 | Eval()
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| 103 |
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| 104 | __legendre3(m)
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| 105 |
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| 106 | __legendre2 = (n, m) ->
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| 107 | i = 0
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| 108 |
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| 109 | push_integer(1)
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| 110 | push_integer(0)
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| 111 |
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| 112 | p6 = pop()
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| 113 |
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| 114 | # i=1 p5 = 0
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| 115 | # p6 = 1
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| 116 | # ((2*i+1)*x*p6 - i*p5) / i = x
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| 117 | #
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| 118 | # i=2 p5 = 1
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| 119 | # p6 = x
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| 120 | # ((2*i+1)*x*p6 - i*p5) / i = -1/2 + 3/2*x^2
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| 121 | #
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| 122 | # i=3 p5 = x
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| 123 | # p6 = -1/2 + 3/2*x^2
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| 124 | # ((2*i+1)*x*p6 - i*p5) / i = -3/2*x + 5/2*x^3
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| 125 |
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| 126 | for i in [0...n]
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| 127 |
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| 128 | p5 = p6
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| 129 |
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| 130 | p6 = pop()
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| 131 |
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| 132 | push_integer(2 * i + 1)
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| 133 | push(p1)
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| 134 | multiply()
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| 135 | push(p6)
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| 136 | multiply()
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| 137 |
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| 138 | push_integer(i)
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| 139 | push(p5)
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| 140 | multiply()
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| 141 |
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| 142 | subtract()
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| 143 |
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| 144 | push_integer(i + 1)
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| 145 | divide()
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| 146 |
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| 147 | for i in [0...m]
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| 148 | push(p1)
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| 149 | derivative()
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| 150 |
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| 151 | # moveTos tos * (-1)^m * (1-x^2)^(m/2)
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| 152 |
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| 153 | __legendre3 = (m) ->
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| 154 | if (m == 0)
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| 155 | return
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| 156 |
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| 157 | if (car(p1) == symbol(COS))
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| 158 | push(cadr(p1))
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| 159 | sine()
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| 160 | square()
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| 161 | else if (car(p1) == symbol(SIN))
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| 162 | push(cadr(p1))
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| 163 | cosine()
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| 164 | square()
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| 165 | else
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| 166 | push_integer(1)
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| 167 | push(p1)
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| 168 | square()
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| 169 | subtract()
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| 170 |
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| 171 | push_integer(m)
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| 172 | push_rational(1, 2)
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| 173 | multiply()
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| 174 | power()
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| 175 | multiply()
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| 176 |
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| 177 | if (m % 2)
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| 178 | negate()
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| 179 |
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