Edit detail for SandBoxPfaffian revision 4 of 7

added:
\begin{axiom}
)spool pfaffian.output
)set message test on
)set message auto off
)clear all

--S 1 of 9
B0 n == matrix [[(if i=j+1 and odd? j then -1 else _
                  if i=j-1 and odd? i then 1 else 0) _
                    for j in 1..n] for i in 1..n]
--R
--R                                                                   Type: Void
--E 1

--S 2 of 9
PfChar(lambda, A) ==
   n := nrows A
   (n = 2) => lambda^2 + A.(1,2)
   M := subMatrix(A, 3, n, 3, n)
   r := subMatrix(A, 1, 1, 3, n)
   s := subMatrix(A, 3, n, 2, 2)

   p := PfChar(lambda, M)
   d := degree(p, lambda)

   B := B0(n-2)
   C := r*B
   g := [(C*s).(1,1), A.(1,2), 1]
   if d >= 4 then
       B := M*B
       for i in 4..d by 2 repeat
           C := C*B
           g := cons((C*s).(1,1), g)
   g := reverse! g

   res := 0
   for i in 0..d by 2 for j in 2..d+2 repeat
       c := coefficient(p, lambda, i)
       for e in first(g, j) for k in 2..-d by -2 repeat
           res := res +  c * e * lambda^(k+i)

   res
--R
--R                                                                   Type: Void
--E 2

--S 3 of 9
pfaffian A == eval(PfChar(l, A), l=0)
--R
--R                                                                   Type: Void
--E 3

--S 4 of 9
m:Matrix(Integer):=[[0,15],[-15,0]]
--R
--R        + 0    15+
--R   (4)  |        |
--R        +- 15  0 +
--R                                                         Type: Matrix Integer
--E 4

--S 5 of 9
pfaffian m
--R
--R   (5)  15
--R                                                     Type: Polynomial Integer
--E 5

--S 6 of 9
(a,b,c,d,e,f):=(3,5,7,11,13,17)
--R
--R   (6)  17
--R                                                        Type: PositiveInteger
--E

--S 7 of 9
m1:Matrix(Integer):=[[0,a,b,c],[-a,0,d,e],[-b,-d,0,f],[-c,-e,-f,0]]
--R
--R        + 0    3     5    7 +
--R        |                   |
--R        |- 3   0     11   13|
--R   (7)  |                   |
--R        |- 5  - 11   0    17|
--R        |                   |
--R        +- 7  - 13  - 17  0 +
--R                                                         Type: Matrix Integer
--E 7

--S 8 of 9
m1ans:=a*f-b*e+d*c
--R
--R
--R   (8)  63
--R                                                        Type: PositiveInteger
--E 8

--S 9 of 9
pfaffian m1
--R
--R   Compiling function B0 with type PositiveInteger -> Matrix Integer
--R
--R   (9)  63
--R                                                     Type: Polynomial Integer
--E 9
)spool
)lisp (bye)
\end{axiom}

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