axiom
-- not allowed: )spool pfaffian.output

axiom
)sys

   Your argument list is not valid.

axiom
)set message test on

axiom
)set message auto off

axiom
)clear all

   All user variables and function definitions have been cleared.

--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]

axiom
--R
--R

axiom
--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

axiom
--R
--R

axiom
--E 2

--S 3 of 9
pfaffian A == eval(PfChar(l, A), l=0)

axiom
--R
--R

axiom
--E 3

--S 4 of 9
m:Matrix(Integer):=[[0,15],[-15,0]]

(1)

axiom
--R
--R        + 0    15+
--R   (4)  |        |
--R        +- 15  0 +
--R

axiom
--E 4

--S 5 of 9
pfaffian m

Compiling function B0 with type Integer -> Matrix Integer 
   The type of the local variable res has changed in the computation.
   We will attempt to interpret the code.
   Cannot compile map: PfChar 
   We will attempt to interpret the code.

(2)

axiom
--R
--R   (5)  15
--R

axiom
--E 5

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

(3)

axiom
--R
--R   (6)  17
--R

axiom
--E

--S 7 of 9
m1:Matrix(Integer):=[[0,a,b,c],[-a,0,d,e],[-b,-d,0,f],[-c,-e,-f,0]]

(4)

axiom
--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

axiom
--E 7

--S 8 of 9
m1ans:=a*f-b*e+d*c

(5)

axiom
--R
--R
--R   (8)  63
--R

axiom
--E 8

--S 9 of 9
pfaffian m1

Compiling function B0 with type PositiveInteger -> Matrix Integer

(6)