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Hall

fricas
x := operator('x)
 (1)
Type: BasicOperator?
fricas
sif(n)==1/(subscript('x,[n])-subscript('x,[n-1]))
Type: Void
fricas
sigma(n)==1/sif(n)^2*(sif(n+1)-2*sif(n)+sif(n-1))
Type: Void
fricas
sigma(n)
fricas
Compiling function sif with type Variable(n) -> Fraction(Polynomial(
Integer))
fricas
Compiling function sif with type Polynomial(Integer) -> Fraction(
Polynomial(Integer))
fricas
Compiling function sigma with type Variable(n) -> Fraction(
Polynomial(Integer))
 (2)
Type: Fraction(Polynomial(Integer))
fricas
r(n)==1/4*(sigma(n+1)-sigma(n))
Type: Void
fricas
numer r(n)
fricas
Compiling function sigma with type Polynomial(Integer) -> Fraction(
Polynomial(Integer))
fricas
Compiling function r with type Variable(n) -> Fraction(Polynomial(
Integer))
 (3)
Type: Polynomial(Integer)
fricas
factor denom(r(n))
 (4)
Type: Factored(Polynomial(Integer))

## Poirier to Hall

1-D

fricas
C:List Symbol := [subscript('x,[0])]
 (5)
Type: List(Symbol)
fricas
d := #C
 (6)
Type: PositiveInteger?
fricas
J := matrix [[D(x(C(1),t),C(1))]]
 (7)
Type: Matrix(Expression(Integer))
fricas
K := inverse J
 (8)
Type: Union(Matrix(Expression(Integer)),...)
fricas
Req18a:=[reduce(+,[reduce(+,[reduce(+,[reduce(+,[ (1/4)*D( K(k,i)*K(m,j)*D( D( K(l,j),C(k) ),C(l) ),C(m) ) for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d]
 (9)
Type: List(Expression(Integer))
fricas
diff1(n)==subscript('x,[n])-subscript('x,[n-1])
Type: Void
fricas
diff1(n)
fricas
Compiling function diff1 with type Variable(n) -> Polynomial(Integer
)
 (10)
Type: Polynomial(Integer)
fricas
Req18b:=eval(Req18a,D(x(C(1),t),C(1))=diff1(n))
 (11)
Type: List(Expression(Integer))
fricas
diff2(n)==diff1(n)-diff1(n-1)
Type: Void
fricas
diff2(n)
fricas
Compiling function diff1 with type Polynomial(Integer) -> Polynomial
(Integer)
fricas
Compiling function diff2 with type Variable(n) -> Polynomial(Integer
)
 (12)
Type: Polynomial(Integer)
fricas
Req18c:=eval(Req18b,D(x(C(1),t),[C(1),C(1)])=diff2(n));
Type: List(Expression(Integer))
fricas
numer Req18c.1
 (13)
Type: SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))
fricas
diff3(n)==diff2(n+1)-diff2(n)
Type: Void
fricas
diff3(n)
fricas
Compiling function diff2 with type Polynomial(Integer) -> Polynomial
(Integer)
fricas
Compiling function diff3 with type Variable(n) -> Polynomial(Integer
)
 (14)
Type: Polynomial(Integer)
fricas
Req18d:=eval(Req18c,D(x(C(1),t),[C(1),C(1),C(1)])=diff3(n));
Type: List(Expression(Integer))
fricas
numer Req18d.1
 (15)
Type: SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))
fricas
diff4(n)==diff3(n+1)-diff3(n)
Type: Void
fricas
diff4(n)
fricas
Compiling function diff3 with type Polynomial(Integer) -> Polynomial
(Integer)
fricas
Compiling function diff4 with type Variable(n) -> Polynomial(Integer
)
 (16)
Type: Polynomial(Integer)
fricas
Req18e:=eval(Req18d,D(x(C(1),t),[C(1),C(1),C(1),C(1)])=diff4(n));
Type: List(Expression(Integer))
fricas
--numer(Req18e.1-r(n))
factor numer Req18e.1
 (17)
Type: Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))
fricas
factor denom Req18e.1
 (18)
Type: Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))

2-D

fricas
x := operator('x)
 (19)
Type: BasicOperator?
fricas
y := operator('y)
 (20)
Type: BasicOperator?
fricas
C:List Symbol := [subscript('x,[0]),subscript('y,[0])]
 (21)
Type: List(Symbol)
fricas
d := #C
 (22)
Type: PositiveInteger?
fricas
J := matrix [[D(x(C(1),C(2),t),C(1)), D(x(C(1),C(2),t),C(2))],
[D(y(C(1),C(2),t),C(1)), D(y(C(1),C(2),t),C(2))]]
 (23)
Type: Matrix(Expression(Integer))
fricas
K := inverse J
 (24)
Type: Union(Matrix(Expression(Integer)),...)
fricas
Req18:=[reduce(+,[reduce(+,[reduce(+,[reduce(+,[
(1/4)*D( K(k,i)*K(m,j)*D( D( K(l,j),C(k) ),C(l) ),C(m) )
for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d];
Type: List(Expression(Integer))
fricas
#Req18
 (25)
Type: PositiveInteger?
fricas
kernels Req18.1
 (26)
Type: List(Kernel(Expression(Integer)))

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