option nolet
!******************************************************
! symbol : general :Levi-Civita symbol
! info:https://en.wikipedia.org/wiki/Levi-Civita_symbol
!******************************************************
! website :https://petervlaschemeng.weebly.com/
! topic :fluid dynamics (2019)
!******************************************************
!filename :fluidgeneralleviCivitasymbol.tru
!******************************************************
! v=epsilon(p,q,r) ? find sign(v) = {-1,+1,0}
!******************************************************
print "------------------------------------------------"
print " fluid:general :Levi-Civita symbol,version 1.0  "
print " peter.vlasschaert@gmail.com,23/07/2019         "
print "------------------------------------------------"
print
print "--------------------------------------------"
print " example : epsilon(1,2,3) => sign = 1   (1) "
print " example : epsilon(2,1,3) => sign = -1  (2) "
print " example : epsilon(1,1,3) => sign = 0   (3) "
print "--------------------------------------------"
print
!**********************************************
! general : formula
!**********************************************
print "**********************************************"
print "* algo : product(sign(ii(p)-ii(q),ii,1,n;p>q)*"
print "**********************************************"
print
!**********************************************
! declare : array
!**********************************************
dim ii(1)
!examples : n = 3  ,{1,2,3}
! example (2)
n = 3
mat redim ii(1:n)
! matrix entries : (2)
print "-------------------------------"
print " matrix entries : sequence (2) "
print "-------------------------------"
print
ii(1) = 2
ii(2) = 1
ii(3) = 3
! output : matrix 'ii'
mat print ii
! copy orginal matrix (jj)
dim jj(1)
mat redim jj(1:n)
mat jj = ii
print "------------------------------"
print " copy array ii into jj        "
print "------------------------------"
print
mat print jj
!*********************************************
! epsilon(2,1,3) ?  sign
!*********************************************
! algo:product(sign(ii(p)-ii(q),ii,1,n;p>q)
!**********************************************
! set initial value : sign
!**********************************************
r=1 
!**********************************************
! algo : start
for p=1 to n
  for q=1 to p-1
! if one of members more than once.
  if ii(p) = ii(q) then r=0
! sign must be alternated stop:ii(p) > ii(q)
  if ii(p) < ii(q) then  r=-r
  next q
next p
! algo : end
print " input : array => "
mat print jj
print "***********************************"
print " sign (epsilon) => epsilon(p,q,...)"
print "***********************************"
print r
end