#C23.txt, April 18, 2019, Computing derivatives and gradients of complicated functions
Help:=proc(): print(`Box(n), IsGG(v) , Fsss(n) , Fss(n) , Fs(n), F(n,K) , EvalC(L,x,x0)`): 
print(`RP(x,d,K), RC(x,d,K,n) , EvalCD(L,x,x0) `):
end:

#Box(n): the set of 0-1 vectors of lenght n
Box:=proc(n) local S,s:
if n=0 then
  RETURN({[]}):
else 
 S:=Box(n-1):
 
RETURN({seq([op(s),0], s in S),seq([op(s),1], s in S)}):
fi:
end:

#IsGG(v): Does v contain 11
IsGG:=proc(v) local i:

for i from 1 to nops(v)-1 do
  if v[i]=1 and v[i+1]=1 then
    RETURN(false):
  fi:
od:
true:
end:
 
#Fsss(n):the n-th Fibonacci number using the extremely stupid way
#One of the Combinatorial definition of fibonacci number, the number
#of 0-1 vectors of length n avoiding 11
Fsss:=proc(n) local S,G,co,s:
S:=Box(n):
co:=0:

for s in S do
 if IsGG(s) then
  co:=co+1:
 fi:
od:
co:
end:


#Fss(n): the n-th Fibonacci number using the very stupid way
Fss:=proc(n):
if n=0 or n=1 then
 n:
else
 Fss(n-1)+Fss(n-2):
fi:
end:

#Fs(n): the n-th Fibonacci number using the usual way mod K
Fs:=proc(n,K)
option remember:
if n=0 or n=1 then
 n:
else
 Fs(n-1,K)+Fs(n-2,K) mod K:
fi:
end:

#F(n,K): the n-th Fibonacci number the clever way mod K
F:=proc(n,K) local a,b,c,i:

a:=0:
b:=1:

for i from 1 to n do
c:=a+b mod K:
a:=b:
b:=c:
od:
a:
end:

#RP(x,d,K): a random poly in x of degree k with coefficients from 1 to K do
RP:=proc(x,d,K) local ra,i:
ra:=rand(1..K):

add(ra()*x^i,i=0..d):

end:
#RC(x,d,K,n): a random chain of length n of poly in x of degree k with coefficients from 1 to K do
RC:=proc(x,d,K,n) local i:
[seq(RP(x,d,K),i=1..n)]:
end:

 #[x0->x1->x2-> ... ->xn] 

  #EvalC(L,x,x0): inputs a listL of expressions in the variable x and a number x0
   #outputs L[n](L[n-1](.... L[1](x0))))))))) mod K:
  #x0->L[1](x0)->L[2](L[1](x0))-....
 EvalC:=proc(L,x,x0) local a,i:
  a:=x0:
   for i from 1 to nops(L) do
     a:=subs(x=a,L[i]):
   od:
a:
end:


 #EvalCD(L,x,x0): inputs a listL of expressions in the variable x and a number x0
   #outputs L[n](L[n-1](.... L[1](x0))))))))) followed by the derivative of the composition
   #at x0:
  #x0->L[1](x0)->L[2](L[1](x0))-....
  #f(g(x))'= f'(g(x))*g'(x) . (F[i](Previous(x)))'(x0)= F[i]'(Previoius(x)|x=x0)* (Previous'(x)|x=x0)
 EvalCD:=proc(L,x,x0) local a,b,i:
  a:=x0:
   b:=1:
   for i from 1 to nops(L) do
      b:=b*subs(x=a,diff(L[i],x)) :
     a:=subs(x=a,L[i]):
     
   od:
a,b:
end: