#Ok to post homework
#Quentin Dubroff, Assignment 21, April 14 2019

with(linalg):

############################THINGS FROM C21.txt########################################

#C21.txt, April 11, 2019, Functional chains (and networks)
Help:=proc(): print(`  EvalFC1(L,x,0), EvalFC(L,x,x0) , RP(d,K,x) , RC(d,K,n,x) `): 
print(`EvalFCdiff(L,x,x0), SumCoeff(A) `):
end:

#y=a*x+b #Data-Set [x1,y1], ..., [xn,yn]  L(a,b):-=Sum(a*xi+b-yi)^2,i=1..n);
#diff(L,a)=0, diff(L,b)=0   a*x_{n+1}+b
# [x1,..., xm; y]: Loss(a1, ..., am,a0):=Sum((a1*x1+...+am*xm+a0-y)^2 , data points in the set)
#[m1,m2,m3, ..., mk] m1xm2 +m2xm3+....
#CNN
#functional chain x->f1(x)->f2(f1(x))
#Def: A functional chain is a list of expressions [f1, ..., fn] in x

#EvalFC1(L,x,x0): inputs L=[f1,..., fn] expressions in x and a number (or symbol) x0
#outputs f1(f2(...fn(x0))))) 
EvalFC1:=proc(L,x,x0) local i,L1:

if nops(L)=1 then
   RETURN(subs(x=x0,L[1])):
fi:

L1:=[op(1..nops(L)-1,L)]:
 expand(subs( x=EvalFC1(L1,x,x0),L[nops(L)])):


end:

#RP(d,K,x): a random polynomial of degree d in x with coeff. between -K and K
RP:=proc(d,K,x) local ra,i:
ra:=rand(-K..K):
add(ra()*x^i,i=0..d):
end:

#RC(d,K,n,x): a random functional chain of length n
RC:=proc(d,K,n,x) local i:
[seq(RP(d,K,x),i=1..n)]:
end:

#f(g(x))'= f'(g(x))*g'(x)
#f1(f2(f3(x))'=f1'(f2(f3(x))*f2'(f3(x))*f3'(x)
#(fk(f(k-1)(f(k-2)))'= fk'(f(k-1)....) [(f(k-1)(f(k-2)....) f1'(x0)]

#x0->f1(x0)->f2(f1(x0)->.... fn(f_{n-1}(x)) ...
#EvalFC(L,x,x0): inputs L=[f1,..., fn] expressions in x and a number (or symbol) x0
#outputs [f1(x0),f2(f1(x0)), f3(f2(f1(x0)), ..., fn(....)]: 
EvalFC:=proc(L,x,x0) local i,L1:

if nops(L)=1 then
   RETURN([subs(x=x0,L[1])]):
fi:

L1:=[op(1..nops(L)-1,L)]:

L1:=EvalFC(L1,x,x0):


[op(L1),subs(x=L1[nops(L1)] , L[nops(L)])]:

end:

#EvalFCdiff(L,x,x0): inputs a functional chain L, variable x, and number or symbol x0
#outputs the derivative of EvalFC(L,x,y) w.r.t y and y=x0
#
EvalFCdiff:=proc(L,x,x0) local i,Ld,M,L0:

L0:=EvalFC(L,x,x0):

Ld:=[seq(diff(L[i],x),i=1..nops(L))]:

M:=[subs(x=x0,Ld[1])]:

for i from 2 to nops(L) do
 M:=[op(M),  subs(x=L0[i-1]  ,Ld[i])  ]:
od:

convert(M,`*`):

end:

#SumCoeff(A): the sum of the coefficient in a + expressin
SumCoeff:=proc(A) local su,i, lauren:
su:=0:
if not type(A,`+`) then
  RETURN(FAIL):
fi:
for i from 1 to nops(A) do
   lauren:=op(1,op(i,A)):
    if type(lauren,integer) then
       su:=su+lauren:
       else
       su:=su+1:
     fi:
od:
su:
end:

      
 
##########################END C21.txt##############################################


#EvalFCdiffk(L,x,x0,k) inputs a functional chain L, variable x, and number or symbol x0 and outputs the 
#k-th derivative of EvalFC(L,x,y) w.r.t. y and y=x0.
#I am not sure how to do this without calling EvalFC(L,x,y) and differentiating 
# w.r.t. y k times.



#Terms of diff(f(g(x)),x$i) in the OEIS is sequence
#A000041: the number of partitions of n.
#It is not mentioned in the notes that this is the number of terms for arbitrary f and g
# but instead says terms of n-th derivative of 1/f or of log(f).

#As found in class, the sum of the coefficients in the above is the sequence of Bell numbers
#A000110

#Terms of diff(f(g(h(x))),x$i) in the OEIS is sequence 
#A022811: Number of terms in n-th derivative of a function composed with itself 3 times.

#We find that if we add the coefficients of the above expansion we get the sequence
#A000258 e.g.f.: exp(exp(exp(x)-1)-1)


#EvalNR2(L,c1,c2,x0,y0)inputs a list L of pairs [A1,B1],[A2,B2], ..., [An,Bn] and ouputs 
#the value in going from left to right and applying to the current vector the transformation
#[x,y] -> [max(a11*x+a12*y+b1,0), max(a21*x+a22*y+b2,0)]  for the current 
#A=[[a11,a12],[a21,a22]], and B=[b1,b2], max(c1*x+c2*y,0) to the final [x,y].
#NOTE It is unclear to me what the role of c1 and c2 are.
EvalNR2:=proc(L,x0,y0) local x,y,A,B,i:

if nops(L)<2 then
  RETURN([x0,y0]):
fi:
A:=L[1][1]:
B:=L[1][2]:
x:=max(dotprod(A[1],[x0,y0])+B[1],0):
y:=max(dotprod(A[2],[x0,y0])+B[2],0):
for i from 2 to nops(L) do
  A:=L[i][1]:
  B:=L[i][2]:
  x:=max(dotprod(A[1],[x,y])+B[1],0):
  y:=max(dotprod(A[2],[x,y])+B[2],0):
od:

[x,y]:
end:

#Using plot3d, we find that the graph looks like a piece of folded paper.