#OK to post homework #TianHao Zhang,04/14/19 #################################Help####################################### Help:=proc():print(`EvalFCdiffk(L,x,x0,k) EvalFCdiffk_sym(L,x,k)`) end: #################################Part1###################################### #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. #By Bruno's formula d^nf(g(x))/dx^n = sum(f^(k)(g(x))*B_{n,k}(g'(x),..,g^(n-k+1)(x))) EvalFCdiffk:=proc(L,x,x0,k) local i, Ld, M, L1, n, g, j: n:=nops(L): if n = 1 then return(subs(x=x0,diff(L[1], [x\$k]))): fi: L1:=[op(2..n,L)]: g:=EvalFC(L1,x,x0): add(subs(x=g,diff(L[1], [x\$i]))*IncompleteBellB(k, i, seq(EvalFCdiffk(L1,x,x0,j),j=1..k-i+1)),i=1..k): end: #################################Part2###################################### #Find out whether the number of terms of diff(f(g(x)),x\$i) and the sum of its coefficients are in the OEIS. #Before dealing with the problem, we transfer the function in Part 1 into a symbolic #function which use 'Diff' instead of 'diff' EvalFCdiffk_sym:=proc(L,x,k) local i, Ld, M, L1, n, g, j: n:=nops(L): if n = 1 then return(Diff(L[1], [x\$k])): fi: L1:=[op(2..n,L)]: g:=EvalFC(L1,x,x): expand(add(Diff(L[1], [x\$i])*IncompleteBellB(k, i, seq(EvalFCdiffk_sym(L1,x,j),j=1..k-i+1)),i=1..k)): end: ##### #seq(SumCoeff(EvalFCdiffk_sym([f, g], x, k)), k = 2 .. 10); # 2, 5, 15, 52, 203, 877, 4140, 21147, 115975 #In OEIS: #A000110 #Bell or exponential numbers: number of ways to partition a set of n labeled elements. #################################Part3###################################### #Find out whether the number of terms of diff(f(g(h(x))),x\$i) and the sum of its coefficients are in the OEIS. #seq(SumCoeff(EvalFCdiffk_sym([f, g, h], x, k)), k = 2 .. 10); # 3, 12, 60, 358, 2471, 19302, 167894, 1606137, 16733779 #In OEIS: #A000258 #E.g.f.: exp(exp(exp(x)-1)-1). (Formerly M2932 N1178) ###############################From C21##################################### #C21.txt, April 11, 2019, Functional chains (and networks) Help21:=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: