#OK to post homework #Yukun Yao, Apr. 14, Assignment 21 ###Problem 1 Write a procedure #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. EvalFCdiffk:=proc(L,x,x0,k) local i,Ld,M,L0,y: L0:=EvalFC(L,x,y): Ld:=[seq(diff(l,y\$k),l in L0)]: subs(y=x0, Ld): end: ###Problem 2 Find out whether the number of terms of diff(f(g(x)),x\$i) and the sum of its coefficients are in the OEIS. #L := [seq(diff(f(g(x)), `\$`(x, i)), i = 1 .. 10)] #[1, seq(nops(L[i]), i=2..10)]=[1, 2, 3, 5, 7, 11, 15, 22, 30, 42]. It is A000041 in OEIS. #[1, seq(SumCoeff(L[i]), i = 2 .. 10)]=[1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]. It is A000110 in OEIS. ###Problem 3 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. #L := [seq(diff(f(g(h(x))), `\$`(x, i)), i = 1 .. 10)] #[1, seq(nops(L[i]), i=2..10)]=[1, 3, 6, 13, 23, 44, 74, 129, 210, 345]. It is A022811 in OEIS. #[1, seq(SumCoeff(L[i]), i = 2 .. 10)]=[1, 3, 12, 60, 358, 2471, 19302, 167894, 1606137, 16733779]. It is A000258 in OEIS. ######Following is 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: