#hw22.txt Cole Franks 20/04/2014

##### Problem 1

Help22:=proc() print(`CubicSplineF(Lx,Lf,x),CubicSplineC(Lx, Lf, Lprimef, x)`): end:


CubicSplineF:=proc(Lx,Lf,x) local a, Lu, i1, var, eq, d, j:

d:=3:


Lu:=[seq(add(a[i1, j1]*x^j1, j1=0..d), i1=0..d)]:

var:={seq(seq(a[i1, j1], j1=0..d), i1=0..d)}:


#free boundary conditions

eq:={subs(x = Lx[1], diff(Lu[1], [x$2]))}:

eq:=eq union {subs(x = Lx[d+2], diff(Lu[d+1], [x$2]))}:

#constraining the values to the Lf’s

eq:=eq union {seq(subs(x = Lx[j], Lu[j]) = Lf[j], j=1..d+1), subs(x = Lx[d+2], Lu[d+1]) = Lf[d+2]}:

#loop for the interior points

for j from 2 to d+1 do 

	
	eq:=eq union {seq(subs(x = Lx[j], diff(Lu[j-1], [x$i1])) = subs(x = Lx[j], 	diff(Lu[j], [x$i1])), i1=0..d-1)}:

od:


var:=solve(eq, var):

if var=NULL then 
	FAIL:
else 
	return subs(var, Lu):

fi:

end:

CubicSplineC:=proc(Lx,Lf, Lprimef,x) local a, Lu, i1, var, eq, d, j:

d:=3:


Lu:=[seq(add(a[i1, j1]*x^j1, j1=0..d), i1=0..d)]:

var:={seq(seq(a[i1, j1], j1=0..d), i1=0..d)}:


#clamped boundary conditions

eq:={subs(x = Lx[1], diff(Lu[1], [x$1]))=Lprimef[1]}:

eq:=eq union {subs(x = Lx[d+2], diff(Lu[d+1], [x$1]))=Lprimef[2]}:

#constraining the values to the Lf’s

eq:=eq union {seq(subs(x = Lx[j], Lu[j]) = Lf[j], j=1..d+1), subs(x = Lx[d+2], Lu[d+1]) = Lf[d+2]}:

#loop for the interior points

for j from 2 to d+1 do 

	
	eq:=eq union {seq(subs(x = Lx[j], diff(Lu[j-1], [x$i1])) = subs(x = Lx[j], 	diff(Lu[j], [x$i1])), i1=0..d-1)}:

od:


var:=solve(eq, var):

if var=NULL then 
	FAIL:
else 
	return subs(var, Lu):

fi:

end: