#hw2.txt
#27 Jan 2014
#Sowmya Srinivasan

#Some examples from pages 30-60

#sequences
x:= 1,2,3; 
y:= 4,5,6; 
seq(i^2, i=1..5); 
f:= 'f' : seq(f(i), i=1..7); 

L:=a+b+2c+4d; 
op(%); 
nops(L); 
op(1,L); 

#sets
a:={1,2,3,4}; b:={2,4,6,8}; 
a union b; 
a minus b; 
a intersect b; 
member(3,a); 
member(5,a); 

#tables
T:=table([a,b]); 
S:=table([(1)=A,(3)=B+C,(5)=A*B*C]); 
S[3]; 
S; 
op(S); 

#arrays
A:=array(1..2,1..3,[]); 
op(A); 
B:=array(1..2,1..2,1..2,[]); 
op(B); 
c:=array(1..2,1..2): 
C[1,1]:=1 : c[1,2]:=0 : C[2,1]:=0 : C[2,2]:= 1 :
op(C); 
F:=array(1..2,1..3,[[1,2,4],[2,4,3]]); 

#data conversions
A:={1,2,3}: 
s:=1,2,3: 
L:=[1,2,3]: 
type(L,list); 
type(A,set); 
convert(s,list); 

#functions
f:=x -> x^2-3*x+5; 
f(2); 
g:=(x,y)-> x*y/(1+x^2+y^2); 
g(sin(t),cos(t)); 
simplify(%); 

f:= x-> x^2; 
g:= x-> sqrt(1-x); 
(f@g)(x); 
(g@f)(x); 
(f@@2)(x); 

#summation and product
Sum(i^2, i=1..10); 
value(%); 
sum(i^2, i=1..10); 

Product(1-q^i, i=1..5); 
value(%); 
i:='i'; 
product(1-q^i, i=1..5); 

#Limits
f:=(x^2-4)/(x-2); 
Limit(f,x=2); value(%); 
Limit(f,x=3,right);value(%); 
Limit(f,x=infinity); value(%); 

#Differentiation
f:=sqrt(1-x^2); 
diff(f,x); 
diff(tan(x),x$5); 
z:=exp(x*y)(1+sqrt(x^2+3y^2-x)); 
diff(z,x); 
diff(z,x,y)-diff(z,y,x); 

#Extrema
maximize(sin(x)+cos(x)); 
minimize(x^2-5*x+1,x=0..3); 

readlib(extrema); 
f:=2*x^2+y+y^2; 
g:=x^2+y^2-1; 
extrema(f,{g},{x,y},'s'); 
s;

#Integration
Int(x^2/sqrt(1-x^3),x); 
value(%); 
Int(r*exp(-r^2),r=0..infinity); 
value(%); 
Int(Int(y*sin(2*x+3*y^2),x),y); 

%Series
y:=1/sqrt(1-x); 
taylor(y,x=0,5); 



#Householder's method Hd(f,d,x,x0,N) inputs a polynomial f in x,
#a positive integer d which is the order of the Householder's method, 
#an initial guess x0,
#and outputs an approximation of root of f.

Hd1:=proc(f,d,x,x0) 
#x0->x0+d*normal(subs(x=x0,(diff(1/f,x$(d-1)))/subs(x=x0,diff(1/f,x$d)))

x0+d*normal(subs(x=x0,(diff(1/f,x$(d-1)))/subs(x=x0,diff(1/f,x$d))): 

end:

#Hd(f,d,x,x0,N): performs N iterations in Householder's of order d
Hd:=proc(f,d,x,x0,N) local x1,i:

x1:=x0:

for i from 1 to N do
 x1:=Hd1(f,d,x,x1):
od:

x1:

end:


#TestHouseholder(f,d,x,x0,eps,N): inputs an expression
#f in the variable x, an intial guess x0, a small eps
#and a pos. integer N , and outputs the successive errors
#abs(xn-alpha) for n from 1 to N where alpha is the
#appx. to the root of f(x)=0 found by the bisection method
TestHouseholder:=proc(f,d,x,x0,eps,N) local i,L,alpha,x1:
#Here (B,f,x,a,b,eps) refers to the bisection method, from C2.txt

alpha:=B(f,x,x0-2,x0+2,eps):

if alpha=FAIL then
 RETURN(FAIL):
fi:

L:=[]:

x1:=x0:


for i from 1 to N do
 x1:=Hd1(f,d,x,x1):
 L:=[ op(L), abs(x1-alpha[1])]:
od:
L:
end: