#hw3.txt, 30 January 2014
#Sowmya Srinivasan


#Some examples from pages 60 to 90

#Differential equations
f:='f' : y:=f(x); 
dy:=diff(y,x); ddy:= diff(%,x); 
dsolve(ddy+5*dy+6*y = sin(x)*exp(-3*x),y); 

#plots
plot(sec(x),x=-Pi..2*Pi,y=-5..5); 
with(plots): 
p1:=plot(sqrt(x),x=0..400): 
p2:=plot(3*log(x),x=0..400,title='Square root and log functions'): 
display(p1,p2); 

plot3d(exp(-(x^2+y^2-1)^2),x=-2..2,y=-2..2); 
 
spaceplot([cos(t),sin(t),t],t=0..8*Pi,numpoints=400); 

#Linear Algebra
with(linalg); 
A:=matrix(2,3,[[a,b,c],[d,e,f]]); 
gausselim(A); 
gaussjord(A); 


#dangers of floating point calculations
#Digits:=75 gives a fairly good agreement for Hil(50): 
#1.393\times 10^{-1466} for evalf(det(Hil(50)))
#and 1.380 \times 10^{-1466} for det(evalf(Hil(50)))
#After repeating Hil(n) for n=10,20,30,... we see that a rough relation might be
#Digits = (n/10)*15 


#counting the number of inversions of a permutation p
#by modifying s1(p)
inv1:=proc(p) local i,j,n,s,k:
s:=1:
n:=nops(p):
k:=0:

for i from 1 to n do
 for j from i+1 to n do

    if p[i] > p[j] then
        s:=-s:
        k:=k+1:
    fi:

 od:
od:

#s:
k:

end:


#gf(n,q) computes the sum of q^inv1(p) over all permutations of length n
with(combinat); 
gf:=proc(n,q) local X,i,p,S : 
X:=permute(n): 
i:=1:  
S:=0: 

for i from 1 to n! do 
   p:=X[i]: 
   S:=S+q^inv1(p): 
od: 

S:

end: