Chapter 5 Assignment

396 days ago by monetta

# Function definition for making a n x n Hilbert matrix def hilbert_matrix(n): return matrix([[1/i for i in [j..j+(n-1)]] for j in [1..n]]) 
       

1)
G = hilbert_matrix(3) f = matrix(RR, [[integral(cos(x), x, 0, 1)],[integral(x*cos(x), x, 0, 1)], [integral(x^2*cos(x), x, 0, 1)]]) a = G\f print "G =" print G print "\nf =" print f print "\na =" print a.change_ring(RR) var('r, s, t, x') p(r,s,t) = r + s*x + t*x^2 print "\np(x) =", p(float(a[0,0]), float(a[1,0]), float(a[2,0])) 
        
G =
[  1 1/2 1/3]
[1/2 1/3 1/4]
[1/3 1/4 1/5]

f =
[0.841470984807897]
[0.381773290676036]
[0.239133626928383]

a =
[   1.00340920678525]
[-0.0365364903942559]
[ -0.431009930340688]

p(x) = -0.43100993034068785*x^2 - 0.036536490394255931*x +
1.0034092067852536
G =
[  1 1/2 1/3]
[1/2 1/3 1/4]
[1/3 1/4 1/5]

f =
[0.841470984807897]
[0.381773290676036]
[0.239133626928383]

a =
[   1.00340920678525]
[-0.0365364903942559]
[ -0.431009930340688]

p(x) = -0.43100993034068785*x^2 - 0.036536490394255931*x + 1.0034092067852536

5)

 
G = hilbert_matrix(7) f = matrix(RR,[[.483789], [.319461], [.237879], [.189239], [.156991], [.134069], [.116951]]) a = G\f print "G =" print G print "\nf =" print f print "\na =" print a.change_ring(RR) var('q, r, s, t, u, v, w, x') p(q,r,s,t,u,v,w) = q + r*x + s*x^2 + t*x^3 + u*x^4 + v*x^5 + w*x^6 print "\np(x) =", p(float(a[0,0]), float(a[1,0]), float(a[2,0]), float(a[3,0]), float(a[4,0]), float(a[5,0]), float(a[6,0])) 
        
G =
[   1  1/2  1/3  1/4  1/5  1/6  1/7]
[ 1/2  1/3  1/4  1/5  1/6  1/7  1/8]
[ 1/3  1/4  1/5  1/6  1/7  1/8  1/9]
[ 1/4  1/5  1/6  1/7  1/8  1/9 1/10]
[ 1/5  1/6  1/7  1/8  1/9 1/10 1/11]
[ 1/6  1/7  1/8  1/9 1/10 1/11 1/12]
[ 1/7  1/8  1/9 1/10 1/11 1/12 1/13]

f =
[0.483789000000000]
[0.319461000000000]
[0.237879000000000]
[0.189239000000000]
[0.156991000000000]
[0.134069000000000]
[0.116951000000000]

a =
[-0.0152250000000000]
[   1.57651200000000]
[  -5.32854000000000]
[   20.0340000000000]
[  -35.7241500000000]
[   30.0484800000000]
[  -9.69368400000000]

p(x) = -9.6936839999999993*x^6 + 30.048479999999998*x^5 -
35.724149999999995*x^4 + 20.033999999999999*x^3 - 5.3285399999999994*x^2
+ 1.5765119999999999*x - 0.015224999999999999
G =
[   1  1/2  1/3  1/4  1/5  1/6  1/7]
[ 1/2  1/3  1/4  1/5  1/6  1/7  1/8]
[ 1/3  1/4  1/5  1/6  1/7  1/8  1/9]
[ 1/4  1/5  1/6  1/7  1/8  1/9 1/10]
[ 1/5  1/6  1/7  1/8  1/9 1/10 1/11]
[ 1/6  1/7  1/8  1/9 1/10 1/11 1/12]
[ 1/7  1/8  1/9 1/10 1/11 1/12 1/13]

f =
[0.483789000000000]
[0.319461000000000]
[0.237879000000000]
[0.189239000000000]
[0.156991000000000]
[0.134069000000000]
[0.116951000000000]

a =
[-0.0152250000000000]
[   1.57651200000000]
[  -5.32854000000000]
[   20.0340000000000]
[  -35.7241500000000]
[   30.0484800000000]
[  -9.69368400000000]

p(x) = -9.6936839999999993*x^6 + 30.048479999999998*x^5 - 35.724149999999995*x^4 + 20.033999999999999*x^3 - 5.3285399999999994*x^2 + 1.5765119999999999*x - 0.015224999999999999