3601.sagews

def skalarA(f,g): 
    return (integral(f*g,x,0,1) + integral(diff(f,x)*diff(g,x),x,0,1)) 

def skalarB(f,g): 
    return integral(f*g,x,0,1) 

var('x')
def gram(base):
    ortho = []
    ortho.append(base[0])
    temp = 0
    for i in range(1,len(base)):
        temp = x^i
        for j in range(i):
            temp = temp + (skalarA(x^i,base[j])/skalarA(base[j],base[j]))*base[j]
        ortho.append(temp)
    return ortho

base = [1,x,x^2,x^3]
ortho = gram(base)
ortho

[1, x + 1/2, x^2 + 15/16*x + 1/3, x^3 + 25/23*x^2 + 9/10*x + 1/4]

var('a,b,c,d')
f(x) = -2/3*x^3 + x^2 + 4*x -2 
g(x) = a*x^3 + b*x^2 + c*x + d

li = []
ri = []

for i in range(len(ortho)):
    print "[%d]" %i
    var('x')
    li.append(skalarA(g,ortho[i]))
    ri.append(skalarB(f,ortho[i]))
    show (skalarA(g,ortho[i]) == skalarB(f,ortho[i]))

[0] x

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{4} \, a + \frac{1}{3} \, b + \frac{1}{2} \, c + d = \left(\frac{1}{6}\right)

[1] x

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{53}{40} \, a + \frac{17}{12} \, b + \frac{19}{12} \, c + d = \left(\frac{8}{15}\right)

[2] x

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{23}{8} \, a + \frac{8111}{2880} \, b + \frac{8}{3} \, c + \frac{109}{96} \, d = \left(\frac{2591}{2880}\right)

[3] x

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{946091}{193200} \, a + \frac{109}{24} \, b + \frac{3573}{920} \, c + \frac{1811}{1380} \, d = \left(\frac{92521}{72450}\right)

#print li
m_list = []
for l in li:
    m = []
    for variable in [a,b,c,d]:
        m.append(l.coefficient(variable))
    m_list.append(m)
m_list

[[x |--> 1/4, x |--> 1/3, x |--> 1/2, x |--> 1], [x |--> 53/40, x |--> 17/12, x |--> 19/12, x |--> 1], [x |--> 23/8, x |--> 8111/2880, x |--> 8/3, x |--> 109/96], [x |--> 946091/193200, x |--> 109/24, x |--> 3573/920, x |--> 1811/1380]]

li[0].coefficients()
k(x) = x^2+3*x
print k.coefficients()

[[3, 1], [1, 2]]