def lanczos(A):
k = len(A)
b = Vector.rand(n=k)
q = [Vector.new(n=k) for i in range(2)]
q[1] = b / abs(b)
b = [0]
a = [0]
for i in range(1, int(2 * sqrt(k))):
z = Vector((A * q[i]).transpose()[0])
a.append(float(Matrix([q[i]]) * z))
z = z - q[i] * a[i] - q[i-1] * b[i-1]
for j in q:
z -= j * (z * j)
for j in q:
z -= j * (z * j)
b.append(abs(z))
if b[i] == 0:
break
q.append(z / b[i])
Q = Matrix(q[-k-1:-1]).transpose()
T = Q.transpose() * A * Q
return (Q, T, )
