In [22]:
from sympy import *
import numpy as np
In [14]:
h = symbols('h')
x = symbols('x')
xi = symbols('x_i')

# (9.21)
M = Matrix([
    [1, xi, xi**2, xi**3], 
    [0, 1, 2*xi, 3*(xi**2)], 
    [1, xi + h, (xi + h)**2, (xi + h)**3], 
    [0, 1, 2*(xi + h), 3*(xi + h)**2]])
M
Out[14]:
$\displaystyle \left[\begin{matrix}1 & x_{i} & x_{i}^{2} & x_{i}^{3}\\0 & 1 & 2 x_{i} & 3 x_{i}^{2}\\1 & h + x_{i} & \left(h + x_{i}\right)^{2} & \left(h + x_{i}\right)^{3}\\0 & 1 & 2 h + 2 x_{i} & 3 \left(h + x_{i}\right)^{2}\end{matrix}\right]$
In [16]:
M_inv = simplify(M.inv())
M_inv
Out[16]:
$\displaystyle \left[\begin{matrix}\frac{h^{3} - 3 h x_{i}^{2} - 2 x_{i}^{3}}{h^{3}} & - x_{i} - \frac{2 x_{i}^{2}}{h} - \frac{x_{i}^{3}}{h^{2}} & \frac{x_{i}^{2} \left(3 h + 2 x_{i}\right)}{h^{3}} & - \frac{x_{i}^{2} \left(h + x_{i}\right)}{h^{2}}\\\frac{6 x_{i} \left(h + x_{i}\right)}{h^{3}} & 1 + \frac{4 x_{i}}{h} + \frac{3 x_{i}^{2}}{h^{2}} & - \frac{6 x_{i} \left(h + x_{i}\right)}{h^{3}} & \frac{x_{i} \left(2 h + 3 x_{i}\right)}{h^{2}}\\- \frac{3 h + 6 x_{i}}{h^{3}} & - \frac{2 h + 3 x_{i}}{h^{2}} & \frac{3 \left(h + 2 x_{i}\right)}{h^{3}} & - \frac{h + 3 x_{i}}{h^{2}}\\\frac{2}{h^{3}} & \frac{1}{h^{2}} & - \frac{2}{h^{3}} & \frac{1}{h^{2}}\end{matrix}\right]$
In [18]:
a = simplify(M_inv.T * Matrix([1, x, x**2, x**3]))
a
Out[18]:
$\displaystyle \left[\begin{matrix}\frac{h^{3} - 3 h x_{i}^{2} + 2 x^{3} - 3 x^{2} \left(h + 2 x_{i}\right) + 6 x x_{i} \left(h + x_{i}\right) - 2 x_{i}^{3}}{h^{3}}\\\frac{- h^{2} x_{i} - 2 h x_{i}^{2} + x^{3} - x^{2} \left(2 h + 3 x_{i}\right) + x \left(h^{2} + 4 h x_{i} + 3 x_{i}^{2}\right) - x_{i}^{3}}{h^{2}}\\\frac{- 2 x^{3} + 3 x^{2} \left(h + 2 x_{i}\right) - 6 x x_{i} \left(h + x_{i}\right) + x_{i}^{2} \left(3 h + 2 x_{i}\right)}{h^{3}}\\\frac{x^{3} - x^{2} \left(h + 3 x_{i}\right) + x x_{i} \left(2 h + 3 x_{i}\right) - x_{i}^{2} \left(h + x_{i}\right)}{h^{2}}\end{matrix}\right]$
In [19]:
M_inv.subs(xi, 0)
Out[19]:
$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\- \frac{3}{h^{2}} & - \frac{2}{h} & \frac{3}{h^{2}} & - \frac{1}{h}\\\frac{2}{h^{3}} & \frac{1}{h^{2}} & - \frac{2}{h^{3}} & \frac{1}{h^{2}}\end{matrix}\right]$
In [20]:
a.subs(xi, 0)
Out[20]:
$\displaystyle \left[\begin{matrix}\frac{h^{3} - 3 h x^{2} + 2 x^{3}}{h^{3}}\\\frac{h^{2} x - 2 h x^{2} + x^{3}}{h^{2}}\\\frac{3 h x^{2} - 2 x^{3}}{h^{3}}\\\frac{- h x^{2} + x^{3}}{h^{2}}\end{matrix}\right]$
In [ ]:
# TBD