integrate_1d.sagews

%typeset_mode True

var('xi')

xi_LGL = [-1.0,           \
          -0.87174014851, \
          -0.591700181433,\
          -0.209299217902,\
          0.209299217902, \
          0.591700181433, \
          0.87174014851,  \
          1.0]

$\displaystyle \xi$

#Assigning L_0 and L_1 as the lagrange basis polynomials.
L_0 =     ((xi - (xi_LGL[1])) / (xi_LGL[0] - xi_LGL[1]))\
        * ((xi - (xi_LGL[2])) / (xi_LGL[0] - xi_LGL[2]))\
        * ((xi - (xi_LGL[3])) / (xi_LGL[0] - xi_LGL[3]))\
        * ((xi - (xi_LGL[4])) / (xi_LGL[0] - xi_LGL[4]))\
        * ((xi - (xi_LGL[5])) / (xi_LGL[0] - xi_LGL[5]))\
        * ((xi - (xi_LGL[6])) / (xi_LGL[0] - xi_LGL[6]))\
        * ((xi - (xi_LGL[7])) / (xi_LGL[0] - xi_LGL[7]))

L_1 =     ((xi - (xi_LGL[0])) / (xi_LGL[1] - xi_LGL[0]))\
        * ((xi - (xi_LGL[2])) / (xi_LGL[1] - xi_LGL[2]))\
        * ((xi - (xi_LGL[3])) / (xi_LGL[1] - xi_LGL[3]))\
        * ((xi - (xi_LGL[4])) / (xi_LGL[1] - xi_LGL[4]))\
        * ((xi - (xi_LGL[5])) / (xi_LGL[1] - xi_LGL[5]))\
        * ((xi - (xi_LGL[6])) / (xi_LGL[1] - xi_LGL[6]))\
        * ((xi - (xi_LGL[7])) / (xi_LGL[1] - xi_LGL[7]))
        
L_2 =   ((xi - (xi_LGL[0])) / (xi_LGL[2] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[2] - xi_LGL[1])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[2] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[2] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[2] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[2] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[2] - xi_LGL[7]))

L_3 =   ((xi - (xi_LGL[0])) / (xi_LGL[3] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[3] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[3] - xi_LGL[2])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[3] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[3] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[3] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[3] - xi_LGL[7]))
        
L_4 =   ((xi - (xi_LGL[0])) / (xi_LGL[4] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[4] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[4] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[4] - xi_LGL[3])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[4] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[4] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[4] - xi_LGL[7]))


L_5 =   ((xi - (xi_LGL[0])) / (xi_LGL[5] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[5] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[5] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[5] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[5] - xi_LGL[4])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[5] - xi_LGL[6])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[5] - xi_LGL[7]))
        
L_6 =   ((xi - (xi_LGL[0])) / (xi_LGL[6] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[6] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[6] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[6] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[6] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[6] - xi_LGL[5])) \
        * ((xi - (xi_LGL[7])) / (xi_LGL[6] - xi_LGL[7]))
        

L_7 =   ((xi - (xi_LGL[0])) / (xi_LGL[7] - xi_LGL[0])) \
        * ((xi - (xi_LGL[1])) / (xi_LGL[7] - xi_LGL[1])) \
        * ((xi - (xi_LGL[2])) / (xi_LGL[7] - xi_LGL[2])) \
        * ((xi - (xi_LGL[3])) / (xi_LGL[7] - xi_LGL[3])) \
        * ((xi - (xi_LGL[4])) / (xi_LGL[7] - xi_LGL[4])) \
        * ((xi - (xi_LGL[5])) / (xi_LGL[7] - xi_LGL[5])) \
        * ((xi - (xi_LGL[6])) / (xi_LGL[7] - xi_LGL[6])) \

print(numerical_integral(L_0 * L_0, -1, 1)[0])
print(numerical_integral(L_1 * L_1, -1, 1)[0])
print(numerical_integral(L_2 * L_2, -1, 1)[0])
print(numerical_integral(L_3 * L_3, -1, 1)[0])
print(numerical_integral(L_4 * L_4, -1, 1)[0])
print(numerical_integral(L_5 * L_5, -1, 1)[0])
print(numerical_integral(L_6 * L_6, -1, 1)[0])
print(numerical_integral(L_7 * L_7, -1, 1)[0])

0.0333333333333 0.196657278667 0.318381179651 0.384961541681 0.384961541681 0.318381179651 0.196657278667 0.0333333333333