CoCalc -- dLp_xi.sagews

Project: PM2.5 dynamics and characterization

Path: PM_2_5/wave_equation/worksheets / dLp_xi.sagews

Views: ²⁷⁸

import numpy as np

var('x, xi')

x_nodes = [-1, 1]

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

(x, 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])) \

numerical_integral(L_0, -1, 1)

Error in lines 1-1
Traceback (most recent call last):
  File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 996, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 1, in <module>
NameError: name 'L_0' is not defined

diff(L_0)(xi_LGL[0]), diff(L_0)(xi_LGL[1]), diff(L_0)(xi_LGL[2]), diff(L_0)(xi_LGL[3]), diff(L_0)(xi_LGL[4]), diff(L_0)(xi_LGL[5]), diff(L_0)(xi_LGL[6]), diff(L_0)(xi_LGL[7])
diff(L_1)(xi_LGL[0]), diff(L_1)(xi_LGL[1]), diff(L_1)(xi_LGL[2]), diff(L_1)(xi_LGL[3]), diff(L_1)(xi_LGL[4]), diff(L_1)(xi_LGL[5]), diff(L_1)(xi_LGL[6]), diff(L_1)(xi_LGL[7])
diff(L_2)(xi_LGL[0]), diff(L_2)(xi_LGL[1]), diff(L_2)(xi_LGL[2]), diff(L_2)(xi_LGL[3]), diff(L_2)(xi_LGL[4]), diff(L_2)(xi_LGL[5]), diff(L_2)(xi_LGL[6]), diff(L_2)(xi_LGL[7])
diff(L_3)(xi_LGL[0]), diff(L_3)(xi_LGL[1]), diff(L_3)(xi_LGL[2]), diff(L_3)(xi_LGL[3]), diff(L_3)(xi_LGL[4]), diff(L_3)(xi_LGL[5]), diff(L_3)(xi_LGL[6]), diff(L_3)(xi_LGL[7])
diff(L_4)(xi_LGL[0]), diff(L_4)(xi_LGL[1]), diff(L_4)(xi_LGL[2]), diff(L_4)(xi_LGL[3]), diff(L_4)(xi_LGL[4]), diff(L_4)(xi_LGL[5]), diff(L_4)(xi_LGL[6]), diff(L_4)(xi_LGL[7])
diff(L_5)(xi_LGL[0]), diff(L_5)(xi_LGL[1]), diff(L_5)(xi_LGL[2]), diff(L_5)(xi_LGL[3]), diff(L_5)(xi_LGL[4]), diff(L_5)(xi_LGL[5]), diff(L_5)(xi_LGL[6]), diff(L_5)(xi_LGL[7])
diff(L_6)(xi_LGL[0]), diff(L_6)(xi_LGL[1]), diff(L_6)(xi_LGL[2]), diff(L_6)(xi_LGL[3]), diff(L_6)(xi_LGL[4]), diff(L_6)(xi_LGL[5]), diff(L_6)(xi_LGL[6]), diff(L_6)(xi_LGL[7])
diff(L_7)(xi_LGL[0]), diff(L_7)(xi_LGL[1]), diff(L_7)(xi_LGL[2]), diff(L_7)(xi_LGL[3]), diff(L_7)(xi_LGL[4]), diff(L_7)(xi_LGL[5]), diff(L_7)(xi_LGL[6]), diff(L_7)(xi_LGL[7])

(-14.0000000000226, -3.20991570302344, 0.792476681323880, -0.372150435728984, 0.243330712724289, -0.203284568901545, 0.219957514771985, -0.500000000000000)
(18.9375986071129, 3.31501492922825e-11, -2.80647579473469, 1.07894468878725, -0.661157350899271, 0.537039586158262, -0.573565414940005, 1.29768738831567)
(-7.56928981931106, 4.54358506455201, -6.49491571635963e-12, -2.37818723350641, 1.13535801687865, -0.845022556506714, 0.869448098330221, -1.94165942553537)
(4.29790816425547, -2.11206121431525, 2.87551740597844, -1.18891563261059e-11, -2.38892435916370, 1.37278583181113, -1.29423205091574, 2.81018898925442)
(-2.81018898925442, 1.29423205091574, -1.37278583181113, 2.38892435916370, 1.18891563261059e-11, -2.87551740597844, 2.11206121431525, -4.29790816425547)
(1.94165942553537, -0.869448098330221, 0.845022556506714, -1.13535801687865, 2.37818723350641, 6.49480469405717e-12, -4.54358506455201, 7.56928981931106)
(-1.29768738831567, 0.573565414940005, -0.537039586158262, 0.661157350899271, -1.07894468878725, 2.80647579473469, -3.31504823591899e-11, -18.9375986071129)
(0.500000000000000, -0.219957514771985, 0.203284568901545, -0.243330712724289, 0.372150435728984, -0.792476681323880, 3.20991570302344, 14.0000000000226)

lobatto_weights = [0.03571428571429,\
                   0.21070422714351,\
                   0.34112269248350,\
                   0.41245879465870,\
                   0.41245879465870,\
                   0.34112269248350,\
                   0.21070422714351,\
                   0.03571428571429]

diff(L_0)(xi_LGL[0]) * lobatto_weights[0]

-0.500000000000866

L_0.simplify_full()

-3.351562500008004*xi^7 + 3.351562500008006*xi^6 + 3.867187500010295*xi^5 - 3.867187500010297*xi^4 - 1.054687500002225*xi^3 + 1.054687500002225*xi^2 + 0.03906249999993106*xi - 0.03906249999993102


(14 - 7 **0.5)/30

0.378474956297847