Contact
CoCalc Logo Icon
StoreFeaturesDocsShareSupport News AboutSign UpSign In
| Download
Views: 278
%typeset_mode True

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]

lobatto_weights = [0.03571428571429,\
                   0.21070422714351,\
                   0.34112269248350,\
                   0.41245879465870,\
                   0.41245879465870,\
                   0.34112269248350,\
                   0.21070422714351,\
                   0.03571428571429]
(x\displaystyle x, ξ\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])) \
        
        
        


L_0.simplify_full()
L_1.simplify_full()
L_2.simplify_full()
L_3.simplify_full()
L_4.simplify_full()
L_5.simplify_full()
L_6.simplify_full()
L_7.simplify_full()
−3.35156250001 ξ7+3.35156250001 ξ6+3.86718750001 ξ5−3.86718750001 ξ4−1.0546875 ξ3+1.0546875 ξ2+0.0390624999999 ξ−0.0390624999999\displaystyle -3.35156250001 \, \xi^{7} + 3.35156250001 \, \xi^{6} + 3.86718750001 \, \xi^{5} - 3.86718750001 \, \xi^{4} - 1.0546875 \, \xi^{3} + 1.0546875 \, \xi^{2} + 0.0390624999999 \, \xi - 0.0390624999999
8.14072271825 ξ7−7.09659483138 ξ6−11.347477684 ξ5+9.89205188146 ξ4+3.33160871212 ξ3−2.90429707348 ξ2−0.124853746365 ξ+0.108840023398\displaystyle 8.14072271825 \, \xi^{7} - 7.09659483138 \, \xi^{6} - 11.347477684 \, \xi^{5} + 9.89205188146 \, \xi^{4} + 3.33160871212 \, \xi^{3} - 2.90429707348 \, \xi^{2} - 0.124853746365 \, \xi + 0.108840023398
−10.3581368289 ξ7+6.12891144098 ξ6+18.6833551584 ξ5−11.054944637 ξ4−8.6700371412 ξ3+5.13006254948 ξ2+0.34481881174 ξ−0.204029353468\displaystyle -10.3581368289 \, \xi^{7} + 6.12891144098 \, \xi^{6} + 18.6833551584 \, \xi^{5} - 11.054944637 \, \xi^{4} - 8.6700371412 \, \xi^{3} + 5.13006254948 \, \xi^{2} + 0.34481881174 \, \xi - 0.204029353468
11.3898137485 ξ7−2.38387910961 ξ6−24.032962502 ξ5+5.03008025554 ξ4+15.6735080469 ξ3−3.280452976 ξ2−3.0303592934 ξ+0.63425183007\displaystyle 11.3898137485 \, \xi^{7} - 2.38387910961 \, \xi^{6} - 24.032962502 \, \xi^{5} + 5.03008025554 \, \xi^{4} + 15.6735080469 \, \xi^{3} - 3.280452976 \, \xi^{2} - 3.0303592934 \, \xi + 0.63425183007
−11.3898137485 ξ7−2.38387910961 ξ6+24.032962502 ξ5+5.03008025554 ξ4−15.6735080469 ξ3−3.280452976 ξ2+3.0303592934 ξ+0.63425183007\displaystyle -11.3898137485 \, \xi^{7} - 2.38387910961 \, \xi^{6} + 24.032962502 \, \xi^{5} + 5.03008025554 \, \xi^{4} - 15.6735080469 \, \xi^{3} - 3.280452976 \, \xi^{2} + 3.0303592934 \, \xi + 0.63425183007
10.3581368289 ξ7+6.12891144098 ξ6−18.6833551584 ξ5−11.054944637 ξ4+8.6700371412 ξ3+5.13006254948 ξ2−0.34481881174 ξ−0.204029353468\displaystyle 10.3581368289 \, \xi^{7} + 6.12891144098 \, \xi^{6} - 18.6833551584 \, \xi^{5} - 11.054944637 \, \xi^{4} + 8.6700371412 \, \xi^{3} + 5.13006254948 \, \xi^{2} - 0.34481881174 \, \xi - 0.204029353468
−8.14072271825 ξ7−7.09659483138 ξ6+11.347477684 ξ5+9.89205188146 ξ4−3.33160871212 ξ3−2.90429707348 ξ2+0.124853746365 ξ+0.108840023398\displaystyle -8.14072271825 \, \xi^{7} - 7.09659483138 \, \xi^{6} + 11.347477684 \, \xi^{5} + 9.89205188146 \, \xi^{4} - 3.33160871212 \, \xi^{3} - 2.90429707348 \, \xi^{2} + 0.124853746365 \, \xi + 0.108840023398
3.35156250001 ξ7+3.35156250001 ξ6−3.86718750001 ξ5−3.86718750001 ξ4+1.0546875 ξ3+1.0546875 ξ2−0.0390624999999 ξ−0.0390624999999\displaystyle 3.35156250001 \, \xi^{7} + 3.35156250001 \, \xi^{6} - 3.86718750001 \, \xi^{5} - 3.86718750001 \, \xi^{4} + 1.0546875 \, \xi^{3} + 1.0546875 \, \xi^{2} - 0.0390624999999 \, \xi - 0.0390624999999
diff(L_0).simplify_full()
diff(L_1).simplify_full()
diff(L_2).simplify_full()
diff(L_3).simplify_full()
diff(L_4).simplify_full()
diff(L_5).simplify_full()
diff(L_6).simplify_full()
diff(L_7).simplify_full()
−23.4609375001 ξ6+20.109375 ξ5+19.3359375001 ξ4−15.46875 ξ3−3.16406250001 ξ2+2.109375 ξ+0.0390624999999\displaystyle -23.4609375001 \, \xi^{6} + 20.109375 \, \xi^{5} + 19.3359375001 \, \xi^{4} - 15.46875 \, \xi^{3} - 3.16406250001 \, \xi^{2} + 2.109375 \, \xi + 0.0390624999999
56.9850590277 ξ6−42.5795689883 ξ5−56.73738842 ξ4+39.5682075259 ξ3+9.99482613636 ξ2−5.80859414696 ξ−0.124853746365\displaystyle 56.9850590277 \, \xi^{6} - 42.5795689883 \, \xi^{5} - 56.73738842 \, \xi^{4} + 39.5682075259 \, \xi^{3} + 9.99482613636 \, \xi^{2} - 5.80859414696 \, \xi - 0.124853746365
−72.5069578025 ξ6+36.7734686459 ξ5+93.4167757919 ξ4−44.219778548 ξ3−26.0101114236 ξ2+10.260125099 ξ+0.34481881174\displaystyle -72.5069578025 \, \xi^{6} + 36.7734686459 \, \xi^{5} + 93.4167757919 \, \xi^{4} - 44.219778548 \, \xi^{3} - 26.0101114236 \, \xi^{2} + 10.260125099 \, \xi + 0.34481881174
79.7286962395 ξ6−14.3032746577 ξ5−120.16481251 ξ4+20.1203210222 ξ3+47.0205241407 ξ2−6.560905952 ξ−3.0303592934\displaystyle 79.7286962395 \, \xi^{6} - 14.3032746577 \, \xi^{5} - 120.16481251 \, \xi^{4} + 20.1203210222 \, \xi^{3} + 47.0205241407 \, \xi^{2} - 6.560905952 \, \xi - 3.0303592934
−79.7286962395 ξ6−14.3032746577 ξ5+120.16481251 ξ4+20.1203210222 ξ3−47.0205241407 ξ2−6.560905952 ξ+3.0303592934\displaystyle -79.7286962395 \, \xi^{6} - 14.3032746577 \, \xi^{5} + 120.16481251 \, \xi^{4} + 20.1203210222 \, \xi^{3} - 47.0205241407 \, \xi^{2} - 6.560905952 \, \xi + 3.0303592934
72.5069578025 ξ6+36.7734686459 ξ5−93.4167757919 ξ4−44.219778548 ξ3+26.0101114236 ξ2+10.260125099 ξ−0.34481881174\displaystyle 72.5069578025 \, \xi^{6} + 36.7734686459 \, \xi^{5} - 93.4167757919 \, \xi^{4} - 44.219778548 \, \xi^{3} + 26.0101114236 \, \xi^{2} + 10.260125099 \, \xi - 0.34481881174
−56.9850590277 ξ6−42.5795689883 ξ5+56.73738842 ξ4+39.5682075259 ξ3−9.99482613636 ξ2−5.80859414696 ξ+0.124853746365\displaystyle -56.9850590277 \, \xi^{6} - 42.5795689883 \, \xi^{5} + 56.73738842 \, \xi^{4} + 39.5682075259 \, \xi^{3} - 9.99482613636 \, \xi^{2} - 5.80859414696 \, \xi + 0.124853746365
23.4609375001 ξ6+20.109375 ξ5−19.3359375001 ξ4−15.46875 ξ3+3.16406250001 ξ2+2.109375 ξ−0.0390624999999\displaystyle 23.4609375001 \, \xi^{6} + 20.109375 \, \xi^{5} - 19.3359375001 \, \xi^{4} - 15.46875 \, \xi^{3} + 3.16406250001 \, \xi^{2} + 2.109375 \, \xi - 0.0390624999999