solving methods for ode
DESCRIPTION
Solving Methods for ODE. 전자 20021267 도영욱 물리 20052317 홍용준. Introduction. Program 9.1. Introduction. Use Runge-Kutta Method to solve a second-order ODE. M, B and K is mass, damping constant and spring constant, respectively. Contents. Problem & Code Analysis Methods - PowerPoint PPT PresentationTRANSCRIPT
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Solving Methods for ODE
20021267 20052317
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Introduction
Program 9.1
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IntroductionUse Runge-Kutta Method to solve a second-order ODE
M, B and K is mass, damping constant and spring constant, respectively
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ContentsProblem & CodeAnalysisMethodsIterations (Modified Euler Method)Time Interval
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Solution (1)Modify equationM = 0.5, B = 10, k = 100
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Solution (2)Solve by using characteristic equation(apply initial values)
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Codefor(count = 0; count < CYCLE; count++) {step++;time = TIME_INTERVAL * step;y = exp(-10 * time) * (cos(10 * time) + sin(10 * time));z = (-20) * exp(-10 * time) * sin(10 * time);}
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Forward Euler Method(1)Equations
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Forward Euler Method(2)for(count = 0; count < CYCLE; count++) {step++;time = TIME_INTERVAL * step;oldy = y;oldz = z;y = oldy + TIME_INTERVAL * oldz;z = oldz - TIME_INTERVAL * (20 * oldz + 200 * oldy);}
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Modified Euler Methodfor(count = 0; count < CYCLE; count++) {step++;time = TIME_INTERVAL * step;oldy2 = y;oldz2 = z;for(iteration = 0; iteration < iter[mode]; iteration++) {oldy = y;oldz = z;y = oldy2 + TIME_INTERVAL / 2 * (oldz + oldz2);z = oldz2 - TIME_INTERVAL / 2 * (20 * (oldz + oldz2) + 200 * (oldy + oldy2));}}
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Runge-Kutta Methodkm = k / m;bm = b / m;for(count = 0; count < CYCLE; count++) {step++;time = TIME_INTERVAL * step;k1 = TIME_INTERVAL * z;l1 = -TIME_INTERVAL * (bm * z + km * y);k2 = TIME_INTERVAL * (z + l1);l2 = -TIME_INTERVAL * (bm * (z + l1) + km * (y + k1));y = y + (k1 + k2) / 2;z = z + (l1 + l2) / 2;}
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Analysis in graphs1 secondMethods (t = 0.01)Forward Euler MethodModified Euler Method (5 iterations)Runge-Kutta MethodIterations (Modified Euler Method, t = 0.01)1, 2, 5, 10Time Interval (Runge-Kutta Method)t = 0.05t = 0.01t = 0.005
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Methods(1) Forward
Chart1
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Forward Euler Method
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Forward Euler Method
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0.642.47E-031.84E-03-2.50E-02-3.87E-03
0.652.22E-031.79E-03-2.50E-02-6.47E-03
0.661.97E-031.72E-03-2.44E-02-8.48E-03
0.671.73E-031.62E-03-2.35E-02-9.97E-03
0.681.49E-031.52E-03-2.22E-02-1.10E-02
0.691.27E-031.41E-03-2.08E-02-1.17E-02
0.71.06E-031.29E-03-1.91E-02-1.20E-02
0.718.69E-041.17E-03-1.74E-02-1.20E-02
0.726.95E-041.05E-03-1.57E-02-1.19E-02
0.735.38E-049.30E-04-1.39E-02-1.15E-02
0.743.99E-048.17E-04-1.22E-02-1.10E-02
0.752.77E-047.11E-04-1.06E-02-1.04E-02
0.761.71E-046.10E-04-9.02E-03-9.69E-03
0.778.05E-055.17E-04-7.56E-03-8.95E-03
0.785.00E-064.31E-04-6.21E-03-8.18E-03
0.79-5.71E-053.53E-04-4.97E-03-7.41E-03
0.8-1.07E-042.83E-04-3.87E-03-6.64E-03
0.81-1.45E-042.20E-04-2.88E-03-5.89E-03
0.82-1.74E-041.65E-04-2.01E-03-5.17E-03
0.83-1.94E-041.17E-04-1.26E-03-4.48E-03
0.84-2.07E-047.54E-05-6.20E-04-3.84E-03
0.85-2.13E-044.00E-05-8.22E-05-3.25E-03
0.86-2.14E-041.03E-053.61E-04-2.70E-03
0.87-2.10E-04-1.43E-057.16E-04-2.21E-03
0.88-2.03E-04-3.41E-059.94E-04-1.76E-03
0.89-1.93E-04-4.97E-051.20E-03-1.37E-03
0.9-1.81E-04-6.16E-051.35E-03-1.02E-03
0.91-1.68E-04-7.02E-051.44E-03-7.13E-04
0.92-1.53E-04-7.60E-051.49E-03-4.50E-04
0.93-1.38E-04-7.93E-051.50E-03-2.28E-04
0.94-1.24E-04-8.06E-051.47E-03-4.10E-05
0.95-1.09E-04-8.03E-051.43E-031.13E-04
0.96-9.45E-05-7.85E-051.36E-032.36E-04
0.97-8.09E-05-7.56E-051.28E-033.33E-04
0.98-6.82E-05-7.19E-051.18E-034.06E-04
0.99-5.63E-05-6.76E-051.08E-034.59E-04
1-4.55E-05-6.28E-059.79E-044.94E-04
Sheet1
y
sol_y
time(sec)
Forward Euler Method
Sheet2
z
sol_z
time(sec)
Forward Euler Method
Sheet3
Chart6
0.09551720.129224
0.07571910.112889
0.05797030.0974727
0.04225680.0830868
0.02852660.0698051
0.01669730.0576676
0.006663390.0466868
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-0.0234603-0.00996669
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-0.0174349-0.0120295
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0.00036057-0.00270414
0.000716431-0.00220883
0.000993909-0.00176333
0.00120156-0.00136667
0.00134781-0.00101719
0.00144077-0.000712648
0.00148819-0.000450413
0.0014973-0.000227563
0.00147483-0.0000409905
0.001426910.000112504
0.001359080.000236139
0.001276270.000333089
0.001182840.000406437
0.001082580.000459134
0.0009787080.00049397
z
sol_z
time(sec)
Forward Euler Method
Sheet1
tysol_yzsol_z
01.00E+001.00E+000.00E+000.00E+00
0.011.00E+009.91E-01-2.00E+00-1.81E+00
0.029.80E-019.65E-01-3.60E+00-3.25E+00
0.039.44E-019.27E-01-4.84E+00-4.38E+00
0.048.96E-018.78E-01-5.76E+00-5.22E+00
0.058.38E-018.23E-01-6.40E+00-5.82E+00
0.067.74E-017.63E-01-6.80E+00-6.20E+00
0.077.06E-017.00E-01-6.98E+00-6.40E+00
0.086.36E-016.35E-01-7.00E+00-6.45E+00
0.095.66E-015.71E-01-6.87E+00-6.37E+00
0.14.97E-015.08E-01-6.63E+00-6.19E+00
0.114.31E-014.48E-01-6.30E+00-5.93E+00
0.123.68E-013.90E-01-5.90E+00-5.61E+00
0.133.09E-013.36E-01-5.46E+00-5.25E+00
0.142.55E-012.85E-01-4.98E+00-4.86E+00
0.152.05E-012.38E-01-4.50E+00-4.45E+00
0.161.60E-011.96E-01-4.01E+00-4.04E+00
0.171.20E-011.58E-01-3.53E+00-3.62E+00
0.188.45E-021.23E-01-3.06E+00-3.22E+00
0.195.39E-029.32E-02-2.62E+00-2.83E+00
0.22.77E-026.67E-02-2.20E+00-2.46E+00
0.215.70E-034.39E-02-1.82E+00-2.11E+00
0.22-1.25E-022.44E-02-1.46E+00-1.79E+00
0.23-2.71E-027.96E-03-1.15E+00-1.50E+00
0.24-3.86E-02-5.62E-03-8.63E-01-1.23E+00
0.25-4.72E-02-1.66E-02-6.13E-01-9.83E-01
0.26-5.33E-02-2.54E-02-3.96E-01-7.66E-01
0.27-5.73E-02-3.20E-02-2.10E-01-5.74E-01
0.28-5.94E-02-3.69E-02-5.37E-02-4.07E-01
0.29-5.99E-02-4.03E-027.59E-02-2.63E-01
0.3-5.92E-02-4.23E-021.81E-01-1.41E-01
0.31-5.74E-02-4.31E-022.63E-01-3.75E-02
0.32-5.47E-02-4.31E-023.25E-014.76E-02
0.33-5.15E-02-4.22E-023.69E-011.16E-01
0.34-4.78E-02-4.08E-023.99E-011.71E-01
0.35-4.38E-02-3.89E-024.14E-012.12E-01
0.36-3.97E-02-3.66E-024.19E-012.42E-01
0.37-3.55E-02-3.41E-024.15E-012.62E-01
0.38-3.13E-02-3.14E-024.03E-012.74E-01
0.39-2.73E-02-2.86E-023.85E-012.78E-01
0.4-2.35E-02-2.58E-023.62E-012.77E-01
0.41-1.98E-02-2.31E-023.37E-012.71E-01
0.42-1.65E-02-2.04E-023.09E-012.61E-01
0.43-1.34E-02-1.79E-022.80E-012.49E-01
0.44-1.06E-02-1.55E-022.51E-012.34E-01
0.45-8.06E-03-1.32E-022.22E-012.17E-01
0.46-5.84E-03-1.11E-021.94E-012.00E-01
0.47-3.90E-03-9.21E-031.67E-011.82E-01
0.48-2.24E-03-7.48E-031.41E-011.64E-01
0.49-8.26E-04-5.93E-031.17E-011.46E-01
0.53.47E-04-4.55E-039.55E-021.29E-01
0.511.30E-03-3.34E-037.57E-021.13E-01
0.522.06E-03-2.29E-035.80E-029.75E-02
0.532.64E-03-1.39E-034.23E-028.31E-02
0.543.06E-03-6.24E-042.85E-026.98E-02
0.553.35E-031.28E-051.67E-025.77E-02
0.563.51E-035.34E-046.66E-034.67E-02
0.573.58E-039.50E-04-1.70E-033.69E-02
0.583.56E-031.27E-03-8.52E-032.81E-02
0.593.48E-031.52E-03-1.39E-022.05E-02
0.63.34E-031.69E-03-1.81E-021.39E-02
0.613.16E-031.80E-03-2.12E-028.17E-03
0.622.95E-031.85E-03-2.33E-023.37E-03
0.632.71E-031.87E-03-2.45E-02-6.18E-04
0.642.47E-031.84E-03-2.50E-02-3.87E-03
0.652.22E-031.79E-03-2.50E-02-6.47E-03
0.661.97E-031.72E-03-2.44E-02-8.48E-03
0.671.73E-031.62E-03-2.35E-02-9.97E-03
0.681.49E-031.52E-03-2.22E-02-1.10E-02
0.691.27E-031.41E-03-2.08E-02-1.17E-02
0.71.06E-031.29E-03-1.91E-02-1.20E-02
0.718.69E-041.17E-03-1.74E-02-1.20E-02
0.726.95E-041.05E-03-1.57E-02-1.19E-02
0.735.38E-049.30E-04-1.39E-02-1.15E-02
0.743.99E-048.17E-04-1.22E-02-1.10E-02
0.752.77E-047.11E-04-1.06E-02-1.04E-02
0.761.71E-046.10E-04-9.02E-03-9.69E-03
0.778.05E-055.17E-04-7.56E-03-8.95E-03
0.785.00E-064.31E-04-6.21E-03-8.18E-03
0.79-5.71E-053.53E-04-4.97E-03-7.41E-03
0.8-1.07E-042.83E-04-3.87E-03-6.64E-03
0.81-1.45E-042.20E-04-2.88E-03-5.89E-03
0.82-1.74E-041.65E-04-2.01E-03-5.17E-03
0.83-1.94E-041.17E-04-1.26E-03-4.48E-03
0.84-2.07E-047.54E-05-6.20E-04-3.84E-03
0.85-2.13E-044.00E-05-8.22E-05-3.25E-03
0.86-2.14E-041.03E-053.61E-04-2.70E-03
0.87-2.10E-04-1.43E-057.16E-04-2.21E-03
0.88-2.03E-04-3.41E-059.94E-04-1.76E-03
0.89-1.93E-04-4.97E-051.20E-03-1.37E-03
0.9-1.81E-04-6.16E-051.35E-03-1.02E-03
0.91-1.68E-04-7.02E-051.44E-03-7.13E-04
0.92-1.53E-04-7.60E-051.49E-03-4.50E-04
0.93-1.38E-04-7.93E-051.50E-03-2.28E-04
0.94-1.24E-04-8.06E-051.47E-03-4.10E-05
0.95-1.09E-04-8.03E-051.43E-031.13E-04
0.96-9.45E-05-7.85E-051.36E-032.36E-04
0.97-8.09E-05-7.56E-051.28E-033.33E-04
0.98-6.82E-05-7.19E-051.18E-034.06E-04
0.99-5.63E-05-6.76E-051.08E-034.59E-04
1-4.55E-05-6.28E-059.79E-044.94E-04
Sheet1
y
sol_y
time(sec)
Forward Euler Method
Sheet2
z
sol_z
time(sec)
Forward Euler Method
Sheet3
Sheet3
11
10.99065
0.980.965067
0.9440.926657
0.89560.878441
0.8380.823067
0.7740080.762836
0.7060540.699718
0.6362110.635379
0.5662160.571205
0.4974950.508326
0.4311940.447646
0.3682040.389865
0.3091870.335502
0.254610.284922
0.2047650.238355
0.1597960.195915
0.1197260.157623
0.0844740.12342
0.05387790.0931828
0.02771150.0667407
0.005700770.0438839
-0.0124620.0243762
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-0.0385724-0.00561821
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-0.0533364-0.0253561
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-0.0594021-0.0369259
-0.0599388-0.040261
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-0.0547461-0.0430722
-0.051496-0.0422395
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-0.0438151-0.0388713
-0.0396704-0.0365941
-0.0354783-0.0340674
-0.0313312-0.0313823
-0.027304-0.028616
-0.0234556-0.0258332
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-0.0164618-0.0204215
-0.01337-0.0178693
-0.0105674-0.0154564
-0.00805785-0.0132011
-0.00583888-0.0111158
-0.00390255-0.00920726
-0.0022367-0.00747809
-0.000825975-0.00592704
0.000347341-0.00454988
0.00130251-0.00334002
0.0020597-0.00228903
0.00263941-0.00138713
0.00306197-0.000623612
0.003347240.0000127893
0.003514210.000533599
0.003580850.000950345
0.003563870.00127435
0.003478670.00151656
0.003339240.00168742
0.003158110.00179677
0.002946430.00185379
0.002713920.00186692
0.002468990.00184389
0.002218760.00179166
0.00196920.00171648
0.001725180.00162386
0.001490570.00151864
0.001268390.00140502
0.001060830.00128656
0.0008694090.0011663
0.000695060.00104673
0.0005381920.000929889
0.0003987970.000817401
0.0002765170.000710512
0.0001707170.00061014
0.00008054660.000516921
0.0000049960.000431246
-0.00005705540.000353296
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-0.0001943530.000117
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-0.000108774-0.0000802654
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-0.0000681513-0.0000719155
-0.0000563228-0.0000675716
-0.0000454971-0.0000627923
y
sol_y
time(sec)
Forward Euler Method
00
-2-1.80666
-3.6-3.25313
-4.84-4.37854
-5.76-5.2207
-6.3992-5.81573
-6.79536-6.19765
-6.9843-6.39818
-6.99955-6.44658
-6.87206-6.36954
-6.63008-6.1912
-6.29906-5.93314
-5.90163-5.6145
-5.45771-5.252
-4.98455-4.86018
-4.49686-4.45142
-4.00702-4.03621
-3.52521-3.62322
-3.05962-3.21952
-2.61664-2.83074
-2.20107-2.4612
-1.81628-2.11411
-1.46442-1.79168
-1.14662-1.49527
-0.86308-1.22553
-0.613319-0.982512
-0.396249-0.765763
-0.210326-0.574446
-0.0536632-0.407413
0.0758737-0.263285
0.180576-0.140519
0.262821-0.0374635
0.3250060.0475892
0.3694970.116363
0.3985890.170565
0.4144730.211855
0.4192090.241826
0.4147080.261988
0.4027230.273755
0.3848410.278434
0.3624810.277226
0.3368960.271221
0.3091780.261396
0.2802660.248621
0.2509530.233663
0.2218970.217188
0.1936330.199768
0.1665850.181892
0.1410730.163964
0.1173320.146318
0.09551720.129224
0.07571910.112889
0.05797030.0974727
0.04225680.0830868
0.02852660.0698051
0.01669730.0576676
0.006663390.0466868
-0.001697710.0368515
-0.008519870.0281322
-0.01394360.0204843
-0.01811230.013852
-0.02116830.00817133
-0.02325080.00337248
-0.0244935-0.000617509
-0.0250227-0.00387306
-0.0249561-0.0064684
-0.0244024-0.00847622
-0.0234603-0.00996669
-0.0222186-0.0110066
-0.020756-0.0116589
-0.0191416-0.0119819
-0.0174349-0.0120295
-0.0156868-0.0118508
-0.0139395-0.0114901
-0.012228-0.0109868
-0.01058-0.0103759
-0.00901703-0.00968794
-0.00755506-0.00894939
-0.00620514-0.00818276
-0.00497411-0.00740702
-0.00386517-0.00663785
-0.00287855-0.00588799
-0.00201194-0.0051675
-0.00126108-0.0044841
-0.000620162-0.00384343
-0.0000822015-0.00324934
0.00036057-0.00270414
0.000716431-0.00220883
0.000993909-0.00176333
0.00120156-0.00136667
0.00134781-0.00101719
0.00144077-0.000712648
0.00148819-0.000450413
0.0014973-0.000227563
0.00147483-0.0000409905
0.001426910.000112504
0.001359080.000236139
0.001276270.000333089
0.001182840.000406437
0.001082580.000459134
0.0009787080.00049397
z
sol_z
time(sec)
Forward Euler Method
Chart1
0.000347341-0.00454988
0.00130251-0.00334002
0.0020597-0.00228903
0.00263941-0.00138713
0.00306197-0.000623612
0.003347240.0000127893
0.003514210.000533599
0.003580850.000950345
0.003563870.00127435
0.003478670.00151656
0.003339240.00168742
0.003158110.00179677
0.002946430.00185379
0.002713920.00186692
0.002468990.00184389
0.002218760.00179166
0.00196920.00171648
0.001725180.00162386
0.001490570.00151864
0.001268390.00140502
0.001060830.00128656
0.0008694090.0011663
0.000695060.00104673
0.0005381920.000929889
0.0003987970.000817401
0.0002765170.000710512
0.0001707170.00061014
0.00008054660.000516921
0.0000049960.000431246
-0.00005705540.000353296
-0.0001067960.000283083
-0.0001454480.000220474
-0.0001742340.000165225
-0.0001943530.000117
-0.0002069640.0000754003
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-0.0002139880.0000102505
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-0.000153377-0.000075977
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-0.000123522-0.0000806491
-0.000108774-0.0000802654
-0.0000945048-0.0000784986
-0.000080914-0.0000756315
-0.0000681513-0.0000719155
-0.0000563228-0.0000675716
-0.0000454971-0.0000627923
y
sol_y
time(sec)
Forward Euler Method
Sheet1
tysol_yzsol_z
01.00E+001.00E+000.00E+000.00E+00
0.011.00E+009.91E-01-2.00E+00-1.81E+00
0.029.80E-019.65E-01-3.60E+00-3.25E+00
0.039.44E-019.27E-01-4.84E+00-4.38E+00
0.048.96E-018.78E-01-5.76E+00-5.22E+00
0.058.38E-018.23E-01-6.40E+00-5.82E+00
0.067.74E-017.63E-01-6.80E+00-6.20E+00
0.077.06E-017.00E-01-6.98E+00-6.40E+00
0.086.36E-016.35E-01-7.00E+00-6.45E+00
0.095.66E-015.71E-01-6.87E+00-6.37E+00
0.14.97E-015.08E-01-6.63E+00-6.19E+00
0.114.31E-014.48E-01-6.30E+00-5.93E+00
0.123.68E-013.90E-01-5.90E+00-5.61E+00
0.133.09E-013.36E-01-5.46E+00-5.25E+00
0.142.55E-012.85E-01-4.98E+00-4.86E+00
0.152.05E-012.38E-01-4.50E+00-4.45E+00
0.161.60E-011.96E-01-4.01E+00-4.04E+00
0.171.20E-011.58E-01-3.53E+00-3.62E+00
0.188.45E-021.23E-01-3.06E+00-3.22E+00
0.195.39E-029.32E-02-2.62E+00-2.83E+00
0.22.77E-026.67E-02-2.20E+00-2.46E+00
0.215.70E-034.39E-02-1.82E+00-2.11E+00
0.22-1.25E-022.44E-02-1.46E+00-1.79E+00
0.23-2.71E-027.96E-03-1.15E+00-1.50E+00
0.24-3.86E-02-5.62E-03-8.63E-01-1.23E+00
0.25-4.72E-02-1.66E-02-6.13E-01-9.83E-01
0.26-5.33E-02-2.54E-02-3.96E-01-7.66E-01
0.27-5.73E-02-3.20E-02-2.10E-01-5.74E-01
0.28-5.94E-02-3.69E-02-5.37E-02-4.07E-01
0.29-5.99E-02-4.03E-027.59E-02-2.63E-01
0.3-5.92E-02-4.23E-021.81E-01-1.41E-01
0.31-5.74E-02-4.31E-022.63E-01-3.75E-02
0.32-5.47E-02-4.31E-023.25E-014.76E-02
0.33-5.15E-02-4.22E-023.69E-011.16E-01
0.34-4.78E-02-4.08E-023.99E-011.71E-01
0.35-4.38E-02-3.89E-024.14E-012.12E-01
0.36-3.97E-02-3.66E-024.19E-012.42E-01
0.37-3.55E-02-3.41E-024.15E-012.62E-01
0.38-3.13E-02-3.14E-024.03E-012.74E-01
0.39-2.73E-02-2.86E-023.85E-012.78E-01
0.4-2.35E-02-2.58E-023.62E-012.77E-01
0.41-1.98E-02-2.31E-023.37E-012.71E-01
0.42-1.65E-02-2.04E-023.09E-012.61E-01
0.43-1.34E-02-1.79E-022.80E-012.49E-01
0.44-1.06E-02-1.55E-022.51E-012.34E-01
0.45-8.06E-03-1.32E-022.22E-012.17E-01
0.46-5.84E-03-1.11E-021.94E-012.00E-01
0.47-3.90E-03-9.21E-031.67E-011.82E-01
0.48-2.24E-03-7.48E-031.41E-011.64E-01
0.49-8.26E-04-5.93E-031.17E-011.46E-01
0.53.47E-04-4.55E-039.55E-021.29E-01
0.511.30E-03-3.34E-037.57E-021.13E-01
0.522.06E-03-2.29E-035.80E-029.75E-02
0.532.64E-03-1.39E-034.23E-028.31E-02
0.543.06E-03-6.24E-042.85E-026.98E-02
0.553.35E-031.28E-051.67E-025.77E-02
0.563.51E-035.34E-046.66E-034.67E-02
0.573.58E-039.50E-04-1.70E-033.69E-02
0.583.56E-031.27E-03-8.52E-032.81E-02
0.593.48E-031.52E-03-1.39E-022.05E-02
0.63.34E-031.69E-03-1.81E-021.39E-02
0.613.16E-031.80E-03-2.12E-028.17E-03
0.622.95E-031.85E-03-2.33E-023.37E-03
0.632.71E-031.87E-03-2.45E-02-6.18E-04
0.642.47E-031.84E-03-2.50E-02-3.87E-03
0.652.22E-031.79E-03-2.50E-02-6.47E-03
0.661.97E-031.72E-03-2.44E-02-8.48E-03
0.671.73E-031.62E-03-2.35E-02-9.97E-03
0.681.49E-031.52E-03-2.22E-02-1.10E-02
0.691.27E-031.41E-03-2.08E-02-1.17E-02
0.71.06E-031.29E-03-1.91E-02-1.20E-02
0.718.69E-041.17E-03-1.74E-02-1.20E-02
0.726.95E-041.05E-03-1.57E-02-1.19E-02
0.735.38E-049.30E-04-1.39E-02-1.15E-02
0.743.99E-048.17E-04-1.22E-02-1.10E-02
0.752.77E-047.11E-04-1.06E-02-1.04E-02
0.761.71E-046.10E-04-9.02E-03-9.69E-03
0.778.05E-055.17E-04-7.56E-03-8.95E-03
0.785.00E-064.31E-04-6.21E-03-8.18E-03
0.79-5.71E-053.53E-04-4.97E-03-7.41E-03
0.8-1.07E-042.83E-04-3.87E-03-6.64E-03
0.81-1.45E-042.20E-04-2.88E-03-5.89E-03
0.82-1.74E-041.65E-04-2.01E-03-5.17E-03
0.83-1.94E-041.17E-04-1.26E-03-4.48E-03
0.84-2.07E-047.54E-05-6.20E-04-3.84E-03
0.85-2.13E-044.00E-05-8.22E-05-3.25E-03
0.86-2.14E-041.03E-053.61E-04-2.70E-03
0.87-2.10E-04-1.43E-057.16E-04-2.21E-03
0.88-2.03E-04-3.41E-059.94E-04-1.76E-03
0.89-1.93E-04-4.97E-051.20E-03-1.37E-03
0.9-1.81E-04-6.16E-051.35E-03-1.02E-03
0.91-1.68E-04-7.02E-051.44E-03-7.13E-04
0.92-1.53E-04-7.60E-051.49E-03-4.50E-04
0.93-1.38E-04-7.93E-051.50E-03-2.28E-04
0.94-1.24E-04-8.06E-051.47E-03-4.10E-05
0.95-1.09E-04-8.03E-051.43E-031.13E-04
0.96-9.45E-05-7.85E-051.36E-032.36E-04
0.97-8.09E-05-7.56E-051.28E-033.33E-04
0.98-6.82E-05-7.19E-051.18E-034.06E-04
0.99-5.63E-05-6.76E-051.08E-034.59E-04
1-4.55E-05-6.28E-059.79E-044.94E-04
Sheet1
y
sol_y
time(sec)
Forward Euler Method
Sheet2
z
sol_z
time(sec)
Forward Euler Method
Sheet3
y
sol_y
time(sec)
Forward Euler Method
11
10.99065
0.980.965067
0.9440.926657
0.89560.878441
0.8380.823067
0.7740080.762836
0.7060540.699718
0.6362110.635379
0.5662160.571205
0.4974950.508326
0.4311940.447646
0.3682040.389865
0.3091870.335502
0.254610.284922
0.2047650.238355
0.1597960.195915
0.1197260.157623
0.0844740.12342
0.05387790.0931828
0.02771150.0667407
0.005700770.0438839
-0.0124620.0243762
-0.02710620.00796348
-0.0385724-0.00561821
-0.0472032-0.0166363
-0.0533364-0.0253561
-0.0572989-0.0320363
-0.0594021-0.0369259
-0.0599388-0.040261
-0.05918-0.0422629
-0.0573743-0.0431371
-0.0547461-0.0430722
-0.051496-0.0422395
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-0.0438151-0.0388713
-0.0396704-0.0365941
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-0.027304-0.028616
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-0.0164618-0.0204215
-0.01337-0.0178693
-0.0105674-0.0154564
-0.00805785-0.0132011
-0.00583888-0.0111158
-0.00390255-0.00920726
-0.0022367-0.00747809
-0.000825975-0.00592704
0.000347341-0.00454988
0.00130251-0.00334002
0.0020597-0.00228903
0.00263941-0.00138713
0.00306197-0.000623612
0.003347240.0000127893
0.003514210.000533599
0.003580850.000950345
0.003563870.00127435
0.003478670.00151656
0.003339240.00168742
0.003158110.00179677
0.002946430.00185379
0.002713920.00186692
0.002468990.00184389
0.002218760.00179166
0.00196920.00171648
0.001725180.00162386
0.001490570.00151864
0.001268390.00140502
0.001060830.00128656
0.0008694090.0011663
0.000695060.00104673
0.0005381920.000929889
0.0003987970.000817401
0.0002765170.000710512
0.0001707170.00061014
0.00008054660.000516921
0.0000049960.000431246
-0.00005705540.000353296
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-0.0001742340.000165225
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-0.000153377-0.000075977
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-0.000108774-0.0000802654
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-0.0000681513-0.0000719155
-0.0000563228-0.0000675716
-0.0000454971-0.0000627923
y
sol_y
time(sec)
Forward Euler Method
00
-2-1.80666
-3.6-3.25313
-4.84-4.37854
-5.76-5.2207
-6.3992-5.81573
-6.79536-6.19765
-6.9843-6.39818
-6.99955-6.44658
-6.87206-6.36954
-6.63008-6.1912
-6.29906-5.93314
-5.90163-5.6145
-5.45771-5.252
-4.98455-4.86018
-4.49686-4.45142
-4.00702-4.03621
-3.52521-3.62322
-3.05962-3.21952
-2.61664-2.83074
-2.20107-2.4612
-1.81628-2.11411
-1.46442-1.79168
-1.14662-1.49527
-0.86308-1.22553
-0.613319-0.982512
-0.396249-0.765763
-0.210326-0.574446
-0.0536632-0.407413
0.0758737-0.263285
0.180576-0.140519
0.262821-0.0374635
0.3250060.0475892
0.3694970.116363
0.3985890.170565
0.4144730.211855
0.4192090.241826
0.4147080.261988
0.4027230.273755
0.3848410.278434
0.3624810.277226
0.3368960.271221
0.3091780.261396
0.2802660.248621
0.2509530.233663
0.2218970.217188
0.1936330.199768
0.1665850.181892
0.1410730.163964
0.1173320.146318
0.09551720.129224
0.07571910.112889
0.05797030.0974727
0.04225680.0830868
0.02852660.0698051
0.01669730.0576676
0.006663390.0466868
-0.001697710.0368515
-0.008519870.0281322
-0.01394360.0204843
-0.01811230.013852
-0.02116830.00817133
-0.02325080.00337248
-0.0244935-0.000617509
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-0.0234603-0.00996669
-0.0222186-0.0110066
-0.020756-0.0116589
-0.0191416-0.0119819
-0.0174349-0.0120295
-0.0156868-0.0118508
-0.0139395-0.0114901
-0.012228-0.0109868
-0.01058-0.0103759
-0.00901703-0.00968794
-0.00755506-0.00894939
-0.00620514-0.00818276
-0.00497411-0.00740702
-0.00386517-0.00663785
-0.00287855-0.00588799
-0.00201194-0.0051675
-0.00126108-0.0044841
-0.000620162-0.00384343
-0.0000822015-0.00324934
0.00036057-0.00270414
0.000716431-0.00220883
0.000993909-0.00176333
0.00120156-0.00136667
0.00134781-0.00101719
0.00144077-0.000712648
0.00148819-0.000450413
0.0014973-0.000227563
0.00147483-0.0000409905
0.001426910.000112504
0.001359080.000236139
0.001276270.000333089
0.001182840.000406437
0.001082580.000459134
0.0009787080.00049397
z
sol_z
time(sec)
Forward Euler Method
-
Methods(2) Modified
Chart1
11
0.990950.99065
0.9656020.965067
0.9273660.926657
0.8792650.878441
0.8239550.823067
0.7637440.762836
0.7006070.699718
0.6362170.635379
0.5719660.571205
0.5089920.508326
0.4482030.447646
0.3903050.389865
0.3358210.335502
0.2851210.284922
0.2384370.238355
0.1958860.195915
0.1574920.157623
0.1231960.12342
0.09287830.0931828
0.06636660.0667407
0.04345220.0438839
0.02389880.0243762
0.007451840.00796348
-0.00615314-0.00561821
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-0.0325836-0.0320363
-0.0374613-0.0369259
-0.0407781-0.040261
-0.0427565-0.0422629
-0.043603-0.0431371
-0.043507-0.0430722
-0.042641-0.0422395
-0.04116-0.0407935
-0.039202-0.0388713
-0.0368889-0.0365941
-0.0343267-0.0340674
-0.0316071-0.0313823
-0.0288075-0.028616
-0.0259933-0.0258332
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-0.0205248-0.0204215
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-0.00590424-0.00592704
-0.00451787-0.00454988
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-0.00133802-0.00138713
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0.00006612960.0000127893
0.000587380.000533599
0.001003650.000950345
0.001326420.00127435
0.001566760.00151656
0.001735240.00168742
0.001841820.00179677
0.001895770.00185379
0.001905650.00186692
0.001879240.00184389
0.001823610.00179166
0.001745030.00171648
0.00164910.00162386
0.001540680.00151864
0.001423990.00140502
0.001302660.00128656
0.00117970.0011663
0.001057650.00104673
0.0009385490.000929889
0.0008240160.000817401
0.0007152980.000710512
0.0006133120.00061014
0.0005186830.000516921
0.0004317940.000431246
0.0003528170.000353296
0.0002817480.000283083
0.0002184430.000220474
0.0001626420.000165225
0.0001139970.000117
0.00007209290.0000754003
0.00003646870.0000399765
0.00000663230.0000102505
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-0.0000826451-0.0000802654
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-0.0000736104-0.0000719155
-0.0000690515-0.0000675716
-0.0000640679-0.0000627923
y
sol_y
time(sec)
Modified Euler Method
Sheet1
tysol_yzsol_z
01.00E+001.00E+000.00E+000.00E+00
0.019.91E-019.91E-01-1.81E+00-1.81E+00
0.029.66E-019.65E-01-3.26E+00-3.25E+00
0.039.27E-019.27E-01-4.39E+00-4.38E+00
0.048.79E-018.78E-01-5.23E+00-5.22E+00
0.058.24E-018.23E-01-5.83E+00-5.82E+00
0.067.64E-017.63E-01-6.21E+00-6.20E+00
0.077.01E-017.00E-01-6.41E+00-6.40E+00
0.086.36E-016.35E-01-6.46E+00-6.45E+00
0.095.72E-015.71E-01-6.39E+00-6.37E+00
0.15.09E-015.08E-01-6.21E+00-6.19E+00
0.114.48E-014.48E-01-5.95E+00-5.93E+00
0.123.90E-013.90E-01-5.63E+00-5.61E+00
0.133.36E-013.36E-01-5.27E+00-5.25E+00
0.142.85E-012.85E-01-4.87E+00-4.86E+00
0.152.38E-012.38E-01-4.46E+00-4.45E+00
0.161.96E-011.96E-01-4.05E+00-4.04E+00
0.171.57E-011.58E-01-3.63E+00-3.62E+00
0.181.23E-011.23E-01-3.23E+00-3.22E+00
0.199.29E-029.32E-02-2.84E+00-2.83E+00
0.26.64E-026.67E-02-2.47E+00-2.46E+00
0.214.35E-024.39E-02-2.12E+00-2.11E+00
0.222.39E-022.44E-02-1.79E+00-1.79E+00
0.237.45E-037.96E-03-1.50E+00-1.50E+00
0.24-6.15E-03-5.62E-03-1.23E+00-1.23E+00
0.25-1.72E-02-1.66E-02-9.81E-01-9.83E-01
0.26-2.59E-02-2.54E-02-7.64E-01-7.66E-01
0.27-3.26E-02-3.20E-02-5.72E-01-5.74E-01
0.28-3.75E-02-3.69E-02-4.04E-01-4.07E-01
0.29-4.08E-02-4.03E-02-2.59E-01-2.63E-01
0.3-4.28E-02-4.23E-02-1.36E-01-1.41E-01
0.31-4.36E-02-4.31E-02-3.30E-02-3.75E-02
0.32-4.35E-02-4.31E-025.22E-024.76E-02
0.33-4.26E-02-4.22E-021.21E-011.16E-01
0.34-4.12E-02-4.08E-021.75E-011.71E-01
0.35-3.92E-02-3.89E-022.16E-012.12E-01
0.36-3.69E-02-3.66E-022.46E-012.42E-01
0.37-3.43E-02-3.41E-022.66E-012.62E-01
0.38-3.16E-02-3.14E-022.78E-012.74E-01
0.39-2.88E-02-2.86E-022.82E-012.78E-01
0.4-2.60E-02-2.58E-022.81E-012.77E-01
0.41-2.32E-02-2.31E-022.74E-012.71E-01
0.42-2.05E-02-2.04E-022.64E-012.61E-01
0.43-1.79E-02-1.79E-022.51E-012.49E-01
0.44-1.55E-02-1.55E-022.36E-012.34E-01
0.45-1.32E-02-1.32E-022.19E-012.17E-01
0.46-1.11E-02-1.11E-022.01E-012.00E-01
0.47-9.21E-03-9.21E-031.83E-011.82E-01
0.48-7.47E-03-7.48E-031.65E-011.64E-01
0.49-5.90E-03-5.93E-031.47E-011.46E-01
0.5-4.52E-03-4.55E-031.30E-011.29E-01
0.51-3.30E-03-3.34E-031.13E-011.13E-01
0.52-2.24E-03-2.29E-039.79E-029.75E-02
0.53-1.34E-03-1.39E-038.33E-028.31E-02
0.54-5.72E-04-6.24E-046.99E-026.98E-02
0.556.61E-051.28E-055.77E-025.77E-02
0.565.87E-045.34E-044.66E-024.67E-02
0.571.00E-039.50E-043.67E-023.69E-02
0.581.33E-031.27E-032.79E-022.81E-02
0.591.57E-031.52E-032.02E-022.05E-02
0.61.74E-031.69E-031.35E-021.39E-02
0.611.84E-031.80E-037.80E-038.17E-03
0.621.90E-031.85E-032.99E-033.37E-03
0.631.91E-031.87E-03-1.01E-03-6.18E-04
0.641.88E-031.84E-03-4.27E-03-3.87E-03
0.651.82E-031.79E-03-6.86E-03-6.47E-03
0.661.75E-031.72E-03-8.86E-03-8.48E-03
0.671.65E-031.62E-03-1.03E-02-9.97E-03
0.681.54E-031.52E-03-1.14E-02-1.10E-02
0.691.42E-031.41E-03-1.20E-02-1.17E-02
0.71.30E-031.29E-03-1.23E-02-1.20E-02
0.711.18E-031.17E-03-1.23E-02-1.20E-02
0.721.06E-031.05E-03-1.21E-02-1.19E-02
0.739.39E-049.30E-04-1.17E-02-1.15E-02
0.748.24E-048.17E-04-1.12E-02-1.10E-02
0.757.15E-047.11E-04-1.06E-02-1.04E-02
0.766.13E-046.10E-04-9.84E-03-9.69E-03
0.775.19E-045.17E-04-9.08E-03-8.95E-03
0.784.32E-044.31E-04-8.30E-03-8.18E-03
0.793.53E-043.53E-04-7.50E-03-7.41E-03
0.82.82E-042.83E-04-6.71E-03-6.64E-03
0.812.18E-042.20E-04-5.95E-03-5.89E-03
0.821.63E-041.65E-04-5.21E-03-5.17E-03
0.831.14E-041.17E-04-4.52E-03-4.48E-03
0.847.21E-057.54E-05-3.86E-03-3.84E-03
0.853.65E-054.00E-05-3.26E-03-3.25E-03
0.866.63E-061.03E-05-2.71E-03-2.70E-03
0.87-1.79E-05-1.43E-05-2.20E-03-2.21E-03
0.88-3.77E-05-3.41E-05-1.75E-03-1.76E-03
0.89-5.32E-05-4.97E-05-1.35E-03-1.37E-03
0.9-6.50E-05-6.16E-05-9.98E-04-1.02E-03
0.91-7.34E-05-7.02E-05-6.91E-04-7.13E-04
0.92-7.90E-05-7.60E-05-4.27E-04-4.50E-04
0.93-8.22E-05-7.93E-05-2.03E-04-2.28E-04
0.94-8.33E-05-8.06E-05-1.54E-05-4.10E-05
0.95-8.26E-05-8.03E-051.38E-041.13E-04
0.96-8.06E-05-7.85E-052.62E-042.36E-04
0.97-7.76E-05-7.56E-053.58E-043.33E-04
0.98-7.36E-05-7.19E-054.30E-044.06E-04
0.99-6.91E-05-6.76E-054.82E-044.59E-04
1-6.41E-05-6.28E-055.15E-044.94E-04
Sheet1
y
sol_y
time(sec)
Modified Euler Method
Sheet2
z
sol_z
time(sec)
Modified Euler Method
Sheet3
Chart2
00
-1.80995-1.80666
-3.25955-3.25313
-4.38778-4.37854
-5.23239-5.2207
-5.82942-5.81573
-6.21289-6.19765
-6.4145-6.39818
-6.46352-6.44658
-6.38668-6.36954
-6.20815-6.1912
-5.94957-5.93314
-5.63011-5.6145
-5.26657-5.252
-4.8735-4.86018
-4.46337-4.45142
-4.04669-4.03621
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-3.22696-3.21952
-2.83667-2.83074
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0.05218750.0475892
0.1210150.116363
0.1751950.170565
0.2163970.211855
0.2462260.241826
0.2661990.261988
0.2777390.273755
0.2821630.278434
0.280680.277226
0.2743840.271221
0.2642630.261396
0.251190.248621
0.2359360.233663
0.2191740.217188
0.2014780.199768
0.1833380.181892
0.1651640.163964
0.1472890.146318
0.1299840.129224
0.1134580.112889
0.09787010.0974727
0.0833320.0830868
0.06991690.0698051
0.05766440.0576676
0.04658590.0466868
0.03666930.0368515
0.0278840.0281322
0.0201840.0204843
0.01351240.013852
0.007803720.00817133
0.002987060.00337248
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0.0001381940.000112504
0.0002615140.000236139
0.0003577810.000333089
0.0004301480.000406437
0.0004816320.000459134
0.000515080.00049397
z
sol_z
time(sec)
Modified Euler Method
Sheet1
tysol_yzsol_z
01.00E+001.00E+000.00E+000.00E+00
0.019.91E-019.91E-01-1.81E+00-1.81E+00
0.029.66E-019.65E-01-3.26E+00-3.25E+00
0.039.27E-019.27E-01-4.39E+00-4.38E+00
0.048.79E-018.78E-01-5.23E+00-5.22E+00
0.058.24E-018.23E-01-5.83E+00-5.82E+00
0.067.64E-017.63E-01-6.21E+00-6.20E+00
0.077.01E-017.00E-01-6.41E+00-6.40E+00
0.086.36E-016.35E-01-6.46E+00-6.45E+00
0.095.72E-015.71E-01-6.39E+00-6.37E+00
0.15.09E-015.08E-01-6.21E+00-6.19E+00
0.114.48E-014.48E-01-5.95E+00-5.93E+00
0.123.90E-013.90E-01-5.63E+00-5.61E+00
0.133.36E-013.36E-01-5.27E+00-5.25E+00
0.142.85E-012.85E-01-4.87E+00-4.86E+00
0.152.38E-012.38E-01-4.46E+00-4.45E+00
0.161.96E-011.96E-01-4.05E+00-4.04E+00
0.171.57E-011.58E-01-3.63E+00-3.62E+00
0.181.23E-011.23E-01-3.23E+00-3.22E+00
0.199.29E-029.32E-02-2.84E+00-2.83E+00
0.26.64E-026.67E-02-2.47E+00-2.46E+00
0.214.35E-024.39E-02-2.12E+00-2.11E+00
0.222.39E-022.44E-02-1.79E+00-1.79E+00
0.237.45E-037.96E-03-1.50E+00-1.50E+00
0.24-6.15E-03-5.62E-03-1.23E+00-1.23E+00
0.25-1.72E-02-1.66E-02-9.81E-01-9.83E-01
0.26-2.59E-02-2.54E-02-7.64E-01-7.66E-01
0.27-3.26E-02-3.20E-02-5.72E-01-5.74E-01
0.28-3.75E-02-3.69E-02-4.04E-01-4.07E-01
0.29-4.08E-02-4.03E-02-2.59E-01-2.63E-01
0.3-4.28E-02-4.23E-02-1.36E-01-1.41E-01
0.31-4.36E-02-4.31E-02-3.30E-02-3.75E-02
0.32-4.35E-02-4.31E-025.22E-024.76E-02
0.33-4.26E-02-4.22E-021.21E-011.16E-01
0.34-4.12E-02-4.08E-021.75E-011.71E-01
0.35-3.92E-02-3.89E-022.16E-012.12E-01
0.36-3.69E-02-3.66E-022.46E-012.42E-01
0.37-3.43E-02-3.41E-022.66E-012.62E-01
0.38-3.16E-02-3.14E-022.78E-012.74E-01
0.39-2.88E-02-2.86E-022.82E-012.78E-01
0.4-2.60E-02-2.58E-022.81E-012.77E-01
0.41-2.32E-02-2.31E-022.74E-012.71E-01
0.42-2.05E-02-2.04E-022.64E-012.61E-01
0.43-1.79E-02-1.79E-022.51E-012.49E-01
0.44-1.55E-02-1.55E-022.36E-012.34E-01
0.45-1.32E-02-1.32E-022.19E-012.17E-01
0.46-1.11E-02-1.11E-022.01E-012.00E-01
0.47-9.21E-03-9.21E-031.83E-011.82E-01
0.48-7.47E-03-7.48E-031.65E-011.64E-01
0.49-5.90E-03-5.93E-031.47E-011.46E-01
0.5-4.52E-03-4.55E-031.30E-011.29E-01
0.51-3.30E-03-3.34E-031.13E-011.13E-01
0.52-2.24E-03-2.29E-039.79E-029.75E-02
0.53-1.34E-03-1.39E-038.33E-028.31E-02
0.54-5.72E-04-6.24E-046.99E-026.98E-02
0.556.61E-051.28E-055.77E-025.77E-02
0.565.87E-045.34E-044.66E-024.67E-02
0.571.00E-039.50E-043.67E-023.69E-02
0.581.33E-031.27E-032.79E-022.81E-02
0.591.57E-031.52E-032.02E-022.05E-02
0.61.74E-031.69E-031.35E-021.39E-02
0.611.84E-031.80E-037.80E-038.17E-03
0.621.90E-031.85E-032.99E-033.37E-03
0.631.91E-031.87E-03-1.01E-03-6.18E-04
0.641.88E-031.84E-03-4.27E-03-3.87E-03
0.651.82E-031.79E-03-6.86E-03-6.47E-03
0.661.75E-031.72E-03-8.86E-03-8.48E-03
0.671.65E-031.62E-03-1.03E-02-9.97E-03
0.681.54E-031.52E-03-1.14E-02-1.10E-02
0.691.42E-031.41E-03-1.20E-02-1.17E-02
0.71.30E-031.29E-03-1.23E-02-1.20E-02
0.711.18E-031.17E-03-1.23E-02-1.20E-02
0.721.06E-031.05E-03-1.21E-02-1.19E-02
0.739.39E-049.30E-04-1.17E-02-1.15E-02
0.748.24E-048.17E-04-1.12E-02-1.10E-02
0.757.15E-047.11E-04-1.06E-02-1.04E-02
0.766.13E-046.10E-04-9.84E-03-9.69E-03
0.775.19E-045.17E-04-9.08E-03-8.95E-03
0.784.32E-044.31E-04-8.30E-03-8.18E-03
0.793.53E-043.53E-04-7.50E-03-7.41E-03
0.82.82E-042.83E-04-6.71E-03-6.64E-03
0.812.18E-042.20E-04-5.95E-03-5.89E-03
0.821.63E-041.65E-04-5.21E-03-5.17E-03
0.831.14E-041.17E-04-4.52E-03-4.48E-03
0.847.21E-057.54E-05-3.86E-03-3.84E-03
0.853.65E-054.00E-05-3.26E-03-3.25E-03
0.866.63E-061.03E-05-2.71E-03-2.70E-03
0.87-1.79E-05-1.43E-05-2.20E-03-2.21E-03
0.88-3.77E-05-3.41E-05-1.75E-03-1.76E-03
0.89-5.32E-05-4.97E-05-1.35E-03-1.37E-03
0.9-6.50E-05-6.16E-05-9.98E-04-1.02E-03
0.91-7.34E-05-7.02E-05-6.91E-04-7.13E-04
0.92-7.90E-05-7.60E-05-4.27E-04-4.50E-04
0.93-8.22E-05-7.93E-05-2.03E-04-2.28E-04
0.94-8.33E-05-8.06E-05-1.54E-05-4.10E-05
0.95-8.26E-05-8.03E-051.38E-041.13E-04
0.96-8.06E-05-7.85E-052.62E-042.36E-04
0.97-7.76E-05-7.56E-053.58E-043.33E-04
0.98-7.36E-05-7.19E-054.30E-044.06E-04
0.99-6.91E-05-6.76E-054.82E-044.59E-04
1-6.41E-05-6.28E-055.15E-044.94E-04
Sheet1
y
sol_y
time(sec)
Modified Euler Method
Sheet2
z
sol_z
time(sec)
Modified Euler Method
Sheet3
Chart3
-0.00451787-0.00454988
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-0.00133802-0.00138713
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0.00006612960.0000127893
0.000587380.000533599
0.001003650.000950345
0.001326420.00127435
0.001566760.00151656
0.001735240.00168742
0.001841820.00179677
0.001895770.00185379
0.001905650.00186692
0.001879240.00184389
0.001823610.00179166
0.001745030.00171648
0.00164910.00162386
0.001540680.00151864
0.001423990.00140502
0.001302660.00128656
0.00117970.0011663
0.001057650.00104673
0.0009385490.000929889
0.0008240160.000817401
0.0007152980.000710512
0.0006133120.00061014
0.0005186830.000516921
0.0004317940.000431246
0.0003528170.000353296
0.0002817480.000283083
0.0002184430.000220474
0.0001626420.000165225
0.0001139970.000117
0.00007209290.0000754003
0.00003646870.0000399765
0.00000663230.0000102505
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y
sol_y
time(sec)
Modified Euler Method
Sheet1
tysol_yzsol_z
01.00E+001.00E+000.00E+000.00E+00
0.019.91E-019.91E-01-1.81E+00-1.81E+00
0.029.66E-019.65E-01-3.26E+00-3.25E+00
0.039.27E-019.27E-01-4.39E+00-4.38E+00
0.048.79E-018.78E-01-5.23E+00-5.22E+00
0.058.24E-018.23E-01-5.83E+00-5.82E+00
0.067.64E-017.63E-01-6.21E+00-6.20E+00
0.077.01E-017.00E-01-6.41E+00-6.40E+00
0.086.36E-016.35E-01-6.46E+00-6.45E+00
0.095.72E-015.71E-01-6.39E+00-6.37E+00
0.15.09E-015.08E-01-6.21E+00-6.19E+00
0.114.48E-014.48E-01-5.95E+00-5.93E+00
0.123.90E-013.90E-01-5.63E+00-5.61E+00
0.133.36E-013.36E-01-5.27E+00-5.25E+00
0.142.85E-012.85E-01-4.87E+00-4.86E+00
0.152.38E-012.38E-01-4.46E+00-4.45E+00
0.161.96E-011.96E-01-4.05E+00-4.04E+00
0.171.57E-011.58E-01-3.63E+00-3.62E+00
0.181.23E-011.23E-01-3.23E+00-3.22E+00
0.199.29E-029.32E-02-2.84E+00-2.83E+00
0.26.64E-026.67E-02-2.47E+00-2.46E+00
0.214.35E-024.39E-02-2.12E+00-2.11E+00
0.222.39E-022.44E-02-1.79E+00-1.79E+00
0.237.45E-037.96E-03-1.50E+00-1.50E+00
0.24-6.15E-03-5.62E-03-1.23E+00-1.23E+00
0.25-1.72E-02-1.66E-02-9.81E-01-9.83E-01
0.26-2.59E-02-2.54E-02-7.64E-01-7.66E-01
0.27-3.26E-02-3.20E-02-5.72E-01-5.74E-01
0.28-3.75E-02-3.69E-02-4.04E-01-4.07E-01
0.29-4.08E-02-4.03E-02-2.59E-01-2.63E-01
0.3-4.28E-02-4.23E-02-1.36E-01-1.41E-01
0.31-4.36E-02-4.31E-02-3.30E-02-3.75E-02
0.32-4.35E-02-4.31E-025.22E-024.76E-02
0.33-4.26E-02-4.22E-021.21E-011.16E-01
0.34-4.12E-02-4.08E-021.75E-011.71E-01
0.35-3.92E-02-3.89E-022.16E-012.12E-01
0.36-3.69E-02-3.66E-022.46E-012.42E-01
0.37-3.43E-02-3.41E-022.66E-012.62E-01
0.38-3.16E-02-3.14E-022.78E-012.74E-01
0.39-2.88E-02-2.86E-022.82E-012.78E-01
0.4-2.60E-02-2.58E-022.81E-012.77E-01
0.41-2.32E-02-2.31E-022.74E-012.71E-01
0.42-2.05E-02-2.04E-022.64E-012.61E-01
0.43-1.79E-02-1.79E-022.51E-012.49E-01
0.44-1.55E-02-1.55E-022.36E-012.34E-01
0.45-1.32E-02-1.32E-022.19E-012.17E-01
0.46-1.11E-02-1.11E-022.01E-012.00E-01
0.47-9.21E-03-9.21E-031.83E-011.82E-01
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Modified Euler Method
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Modified Euler Method
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time(sec)
Modified Euler Method
z
sol_z
time(sec)
Modified Euler Method
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sol_z
time(sec)
Modified Euler Method
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Modified Euler Method
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00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
z
sol_z
time(sec)
Modified Euler Method
Sheet3
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y
sol_y
time(sec)
Modified Euler Method
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
z
sol_z
time(sec)
Modified Euler Method
-
Methods(3) Runge-Kutta
Chart1
11
0.990.99065
0.96390.965067
0.9251010.926657
0.8766150.878441
0.8210820.823067
0.7607890.762836
0.6976930.699718
0.6334460.635379
0.569420.571205
0.5067340.508326
0.4462780.447646
0.3887420.389865
0.3346360.335502
0.2843140.284922
0.2380.238355
0.1958030.195915
0.1577370.157623
0.123740.12342
0.0936880.0931828
0.06740640.0667407
0.04468540.0438839
0.02528860.0243762
0.008962310.00796348
-0.00455645-0.00561821
-0.0155337-0.0166363
-0.024233-0.0253561
-0.0309113-0.0320363
-0.0358153-0.0369259
-0.039179-0.040261
-0.0412217-0.0422629
-0.0421467-0.0431371
-0.0421406-0.0430722
-0.0413729-0.0422395
-0.039996-0.0407935
-0.0381456-0.0388713
-0.0359414-0.0365941
-0.0334875-0.0340674
-0.0308739-0.0313823
-0.0281769-0.028616
-0.0254605-0.0258332
-0.0227774-0.0230874
-0.02017-0.0204215
-0.0176719-0.0178693
-0.0153083-0.0154564
-0.0130975-0.0132011
-0.0110519-0.0111158
-0.00917827-0.00920726
-0.00747935-0.00747809
-0.00595408-0.00592704
-0.0045985-0.00454988
-0.00340626-0.00334002
-0.00236924-0.00228903
-0.00147797-0.00138713
-0.000722065-0.000623612
-0.00009059360.0000127893
0.0004276530.000533599
0.000843890.000950345
0.001169140.00127435
0.001414060.00151656
0.001588840.00168742
0.001703070.00179677
0.00176570.00185379
0.001784970.00186692
0.001768430.00184389
0.001722890.00179166
0.001654450.00171648
0.001568510.00162386
0.001469820.00151864
0.001362470.00140502
0.001249990.00128656
0.001135350.0011663
0.001021010.00104673
0.0009089830.000929889
0.0008008850.000817401
0.0006979540.000710512
0.0006011130.00061014
0.0005110070.000516921
0.0004280420.000431246
0.0003524210.000353296
0.0002841770.000283083
0.0002232020.000220474
0.0001692790.000165225
0.0001221010.000117
0.00008129440.0000754003
0.0000464390.0000399765
0.00001708330.0000102505
-0.0000072419-0.0000142726
-0.0000270112-0.0000340922
-0.0000426956-0.0000497021
-0.0000547541-0.0000615829
-0.0000636282-0.0000701957
-0.0000697364-0.000075977
-0.0000734714-0.0000793353
-0.000075197-0.0000806491
-0.0000752478-0.0000802654
-0.0000739273-0.0000784986
-0.0000715089-0.0000756315
-0.0000682361-0.0000719155
-0.0000643236-0.0000675716
-0.0000599585-0.0000627923
y
sol_y
time(sec)
Runge-Kutta Method
Sheet1
tysol_yzsol_z
01.00E+001.00E+000.00E+000.00E+00
0.019.90E-019.91E-01-1.80E+00-1.81E+00
0.029.64E-019.65E-01-3.24E+00-3.25E+00
0.039.25E-019.27E-01-4.36E+00-4.38E+00
0.048.77E-018.78E-01-5.20E+00-5.22E+00
0.058.21E-018.23E-01-5.79E+00-5.82E+00
0.067.61E-017.63E-01-6.17E+00-6.20E+00
0.076.98E-017.00E-01-6.36E+00-6.40E+00
0.086.33E-016.35E-01-6.41E+00-6.45E+00
0.095.69E-015.71E-01-6.33E+00-6.37E+00
0.15.07E-015.08E-01-6.15E+00-6.19E+00
0.114.46E-014.48E-01-5.90E+00-5.93E+00
0.123.89E-013.90E-01-5.58E+00-5.61E+00
0.133.35E-013.36E-01-5.22E+00-5.25E+00
0.142.84E-012.85E-01-4.83E+00-4.86E+00
0.152.38E-012.38E-01-4.42E+00-4.45E+00
0.161.96E-011.96E-01-4.01E+00-4.04E+00
0.171.58E-011.58E-01-3.60E+00-3.62E+00
0.181.24E-011.23E-01-3.20E+00-3.22E+00
0.199.37E-029.32E-02-2.82E+00-2.83E+00
0.26.74E-026.67E-02-2.45E+00-2.46E+00
0.214.47E-024.39E-02-2.11E+00-2.11E+00
0.222.53E-022.44E-02-1.79E+00-1.79E+00
0.238.96E-037.96E-03-1.49E+00-1.50E+00
0.24-4.56E-03-5.62E-03-1.22E+00-1.23E+00
0.25-1.55E-02-1.66E-02-9.84E-01-9.83E-01
0.26-2.42E-02-2.54E-02-7.69E-01-7.66E-01
0.27-3.09E-02-3.20E-02-5.79E-01-5.74E-01
0.28-3.58E-02-3.69E-02-4.14E-01-4.07E-01
0.29-3.92E-02-4.03E-02-2.71E-01-2.63E-01
0.3-4.12E-02-4.23E-02-1.49E-01-1.41E-01
0.31-4.21E-02-4.31E-02-4.62E-02-3.75E-02
0.32-4.21E-02-4.31E-023.85E-024.76E-02
0.33-4.14E-02-4.22E-021.07E-011.16E-01
0.34-4.00E-02-4.08E-021.61E-011.71E-01
0.35-3.81E-02-3.89E-022.03E-012.12E-01
0.36-3.59E-02-3.66E-022.33E-012.42E-01
0.37-3.35E-02-3.41E-022.53E-012.62E-01
0.38-3.09E-02-3.14E-022.65E-012.74E-01
0.39-2.82E-02-2.86E-022.71E-012.78E-01
0.4-2.55E-02-2.58E-022.70E-012.77E-01
0.41-2.28E-02-2.31E-022.64E-012.71E-01
0.42-2.02E-02-2.04E-022.55E-012.61E-01
0.43-1.77E-02-1.79E-022.43E-012.49E-01
0.44-1.53E-02-1.55E-022.29E-012.34E-01
0.45-1.31E-02-1.32E-022.13E-012.17E-01
0.46-1.11E-02-1.11E-021.96E-012.00E-01
0.47-9.18E-03-9.21E-031.79E-011.82E-01
0.48-7.48E-03-7.48E-031.61E-011.64E-01
0.49-5.95E-03-5.93E-031.44E-011.46E-01
0.5-4.60E-03-4.55E-031.27E-011.29E-01
0.51-3.41E-03-3.34E-031.11E-011.13E-01
0.52-2.37E-03-2.29E-039.64E-029.75E-02
0.53-1.48E-03-1.39E-038.23E-028.31E-02
0.54-7.22E-04-6.24E-046.94E-026.98E-02
0.55-9.06E-051.28E-055.75E-025.77E-02
0.564.28E-045.34E-044.67E-024.67E-02
0.578.44E-049.50E-043.71E-023.69E-02
0.581.17E-031.27E-032.85E-022.81E-02
0.591.41E-031.52E-032.10E-022.05E-02
0.61.59E-031.69E-031.45E-021.39E-02
0.611.70E-031.80E-038.85E-038.17E-03
0.621.77E-031.85E-034.10E-033.37E-03
0.631.78E-031.87E-031.46E-04-6.18E-04
0.641.77E-031.84E-03-3.10E-03-3.87E-03
0.651.72E-031.79E-03-5.69E-03-6.47E-03
0.661.65E-031.72E-03-7.71E-03-8.48E-03
0.671.57E-031.62E-03-9.22E-03-9.97E-03
0.681.47E-031.52E-03-1.03E-02-1.10E-02
0.691.36E-031.41E-03-1.10E-02-1.17E-02
0.71.25E-031.29E-03-1.13E-02-1.20E-02
0.711.14E-031.17E-03-1.14E-02-1.20E-02
0.721.02E-031.05E-03-1.13E-02-1.19E-02
0.739.09E-049.30E-04-1.10E-02-1.15E-02
0.748.01E-048.17E-04-1.05E-02-1.10E-02
0.756.98E-047.11E-04-9.98E-03-1.04E-02
0.766.01E-046.10E-04-9.34E-03-9.69E-03
0.775.11E-045.17E-04-8.65E-03-8.95E-03
0.784.28E-044.31E-04-7.93E-03-8.18E-03
0.793.52E-043.53E-04-7.19E-03-7.41E-03
0.82.84E-042.83E-04-6.46E-03-6.64E-03
0.812.23E-042.20E-04-5.74E-03-5.89E-03
0.821.69E-041.65E-04-5.05E-03-5.17E-03
0.831.22E-041.17E-04-4.40E-03-4.48E-03
0.848.13E-057.54E-05-3.78E-03-3.84E-03
0.854.64E-054.00E-05-3.21E-03-3.25E-03
0.861.71E-051.03E-05-2.68E-03-2.70E-03
0.87-7.24E-06-1.43E-05-2.20E-03-2.21E-03
0.88-2.70E-05-3.41E-05-1.77E-03-1.76E-03
0.89-4.27E-05-4.97E-05-1.39E-03-1.37E-03
0.9-5.48E-05-6.16E-05-1.05E-03-1.02E-03
0.91-6.36E-05-7.02E-05-7.49E-04-7.13E-04
0.92-6.97E-05-7.60E-05-4.92E-04-4.50E-04
0.93-7.35E-05-7.93E-05-2.73E-04-2.28E-04
0.94-7.52E-05-8.06E-05-8.92E-05-4.10E-05
0.95-7.52E-05-8.03E-056.31E-051.13E-04
0.96-7.39E-05-7.85E-051.87E-042.36E-04
0.97-7.15E-05-7.56E-052.84E-043.33E-04
0.98-6.82E-05-7.19E-053.59E-044.06E-04
0.99-6.43E-05-6.76E-054.14E-044.59E-04
1-6.00E-05-6.28E-054.51E-044.94E-04
Sheet1
y
sol_y
time(sec)
Runge-Kutta Method
Sheet2
z
sol_z
time(sec)
Runge-Kutta Method
Sheet3
Chart2
00
-1.8-1.80666
-3.24-3.25313
-4.35942-4.37854
-5.19631-5.2207
-5.78692-5.81573
-6.16535-6.19765
-6.36336-6.39818
-6.41017-6.44658
-6.33244-6.36954
-6.15423-6.1912
-5.89705-5.93314
-5.57991-5.6145
-5.21946-5.252
-4.83011-4.86018
-4.42415-4.45142
-4.01196-4.03621
-3.60214-3.62322
-3.20166-3.21952
-2.81607-2.83074
-2.44966-2.4612
-2.10555-2.11411
-1.78593-1.79168
-1.49213-1.49527
-1.22475-1.22553
-0.983849-0.982512
-0.768957-0.765763
-0.579236-0.574446
-0.413541-0.407413
-0.270501-0.263285
-0.148583-0.140519
-0.0461534-0.0374635
0.03847980.0475892
0.1070220.116363
0.1611590.170565
0.2025310.211855
0.2327130.241826
0.2531920.261988
0.2653630.273755
0.2705170.278434
0.2698370.277226
0.2643970.271221
0.2551610.261396
0.2429860.248621
0.2286280.233663
0.2127440.217188
0.1958980.199768
0.1785710.181892
0.1611630.163964
0.1440050.146318
0.1273610.129224
0.111440.112889
0.09639770.0974727
0.08234680.0830868
0.06936120.0698051
0.05748230.0576676
0.04672370.0466868
0.03707650.0368515
0.02851290.0281322
0.0209910.0204843
0.01445740.013852
0.008850580.00817133
0.004103450.00337248
0.000145538-0.000617509
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-0.0092233-0.00996669
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-0.0110009-0.0114901
-0.0105469-0.0109868
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-0.00865052-0.00894939
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-0.00378249-0.00384343
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-0.00138728-0.00136667
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0.00006311230.000112504
0.0001865670.000236139
0.0002841880.000333089
0.0003589090.000406437
0.0004135410.000459134
0.0004507510.00049397
z
sol_z
time(sec)
Runge-Kutta Method
Sheet1
tysol_yzsol_z
01.00E+001.00E+000.00E+000.00E+00
0.019.90E-019.91E-01-1.80E+00-1.81E+00
0.029.64E-019.65E-01-3.24E+00-3.25E+00
0.039.25E-019.27E-01-4.36E+00-4.38E+00
0.048.77E-018.78E-01-5.20E+00-5.22E+00
0.058.21E-018.23E-01-5.79E+00-5.82E+00
0.067.61E-017.63E-01-6.17E+00-6.20E+00
0.076.98E-017.00E-01-6.36E+00-6.40E+00
0.086.33E-016.35E-01-6.41E+00-6.45E+00
0.095.69E-015.71E-01-6.33E+00-6.37E+00
0.15.07E-015.08E-01-6.15E+00-6.19E+00
0.114.46E-014.48E-01-5.90E+00-5.93E+00
0.123.89E-013.90E-01-5.58E+00-5.61E+00
0.133.35E-013.36E-01-5.22E+00-5.25E+00
0.142.84E-012.85E-01-4.83E+00-4.86E+00
0.152.38E-012.38E-01-4.42E+00-4.45E+00
0.161.96E-011.96E-01-4.01E+00-4.04E+00
0.171.58E-011.58E-01-3.60E+00-3.62E+00
0.181.24E-011.23E-01-3.20E+00-3.22E+00
0.199.37E-029.32E-02-2.82E+00-2.83E+00
0.26.74E-026.67E-02-2.45E+00-2.46E+00
0.214.47E-024.39E-02-2.11E+00-2.11E+00
0.222.53E-022.44E-02-1.79E+00-1.79E+00
0.238.96E-037.96E-03-1.49E+00-1.50E+00
0.24-4.56E-03-5.62E-03-1.22E+00-1.23E+00
0.25-1.55E-02-1.66E-02-9.84E-01-9.83E-01
0.26-2.42E-02-2.54E-02-7.69E-01-7.66E-01
0.27-3.09E-02-3.20E-02-5.79E-01-5.74E-01
0.28-3.58E-02-3.69E-02-4.14E-01-4.07E-01
0.29-3.92E-02-4.03E-02-2.71E-01-2.63E-01
0.3-4.12E-02-4.23E-02-1.49E-01-1.41E-01
0.31-4.21E-02-4.31E-02-4.62E-02-3.75E-02
0.32-4.21E-02-4.31E-023.85E-024.76E-02
0.33-4.14E-02-4.22E-021.07E-011.16E-01
0.34-4.00E-02-4.08E-021.61E-011.71E-01
0.35-3.81E-02-3.89E-022.03E-012.12E-01
0.36-3.59E-02-3.66E-022.33E-012.42E-01
0.37-3.35E-02-3.41E-022.53E-012.62E-01
0.38-3.09E-02-3.14E-022.65E-012.74E-01
0.39-2.82E-02-2.86E-022.71E-012.78E-01
0.4-2.55E-02-2.58E-022.70E-012.77E-01
0.41-2.28E-02-2.31E-022.64E-012.71E-01
0.42-2.02E-02-2.04E-022.55E-012.61E-01
0.43-1.77E-02-1.79E-022.43E-012.49E-01
0.44-1.53E-02-1.55E-022.29E-012.34E-01
0.45-1.31E-02-1.32E-022.13E-012.17E-01
0.46-1.11E-02-1.11E-021.96E-012.00E-01
0.47-9.18E-03-9.21E-031.79E-011.82E-01
0.48-7.48E-03-7.48E-031.61E-011.64E-01
0.49-5.95E-03-5.93E-031.44E-011.46E-01
0.5-4.60E-03-4.55E-031.27E-011.29E-01
0.51-3.41E-03-3.34E-031.11E-011.13E-01
0.52-2.37E-03-2.29E-039.64E-029.75E-02
0.53-1.48E-03-1.39E-038.23E-028.31E-02
0.54-7.22E-04-6.24E-046.94E-026.98E-02
0.55-9.06E-051.28E-055.75E-025.77E-02
0.564.28E-045.34E-044.67E-024.67E-02
0.578.44E-049.50E-043.71E-023.69E-02
0.581.17E-031.27E-032.85E-022.81E-02
0.591.41E-031.52E-032.10E-022.05E-02
0.61.59E-031.69E-031.45E-021.39E-02
0.611.70E-031.80E-038.85E-038.17E-03
0.621.77E-031.85E-034.10E-033.37E-03
0.631.78E-031.87E-031.46E-04-6.18E-04
0.641.77E-031.84E-03-3.10E-03-3.87E-03
0.651.72E-031.79E-03-5.69E-03-6.47E-03
0.661.65E-031.72E-03-7.71E-03-8.48E-03
0.671.57E-031.62E-03-9.22E-03-9.97E-03
0.681.47E-031.52E-03-1.03E-02-1.10E-02
0.691.36E-031.41E-03-1.10E-02-1.17E-02
0.71.25E-031.29E-03-1.13E-02-1.20E-02
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