03-Jan-2023 15:23:10 bdf2_test(): MATLAB/Octave version 4.2.2 Test bdf2(). predator_prey_bdf2(): A pair of ordinary differential equations for a population of predators and prey are solved using the BDF2, solving the implicit function using fsolve(). The exact solution shows periodic behavior, with a fixed period and amplitude. t1 = 0 y1 = 5000 100 t2 = 0.025000 t3 = 0.050000 t1 = 0.025000 y1 = 5243.24 100.61 t2 = 0.050000 t3 = 0.075000 t1 = 0.050000 y1 = 5498.20 102.52 t2 = 0.075000 t3 = 0.10000 t1 = 0.075000 y1 = 5765.18 105.85 t2 = 0.10000 t3 = 0.12500 t1 = 0.10000 y1 = 6044.50 110.82 t2 = 0.12500 t3 = 0.15000 t1 = 0.12500 y1 = 6336.39 117.72 t2 = 0.15000 t3 = 0.17500 t1 = 0.15000 y1 = 6640.99 126.97 t2 = 0.17500 t3 = 0.20000 t1 = 0.17500 y1 = 6958.34 139.13 t2 = 0.20000 t3 = 0.22500 t1 = 0.20000 y1 = 7288.24 155.01 t2 = 0.22500 t3 = 0.25000 t1 = 0.22500 y1 = 7630.20 175.70 t2 = 0.25000 t3 = 0.27500 t1 = 0.25000 y1 = 7983.31 202.77 t2 = 0.27500 t3 = 0.30000 t1 = 0.27500 y1 = 8346.06 238.37 t2 = 0.30000 t3 = 0.32500 t1 = 0.30000 y1 = 8716.02 285.62 t2 = 0.32500 t3 = 0.35000 t1 = 0.32500 y1 = 9089.46 348.94 t2 = 0.35000 t3 = 0.37500 t1 = 0.35000 y1 = 9460.76 434.68 t2 = 0.37500 t3 = 0.40000 t1 = 0.37500 y1 = 9821.56 551.97 t2 = 0.40000 t3 = 0.42500 t1 = 0.40000 y1 = 1.0160e+04 7.1383e+02 t2 = 0.42500 t3 = 0.45000 t1 = 0.42500 y1 = 1.0457e+04 9.3851e+02 t2 = 0.45000 t3 = 0.47500 t1 = 0.45000 y1 = 1.0690e+04 1.2506e+03 t2 = 0.47500 t3 = 0.50000 t1 = 0.47500 y1 = 1.0825e+04 1.6811e+03 t2 = 0.50000 t3 = 0.52500 t1 = 0.50000 y1 = 1.0823e+04 2.2639e+03 t2 = 0.52500 t3 = 0.55000 t1 = 0.52500 y1 = 1.0640e+04 3.0273e+03 t2 = 0.55000 t3 = 0.57500 t1 = 0.55000 y1 = 1.0241e+04 3.9765e+03 t2 = 0.57500 t3 = 0.60000 t1 = 0.57500 y1 = 9615.8 5073.5 t2 = 0.60000 t3 = 0.62500 t1 = 0.60000 y1 = 8788.5 6225.4 t2 = 0.62500 t3 = 0.65000 t1 = 0.62500 y1 = 7822.3 7295.6 t2 = 0.65000 t3 = 0.67500 t1 = 0.65000 y1 = 6803.8 8141.9 t2 = 0.67500 t3 = 0.70000 t1 = 0.67500 y1 = 5818.4 8660.3 t2 = 0.70000 t3 = 0.72500 t1 = 0.70000 y1 = 4930.1 8812.5 t2 = 0.72500 t3 = 0.75000 t1 = 0.72500 y1 = 4173.2 8625.6 t2 = 0.75000 t3 = 0.77500 t1 = 0.75000 y1 = 3555.3 8169.8 t2 = 0.77500 t3 = 0.80000 t1 = 0.77500 y1 = 3066.6 7532.5 t2 = 0.80000 t3 = 0.82500 t1 = 0.80000 y1 = 2688.7 6796.4 t2 = 0.82500 t3 = 0.85000 t1 = 0.82500 y1 = 2401.5 6028.9 t2 = 0.85000 t3 = 0.87500 t1 = 0.85000 y1 = 2186.4 5278.0 t2 = 0.87500 t3 = 0.90000 t1 = 0.87500 y1 = 2027.6 4574.5 t2 = 0.90000 t3 = 0.92500 t1 = 0.90000 y1 = 1913.0 3935.1 t2 = 0.92500 t3 = 0.95000 t1 = 0.92500 y1 = 1833.1 3366.5 t2 = 0.95000 t3 = 0.97500 t1 = 0.95000 y1 = 1780.6 2869.1 t2 = 0.97500 t3 = 1 t1 = 0.97500 y1 = 1750.3 2439.1 t2 = 1 t3 = 1.0250 t1 = 1 y1 = 1738.2 2070.6 t2 = 1.0250 t3 = 1.0500 t1 = 1.0250 y1 = 1741.2 1757.0 t2 = 1.0500 t3 = 1.0750 t1 = 1.0500 y1 = 1757.1 1491.2 t2 = 1.0750 t3 = 1.1000 t1 = 1.0750 y1 = 1784.2 1266.8 t2 = 1.1000 t3 = 1.1250 t1 = 1.1000 y1 = 1821.2 1077.8 t2 = 1.1250 t3 = 1.1500 t1 = 1.1250 y1 = 1867.21 918.88 t2 = 1.1500 t3 = 1.1750 t1 = 1.1500 y1 = 1921.44 785.30 t2 = 1.1750 t3 = 1.2000 t1 = 1.1750 y1 = 1983.38 673.10 t2 = 1.2000 t3 = 1.2250 t1 = 1.2000 y1 = 2052.64 578.83 t2 = 1.2250 t3 = 1.2500 t1 = 1.2250 y1 = 2128.97 499.59 t2 = 1.2500 t3 = 1.2750 t1 = 1.2500 y1 = 2212.19 432.95 t2 = 1.2750 t3 = 1.3000 t1 = 1.2750 y1 = 2302.21 376.85 t2 = 1.3000 t3 = 1.3250 t1 = 1.3000 y1 = 2399.00 329.57 t2 = 1.3250 t3 = 1.3500 t1 = 1.3250 y1 = 2502.61 289.70 t2 = 1.3500 t3 = 1.3750 t1 = 1.3500 y1 = 2613.09 256.03 t2 = 1.3750 t3 = 1.4000 t1 = 1.3750 y1 = 2730.58 227.60 t2 = 1.4000 t3 = 1.4250 t1 = 1.4000 y1 = 2855.23 203.56 t2 = 1.4250 t3 = 1.4500 t1 = 1.4250 y1 = 2987.23 183.26 t2 = 1.4500 t3 = 1.4750 t1 = 1.4500 y1 = 3126.80 166.12 t2 = 1.4750 t3 = 1.5000 t1 = 1.4750 y1 = 3274.19 151.68 t2 = 1.5000 t3 = 1.5250 t1 = 1.5000 y1 = 3429.66 139.56 t2 = 1.5250 t3 = 1.5500 t1 = 1.5250 y1 = 3593.52 129.45 t2 = 1.5500 t3 = 1.5750 t1 = 1.5500 y1 = 3766.07 121.10 t2 = 1.5750 t3 = 1.6000 t1 = 1.5750 y1 = 3947.66 114.31 t2 = 1.6000 t3 = 1.6250 t1 = 1.6000 y1 = 4138.63 108.91 t2 = 1.6250 t3 = 1.6500 t1 = 1.6250 y1 = 4339.35 104.81 t2 = 1.6500 t3 = 1.6750 t1 = 1.6500 y1 = 4550.20 101.91 t2 = 1.6750 t3 = 1.7000 t1 = 1.6750 y1 = 4771.57 100.18 t2 = 1.7000 t3 = 1.7250 t1 = 1.7000 y1 = 5003.849 99.621 t2 = 1.7250 t3 = 1.7500 t1 = 1.7250 y1 = 5247.42 100.26 t2 = 1.7500 t3 = 1.7750 t1 = 1.7500 y1 = 5502.66 102.18 t2 = 1.7750 t3 = 1.8000 t1 = 1.7750 y1 = 5769.92 105.53 t2 = 1.8000 t3 = 1.8250 t1 = 1.8000 y1 = 6049.52 110.52 t2 = 1.8250 t3 = 1.8500 t1 = 1.8250 y1 = 6341.70 117.43 t2 = 1.8500 t3 = 1.8750 t1 = 1.8500 y1 = 6646.61 126.69 t2 = 1.8750 t3 = 1.9000 t1 = 1.8750 y1 = 6964.27 138.86 t2 = 1.9000 t3 = 1.9250 t1 = 1.9000 y1 = 7294.49 154.76 t2 = 1.9250 t3 = 1.9500 t1 = 1.9250 y1 = 7636.79 175.48 t2 = 1.9500 t3 = 1.9750 t1 = 1.9500 y1 = 7990.25 202.58 t2 = 1.9750 t3 = 2 t1 = 1.9750 y1 = 8353.35 238.25 t2 = 2 t3 = 2.0250 t1 = 2 y1 = 8723.65 285.58 t2 = 2.0250 t3 = 2.0500 t1 = 2.0250 y1 = 9097.41 349.03 t2 = 2.0500 t3 = 2.0750 t1 = 2.0500 y1 = 9468.98 434.98 t2 = 2.0750 t3 = 2.1000 t1 = 2.0750 y1 = 9829.97 552.59 t2 = 2.1000 t3 = 2.1250 t1 = 2.1000 y1 = 1.0168e+04 7.1497e+02 t2 = 2.1250 t3 = 2.1500 t1 = 2.1250 y1 = 1.0466e+04 9.4042e+02 t2 = 2.1500 t3 = 2.1750 t1 = 2.1500 y1 = 1.0698e+04 1.2537e+03 t2 = 2.1750 t3 = 2.2000 t1 = 2.1750 y1 = 1.0832e+04 1.6858e+03 t2 = 2.2000 t3 = 2.2250 t1 = 2.2000 y1 = 1.0828e+04 2.2710e+03 t2 = 2.2250 t3 = 2.2500 t1 = 2.2250 y1 = 1.0642e+04 3.0374e+03 t2 = 2.2500 t3 = 2.2750 t1 = 2.2500 y1 = 1.0241e+04 3.9898e+03 t2 = 2.2750 t3 = 2.3000 t1 = 2.2750 y1 = 9612.1 5089.8 t2 = 2.3000 t3 = 2.3250 t1 = 2.3000 y1 = 8781.5 6243.5 t2 = 2.3250 t3 = 2.3500 t1 = 2.3250 y1 = 7812.7 7313.7 t2 = 2.3500 t3 = 2.3750 t1 = 2.3500 y1 = 6792.6 8157.9 t2 = 2.3750 t3 = 2.4000 t1 = 2.3750 y1 = 5806.8 8672.6 t2 = 2.4000 t3 = 2.4250 t1 = 2.4000 y1 = 4919.0 8820.3 t2 = 2.4250 t3 = 2.4500 t1 = 2.4250 y1 = 4163.2 8628.9 t2 = 2.4500 t3 = 2.4750 t1 = 2.4500 y1 = 3546.6 8169.3 t2 = 2.4750 t3 = 2.5000 t1 = 2.4750 y1 = 3059.1 7529.0 t2 = 2.5000 t3 = 2.5250 t1 = 2.5000 y1 = 2682.5 6790.9 t2 = 2.5250 t3 = 2.5500 t1 = 2.5250 y1 = 2396.3 6022.2 t2 = 2.5500 t3 = 2.5750 t1 = 2.5500 y1 = 2181.9 5270.9 t2 = 2.5750 t3 = 2.6000 t1 = 2.5750 y1 = 2023.9 4567.3 t2 = 2.6000 t3 = 2.6250 t1 = 2.6000 y1 = 1909.8 3928.2 t2 = 2.6250 t3 = 2.6500 t1 = 2.6250 y1 = 1830.3 3360.1 t2 = 2.6500 t3 = 2.6750 t1 = 2.6500 y1 = 1778.2 2863.3 t2 = 2.6750 t3 = 2.7000 t1 = 2.6750 y1 = 1748.2 2433.8 t2 = 2.7000 t3 = 2.7250 t1 = 2.7000 y1 = 1736.3 2065.9 t2 = 2.7250 t3 = 2.7500 t1 = 2.7250 y1 = 1739.5 1752.8 t2 = 2.7500 t3 = 2.7750 t1 = 2.7500 y1 = 1755.6 1487.6 t2 = 2.7750 t3 = 2.8000 t1 = 2.7750 y1 = 1782.8 1263.6 t2 = 2.8000 t3 = 2.8250 t1 = 2.8000 y1 = 1819.9 1075.0 t2 = 2.8250 t3 = 2.8500 t1 = 2.8250 y1 = 1866.00 916.43 t2 = 2.8500 t3 = 2.8750 t1 = 2.8500 y1 = 1920.31 783.16 t2 = 2.8750 t3 = 2.9000 t1 = 2.8750 y1 = 1982.31 671.22 t2 = 2.9000 t3 = 2.9250 t1 = 2.9000 y1 = 2051.63 577.18 t2 = 2.9250 t3 = 2.9500 t1 = 2.9250 y1 = 2128.00 498.15 t2 = 2.9500 t3 = 2.9750 t1 = 2.9500 y1 = 2211.26 431.68 t2 = 2.9750 t3 = 3 t1 = 2.9750 y1 = 2301.31 375.72 t2 = 3 t3 = 3.0250 t1 = 3 y1 = 2398.13 328.57 t2 = 3.0250 t3 = 3.0500 t1 = 3.0250 y1 = 2501.76 288.80 t2 = 3.0500 t3 = 3.0750 t1 = 3.0500 y1 = 2612.26 255.23 t2 = 3.0750 t3 = 3.1000 t1 = 3.0750 y1 = 2729.76 226.88 t2 = 3.1000 t3 = 3.1250 t1 = 3.1000 y1 = 2854.43 202.91 t2 = 3.1250 t3 = 3.1500 t1 = 3.1250 y1 = 2986.44 182.66 t2 = 3.1500 t3 = 3.1750 t1 = 3.1500 y1 = 3126.02 165.57 t2 = 3.1750 t3 = 3.2000 t1 = 3.1750 y1 = 3273.41 151.17 t2 = 3.2000 t3 = 3.2250 t1 = 3.2000 y1 = 3428.89 139.09 t2 = 3.2250 t3 = 3.2500 t1 = 3.2250 y1 = 3592.74 129.01 t2 = 3.2500 t3 = 3.2750 t1 = 3.2500 y1 = 3765.30 120.68 t2 = 3.2750 t3 = 3.3000 t1 = 3.2750 y1 = 3946.89 113.91 t2 = 3.3000 t3 = 3.3250 t1 = 3.3000 y1 = 4137.86 108.53 t2 = 3.3250 t3 = 3.3500 t1 = 3.3250 y1 = 4338.59 104.44 t2 = 3.3500 t3 = 3.3750 t1 = 3.3500 y1 = 4549.45 101.55 t2 = 3.3750 t3 = 3.4000 t1 = 3.3750 y1 = 4770.823 99.820 t2 = 3.4000 t3 = 3.4250 t1 = 3.4000 y1 = 5003.109 99.255 t2 = 3.4250 t3 = 3.4500 t1 = 3.4250 y1 = 5246.693 99.887 t2 = 3.4500 t3 = 3.4750 t1 = 3.4500 y1 = 5501.95 101.80 t2 = 3.4750 t3 = 3.5000 t1 = 3.4750 y1 = 5769.24 105.14 t2 = 3.5000 t3 = 3.5250 t1 = 3.5000 y1 = 6048.86 110.10 t2 = 3.5250 t3 = 3.5500 t1 = 3.5250 y1 = 6341.08 116.98 t2 = 3.5500 t3 = 3.5750 t1 = 3.5500 y1 = 6646.04 126.20 t2 = 3.5750 t3 = 3.6000 t1 = 3.5750 y1 = 6963.76 138.32 t2 = 3.6000 t3 = 3.6250 t1 = 3.6000 y1 = 7294.07 154.15 t2 = 3.6250 t3 = 3.6500 t1 = 3.6250 y1 = 7636.47 174.79 t2 = 3.6500 t3 = 3.6750 t1 = 3.6500 y1 = 7990.07 201.78 t2 = 3.6750 t3 = 3.7000 t1 = 3.6750 y1 = 8353.34 237.30 t2 = 3.7000 t3 = 3.7250 t1 = 3.7000 y1 = 8723.87 284.45 t2 = 3.7250 t3 = 3.7500 t1 = 3.7250 y1 = 9097.93 347.66 t2 = 3.7500 t3 = 3.7750 t1 = 3.7500 y1 = 9469.89 433.28 t2 = 3.7750 t3 = 3.8000 t1 = 3.7750 y1 = 9831.39 550.48 t2 = 3.8000 t3 = 3.8250 t1 = 3.8000 y1 = 1.0170e+04 7.1230e+02 t2 = 3.8250 t3 = 3.8500 t1 = 3.8250 y1 = 1.0469e+04 9.3705e+02 t2 = 3.8500 t3 = 3.8750 t1 = 3.8500 y1 = 1.0702e+04 1.2494e+03 t2 = 3.8750 t3 = 3.9000 t1 = 3.8750 y1 = 1.0837e+04 1.6806e+03 t2 = 3.9000 t3 = 3.9250 t1 = 3.9000 y1 = 1.0835e+04 2.2648e+03 t2 = 3.9250 t3 = 3.9500 t1 = 3.9250 y1 = 1.0651e+04 3.0303e+03 t2 = 3.9500 t3 = 3.9750 t1 = 3.9500 y1 = 1.0251e+04 3.9825e+03 t2 = 3.9750 t3 = 4 t1 = 3.9750 y1 = 9623.1 5083.3 t2 = 4 t3 = 4.0250 t1 = 4 y1 = 8792.6 6239.0 t2 = 4.0250 t3 = 4.0500 t1 = 4.0250 y1 = 7823.2 7312.3 t2 = 4.0500 t3 = 4.0750 t1 = 4.0500 y1 = 6801.7 8160.2 t2 = 4.0750 t3 = 4.1000 t1 = 4.0750 y1 = 5814.1 8678.3 t2 = 4.1000 t3 = 4.1250 t1 = 4.1000 y1 = 4924.4 8828.7 t2 = 4.1250 t3 = 4.1500 t1 = 4.1250 y1 = 4166.8 8638.9 t2 = 4.1500 t3 = 4.1750 t1 = 4.1500 y1 = 3548.7 8180.0 t2 = 4.1750 t3 = 4.2000 t1 = 4.1750 y1 = 3060.2 7539.5 t2 = 4.2000 t3 = 4.2250 t1 = 4.2000 y1 = 2682.7 6800.7 t2 = 4.2250 t3 = 4.2500 t1 = 4.2250 y1 = 2395.9 6030.9 t2 = 4.2500 t3 = 4.2750 t1 = 4.2500 y1 = 2181.1 5278.3 t2 = 4.2750 t3 = 4.3000 t1 = 4.2750 y1 = 2022.8 4573.6 t2 = 4.3000 t3 = 4.3250 t1 = 4.3000 y1 = 1908.5 3933.4 t2 = 4.3250 t3 = 4.3500 t1 = 4.3250 y1 = 1828.8 3364.3 t2 = 4.3500 t3 = 4.3750 t1 = 4.3500 y1 = 1776.6 2866.6 t2 = 4.3750 t3 = 4.4000 t1 = 4.3750 y1 = 1746.5 2436.5 t2 = 4.4000 t3 = 4.4250 t1 = 4.4000 y1 = 1734.5 2068.0 t2 = 4.4250 t3 = 4.4500 t1 = 4.4250 y1 = 1737.6 1754.4 t2 = 4.4500 t3 = 4.4750 t1 = 4.4500 y1 = 1753.6 1488.8 t2 = 4.4750 t3 = 4.5000 t1 = 4.4750 y1 = 1780.7 1264.5 t2 = 4.5000 t3 = 4.5250 t1 = 4.5000 y1 = 1817.8 1075.7 t2 = 4.5250 t3 = 4.5500 t1 = 4.5250 y1 = 1863.78 916.87 t2 = 4.5500 t3 = 4.5750 t1 = 4.5500 y1 = 1918.00 783.45 t2 = 4.5750 t3 = 4.6000 t1 = 4.5750 y1 = 1979.92 671.39 t2 = 4.6000 t3 = 4.6250 t1 = 4.6000 y1 = 2049.15 577.25 t2 = 4.6250 t3 = 4.6500 t1 = 4.6250 y1 = 2125.43 498.14 t2 = 4.6500 t3 = 4.6750 t1 = 4.6500 y1 = 2208.58 431.61 t2 = 4.6750 t3 = 4.7000 t1 = 4.6750 y1 = 2298.53 375.61 t2 = 4.7000 t3 = 4.7250 t1 = 4.7000 y1 = 2395.25 328.43 t2 = 4.7250 t3 = 4.7500 t1 = 4.7250 y1 = 2498.75 288.64 t2 = 4.7500 t3 = 4.7750 t1 = 4.7500 y1 = 2609.14 255.05 t2 = 4.7750 t3 = 4.8000 t1 = 4.7750 y1 = 2726.52 226.67 t2 = 4.8000 t3 = 4.8250 t1 = 4.8000 y1 = 2851.04 202.69 t2 = 4.8250 t3 = 4.8500 t1 = 4.8250 y1 = 2982.92 182.44 t2 = 4.8500 t3 = 4.8750 t1 = 4.8500 y1 = 3122.35 165.33 t2 = 4.8750 t3 = 4.9000 t1 = 4.8750 y1 = 3269.59 150.93 t2 = 4.9000 t3 = 4.9250 t1 = 4.9000 y1 = 3424.90 138.84 t2 = 4.9250 t3 = 4.9500 t1 = 4.9250 y1 = 3588.60 128.75 t2 = 4.9500 t3 = 4.9750 t1 = 4.9500 y1 = 3760.98 120.41 t2 = 4.9750 t3 = 5 Graphics saved as "predator_prey_bdf2.png" predator_prey_bdf2(): Normal end of execution. stiff_bdf2() Use bdf2() to solve the stiff ODE. t1 = 0 y1 = 0 t2 = 0.037037 t3 = 0.074074 t1 = 0.037037 y1 = 0.96137 t2 = 0.074074 t3 = 0.11111 t1 = 0.074074 y1 = 1.1246 t2 = 0.11111 t3 = 0.14815 t1 = 0.11111 y1 = 1.0767 t2 = 0.14815 t3 = 0.18519 t1 = 0.14815 y1 = 1.0211 t2 = 0.18519 t3 = 0.22222 t1 = 0.18519 y1 = 0.99172 t2 = 0.22222 t3 = 0.25926 t1 = 0.22222 y1 = 0.97832 t2 = 0.25926 t3 = 0.29630 t1 = 0.25926 y1 = 0.96984 t2 = 0.29630 t3 = 0.33333 t1 = 0.29630 y1 = 0.96116 t2 = 0.33333 t3 = 0.37037 t1 = 0.33333 y1 = 0.95091 t2 = 0.37037 t3 = 0.40741 t1 = 0.37037 y1 = 0.93904 t2 = 0.40741 t3 = 0.44444 t1 = 0.40741 y1 = 0.92573 t2 = 0.44444 t3 = 0.48148 t1 = 0.44444 y1 = 0.91110 t2 = 0.48148 t3 = 0.51852 t1 = 0.48148 y1 = 0.89522 t2 = 0.51852 t3 = 0.55556 t1 = 0.51852 y1 = 0.87812 t2 = 0.55556 t3 = 0.59259 t1 = 0.55556 y1 = 0.85982 t2 = 0.59259 t3 = 0.62963 t1 = 0.59259 y1 = 0.84033 t2 = 0.62963 t3 = 0.66667 t1 = 0.62963 y1 = 0.81970 t2 = 0.66667 t3 = 0.70370 t1 = 0.66667 y1 = 0.79794 t2 = 0.70370 t3 = 0.74074 t1 = 0.70370 y1 = 0.77509 t2 = 0.74074 t3 = 0.77778 t1 = 0.74074 y1 = 0.75117 t2 = 0.77778 t3 = 0.81481 t1 = 0.77778 y1 = 0.72622 t2 = 0.81481 t3 = 0.85185 t1 = 0.81481 y1 = 0.70028 t2 = 0.85185 t3 = 0.88889 t1 = 0.85185 y1 = 0.67338 t2 = 0.88889 t3 = 0.92593 t1 = 0.88889 y1 = 0.64555 t2 = 0.92593 t3 = 0.96296 t1 = 0.92593 y1 = 0.61684 t2 = 0.96296 t3 = 1 Graphics saved as "stiff_bdf2.png" stiff_bdf2() Normal end of execution. bdf2_test(): Normal end of execution. 03-Jan-2023 15:23:11