------------------------ TERMINATION CRITERIA ------------------------- EDU> eta1 = 100*sqrt(eps), eta2 = 10000*eps eta1 = 1.4901e-006 eta2 = 2.2204e-012 -------------------- MATLAB TRANSCRIPT: PART C(a), BFGS -------------------- EDU> fname='hw03ca'; dfname='hw03cad'; x0=[-1;3]; bfgs BFGS/Armijo/Skipping minimization of function "hw03ca", using gradient "hw03cad": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 -1.0000e+000 3.0000e+000 3.7026e+002 1 -1.0000e+000 9.1851e-001 4.2734e+000 6.9383e-001 1.5866e+000 1.000000 2 -9.8865e-001 8.9832e-001 4.0819e+000 2.1982e-002 1.5333e+000 1.000000 3 -7.6468e-001 5.1129e-001 1.8516e+000 4.3084e-001 9.4794e-001 1.000000 4 -6.5134e-001 3.2437e-001 1.4337e+000 3.6559e-001 7.9923e-001 1.000000 5 -5.4566e-001 1.5786e-001 1.2683e+000 5.1332e-001 6.1329e-001 1.000000 6 -4.8552e-001 7.0667e-002 1.2330e+000 5.5235e-001 5.0697e-001 1.000000 7 -4.4837e-001 2.4719e-002 1.2205e+000 6.5021e-001 4.4372e-001 1.000000 8 -4.0672e-001 -1.5588e-002 1.2059e+000 1.6306e+000 3.7562e-001 1.000000 9 -3.3202e-001 -6.7299e-002 1.1758e+000 3.3173e+000 2.6684e-001 1.000000 10 -2.0936e-001 -1.1836e-001 1.1218e+000 7.5874e-001 1.2917e-001 1.000000 11 -3.8976e-002 -1.4517e-001 1.0522e+000 8.1383e-001 8.3065e-002 1.000000 12 9.2064e-002 -1.2831e-001 1.0159e+000 3.3621e+000 3.5913e-002 1.000000 13 9.6016e-002 -8.8328e-002 1.0073e+000 3.1161e-001 1.2225e-002 1.000000 14 4.4328e-002 -1.8893e-002 1.0010e+000 7.8610e-001 2.2681e-003 1.000000 15 1.1432e-002 -1.1252e-003 1.0001e+000 9.4044e-001 2.3570e-004 1.000000 16 1.0837e-004 1.6872e-004 1.0000e+000 1.1499e+000 1.5045e-007 1.000000 17 5.1397e-007 -5.1096e-007 1.0000e+000 1.0030e+000 5.1907e-013 1.000000 Best point: x' = 5.1397365538380280e-007 -5.1095608624723450e-007. Minimum value: hw03ca(x) = 1.0000000000002610e+000. Number of BFGS Quasi-Newton steps: 17. Termination criterion: e2 < eta2. Flop count: 2438 total--average 143 per step. -------------------- MATLAB TRANSCRIPT: PART C(a), DFP -------------------- EDU> fname='hw03ca'; dfname='hw03cad'; x0=[-1;3]; dfp DFP/Armijo/Skipping minimization of function "hw03ca", using gradient "hw03cad": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 -1.0000e+000 3.0000e+000 3.7026e+002 1 -1.0000e+000 9.1851e-001 4.2734e+000 6.9383e-001 1.5866e+000 1.000000 2 -9.8865e-001 8.9832e-001 4.0819e+000 2.1982e-002 1.5333e+000 1.000000 3 -7.6499e-001 5.1121e-001 1.8522e+000 4.3093e-001 9.4925e-001 1.000000 4 -6.5206e-001 3.2422e-001 1.4349e+000 3.6577e-001 8.0192e-001 1.000000 5 -5.4712e-001 1.5777e-001 1.2704e+000 5.1339e-001 6.1749e-001 1.000000 6 -4.8797e-001 7.0946e-002 1.2360e+000 5.5032e-001 5.1278e-001 1.000000 7 -4.5292e-001 2.6008e-002 1.2250e+000 6.3340e-001 4.5307e-001 1.000000 8 -4.1947e-001 -1.0997e-002 1.2164e+000 1.4228e+000 3.9871e-001 1.000000 9 -3.7941e-001 -5.0017e-002 1.2075e+000 3.5482e+000 3.3851e-001 1.000000 10 -3.3588e-001 -8.7495e-002 1.1990e+000 7.4931e-001 2.7966e-001 1.000000 11 -2.8460e-001 -1.2700e-001 1.1902e+000 4.5147e-001 2.1889e-001 1.000000 12 -2.2666e-001 -1.6716e-001 1.1814e+000 3.1623e-001 1.6005e-001 1.000000 13 -1.5777e-001 -2.1058e-001 1.1724e+000 3.0393e-001 1.8758e-001 1.000000 14 -7.6311e-002 -2.5767e-001 1.1632e+000 5.1632e-001 2.2574e-001 1.000000 15 1.8685e-002 -3.0837e-001 1.1543e+000 1.2448e+000 2.5893e-001 1.000000 16 1.0770e-001 -3.5169e-001 1.1476e+000 4.7640e+000 2.7739e-001 1.000000 17 1.6154e-001 -3.7355e-001 1.1428e+000 4.9989e-001 2.7887e-001 1.000000 18 2.1679e-001 -3.9132e-001 1.1375e+000 3.4206e-001 2.7073e-001 1.000000 19 2.5181e-001 -3.9753e-001 1.1329e+000 1.6153e-001 2.5697e-001 1.000000 20 2.9562e-001 -3.9962e-001 1.1270e+000 1.7398e-001 2.3042e-001 1.000000 21 3.1842e-001 -3.9446e-001 1.1223e+000 7.7100e-002 2.0790e-001 1.000000 22 3.5150e-001 -3.7951e-001 1.1158e+000 1.0391e-001 1.6603e-001 1.000000 23 3.6500e-001 -3.6492e-001 1.1112e+000 3.8442e-002 1.3936e-001 1.000000 24 3.8437e-001 -3.3242e-001 1.1045e+000 8.9044e-002 1.1520e-001 1.000000 25 3.8911e-001 -3.1001e-001 1.1002e+000 6.7416e-002 1.3928e-001 1.000000 26 3.9225e-001 -2.6426e-001 1.0940e+000 1.4759e-001 1.7835e-001 1.000000 27 3.8831e-001 -2.3820e-001 1.0902e+000 9.8592e-002 1.9035e-001 1.000000 28 3.7554e-001 -1.8634e-001 1.0846e+000 2.1773e-001 2.0437e-001 1.000000 29 3.6501e-001 -1.6220e-001 1.0813e+000 1.2957e-001 2.0259e-001 1.000000 30 3.3728e-001 -1.0847e-001 1.0759e+000 3.3124e-001 1.9158e-001 1.000000 31 3.2284e-001 -8.8612e-002 1.0729e+000 1.8308e-001 1.8146e-001 1.000000 32 2.7970e-001 -3.5163e-002 1.0671e+000 6.0318e-001 1.5113e-001 1.000000 33 2.6187e-001 -1.8409e-002 1.0643e+000 4.7648e-001 1.3748e-001 1.000000 34 2.0542e-001 3.0225e-002 1.0583e+000 2.6419e+000 9.8214e-002 1.000000 35 1.8205e-001 4.6176e-002 1.0554e+000 5.2774e-001 8.2899e-002 1.000000 36 1.2027e-001 8.4593e-002 1.0500e+000 8.3197e-001 4.8489e-002 1.000000 37 9.0166e-002 9.9683e-002 1.0471e+000 2.5027e-001 5.8197e-002 1.000000 38 3.0234e-002 1.2627e-001 1.0426e+000 6.6468e-001 7.4901e-002 1.000000 39 -2.0433e-003 1.3722e-001 1.0399e+000 1.0676e+000 8.0200e-002 1.000000 40 -5.4062e-002 1.5147e-001 1.0362e+000 2.5458e+001 8.4112e-002 1.000000 41 -8.0656e-002 1.5535e-001 1.0339e+000 4.9192e-001 8.1443e-002 1.000000 42 -1.2093e-001 1.5753e-001 1.0307e+000 4.9936e-001 7.2236e-002 1.000000 43 -1.3853e-001 1.5456e-001 1.0286e+000 1.4549e-001 6.3443e-002 1.000000 44 -1.6461e-001 1.4533e-001 1.0256e+000 1.8831e-001 4.5803e-002 1.000000 45 -1.7309e-001 1.3674e-001 1.0237e+000 5.9100e-002 3.5174e-002 1.000000 46 -1.8257e-001 1.1807e-001 1.0210e+000 1.3658e-001 2.7422e-002 1.000000 47 -1.8200e-001 1.0502e-001 1.0193e+000 1.1047e-001 3.1798e-002 1.000000 48 -1.7533e-001 8.0991e-002 1.0170e+000 2.2882e-001 3.6248e-002 1.000000 49 -1.6588e-001 6.4944e-002 1.0154e+000 1.9814e-001 3.5960e-002 1.000000 50 -1.4657e-001 3.9816e-002 1.0135e+000 3.8691e-001 3.2700e-002 1.000000 51 -1.2886e-001 2.2313e-002 1.0121e+000 4.3960e-001 2.8218e-002 1.000000 52 -1.0116e-001 -8.9485e-004 1.0105e+000 1.0401e+000 2.0849e-002 1.000000 53 -7.6699e-002 -1.7864e-002 1.0093e+000 1.8964e+001 1.4509e-002 1.000000 54 -4.5595e-002 -3.6337e-002 1.0080e+000 1.0341e+000 8.3882e-003 1.000000 55 -1.9409e-002 -4.9023e-002 1.0070e+000 5.7430e-001 1.1099e-002 1.000000 56 7.7033e-003 -5.9389e-002 1.0060e+000 1.3969e+000 1.2518e-002 1.000000 57 2.8127e-002 -6.4408e-002 1.0052e+000 2.6513e+000 1.2144e-002 1.000000 58 4.6303e-002 -6.5862e-002 1.0044e+000 6.4623e-001 1.0391e-002 1.000000 59 5.8035e-002 -6.3369e-002 1.0038e+000 2.5335e-001 7.9501e-003 1.000000 60 6.6023e-002 -5.7236e-002 1.0032e+000 1.3764e-001 4.9919e-003 1.000000 61 6.8304e-002 -4.8865e-002 1.0027e+000 1.4625e-001 2.7353e-003 1.000000 62 6.6043e-002 -3.7843e-002 1.0022e+000 2.2557e-001 3.7926e-003 1.000000 63 5.9875e-002 -2.6875e-002 1.0018e+000 2.8983e-001 3.9960e-003 1.000000 64 5.0094e-002 -1.5149e-002 1.0015e+000 4.3631e-001 3.5212e-003 1.000000 65 3.9050e-002 -5.2924e-003 1.0012e+000 6.5064e-001 2.6427e-003 1.000000 66 2.6156e-002 3.8041e-003 1.0009e+000 1.7188e+000 1.5678e-003 1.000000 67 1.4355e-002 1.0174e-002 1.0007e+000 1.6746e+000 7.0887e-004 1.000000 68 2.8770e-003 1.4645e-002 1.0005e+000 7.9958e-001 9.5119e-004 1.000000 69 -5.6541e-003 1.6322e-002 1.0004e+000 2.9653e+000 8.9393e-004 1.000000 70 -1.1830e-002 1.5787e-002 1.0003e+000 1.0923e+000 6.3517e-004 1.000000 71 -1.4524e-002 1.3436e-002 1.0002e+000 2.2767e-001 3.3909e-004 1.000000 72 -1.4442e-002 9.8174e-003 1.0001e+000 2.6930e-001 1.3375e-004 1.000000 73 -1.2100e-002 5.9763e-003 1.0001e+000 3.9126e-001 1.4826e-004 1.000000 74 -8.3546e-003 2.3682e-003 1.0000e+000 6.0374e-001 1.0004e-004 1.000000 75 -4.4989e-003 -1.5206e-004 1.0000e+000 1.0642e+000 4.1849e-005 1.000000 76 -1.3143e-003 -1.4702e-003 1.0000e+000 8.6689e+000 1.2501e-005 1.000000 77 4.8022e-004 -1.6266e-003 1.0000e+000 1.3654e+000 9.0080e-006 1.000000 78 1.0075e-003 -1.1136e-003 1.0000e+000 1.0980e+000 2.7126e-006 1.000000 79 7.2714e-004 -4.8676e-004 1.0000e+000 5.6291e-001 3.4958e-007 1.000000 80 2.8605e-004 -1.0147e-004 1.0000e+000 7.9154e-001 1.0560e-007 1.000000 81 5.1528e-005 8.8211e-006 1.0000e+000 1.0869e+000 6.2194e-009 1.000000 82 -9.7604e-007 8.6960e-006 1.0000e+000 1.0189e+000 2.8551e-010 1.000000 83 -1.3560e-006 1.3528e-006 1.0000e+000 8.4443e-001 3.6515e-012 1.000000 84 -1.2336e-007 6.3242e-008 1.0000e+000 9.5325e-001 1.4831e-014 1.000000 Best point: x' = -1.2335680672906050e-007 6.3242146211745820e-008. Minimum value: hw03ca(x) = 1.0000000000000080e+000. Number of DFP Quasi-Newton steps: 84. Termination criterion: e2 < eta2. Flop count: 11805 total--average 141 per step. -------------------- MATLAB TRANSCRIPT: PART C(b), BFGS -------------------- EDU> fname='hw02cb'; dfname='hw02cbd'; x0=[1;1]; bfgs BFGS/Armijo/Skipping minimization of function "hw02cb", using gradient "hw02cbd": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 1.0000e+000 1.0000e+000 7.0000e+000 1 1.8571e+000 1.4286e-001 2.3070e+000 8.5714e-001 1.4190e-001 1.000000 2 1.9074e+000 -3.0970e-001 1.4774e+000 3.1679e+000 2.8830e-001 1.000000 3 1.9864e+000 -9.9043e-001 1.0003e+000 2.1980e+000 5.2910e-002 1.000000 4 1.9911e+000 -1.0014e+000 1.0001e+000 1.1097e-002 3.5448e-002 1.000000 5 1.9944e+000 -1.0024e+000 1.0000e+000 1.6499e-003 2.2407e-002 1.000000 6 1.9997e+000 -1.0011e+000 1.0000e+000 2.6381e-003 2.1770e-003 1.000000 7 2.0000e+000 -1.0002e+000 1.0000e+000 8.9422e-004 3.8407e-004 1.000000 8 2.0000e+000 -1.0000e+000 1.0000e+000 1.8892e-004 2.9397e-005 1.000000 9 2.0000e+000 -1.0000e+000 1.0000e+000 3.5059e-006 1.3494e-006 1.000000 10 2.0000e+000 -1.0000e+000 1.0000e+000 1.6804e-007 5.7143e-009 1.000000 Best point: x' = 2.0000000012668700e+000 -9.9999999714282720e-001. Minimum value: hw02cb(x) = 1.0000000000000000e+000. Number of BFGS Quasi-Newton steps: 10. Termination criterion: e1 < eta1. Flop count: 1493 total--average 149 per step. -------------------- MATLAB TRANSCRIPT: PART C(b), DFP -------------------- EDU> fname='hw02cb'; dfname='hw02cbd'; x0=[1;1]; dfp DFP/Armijo/Skipping minimization of function "hw02cb", using gradient "hw02cbd": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 1.0000e+000 1.0000e+000 7.0000e+000 1 1.8571e+000 1.4286e-001 2.3070e+000 8.5714e-001 1.4190e-001 1.000000 2 1.9155e+000 -2.9674e-001 1.4952e+000 3.0772e+000 2.7829e-001 1.000000 3 2.0097e+000 -9.9105e-001 1.0002e+000 2.3398e+000 3.8353e-002 1.000000 4 2.0083e+000 -9.9867e-001 1.0001e+000 7.6852e-003 3.3306e-002 1.000000 5 2.0057e+000 -1.0010e+000 1.0000e+000 2.3713e-003 2.2994e-002 1.000000 6 2.0020e+000 -1.0022e+000 1.0000e+000 1.8760e-003 7.8713e-003 1.000000 7 2.0005e+000 -1.0015e+000 1.0000e+000 7.4708e-004 3.0130e-003 1.000000 8 1.9998e+000 -1.0005e+000 1.0000e+000 1.0332e-003 9.3919e-004 1.000000 9 1.9999e+000 -1.0001e+000 1.0000e+000 3.8381e-004 5.2383e-004 1.000000 10 2.0000e+000 -9.9999e-001 1.0000e+000 9.0352e-005 9.5217e-005 1.000000 11 2.0000e+000 -1.0000e+000 1.0000e+000 1.0977e-005 7.4035e-006 1.000000 12 2.0000e+000 -1.0000e+000 1.0000e+000 2.0473e-006 2.8524e-007 1.000000 13 2.0000e+000 -1.0000e+000 1.0000e+000 1.3931e-007 1.4498e-008 1.000000 Best point: x' = 2.0000000036244220e+000 -9.9999999669079720e-001. Minimum value: hw02cb(x) = 1.0000000000000000e+000. Number of DFP Quasi-Newton steps: 13. Termination criterion: e1 < eta1. Flop count: 1892 total--average 146 per step. -------------------- MATLAB TRANSCRIPT: PART C(c), BFGS -------------------- EDU> fname='banana'; dfname='bananad'; x0=[-1.2;1]; bfgs BFGS/Armijo/Skipping minimization of function "banana", using gradient "bananad": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 -1.2000e+000 1.0000e+000 2.5200e+001 1 1.4848e+000 2.0958e+000 2.4200e+000 2.2374e+000 4.0263e+001 0.313811 2 1.4364e+000 2.1675e+000 2.2751e+000 3.4199e-002 3.7230e+001 0.058150 3 1.4548e+000 2.1193e+000 1.2076e+000 2.2254e-002 9.6465e-001 1.000000 4 1.4530e+000 2.1120e+000 1.2052e+000 3.4519e-003 4.7595e-001 1.000000 5 1.4389e+000 2.0656e+000 1.1948e+000 2.1935e-002 4.2983e+000 1.000000 6 1.4079e+000 1.9710e+000 1.1791e+000 4.5822e-002 8.5587e+000 1.000000 7 1.3689e+000 1.8585e+000 1.1600e+000 5.7061e-002 1.0861e+001 1.000000 8 1.3171e+000 1.7191e+000 1.1254e+000 7.5002e-002 1.0448e+001 1.000000 9 1.2601e+000 1.5807e+000 1.0727e+000 8.0543e-002 4.8254e+000 1.000000 10 1.1890e+000 1.4038e+000 1.0458e+000 1.1190e-001 5.8504e+000 1.000000 11 1.1333e+000 1.2855e+000 1.0179e+000 8.4255e-002 2.7835e-001 1.000000 12 1.0623e+000 1.1177e+000 1.0155e+000 1.3053e-001 4.9173e+000 0.656100 13 1.0871e+000 1.1808e+000 1.0077e+000 5.6453e-002 6.2000e-001 1.000000 14 1.0664e+000 1.1362e+000 1.0045e+000 3.7802e-002 6.0974e-001 1.000000 15 1.0127e+000 1.0215e+000 1.0019e+000 1.0096e-001 1.7084e+000 1.000000 16 1.0216e+000 1.0433e+000 1.0005e+000 2.1338e-002 1.7288e-001 1.000000 17 1.0108e+000 1.0216e+000 1.0001e+000 2.0726e-002 9.0842e-002 1.000000 18 1.0007e+000 1.0011e+000 1.0000e+000 2.0087e-002 8.6608e-002 1.000000 19 1.0002e+000 1.0005e+000 1.0000e+000 6.5692e-004 8.8227e-004 1.000000 20 1.0000e+000 1.0000e+000 1.0000e+000 4.6422e-004 1.0826e-004 1.000000 21 1.0000e+000 1.0000e+000 1.0000e+000 1.8989e-006 2.8364e-007 1.000000 22 1.0000e+000 1.0000e+000 1.0000e+000 1.3660e-008 6.4964e-011 1.000000 Best point: x' = 1.0000000000050830e+000 1.0000000000100300e+000. Minimum value: banana(x) = 1.0000000000000000e+000. Number of BFGS Quasi-Newton steps: 22. Termination criterion: e1 < eta1. Flop count: 3882 total--average 176 per step. -------------------- MATLAB TRANSCRIPT: PART C(c), DFP -------------------- EDU> fname='banana'; dfname='bananad'; x0=[-1.2;1]; dfp DFP/Armijo/Skipping minimization of function "banana", using gradient "bananad": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 -1.2000e+000 1.0000e+000 2.5200e+001 1 1.4848e+000 2.0958e+000 2.4200e+000 2.2374e+000 4.0263e+001 0.313811 2 1.4369e+000 2.1655e+000 2.2061e+000 3.3235e-002 3.7151e+001 0.058150 3 1.4548e+000 2.1191e+000 1.2076e+000 2.1415e-002 9.5183e-001 1.000000 4 1.4529e+000 2.1119e+000 1.2052e+000 3.4151e-003 4.7448e-001 1.000000 5 1.4452e+000 2.0839e+000 1.2003e+000 1.3246e-002 4.2920e+000 1.000000 6 1.4409e+000 2.0700e+000 1.1982e+000 6.6868e-003 5.3464e+000 1.000000 7 1.4250e+000 2.0190e+000 1.1938e+000 2.4615e-002 8.8224e+000 1.000000 8 1.4214e+000 2.0085e+000 1.1921e+000 5.2323e-003 9.1540e+000 1.000000 9 1.3798e+000 1.8841e+000 1.1838e+000 6.1927e-002 1.3671e+001 1.000000 10 1.3889e+000 1.9116e+000 1.1817e+000 1.4585e-002 1.2310e+001 1.000000 11 1.3931e+000 1.9247e+000 1.1799e+000 6.8734e-003 1.1408e+001 1.000000 12 1.4021e+000 1.9539e+000 1.1765e+000 1.5143e-002 9.0767e+000 1.000000 13 1.4052e+000 1.9645e+000 1.1747e+000 5.4444e-003 7.8478e+000 1.000000 14 1.4092e+000 1.9791e+000 1.1722e+000 7.4160e-003 5.6329e+000 1.000000 15 1.4101e+000 1.9834e+000 1.1705e+000 2.1749e-003 4.3145e+000 1.000000 16 1.4099e+000 1.9861e+000 1.1683e+000 1.4011e-003 2.1694e+000 1.000000 17 1.4084e+000 1.9837e+000 1.1668e+000 1.2308e-003 8.6854e-001 1.000000 18 1.4044e+000 1.9756e+000 1.1646e+000 4.0871e-003 1.2250e+000 1.000000 19 1.4004e+000 1.9662e+000 1.1630e+000 4.7360e-003 2.5284e+000 1.000000 20 1.3922e+000 1.9465e+000 1.1607e+000 1.0018e-002 4.5797e+000 1.000000 21 1.3844e+000 1.9271e+000 1.1588e+000 9.9776e-003 6.0231e+000 1.000000 22 1.3698e+000 1.8903e+000 1.1560e+000 1.9096e-002 8.1241e+000 1.000000 23 1.3495e+000 1.8385e+000 1.1525e+000 2.7412e-002 1.0201e+001 1.000000 24 1.2590e+000 1.6080e+000 1.1195e+000 1.2539e-001 1.2388e+001 1.000000 24 ** Note: s'y is very small. Hessian update skipped. 25 1.0502e+000 1.0761e+000 1.0737e+000 3.3075e-001 1.1061e+001 0.590490 26 1.1969e+000 1.4497e+000 1.0682e+000 3.4715e-001 8.7591e+000 0.900000 27 1.1550e+000 1.3429e+000 1.0321e+000 7.3658e-002 4.2907e+000 1.000000 28 1.0819e+000 1.1569e+000 1.0252e+000 1.3852e-001 6.3757e+000 0.430467 29 1.1216e+000 1.2580e+000 1.0148e+000 8.7354e-002 2.6478e-001 1.000000 30 1.1129e+000 1.2358e+000 1.0135e+000 1.7647e-002 1.5574e+000 1.000000 31 1.1101e+000 1.2289e+000 1.0134e+000 5.5658e-003 1.9491e+000 1.000000 32 1.1099e+000 1.2282e+000 1.0133e+000 5.2331e-004 1.9709e+000 1.000000 33 1.1003e+000 1.2054e+000 1.0129e+000 1.8597e-002 2.7665e+000 1.000000 34 1.1010e+000 1.2070e+000 1.0129e+000 1.2808e-003 2.7009e+000 1.000000 35 1.1034e+000 1.2129e+000 1.0128e+000 4.8903e-003 2.4404e+000 1.000000 36 1.1040e+000 1.2144e+000 1.0128e+000 1.2890e-003 2.3603e+000 1.000000 37 1.1059e+000 1.2191e+000 1.0127e+000 3.8758e-003 2.1070e+000 1.000000 38 1.1065e+000 1.2205e+000 1.0127e+000 1.1672e-003 2.0171e+000 1.000000 39 1.1079e+000 1.2243e+000 1.0126e+000 3.0641e-003 1.7658e+000 1.000000 40 1.1084e+000 1.2255e+000 1.0126e+000 9.8273e-004 1.6686e+000 1.000000 41 1.1094e+000 1.2284e+000 1.0126e+000 2.3482e-003 1.4171e+000 1.000000 42 1.1097e+000 1.2293e+000 1.0125e+000 7.5700e-004 1.3151e+000 1.000000 43 1.1104e+000 1.2314e+000 1.0125e+000 1.6816e-003 1.0626e+000 1.000000 44 1.1106e+000 1.2320e+000 1.0125e+000 5.0811e-004 9.5780e-001 1.000000 45 1.1110e+000 1.2333e+000 1.0124e+000 1.0411e-003 7.0441e-001 1.000000 46 1.1110e+000 1.2336e+000 1.0124e+000 2.4963e-004 5.9847e-001 1.000000 47 1.1110e+000 1.2341e+000 1.0123e+000 4.1363e-004 3.4478e-001 1.000000 48 1.1109e+000 1.2341e+000 1.0123e+000 9.3133e-005 2.3871e-001 1.000000 49 1.1105e+000 1.2338e+000 1.0122e+000 3.1375e-004 1.2846e-001 1.000000 50 1.1103e+000 1.2335e+000 1.0122e+000 2.7036e-004 1.8109e-001 1.000000 51 1.1096e+000 1.2325e+000 1.0122e+000 8.2673e-004 3.7076e-001 1.000000 52 1.1092e+000 1.2318e+000 1.0121e+000 5.3426e-004 4.7637e-001 1.000000 53 1.1082e+000 1.2300e+000 1.0121e+000 1.4404e-003 7.2294e-001 1.000000 54 1.1076e+000 1.2290e+000 1.0121e+000 8.1226e-004 8.2911e-001 1.000000 55 1.1063e+000 1.2265e+000 1.0120e+000 2.0444e-003 1.0693e+000 1.000000 56 1.1055e+000 1.2251e+000 1.0120e+000 1.1205e-003 1.1772e+000 1.000000 57 1.1039e+000 1.2219e+000 1.0119e+000 2.6352e-003 1.4089e+000 1.000000 58 1.1029e+000 1.2201e+000 1.0119e+000 1.4825e-003 1.5200e+000 1.000000 59 1.1010e+000 1.2162e+000 1.0119e+000 3.2166e-003 1.7416e+000 1.000000 60 1.0998e+000 1.2138e+000 1.0118e+000 1.9323e-003 1.8577e+000 1.000000 61 1.0975e+000 1.2092e+000 1.0118e+000 3.8102e-003 2.0685e+000 1.000000 62 1.0959e+000 1.2062e+000 1.0118e+000 2.5251e-003 2.1921e+000 1.000000 63 1.0933e+000 1.2008e+000 1.0117e+000 4.4736e-003 2.3929e+000 1.000000 64 1.0913e+000 1.1967e+000 1.0117e+000 3.3664e-003 2.5274e+000 1.000000 65 1.0882e+000 1.1903e+000 1.0116e+000 5.3519e-003 2.7216e+000 1.000000 66 1.0854e+000 1.1847e+000 1.0116e+000 4.7184e-003 2.8736e+000 1.000000 67 1.0814e+000 1.1765e+000 1.0116e+000 6.9124e-003 3.0715e+000 1.000000 68 1.0771e+000 1.1675e+000 1.0115e+000 7.6352e-003 3.2604e+000 1.000000 69 1.0701e+000 1.1532e+000 1.0114e+000 1.2256e-002 3.5114e+000 1.000000 70 1.0519e+000 1.1158e+000 1.0112e+000 3.2477e-002 3.9282e+000 1.000000 70 ** Note: s'y is very small. Hessian update skipped. 71 1.0074e+000 1.0242e+000 1.0086e+000 8.2096e-002 3.7034e+000 1.000000 71 ** Note: s'y is very small. Hessian update skipped. 72 9.1794e-001 8.3993e-001 1.0075e+000 1.7989e-001 7.5210e-001 0.430467 73 9.8492e-001 9.7781e-001 1.0062e+000 1.6416e-001 3.0119e+000 1.000000 74 9.6663e-001 9.4015e-001 1.0044e+000 3.8515e-002 2.2107e+000 1.000000 75 9.3469e-001 8.7437e-001 1.0043e+000 6.9964e-002 3.7588e-001 0.656100 76 9.5286e-001 9.1178e-001 1.0037e+000 4.2779e-002 1.4792e+000 1.000000 77 9.4954e-001 9.0494e-001 1.0036e+000 7.4968e-003 1.2873e+000 1.000000 78 9.4849e-001 9.0276e-001 1.0036e+000 2.4063e-003 1.2233e+000 1.000000 79 9.4809e-001 9.0195e-001 1.0036e+000 9.0590e-004 1.1974e+000 1.000000 80 9.4700e-001 8.9967e-001 1.0036e+000 2.5282e-003 1.1229e+000 1.000000 81 9.4667e-001 8.9898e-001 1.0036e+000 7.6432e-004 1.0984e+000 1.000000 82 9.4565e-001 8.9683e-001 1.0036e+000 2.3940e-003 1.0195e+000 1.000000 83 9.4538e-001 8.9625e-001 1.0036e+000 6.4159e-004 9.9617e-001 1.000000 84 9.4445e-001 8.9425e-001 1.0036e+000 2.2316e-003 9.1275e-001 1.000000 85 9.4423e-001 8.9377e-001 1.0036e+000 5.3414e-004 8.9044e-001 1.000000 86 9.4339e-001 8.9195e-001 1.0036e+000 2.0348e-003 8.0277e-001 1.000000 87 9.4322e-001 8.9156e-001 1.0036e+000 4.3916e-004 7.8120e-001 1.000000 88 9.4250e-001 8.8996e-001 1.0036e+000 1.8006e-003 6.8971e-001 1.000000 89 9.4237e-001 8.8964e-001 1.0036e+000 3.5386e-004 6.6864e-001 1.000000 90 9.4179e-001 8.8828e-001 1.0036e+000 1.5298e-003 5.7398e-001 1.000000 91 9.4169e-001 8.8804e-001 1.0036e+000 2.7546e-004 5.5321e-001 1.000000 92 9.4126e-001 8.8695e-001 1.0035e+000 1.2274e-003 4.5618e-001 1.000000 93 9.4119e-001 8.8677e-001 1.0035e+000 2.0136e-004 4.3554e-001 1.000000 94 9.4092e-001 8.8597e-001 1.0035e+000 9.0195e-004 3.3698e-001 1.000000 95 9.4089e-001 8.8585e-001 1.0035e+000 1.2930e-004 3.1633e-001 1.000000 96 9.4077e-001 8.8536e-001 1.0035e+000 5.6289e-004 2.1698e-001 1.000000 97 9.4078e-001 8.8530e-001 1.0035e+000 5.7729e-005 1.9629e-001 1.000000 98 9.4082e-001 8.8511e-001 1.0035e+000 2.1830e-004 9.6701e-002 1.000000 99 9.4086e-001 8.8512e-001 1.0035e+000 4.0085e-005 7.5980e-002 1.000000 100 9.4107e-001 8.8524e-001 1.0035e+000 2.2197e-004 6.6990e-002 1.000000 Best point: x' = 9.4107142179853940e-001 8.8523572516384870e-001. Minimum value: banana(x) = 1.0034869942160200e+000. Number of DFP Quasi-Newton steps: 100. Termination criterion: iteration limit reached. Flop count: 15084 total--average 151 per step. -------------------- MATLAB TRANSCRIPT: PART C(d), BFGS -------------------- EDU> fname='hw03cd'; dfname='hw03cdd'; x0=[9;1]; bfgs BFGS/Armijo/Skipping minimization of function "hw03cd", using gradient "hw03cdd": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 9.0000e+000 1.0000e+000 9.1000e+001 1 8.8022e+000 8.0220e-001 8.4270e+001 1.9780e-001 1.8388e+000 1.000000 2 6.9152e+000 -7.6835e-001 5.4133e+001 1.9578e+000 1.7667e+000 1.000000 3 6.4649e+000 -8.6308e-001 4.9499e+001 1.2328e-001 1.6887e+000 1.000000 4 1.8897e+000 -9.9804e-001 1.3536e+001 7.0769e-001 1.3246e+000 1.000000 5 1.3786e-001 -5.1923e-001 3.4454e+000 9.2705e-001 1.4085e+000 1.000000 6 -1.7951e-001 -7.2978e-002 1.0802e+000 2.3022e+000 8.8751e-002 1.000000 7 -3.5123e-002 -2.7755e-003 1.0013e+000 9.6197e-001 2.4641e-003 1.000000 8 -1.4490e-003 3.3114e-004 1.0000e+000 1.1193e+000 4.1992e-006 1.000000 9 -1.5190e-005 1.8295e-005 1.0000e+000 9.8952e-001 6.0249e-009 1.000000 10 3.8048e-007 1.9376e-007 1.0000e+000 1.0250e+000 6.7575e-013 1.000000 Best point: x' = 3.8048037209058210e-007 1.9375710972012190e-007. Minimum value: hw03cd(x) = 1.0000000000004830e+000. Number of BFGS Quasi-Newton steps: 10. Termination criterion: e2 < eta2. Flop count: 1284 total--average 128 per step. -------------------- MATLAB TRANSCRIPT: PART C(d), DFP -------------------- EDU> fname='hw03cd'; dfname='hw03cdd'; x0=[9;1]; dfp DFP/Armijo/Skipping minimization of function "hw03cd", using gradient "hw03cdd": k x_k(1) x_k(2) f(x_k) e_1 e_2 lambda 0 9.0000e+000 1.0000e+000 9.1000e+001 1 8.8022e+000 8.0220e-001 8.4270e+001 1.9780e-001 1.8388e+000 1.000000 2 7.0263e+000 -7.8070e-001 5.5855e+001 1.9732e+000 1.7678e+000 1.000000 3 6.7652e+000 -8.4333e-001 5.3169e+001 8.0220e-002 1.7216e+000 1.000000 4 3.3858e+000 -1.5686e+000 3.4608e+001 8.5998e-001 1.2797e+000 1.000000 5 3.0255e+000 -1.5664e+000 3.2235e+001 1.0644e-001 1.3700e+000 1.000000 6 -8.1140e-002 -1.4708e+000 2.0475e+001 1.0268e+000 1.9017e+000 1.000000 7 -3.8445e-001 -1.3882e+000 1.8492e+001 3.7381e+000 1.8759e+000 1.000000 8 -2.0391e+000 -8.3222e-001 1.1391e+001 4.3039e+000 1.0944e+000 1.000000 9 -2.1135e+000 -6.9888e-001 9.8628e+000 1.6022e-001 9.0580e-001 1.000000 10 -2.1750e+000 -1.3483e-001 5.8944e+000 8.0708e-001 1.6052e+000 1.000000 11 -1.9613e+000 -2.3778e-002 4.8519e+000 8.2365e-001 1.5857e+000 1.000000 12 -1.1596e+000 2.4794e-001 2.8980e+000 1.1428e+001 9.2804e-001 1.000000 13 -8.1617e-001 2.6757e-001 2.3104e+000 2.9618e-001 5.7662e-001 1.000000 14 -1.2358e-001 2.4033e-001 1.5351e+000 8.4858e-001 6.7726e-001 1.000000 15 6.3535e-002 1.7953e-001 1.2941e+000 1.5141e+000 4.4831e-001 1.000000 16 2.0462e-001 6.9971e-002 1.0859e+000 2.2206e+000 8.1153e-002 1.000000 17 1.6086e-001 2.3016e-002 1.0306e+000 6.7106e-001 5.0216e-002 1.000000 18 6.5346e-002 -6.7659e-003 1.0047e+000 1.2940e+000 8.5003e-003 1.000000 19 1.8034e-002 -6.4267e-003 1.0007e+000 7.2402e-001 7.4292e-004 1.000000 20 -1.1786e-004 -1.8262e-003 1.0000e+000 1.0065e+000 6.0029e-005 1.000000 21 -5.6604e-004 -2.4816e-004 1.0000e+000 3.8027e+000 1.1085e-006 1.000000 22 -6.8981e-005 -2.1768e-006 1.0000e+000 9.9123e-001 9.5169e-009 1.000000 23 -3.0368e-006 6.8182e-007 1.0000e+000 1.3132e+000 1.8444e-011 1.000000 24 -9.6921e-009 2.4843e-008 1.0000e+000 9.9681e-001 1.1109e-014 1.000000 Best point: x' = -9.6920546198212980e-009 2.4842626375785000e-008. Minimum value: hw03cd(x) = 1.0000000000000060e+000. Number of DFP Quasi-Newton steps: 24. Termination criterion: e2 < eta2. Flop count: 3045 total--average 127 per step.