------------------------ TERMINATION CRITERIA ------------------------- EDU> eta1 = 100*sqrt(eps), eta2 = 10000*eps eta1 = 1.4901e-006 eta2 = 2.2204e-012 ------------------- MATLAB TRANSCRIPT FOR PART C(a) ------------------- EDU> fname='banana'; dfname='bananad'; hfname='bananah'; x0=[-1.2;1]; EDU> dogleg Newton/Trust Region minimization of function "banana", using gradient "bananad", Hessian "bananah": k x_k(1) x_k(2) f(x_k) Delta |dx|/Delta RR e_1 e_2 0 -1.2000e+000 1.0000e+000 2.5200e+001 1.00e-001 1 -1.1074e+000 1.0378e+000 8.9974e+000 1.00e-001 1.0000 1.028 3.8e-001 1.1e+001 2 -1.0530e+000 1.1058e+000 5.2156e+000 2.00e-001 0.4356 1.030 6.6e-002 1.1e+000 3 -9.6274e-001 9.2733e-001 4.8524e+000 2.00e-001 1.0000 1.035 2.5e+000 7.4e-001 4 -7.7472e-001 5.7427e-001 4.2168e+000 4.00e-001 1.0000 0.912 4.5e+000 2.1e+000 5 -4.8772e-001 1.5551e-001 3.8917e+000 8.00e-001 0.6346 0.564 7.3e-001 2.4e+000 6 -4.0258e-001 1.5482e-001 2.9725e+000 8.00e-001 0.1064 1.142 1.7e-001 5.4e-001 7 -4.0258e-001 1.5482e-001 2.9725e+000 8.00e-001 0.9131 -11.704 2.9e+000 5.4e-001 8 -2.3937e-001 3.9228e-002 2.5687e+000 2.00e-001 1.0000 1.034 2.9e+000 3.9e-001 9 2.9256e-002 -7.1303e-002 2.4630e+000 4.00e-001 0.7262 0.289 2.8e+000 4.2e-001 10 9.2161e-002 4.5366e-003 1.8257e+000 4.00e-001 0.2463 1.095 2.2e+000 8.4e-002 11 9.2161e-002 4.5366e-003 1.8257e+000 4.00e-001 1.0000 -3.631 2.1e+001 8.4e-002 12 1.8253e-001 4.7366e-002 1.6880e+000 1.00e-001 1.0000 1.355 2.1e+001 2.9e-001 13 1.9659e-001 3.2517e-002 1.6492e+000 2.00e-001 0.1022 0.980 3.8e+000 1.3e-001 14 3.7840e-001 1.1585e-001 1.4611e+000 2.00e-001 1.0000 0.859 4.6e+000 7.5e-001 15 4.7453e-001 2.1594e-001 1.2847e+000 4.00e-001 0.3469 1.312 8.6e-001 3.1e-001 16 6.5901e-001 4.0026e-001 1.2321e+000 4.00e-001 0.6520 0.498 8.5e-001 4.4e+000 17 7.0269e-001 4.9187e-001 1.0888e+000 4.00e-001 0.2537 1.097 2.3e-001 1.7e-001 18 7.0269e-001 4.9187e-001 1.0888e+000 4.00e-001 0.9319 -2.059 6.2e-001 1.7e-001 19 7.5956e-001 5.7412e-001 1.0586e+000 1.00e-001 1.0000 1.016 6.2e-001 3.0e-001 20 8.1349e-001 6.5834e-001 1.0360e+000 1.00e-001 1.0000 1.022 4.1e-001 5.8e-001 21 8.6449e-001 7.4435e-001 1.0193e+000 1.00e-001 1.0000 1.057 2.8e-001 6.5e-001 22 9.4923e-001 8.9385e-001 1.0077e+000 2.00e-001 0.8592 0.931 2.0e-001 2.5e+000 23 9.7007e-001 9.4060e-001 1.0009e+000 2.00e-001 0.2559 1.097 5.2e-002 1.1e-001 24 9.9761e-001 9.9446e-001 1.0001e+000 2.00e-001 0.3025 1.010 5.7e-002 3.0e-001 25 9.9968e-001 9.9937e-001 1.0000e+000 2.00e-001 0.0266 1.010 4.9e-003 1.1e-003 26 1.0000e+000 1.0000e+000 1.0000e+000 2.00e-001 0.0035 1.001 6.3e-004 3.9e-005 27 1.0000e+000 1.0000e+000 1.0000e+000 2.00e-001 0.0000 1.000 6.4e-007 1.9e-011 Best point: x' = 9.9999999999463860e-001 9.9999999998920340e-001. Best value: banana(x) = 1.0000000000000000e+000. Step count: 27 Trust-Region passes. Stopped by: e1 < eta1. Flop count: 4873 total--average 180 per step. ------------------- MATLAB TRANSCRIPT FOR PART C(b), sigma=1 ------------------- EDUğ fname='cq4'; dfname='cq4d'; hfname='cq4h'; CQ4_SIGMA=1; EDUğ th = 70*pi/180; c = cos(th); s = sin(th); x0 = [c;s;c;s]; EDUğ dogleg Newton/Trust Region minimization of function "cq4", using gradient "cq4d", Hessian "cq4h": k x_k(1) x_k(2) x_k(3) x_k(4) f(x_k) Delta |dx|/Delta RR e_1 e_2 0 3.4202e-001 9.3969e-001 3.4202e-001 9.3969e-001 1.6015e+001 1.00e-001 1 2.8939e-001 8.6714e-001 3.1639e-001 9.0351e-001 1.2019e+001 1.00e-001 1.0000 1.007 4.3e-001 1.8e+000 2 1.8758e-001 7.2155e-001 2.6452e-001 8.2770e-001 6.5756e+000 2.00e-001 1.0000 1.039 4.3e-001 1.6e+000 3 1.2349e-001 4.8325e-001 1.5322e-001 5.3321e-001 2.1820e+000 4.00e-001 1.0000 1.203 4.3e-001 1.0e+000 4 9.5446e-002 3.1675e-001 8.5487e-002 3.0700e-001 1.2543e+000 8.00e-001 0.3629 1.210 4.4e-001 3.5e-001 5 7.7579e-002 1.9480e-001 4.5805e-002 1.4876e-001 1.0540e+000 8.00e-001 0.2556 1.213 5.2e-001 8.2e-002 6 5.7576e-002 1.0145e-001 2.1840e-002 5.1991e-002 1.0100e+000 8.00e-001 0.1725 1.208 6.5e-001 1.4e-002 7 2.7830e-002 3.4745e-002 7.1369e-003 1.0685e-002 1.0011e+000 8.00e-001 0.1065 1.184 7.9e-001 1.3e-003 8 3.4526e-003 3.3228e-003 6.4980e-004 6.9136e-004 1.0000e+000 8.00e-001 0.0519 1.072 9.4e-001 1.2e-005 9 5.5662e-006 4.5480e-006 8.2766e-007 8.2682e-007 1.0000e+000 8.00e-001 0.0061 1.001 1.0e+000 3.1e-011 10 1.9655e-014 1.4513e-014 2.4423e-015 2.6593e-015 1.0000e+000 8.00e-001 0.0000 1.000 1.0e+000 3.9e-028 Best point: x' = 1.9654927743684420e-014 1.4512910899357620e-014 2.4423370235745360e-015 2.6592721140034920e-015. Best value: cq4(x) = 1.0000000000000000e+000. Step count: 10 Trust-Region passes. Stopped by: e2 < eta2. Flop count: 9341 total--average 934 per step. ------------------- MATLAB TRANSCRIPT FOR PART C(b), sigma=2 ------------------- EDUğ fname='cq4'; dfname='cq4d'; hfname='cq4h'; CQ4_SIGMA=2; EDUğ th = 70*pi/180; c = cos(th); s = sin(th); x0 = [c;s;c;s]; EDUğ dogleg Newton/Trust Region minimization of function "cq4", using gradient "cq4d", Hessian "cq4h": k x_k(1) x_k(2) x_k(3) x_k(4) f(x_k) Delta |dx|/Delta RR e_1 e_2 0 3.4202e-001 9.3969e-001 3.4202e-001 9.3969e-001 3.0031e+001 1.00e-001 1 2.8901e-001 8.6715e-001 3.1640e-001 9.0405e-001 2.2163e+001 1.00e-001 1.0000 1.008 4.3e-001 2.0e+000 2 2.3767e-001 7.9434e-001 2.9047e-001 8.6677e-001 1.6090e+001 1.00e-001 1.0000 1.009 4.3e-001 1.9e+000 3 1.3886e-001 6.4824e-001 2.3796e-001 7.8842e-001 8.1065e+000 2.00e-001 1.0000 1.048 4.3e-001 1.7e+000 4 9.7266e-002 4.3784e-001 1.3602e-001 5.0482e-001 2.4653e+000 4.00e-001 0.9247 1.205 4.3e-001 1.1e+000 5 7.2821e-002 2.9141e-001 7.5986e-002 3.0711e-001 1.3109e+000 4.00e-001 0.6361 1.208 4.4e-001 4.4e-001 6 5.8946e-002 1.8705e-001 4.1440e-002 1.6703e-001 1.0671e+000 4.00e-001 0.4465 1.212 4.6e-001 1.1e-001 7 4.8222e-002 1.0901e-001 2.1593e-002 7.2136e-002 1.0137e+000 4.00e-001 0.3123 1.212 5.7e-001 2.1e-002 8 3.2064e-002 4.9665e-002 9.4650e-003 2.0707e-002 1.0022e+000 4.00e-001 0.2027 1.202 7.1e-001 2.9e-003 9 1.0548e-002 1.1789e-002 2.2321e-003 2.9931e-003 1.0001e+000 4.00e-001 0.1189 1.150 8.6e-001 1.4e-004 10 3.6461e-004 3.2618e-004 5.9797e-005 6.2369e-005 1.0000e+000 4.00e-001 0.0394 1.022 9.8e-001 1.3e-007 11 1.2096e-008 9.4429e-009 1.6285e-009 1.7033e-009 1.0000e+000 4.00e-001 0.0012 1.000 1.0e+000 1.5e-016 Best point: x' = 1.2095813863434270e-008 9.4429338821509960e-009 1.6284785583347700e-009 1.7033188387310590e-009. Best value: cq4(x) = 1.0000000000000000e+000. Step count: 11 Trust-Region passes. Stopped by: e2 < eta2. Flop count: 10212 total--average 928 per step. ------------------- MATLAB TRANSCRIPT FOR PART C(b), sigma=4 ------------------- EDUğ fname='cq4'; dfname='cq4d'; hfname='cq4h'; CQ4_SIGMA=4; EDUğ th = 70*pi/180; c = cos(th); s = sin(th); x0 = [c;s;c;s]; EDUğ dogleg Newton/Trust Region minimization of function "cq4", using gradient "cq4d", Hessian "cq4h": k x_k(1) x_k(2) x_k(3) x_k(4) f(x_k) Delta |dx|/Delta RR e_1 e_2 0 3.4202e-001 9.3969e-001 3.4202e-001 9.3969e-001 5.8061e+001 1.00e-001 1 2.8882e-001 8.6715e-001 3.1641e-001 9.0432e-001 4.2451e+001 1.00e-001 1.0000 1.008 4.3e-001 2.0e+000 2 1.8568e-001 7.2152e-001 2.6455e-001 8.3040e-001 2.1351e+001 2.00e-001 1.0000 1.040 4.3e-001 1.9e+000 3 1.2461e-001 4.8958e-001 1.5099e-001 5.4450e-001 5.1060e+000 4.00e-001 0.9752 1.204 4.3e-001 1.7e+000 4 8.6093e-002 3.2980e-001 8.3849e-002 3.4933e-001 1.8434e+000 4.00e-001 0.6596 1.205 4.4e-001 9.3e-001 5 6.2575e-002 2.1893e-001 4.5203e-002 2.1353e-001 1.1781e+000 4.00e-001 0.4526 1.207 4.6e-001 3.0e-001 6 4.8488e-002 1.4039e-001 2.3827e-002 1.1750e-001 1.0384e+000 4.00e-001 0.3167 1.211 4.7e-001 6.7e-002 7 3.8194e-002 8.2377e-002 1.2394e-002 5.2189e-002 1.0080e+000 4.00e-001 0.2217 1.213 5.6e-001 1.3e-002 8 2.5481e-002 3.8782e-002 5.8601e-003 1.5908e-002 1.0014e+000 4.00e-001 0.1462 1.206 7.0e-001 1.9e-003 9 9.3121e-003 1.0335e-002 1.6717e-003 2.6307e-003 1.0001e+000 4.00e-001 0.0889 1.163 8.3e-001 1.1e-004 10 4.8260e-004 4.2903e-004 7.1539e-005 8.2599e-005 1.0000e+000 4.00e-001 0.0340 1.033 9.7e-001 2.3e-007 11 5.5614e-008 4.3168e-008 7.0358e-009 7.8405e-009 1.0000e+000 4.00e-001 0.0016 1.000 1.0e+000 3.1e-015 Best point: x' = 5.5613707576076220e-008 4.3168031041310260e-008 7.0358097098947990e-009 7.8405331032038940e-009. Best value: cq4(x) = 1.0000000000000020e+000. Step count: 11 Trust-Region passes. Stopped by: e2 < eta2. Flop count: 10173 total--average 925 per step. ------------------- MATLAB TRANSCRIPT FOR PART C(b), sigma=8 ------------------- EDUğ fname='cq4'; dfname='cq4d'; hfname='cq4h'; CQ4_SIGMA=8; EDUğ th = 70*pi/180; c = cos(th); s = sin(th); x0 = [c;s;c;s]; EDUğ dogleg Newton/Trust Region minimization of function "cq4", using gradient "cq4d", Hessian "cq4h": k x_k(1) x_k(2) x_k(3) x_k(4) f(x_k) Delta |dx|/Delta RR e_1 e_2 0 3.4202e-001 9.3969e-001 3.4202e-001 9.3969e-001 1.1412e+002 1.00e-001 1 2.8872e-001 8.6716e-001 3.1641e-001 9.0446e-001 8.3027e+001 1.00e-001 1.0000 1.008 4.3e-001 2.1e+000 2 1.8535e-001 7.2152e-001 2.6455e-001 8.3087e-001 4.1050e+001 2.00e-001 1.0000 1.041 4.3e-001 2.0e+000 3 9.0462e-002 5.7486e-001 2.1126e-001 7.4933e-001 1.8301e+001 2.00e-001 1.0000 1.060 4.3e-001 1.8e+000 4 6.0643e-002 3.9061e-001 1.2075e-001 4.9362e-001 4.4815e+000 4.00e-001 0.8232 1.203 4.3e-001 1.5e+000 5 4.2398e-002 2.6397e-001 6.7205e-002 3.1999e-001 1.7118e+000 4.00e-001 0.5556 1.204 4.4e-001 8.2e-001 6 3.1932e-002 1.7658e-001 3.6347e-002 2.0012e-001 1.1494e+000 4.00e-001 0.3797 1.207 4.6e-001 2.5e-001 7 2.6567e-002 1.1534e-001 1.9270e-002 1.1554e-001 1.0322e+000 4.00e-001 0.2649 1.210 4.7e-001 5.5e-002 8 2.3507e-002 7.0664e-002 1.0243e-002 5.6112e-002 1.0068e+000 4.00e-001 0.1874 1.213 5.1e-001 1.1e-002 9 1.8639e-002 3.6428e-002 5.2548e-003 1.9458e-002 1.0012e+000 4.00e-001 0.1266 1.207 6.5e-001 1.8e-003 10 9.1978e-003 1.2106e-002 1.9119e-003 3.8472e-003 1.0001e+000 4.00e-001 0.0765 1.179 8.0e-001 1.5e-004 11 1.0794e-003 1.0659e-003 1.7943e-004 2.2488e-004 1.0000e+000 4.00e-001 0.0357 1.067 9.4e-001 1.2e-006 12 1.3950e-006 1.1546e-006 1.9335e-007 2.0993e-007 1.0000e+000 4.00e-001 0.0039 1.001 1.0e+000 1.9e-012 Best point: x' = 1.3949689684317980e-006 1.1545906493937160e-006 1.9334919511556170e-007 2.0992880317893730e-007. Best value: cq4(x) = 1.0000000000016800e+000. Step count: 12 Trust-Region passes. Stopped by: e2 < eta2. Flop count: 11117 total--average 926 per step. ------------------- MATLAB TRANSCRIPT FOR PART C(b), sigma=128 ------------------- EDUğ fname='cq4'; dfname='cq4d'; hfname='cq4h'; CQ4_SIGMA=128; EDUğ th = 70*pi/180; c = cos(th); s = sin(th); x0 = [c;s;c;s]; EDUğ dogleg Newton/Trust Region minimization of function "cq4", using gradient "cq4d", Hessian "cq4h": k x_k(1) x_k(2) x_k(3) x_k(4) f(x_k) Delta |dx|/Delta RR e_1 e_2 0 3.4202e-001 9.3969e-001 3.4202e-001 9.3969e-001 1.7960e+003 1.00e-001 1 2.8863e-001 8.6716e-001 3.1641e-001 9.0459e-001 1.3003e+003 1.00e-001 1.0000 1.008 4.3e-001 2.1e+000 2 1.8505e-001 7.2151e-001 2.6456e-001 8.3131e-001 6.3202e+002 2.00e-001 1.0000 1.041 4.3e-001 2.1e+000 3 1.2134e-001 4.9128e-001 1.5095e-001 5.5443e-001 1.2616e+002 4.00e-001 0.9573 1.203 4.3e-001 2.1e+000 4 7.9757e-002 3.3368e-001 8.3438e-002 3.6948e-001 2.5846e+001 4.00e-001 0.6390 1.203 4.5e-001 2.1e+000 5 5.2624e-002 2.2605e-001 4.4043e-002 2.4582e-001 5.9402e+000 4.00e-001 0.4269 1.203 4.7e-001 1.9e+000 6 3.4961e-002 1.5268e-001 2.1614e-002 1.6296e-001 1.9859e+000 4.00e-001 0.2857 1.203 5.1e-001 1.1e+000 7 2.3539e-002 1.0272e-001 9.3046e-003 1.0719e-001 1.1983e+000 4.00e-001 0.1918 1.203 5.7e-001 3.8e-001 8 1.6266e-002 6.8679e-002 2.9572e-003 6.9275e-002 1.0405e+000 4.00e-001 0.1297 1.204 6.8e-001 8.7e-002 9 1.1777e-002 4.5359e-002 8.8855e-005 4.3066e-002 1.0085e+000 4.00e-001 0.0887 1.206 9.7e-001 1.8e-002 10 9.0985e-003 2.9129e-002 -7.7000e-004 2.4591e-002 1.0018e+000 4.00e-001 0.0619 1.210 9.7e+000 3.5e-003 11 7.3058e-003 1.7432e-002 -5.4798e-004 1.1755e-002 1.0004e+000 4.00e-001 0.0437 1.213 5.2e-001 6.6e-004 12 5.2798e-003 8.7461e-003 -2.6068e-005 4.0422e-003 1.0001e+000 4.00e-001 0.0295 1.209 9.5e-001 1.0e-004 13 2.4272e-003 2.8406e-003 1.4532e-004 8.2041e-004 1.0000e+000 4.00e-001 0.0183 1.182 6.6e+000 8.5e-006 14 2.6608e-004 2.4183e-004 2.4241e-005 5.0002e-005 1.0000e+000 4.00e-001 0.0087 1.066 9.4e-001 7.1e-008 15 3.0288e-007 2.3643e-007 2.9837e-008 4.4194e-008 1.0000e+000 4.00e-001 0.0009 1.001 1.0e+000 9.2e-014 Best point: x' = 3.0288043510810460e-007 2.3642641596295340e-007 2.9837039604901280e-008 4.4193853325991610e-008. Best value: cq4(x) = 1.0000000000000750e+000. Step count: 15 Trust-Region passes. Stopped by: e2 < eta2. Flop count: 13793 total--average 920 per step.