Research Article
A Limited Memory BFGS Method for Solving Large-Scale Symmetric Nonlinear Equations
| | Nr. | Dim | Algorithm 1 | Normal BFGS algorithm | | NI/NG | GN | Time | NI/NG | GN | Time |
| | Problem 1 | 500 | 24/25 | 9.908338e − 05 | 2.901619e + 01 | 23/87 | NaN | 1.950013e + 00 | | 1000 | 9/10 | 5.697414e − 05 | 5.366434e + 01 | 23/87 | NaN | 1.124767e + 01 | | 1500 | 9/10 | 3.363290e − 05 | 1.689803e + 02 | NI > 1000 | Inf | 8.094892e + 01 | | 2000 | 9/10 | 2.706880e − 05 | 3.880213e + 02 | NI > 1000 | Inf | 1.550962e + 02 |
| | Problem 2 | 500 | 8/16 | 3.660478e − 05 | 6.926444e + 00 | 60/390 | NaN | 7.269647e + 00 | | 1000 | 8/16 | 6.406401e − 05 | 4.513109e + 01 | 40/314 | NaN | 2.857938e + 01 | | 1500 | 9/17 | 7.655321e − 05 | 1.689959e + 02 | 33/258 | NaN | 7.102726e + 01 | | 2000 | 9/17 | 9.651851e − 05 | 3.881617e + 02 | 29/226 | NaN | 1.410717e + 02 |
| | Problem 3 | 500 | 18/33 | 7.747640e − 06 | 2.110694e + 01 | NI > 1000 | 5.905028e + 04 | 1.220396e + 02 | | 1000 | 17/32 | 8.381440e − 05 | 1.270472e + 02 | NI > 1000 | 1.577282e + 05 | 5.830069e + 02 | | 1500 | 17/32 | 6.658002e − 05 | 3.975217e + 02 | NI > 1000 | 2.929920e + 05 | 1.645561e + 03 | | 2000 | 17/32 | 5.689277e − 05 | 9.154451e + 02 | NI > 1000 | 4.345902e + 05 | 3.776363e + 03 |
| | Problem 4 | 500 | 809/3134 | 9.614778e − 05 | 1.113785e + 03 | 9/73 | NaN | 1.107607e + 00 | | 1000 | 960/3894 | 9.865549e − 05 | 8.674373e + 03 | 8/65 | NaN | 5.725237e + 00 | | 1500 | 197/695 | 8.562547e − 05 | 5.509394e + 03 | 8/65 | NaN | 1.584970e + 01 | | 2000 | 220/676 | 9.847745e − 05 | 1.418851e + 04 | 8/65 | NaN | 3.584903e + 01 |
| | Problem 5 | 500 | 6/7 | 4.925073e − 06 | 4.305628e + 00 | 6/7 | 4.925073e − 06 | 7.488048e − 01 | | 1000 | 6/7 | 6.747222e − 06 | 2.725337e + 01 | 6/7 | 6.747222e − 06 | 4.477229e + 00 | | 1500 | 6/7 | 8.176417e − 06 | 8.352294e + 01 | 6/7 | 8.176417e − 06 | 1.340049e + 01 | | 2000 | 6/7 | 9.391335e − 06 | 1.925364e + 02 | 6/7 | 9.391335e − 06 | 3.001459e + 01 |
| | Problem 6 | 500 | 96/97 | 9.183793e − 05 | 1.272812e + 02 | 65/66 | 9.473359e − 05 | 8.346053e + 00 | | 1000 | 17/18 | 7.206980e − 05 | 1.263140e + 02 | 63/64 | 8.700852e − 05 | 4.623870e + 01 | | 1500 | 17/18 | 6.663689e − 05 | 3.956965e + 02 | 63/64 | 9.605270e − 05 | 1.402917e + 02 | | 2000 | 17/18 | 6.136239e − 05 | 9.107026e + 02 | 64/65 | 8.724048e − 05 | 3.194588e + 02 |
| | Problem 7 | 500 | 14/15 | 7.551200e − 05 | 1.533490e + 01 | 52/53 | 9.761790e − 05 | 6.333641e + 00 | | 1000 | 15/16 | 5.530847e − 05 | 1.084051e + 02 | 55/63 | 8.871231e − 05 | 3.985826e + 01 | | 1500 | 14/15 | 8.445286e − 05 | 3.119708e + 02 | 60/68 | 7.722045e − 05 | 1.336617e + 02 | | 2000 | 15/16 | 4.705617e − 05 | 7.805354e + 02 | 13/63 | NaN | 5.564556e + 01 |
| | Problem 8 | 500 | 6/7 | 4.600878e − 05 | 4.196427e + 00 | 6/7 | 2.375490e − 05 | 7.020045e − 01 | | 1000 | 6/7 | 6.434846e − 05 | 2.658257e + 01 | 6/7 | 3.327487e − 05 | 4.461629e + 00 | | 1500 | 6/7 | 7.851881e − 05 | 8.319533e + 01 | 6/7 | 4.062329e − 05 | 1.335369e + 01 | | 2000 | 6/7 | 9.049762e − 05 | 1.928484e + 02 | 6/7 | 4.683284e − 05 | 2.985859e + 01 |
| | Problem 9 | 500 | 2/10 | 2.027911e − 10 | 5.460035e − 01 | NI > 1000 | 4.514037e + 117 | 1.037407e + 01 | | 1000 | 2/10 | 2.666034e − 10 | 2.698817e + 00 | NI > 1000 | 6.383813e + 117 | 2.861058e + 01 | | 1500 | 2/10 | 1.379742e − 09 | 8.330453e + 00 | NI > 1000 | 7.818542e + 117 | 5.319634e + 01 | | 2000 | 2/10 | 1.701735e − 09 | 1.942212e + 01 | NI > 1000 | 9.028075e + 117 | 8.375694e + 01 |
| | Problem 10 | 500 | 435/2865 | 5.154286e − 05 | 5.901362e + 02 | 41/210 | NaN | 3.806424e + 00 | | 1000 | 637/4215 | 5.701191e − 05 | 5.749432e + 03 | 67/425 | NaN | 3.689424e + 01 | | 1500 | 303/1914 | 4.003848e − 05 | 8.526469e + 03 | 63/491 | NaN | 1.146919e + 02 | | 2000 | 473/3281 | 3.558910e − 05 | 3.074095e + 04 | 64/499 | NaN | 2.622845e + 02 |
| | Problem 11 | 500 | 1/2 | 0.000000e + 00 | 1.092007e − 01 | 1/2 | 0.000000e + 00 | 1.560010e − 01 | | 1000 | 1/2 | 0.000000e + 00 | 7.956051e − 01 | 1/2 | 0.000000e + 00 | 6.864044e − 01 | | 1500 | 1/2 | 0.000000e + 00 | 2.277615e + 00 | 1/2 | 0.000000e + 00 | 2.324415e + 00 | | 2000 | 1/2 | 0.000000e + 00 | 5.101233e + 00 | 1/2 | 0.000000e + 00 | 5.210433e + 00 |
| | Problem 12 | 500 | 260/800 | 1.654459e − 05 | 3.509398e + 02 | 8/51 | NaN | 7.800050e − 01 | | 1000 | 324/1053 | 8.997051e − 05 | 2.902289e + 03 | 8/51 | NaN | 5.226033e + 00 | | 1500 | 254/829 | 1.257117e − 05 | 7.123224e + 03 | 8/51 | NaN | 1.583410e + 01 | | 2000 | 372/1353 | 9.331122e − 05 | 2.414117e + 04 | 8/51 | NaN | 3.046700e + 01 |
| | Problem 13 | 500 | 96/209 | 9.194537e − 05 | 1.287632e + 02 | 10/74 | NaN | 1.185608e + 00 | | 1000 | 53/89 | 5.718492e − 05 | 4.531049e + 02 | 10/74 | NaN | 6.021639e + 00 | | 1500 | | 1.180195e + 10 | 2.835008e + 04 | 10/74 | NaN | 1.790891e + 01 | | 2000 | 54/132 | 3.919816e − 05 | 3.337298e + 03 | 10/74 | NaN | 4.031066e + 01 |
| | Problem 14 | 500 | 18/68 | 5.826508e − 05 | 2.076373e + 01 | 16/115 | NaN | 6.240040e − 01 | | 1000 | 18/68 | 8.239959e − 05 | 1.357989e + 02 | 16/115 | NaN | 3.354021e + 00 | | 1500 | 19/69 | 5.002681e − 05 | 4.532921e + 02 | 16/115 | NaN | 9.765663e + 00 | | 2000 | 19/69 | 5.774900e − 05 | 1.042820e + 03 | 16/115 | NaN | 2.148134e + 01 |
| | Problem 15 | 500 | 8/9 | 6.870902e − 05 | 6.910844e + 00 | 22/23 | 8.899350e − 05 | 2.667617e + 00 | | 1000 | 7/8 | 7.203786e − 05 | 3.561503e + 01 | 16/17 | 9.828207e − 05 | 1.174688e + 01 | | 1500 | 7/8 | 4.809670e − 05 | 1.122739e + 02 | 13/14 | 9.654107e − 05 | 2.903179e + 01 | | 2000 | 7/8 | 3.609946e − 05 | 2.583845e + 02 | 11/12 | 9.465949e − 05 | 5.519315e + 01 |
| | Problem 16 | 500 | 0/1 | 0.000000e + 00 | 0.000000e + 00 | 0/1 | 0.000000e + 00 | 0.000000e + 00 | | 1000 | 0/1 | 0.000000e + 00 | 0.000000e + 00 | 0/1 | 0.000000e + 00 | 0.000000e + 00 | | 1500 | 0/1 | 0.000000e + 00 | 1.560010e − 02 | 0/1 | 0.000000e + 00 | 0.000000e + 00 | | 2000 | 0/1 | 0.000000e + 00 | 1.560010e − 02 | 0/1 | 0.000000e + 00 | 1.560010e − 02 |
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