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Table 3 Results of the numerical experiments

From: RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations

Example Method No. iterations Final subspace dim. Time
CUBE \(n=10648, \; m=p=1\) RADI—Penzl 97 97 18.96
RADI—Ham, \(\ell =2p\) 119 119 17.10
RADI—Ham, \(\ell =6p\) 99 99 14.15
RADI—Ham, \(\ell =\infty \) 75 75 11.60
RADI—Ham + Opt, \(\ell =2p\) 122 122 17.87
RADI—Ham + Opt, \(\ell =6p\) 103 103 16.70
RADI—Ham + Opt, \(\ell =\infty \) 108 108 18.44
RKSM—adaptive 83 83 14.80
EBA 111 222 6.23
Newton-ADI 2 outer, 296 inner 192 42.11
CUBE \(n=10648, \; m=p=10\) RADI—Penzl 135 1350 49.19
RADI—Ham, \(\ell =2p\) 139 1390 48.79
RADI—Ham, \(\ell =6p\) 100 1000 35.92
RADI—Ham, \(\ell =\infty \) 74 740 57.51
RADI—Ham + Opt, \(\ell =2p\) 87 870 30.59
RADI—Ham + Opt, \(\ell =6p\) 90 900 33.20
RADI—Ham + Opt, \(\ell =\infty \) 90 900 119.55
RKSM—adaptive 79 790 53.82
EBA 91 1820 230.57
Newton-ADI 2 outer, 202 inner 1960 75.60
CUBE \(n=74088, \; m=10, \; p=1\) RADI—Penzl 139 139 1048.60
RADI—Ham, \(\ell =2p\) 97 97 617.62
RADI—Ham, \(\ell =6p\) 81 81 506.37
RADI—Ham, \(\ell =\infty \) 72 72 446.64
RADI—Ham + Opt, \(\ell =2p\) 101 101 621.38
RADI—Ham + Opt, \(\ell =6p\) 93 93 571.34
RADI—Ham + Opt, \(\ell =\infty \) 63 63 387.43
RKSM—adaptive 73 73 338.78
EBA 81 162 30.45
Newton-ADI 2 outer, 288 inner 968 1546.29
CHIP \(n=20082, \; m=1, \; p=5\) RADI—Penzl 33 165 51.57
RADI—Ham, \(\ell =2p\) 36 180 30.32
RADI—Ham, \(\ell =6p\) 29 145 24.36
RADI—Ham, \(\ell =\infty \) 26 130 22.64
RADI—Ham + Opt, \(\ell =2p\) 29 145 23.97
RADI—Ham + Opt, \(\ell =6p\) 26 130 22.26
RADI—Ham + Opt, \(\ell =\infty \) 25 125 22.33
RKSM—adaptive 26 130 23.33
EBA 26 260 6.69
Newton-ADI 2 outer, 64 inner 204 54.04
IFISS \(n=66049, \; m=p=5\) RADI—Penzl >50 >250  
RADI—Ham, \(\ell =2p\) 22 110 17.21
RADI—Ham, \(\ell =6p\) 19 95 15.37
RADI—Ham, \(\ell =\infty \) 20 100 17.46
RADI—Ham + Opt, \(\ell =2p\) 27 135 21.12
RADI—Ham + Opt, \(\ell =6p\)   Did not converge  
RADI—Ham + Opt, \(\ell =\infty \)   Did not converge  
RKSM—adaptive 26 130 22.28
EBA 11 110 9.26
Newton-ADI 2 outer, 46 inner 250 38.05
RAIL \(n=317377, \; m=7, \; p=6\) RADI—Penzl 66 396 182.60
RADI—Ham, \(\ell =2p\) 49 294 131.34
RADI—Ham, \(\ell =6p\) 43 258 127.11
RADI—Ham, \(\ell =\infty \) 46 276 197.06
RADI—Ham + Opt, \(\ell =2p\) 46 276 124.13
RADI—Ham + Opt, \(\ell =6p\) 40 240 120.04
RADI—Ham + Opt, \(\ell =\infty \) 39 234 158.89
RKSM—adaptive 41 246 188.60
EBA 91 1092 916.21
Newton-ADI 1 outer, 62 inner 372 279.90
LUNG \(n=109460, \; m=p=10\) RADI—Penzl   Did not converge  
RADI—Ham, \(\ell =2p\) 31 310 30.03
RADI—Ham, \(\ell =6p\) 28 280 30.22
RADI—Ham, \(\ell =\infty \) 26 260 34.83
RADI—Ham + Opt, \(\ell =2p\) 25 250 22.33
RADI—Ham + Opt, \(\ell =6p\) 17 170 17.74
RADI—Ham + Opt, \(\ell =\infty \) 17 170 19.02
RKSM—adaptive 61 610 114.22
EBA   Did not converge  
Newton-ADI   Did not converge  
  1. Numbers in bold indicate the smallest subspace dimensions and execution times