31:
101:
method is essentially a generalization of MINRES for arbitrary matrices. Both minimize the 2-norm of the residual and do the same calculations in exact arithmetic when the matrix is symmetric. MINRES is a short-recurrence method with a constant memory requirement, whereas GMRES requires storing the
1835:
In the case of positive definite matrices, the convergence rate of the MINRES method can be estimated in a way similar to that of the CG method. In contrast to the CG method, however, the estimation does not apply to the errors of the iterates, but to the residual. The following applies:
2133:
1961:
1154:
942:
1676:
407:
619:
2041:
1841:
1824:
1750:
826:
1314:
1234:
1033:
821:
181:
1561:
816:
449:
2205:
2169:
1496:
1447:
220:
678:
498:
102:
whole Krylov space, so its memory requirement is roughly proportional to the number of iterations. On the other hand, GMRES tends to suffer less from loss of orthogonality.
982:
299:
1027:
1992:
645:
271:
1347:
63:
1530:
1401:
1374:
771:
744:
528:
140:
1556:
719:(CR) method was therefore produced below as a substitute. It differs from MINRES in that in MINRES, the columns of a basis of the Krylov space (denoted below by
2229:
2036:
2016:
701:
291:
773:) can be orthogonalized via the Lanczos recursion. There are more efficient and preconditioned variants with fewer AXPYs. Compare with the article.
533:
1754:
1680:
707:, it is possible to carry out this minimization process recursively, storing only two previous steps (short recurrence). This saves memory.
1238:
1158:
1452:
2814:
1406:
704:
2862:
145:
2128:{\displaystyle \kappa (A)={\frac {\left|\lambda _{\text{max}}(A)\right|}{\left|\lambda _{\text{min}}(A)\right|}},}
1956:{\displaystyle \|r_{k}\|\leq 2\left({\frac {{\sqrt {\kappa (A)}}-1}{{\sqrt {\kappa (A)}}+1}}\right)^{k}\|r_{0}\|,}
779:
412:
2174:
2138:
715:
Note: The MINRES method is more complicated than the algebraically equivalent
Conjugate Residual method. The
110:
The MINRES method iteratively calculates an approximate solution of a linear system of equations of the form
2830:
190:
113:
1149:{\displaystyle \alpha _{k-1}={\frac {\langle r_{k-1},s_{k-1}\rangle }{\langle s_{k-1},s_{k-1}\rangle }}}
59:
937:{\displaystyle {\begin{aligned}r_{0}&=b-Ax_{0}\\p_{0}&=r_{0}\\s_{0}&=Ap_{0}\end{aligned}}}
654:
1671:{\displaystyle \beta _{k,l}={\frac {\langle s_{k},s_{k-l}\rangle }{\langle s_{k-l},s_{k-l}\rangle }}}
716:
454:
74:
35:
1349:
is smaller than a specified tolerance, the algorithm is interrupted with the approximate solution
949:
992:
1968:
624:
232:
1319:
55:
39:
17:
67:
2846:
1503:
1379:
1352:
749:
722:
506:
78:
8:
1535:
2824:
2214:
2021:
2001:
686:
276:
2810:
402:{\displaystyle V_{k}=x_{0}+\operatorname {span} \{r_{0},Ar_{0}\ldots ,A^{k-1}r_{0}\}}
2789:"Effcient solvers for constrained optimization in parameter identification problems"
2769:
1995:
184:
86:
294:
82:
648:
2856:
223:
2788:
2755:
30:
746:) can be orthogonalized, whereas in CR their images (below labeled with
2208:
38:(blue) and the MINRES method (green). The matrix used comes from a 2D
2773:
2757:
614:{\displaystyle x_{k}:=\mathrm {argmin} _{x\in V_{k}}\|r(x)\|,}
98:
2758:"Solution of sparse indefinite systems of linear equations"
1558:
is not carried out in the first iteration step) calculate:
1819:{\displaystyle s_{k}\leftarrow s_{k}-\beta _{k,l}s_{k-l}}
1745:{\displaystyle p_{k}\leftarrow p_{k}-\beta _{k,l}p_{k-l}}
2234:
1830:
2804:
34:
A comparison of the norm of error and residual in the
2217:
2177:
2141:
2044:
2024:
2004:
1971:
1844:
1757:
1683:
1564:
1538:
1506:
1455:
1409:
1382:
1355:
1322:
1241:
1161:
1036:
995:
952:
824:
782:
752:
725:
689:
657:
627:
536:
509:
457:
415:
302:
279:
235:
193:
148:
116:
2807:
Numerical
Methods for Two-phase Incompressible Flows
503:
More precisely, we define the approximate solutions
2756:Christopher C. Paige, Michael A. Saunders (1975).
2223:
2199:
2163:
2127:
2030:
2010:
1986:
1955:
1818:
1744:
1670:
1550:
1524:
1490:
1441:
1395:
1368:
1341:
1309:{\displaystyle r_{k}=r_{k-1}-\alpha _{k-1}s_{k-1}}
1308:
1229:{\displaystyle x_{k}=x_{k-1}+\alpha _{k-1}p_{k-1}}
1228:
1148:
1021:
976:
936:
810:
765:
738:
695:
672:
639:
613:
522:
492:
443:
401:
285:
265:
214:
175:
134:
105:
2854:
176:{\displaystyle A\in \mathbb {R} ^{n\times n}}
77:, the MINRES method does not assume that the
1947:
1934:
1858:
1845:
1662:
1624:
1619:
1587:
1336:
1323:
1140:
1102:
1097:
1059:
634:
628:
605:
590:
396:
335:
811:{\displaystyle x_{0}\in \mathbb {R} ^{n}}
798:
660:
444:{\displaystyle x_{0}\in \mathbb {R} ^{n}}
431:
202:
157:
2200:{\displaystyle \lambda _{\text{min}}(A)}
2164:{\displaystyle \lambda _{\text{max}}(A)}
1491:{\displaystyle s_{k}\leftarrow As_{k-1}}
58:for the iterative solution of symmetric
29:
1442:{\displaystyle p_{k}\leftarrow s_{k-1}}
14:
2855:
2235:Implementation in GNU Octave / MATLAB
1831:Convergence rate of the MINRES method
1376:. Otherwise, a new descent direction
451:is an initial value (often zero) and
215:{\displaystyle b\in \mathbb {R} ^{n}}
2751:
2749:
62:. It was proposed by mathematicians
710:
229:For this, the norm of the residual
92:
24:
2849:, Wolfram MathWorld, Jul 26, 2022.
2794:(Doctoral Thesis). pp. 51–52.
2762:SIAM Journal on Numerical Analysis
567:
564:
561:
558:
555:
552:
25:
2874:
2840:
2746:
2786:
673:{\displaystyle \mathbb {R} ^{n}}
493:{\displaystyle r_{0}:=r(x_{0})}
106:Properties of the MINRES method
2798:
2780:
2194:
2188:
2158:
2152:
2112:
2106:
2083:
2077:
2054:
2048:
1981:
1975:
1910:
1904:
1886:
1880:
1768:
1694:
1466:
1420:
602:
596:
487:
474:
245:
239:
13:
1:
2740:
2805:Sven Gross, Arnold Reusken.
977:{\displaystyle k=1,2,\dots }
89:of the matrix is mandatory.
7:
1022:{\displaystyle x_{k},r_{k}}
683:Because of the symmetry of
73:In contrast to the popular
10:
2879:
1987:{\displaystyle \kappa (A)}
640:{\displaystyle \|\cdot \|}
266:{\displaystyle r(x):=b-Ax}
2809:. section 5.2: Springer.
1342:{\displaystyle \|r_{k}\|}
2863:Numerical linear algebra
2829:: CS1 maint: location (
2238:
2207:are maximal and minimal
984:in the following steps:
64:Christopher Conway Paige
2847:Minimal Residual Method
60:linear equation systems
48:Minimal Residual Method
2225:
2201:
2165:
2129:
2032:
2012:
1988:
1957:
1820:
1746:
1672:
1552:
1526:
1492:
1443:
1403:is calculated through
1397:
1370:
1343:
1310:
1230:
1150:
1023:
978:
938:
818:arbitrary and compute
812:
767:
740:
697:
674:
641:
615:
524:
494:
445:
403:
287:
267:
216:
177:
136:
56:Krylov subspace method
43:
40:boundary-value problem
2226:
2202:
2166:
2130:
2033:
2013:
1989:
1958:
1821:
1747:
1673:
1553:
1527:
1525:{\displaystyle l=1,2}
1493:
1444:
1398:
1396:{\displaystyle p_{k}}
1371:
1369:{\displaystyle x_{k}}
1344:
1311:
1231:
1151:
1024:
979:
939:
813:
768:
766:{\displaystyle s_{k}}
741:
739:{\displaystyle p_{k}}
698:
675:
642:
616:
525:
523:{\displaystyle x_{k}}
495:
446:
404:
288:
268:
217:
178:
137:
135:{\displaystyle Ax=b,}
68:Michael Alan Saunders
33:
2249:A, b, x0, maxit, tol
2215:
2175:
2139:
2042:
2022:
2002:
1969:
1842:
1755:
1681:
1562:
1536:
1504:
1453:
1407:
1380:
1353:
1320:
1239:
1159:
1034:
993:
950:
946:Then we iterate for
822:
780:
750:
723:
687:
655:
625:
534:
507:
455:
413:
300:
277:
233:
191:
146:
114:
27:Computational method
2038:is normal, we have
1551:{\displaystyle l=2}
409:is minimized. Here
2787:Nifa, M. Naoufal.
2221:
2197:
2161:
2125:
2028:
2008:
1984:
1953:
1816:
1742:
1668:
1548:
1522:
1488:
1439:
1393:
1366:
1339:
1306:
1226:
1146:
1019:
974:
934:
932:
808:
763:
736:
717:Conjugate Residual
693:
670:
637:
611:
520:
490:
441:
399:
283:
263:
212:
173:
132:
44:
2816:978-3-642-19685-0
2224:{\displaystyle A}
2185:
2149:
2120:
2103:
2074:
2031:{\displaystyle A}
2011:{\displaystyle A}
1922:
1913:
1889:
1666:
1144:
776:First you choose
696:{\displaystyle A}
286:{\displaystyle k}
83:positive definite
16:(Redirected from
2870:
2835:
2834:
2828:
2820:
2802:
2796:
2795:
2793:
2784:
2778:
2777:
2753:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2248:
2245:
2242:
2231:, respectively.
2230:
2228:
2227:
2222:
2206:
2204:
2203:
2198:
2187:
2186:
2183:
2170:
2168:
2167:
2162:
2151:
2150:
2147:
2134:
2132:
2131:
2126:
2121:
2119:
2115:
2105:
2104:
2101:
2090:
2086:
2076:
2075:
2072:
2061:
2037:
2035:
2034:
2029:
2017:
2015:
2014:
2009:
1996:condition number
1993:
1991:
1990:
1985:
1962:
1960:
1959:
1954:
1946:
1945:
1933:
1932:
1927:
1923:
1921:
1914:
1900:
1897:
1890:
1876:
1873:
1857:
1856:
1825:
1823:
1822:
1817:
1815:
1814:
1799:
1798:
1780:
1779:
1767:
1766:
1751:
1749:
1748:
1743:
1741:
1740:
1725:
1724:
1706:
1705:
1693:
1692:
1677:
1675:
1674:
1669:
1667:
1665:
1661:
1660:
1642:
1641:
1622:
1618:
1617:
1599:
1598:
1585:
1580:
1579:
1557:
1555:
1554:
1549:
1531:
1529:
1528:
1523:
1497:
1495:
1494:
1489:
1487:
1486:
1465:
1464:
1448:
1446:
1445:
1440:
1438:
1437:
1419:
1418:
1402:
1400:
1399:
1394:
1392:
1391:
1375:
1373:
1372:
1367:
1365:
1364:
1348:
1346:
1345:
1340:
1335:
1334:
1315:
1313:
1312:
1307:
1305:
1304:
1289:
1288:
1270:
1269:
1251:
1250:
1235:
1233:
1232:
1227:
1225:
1224:
1209:
1208:
1190:
1189:
1171:
1170:
1155:
1153:
1152:
1147:
1145:
1143:
1139:
1138:
1120:
1119:
1100:
1096:
1095:
1077:
1076:
1057:
1052:
1051:
1028:
1026:
1025:
1020:
1018:
1017:
1005:
1004:
983:
981:
980:
975:
943:
941:
940:
935:
933:
929:
928:
909:
908:
895:
894:
878:
877:
864:
863:
838:
837:
817:
815:
814:
809:
807:
806:
801:
792:
791:
772:
770:
769:
764:
762:
761:
745:
743:
742:
737:
735:
734:
711:MINRES algorithm
703:, unlike in the
702:
700:
699:
694:
679:
677:
676:
671:
669:
668:
663:
647:is the standard
646:
644:
643:
638:
620:
618:
617:
612:
589:
588:
587:
586:
570:
546:
545:
529:
527:
526:
521:
519:
518:
499:
497:
496:
491:
486:
485:
467:
466:
450:
448:
447:
442:
440:
439:
434:
425:
424:
408:
406:
405:
400:
395:
394:
385:
384:
363:
362:
347:
346:
325:
324:
312:
311:
292:
290:
289:
284:
272:
270:
269:
264:
221:
219:
218:
213:
211:
210:
205:
185:symmetric matrix
182:
180:
179:
174:
172:
171:
160:
141:
139:
138:
133:
93:GMRES vs. MINRES
21:
2878:
2877:
2873:
2872:
2871:
2869:
2868:
2867:
2853:
2852:
2843:
2838:
2822:
2821:
2817:
2803:
2799:
2791:
2785:
2781:
2774:10.1137/0712047
2754:
2747:
2743:
2738:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2246:
2243:
2240:
2237:
2216:
2213:
2212:
2182:
2178:
2176:
2173:
2172:
2146:
2142:
2140:
2137:
2136:
2100:
2096:
2095:
2091:
2071:
2067:
2066:
2062:
2060:
2043:
2040:
2039:
2023:
2020:
2019:
2003:
2000:
1999:
1970:
1967:
1966:
1941:
1937:
1928:
1899:
1898:
1875:
1874:
1872:
1868:
1867:
1852:
1848:
1843:
1840:
1839:
1833:
1828:
1804:
1800:
1788:
1784:
1775:
1771:
1762:
1758:
1756:
1753:
1752:
1730:
1726:
1714:
1710:
1701:
1697:
1688:
1684:
1682:
1679:
1678:
1650:
1646:
1631:
1627:
1623:
1607:
1603:
1594:
1590:
1586:
1584:
1569:
1565:
1563:
1560:
1559:
1537:
1534:
1533:
1505:
1502:
1501:
1476:
1472:
1460:
1456:
1454:
1451:
1450:
1427:
1423:
1414:
1410:
1408:
1405:
1404:
1387:
1383:
1381:
1378:
1377:
1360:
1356:
1354:
1351:
1350:
1330:
1326:
1321:
1318:
1317:
1294:
1290:
1278:
1274:
1259:
1255:
1246:
1242:
1240:
1237:
1236:
1214:
1210:
1198:
1194:
1179:
1175:
1166:
1162:
1160:
1157:
1156:
1128:
1124:
1109:
1105:
1101:
1085:
1081:
1066:
1062:
1058:
1056:
1041:
1037:
1035:
1032:
1031:
1013:
1009:
1000:
996:
994:
991:
990:
951:
948:
947:
931:
930:
924:
920:
910:
904:
900:
897:
896:
890:
886:
879:
873:
869:
866:
865:
859:
855:
839:
833:
829:
825:
823:
820:
819:
802:
797:
796:
787:
783:
781:
778:
777:
757:
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2841:External links
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293:-dimensional
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1964:
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1834:
1030:
945:
775:
714:
705:GMRES method
682:
502:
228:
109:
96:
72:
51:
47:
45:
2209:eigenvalues
85:, only the
2741:References
2018:. Because
1998:of matrix
1532:(the step
2825:cite book
2180:λ
2144:λ
2098:λ
2069:λ
2046:κ
1973:κ
1948:‖
1935:‖
1902:κ
1892:−
1878:κ
1862:≤
1859:‖
1846:‖
1809:−
1786:β
1782:−
1769:←
1735:−
1712:β
1708:−
1695:←
1663:⟩
1655:−
1636:−
1625:⟨
1620:⟩
1612:−
1588:⟨
1567:β
1481:−
1467:←
1432:−
1421:←
1337:‖
1324:‖
1299:−
1283:−
1276:α
1272:−
1264:−
1219:−
1203:−
1196:α
1184:−
1141:⟩
1133:−
1114:−
1103:⟨
1098:⟩
1090:−
1071:−
1060:⟨
1046:−
1039:α
972:…
850:−
794:∈
635:‖
632:⋅
629:‖
606:‖
591:‖
576:∈
427:∈
379:−
365:…
333:
255:−
198:∈
166:×
153:∈
75:CG method
70:in 1975.
36:CG method
2857:Category
2244:= minres
2241:function
1029:through
989:Compute
530:through
87:symmetry
1994:is the
2813:
2672:'*
2657:'*
2579:'*
2564:'*
2500:'*
2434:'*
2419:'*
2135:where
1965:where
621:where
224:vector
142:where
79:matrix
52:MINRES
18:MINRES
2792:(PDF)
2717:beta2
2693:beta2
2648:beta2
2624:beta1
2600:beta1
2555:beta1
2519:break
2479:alpha
2455:alpha
2410:alpha
2359:maxit
273:in a
183:is a
99:GMRES
54:is a
2831:link
2811:ISBN
2642:>
2639:iter
2506:<
2347:iter
2171:and
1500:for
330:span
187:and
97:The
66:and
46:The
2770:doi
2735:end
2732:end
2729:end
2522:end
2509:tol
2344:for
2211:of
2184:min
2148:max
2102:min
2073:max
1316:if
651:on
81:is
50:or
2859::
2827:}}
2823:{{
2766:12
2764:.
2760:.
2748:^
2723:s2
2711:s0
2705:s0
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2260:x0
680:.
548::=
500:.
469::=
249::=
226:.
222:a
2833:)
2819:.
2776:.
2772::
2726:;
2720:*
2714:-
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2702:;
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2666:(
2663:/
2651:=
2645:1
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2603:*
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2558:=
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2428:(
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2413:=
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2401:=
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2371:;
2365:=
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2329:;
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2317:;
2311:*
2308:A
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2278:A
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2269:=
2266:r
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2257:=
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2251:)
2247:(
2219:A
2195:)
2192:A
2189:(
2159:)
2156:A
2153:(
2123:,
2117:|
2113:)
2110:A
2107:(
2093:|
2088:|
2084:)
2081:A
2078:(
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2058:=
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2052:A
2049:(
2026:A
2006:A
1982:)
1979:A
1976:(
1951:,
1943:0
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1930:k
1925:)
1919:1
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1911:)
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957:=
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314:=
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240:(
237:r
208:n
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195:b
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158:R
150:A
130:,
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124:=
121:x
118:A
42:.
20:)
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