20:
90:
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
1824:
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:
2122:
1950:
1143:
931:
1665:
396:
608:
2030:
1830:
1813:
1739:
815:
1303:
1223:
1022:
810:
170:
1550:
805:
438:
2194:
2158:
1485:
1436:
209:
667:
487:
91:
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.
971:
288:
1016:
1981:
634:
260:
1336:
52:
1519:
1390:
1363:
760:
733:
517:
129:
1545:
708:(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
2218:
2025:
2005:
690:
280:
762:) can be orthogonalized via the Lanczos recursion. There are more efficient and preconditioned variants with fewer AXPYs. Compare with the article.
522:
1743:
1669:
696:, it is possible to carry out this minimization process recursively, storing only two previous steps (short recurrence). This saves memory.
1227:
1147:
1441:
2803:
1395:
693:
2851:
134:
2117:{\displaystyle \kappa (A)={\frac {\left|\lambda _{\text{max}}(A)\right|}{\left|\lambda _{\text{min}}(A)\right|}},}
1945:{\displaystyle \|r_{k}\|\leq 2\left({\frac {{\sqrt {\kappa (A)}}-1}{{\sqrt {\kappa (A)}}+1}}\right)^{k}\|r_{0}\|,}
768:
401:
2163:
2127:
704:
Note: The MINRES method is more complicated than the algebraically equivalent
Conjugate Residual method. The
99:
The MINRES method iteratively calculates an approximate solution of a linear system of equations of the form
2819:
179:
102:
1138:{\displaystyle \alpha _{k-1}={\frac {\langle r_{k-1},s_{k-1}\rangle }{\langle s_{k-1},s_{k-1}\rangle }}}
48:
926:{\displaystyle {\begin{aligned}r_{0}&=b-Ax_{0}\\p_{0}&=r_{0}\\s_{0}&=Ap_{0}\end{aligned}}}
643:
1660:{\displaystyle \beta _{k,l}={\frac {\langle s_{k},s_{k-l}\rangle }{\langle s_{k-l},s_{k-l}\rangle }}}
705:
443:
63:
24:
1338:
is smaller than a specified tolerance, the algorithm is interrupted with the approximate solution
938:
981:
1957:
613:
221:
1308:
44:
28:
56:
2835:
1492:
1368:
1341:
738:
711:
495:
67:
8:
1524:
2813:
2203:
2010:
1990:
675:
265:
2799:
391:{\displaystyle V_{k}=x_{0}+\operatorname {span} \{r_{0},Ar_{0}\ldots ,A^{k-1}r_{0}\}}
2778:"Effcient solvers for constrained optimization in parameter identification problems"
2758:
1984:
173:
75:
283:
71:
637:
2845:
212:
2777:
2744:
19:
735:) can be orthogonalized, whereas in CR their images (below labeled with
2197:
27:(blue) and the MINRES method (green). The matrix used comes from a 2D
2762:
2746:
603:{\displaystyle x_{k}:=\mathrm {argmin} _{x\in V_{k}}\|r(x)\|,}
87:
2747:"Solution of sparse indefinite systems of linear equations"
1547:
is not carried out in the first iteration step) calculate:
1808:{\displaystyle s_{k}\leftarrow s_{k}-\beta _{k,l}s_{k-l}}
1734:{\displaystyle p_{k}\leftarrow p_{k}-\beta _{k,l}p_{k-l}}
2223:
1819:
2793:
23:
A comparison of the norm of error and residual in the
2206:
2166:
2130:
2033:
2013:
1993:
1960:
1833:
1746:
1672:
1553:
1527:
1495:
1444:
1398:
1371:
1344:
1311:
1230:
1150:
1025:
984:
941:
813:
771:
741:
714:
678:
646:
616:
525:
498:
446:
404:
291:
268:
224:
182:
137:
105:
2796:
Numerical
Methods for Two-phase Incompressible Flows
492:
More precisely, we define the approximate solutions
2745:Christopher C. Paige, Michael A. Saunders (1975).
2212:
2188:
2152:
2116:
2019:
1999:
1975:
1944:
1807:
1733:
1659:
1539:
1513:
1479:
1430:
1384:
1357:
1330:
1298:{\displaystyle r_{k}=r_{k-1}-\alpha _{k-1}s_{k-1}}
1297:
1218:{\displaystyle x_{k}=x_{k-1}+\alpha _{k-1}p_{k-1}}
1217:
1137:
1010:
965:
925:
799:
754:
727:
684:
661:
628:
602:
511:
481:
432:
390:
274:
254:
203:
164:
123:
94:
2843:
165:{\displaystyle A\in \mathbb {R} ^{n\times n}}
66:, the MINRES method does not assume that the
1936:
1923:
1847:
1834:
1651:
1613:
1608:
1576:
1325:
1312:
1129:
1091:
1086:
1048:
623:
617:
594:
579:
385:
324:
800:{\displaystyle x_{0}\in \mathbb {R} ^{n}}
787:
649:
433:{\displaystyle x_{0}\in \mathbb {R} ^{n}}
420:
191:
146:
2189:{\displaystyle \lambda _{\text{min}}(A)}
2153:{\displaystyle \lambda _{\text{max}}(A)}
1480:{\displaystyle s_{k}\leftarrow As_{k-1}}
47:for the iterative solution of symmetric
18:
1431:{\displaystyle p_{k}\leftarrow s_{k-1}}
2844:
2224:Implementation in GNU Octave / MATLAB
1820:Convergence rate of the MINRES method
1365:. Otherwise, a new descent direction
440:is an initial value (often zero) and
204:{\displaystyle b\in \mathbb {R} ^{n}}
2740:
2738:
51:. It was proposed by mathematicians
699:
218:For this, the norm of the residual
81:
13:
2838:, Wolfram MathWorld, Jul 26, 2022.
2783:(Doctoral Thesis). pp. 51–52.
2751:SIAM Journal on Numerical Analysis
556:
553:
550:
547:
544:
541:
14:
2863:
2829:
2735:
2775:
662:{\displaystyle \mathbb {R} ^{n}}
482:{\displaystyle r_{0}:=r(x_{0})}
95:Properties of the MINRES method
2787:
2769:
2183:
2177:
2147:
2141:
2101:
2095:
2072:
2066:
2043:
2037:
1970:
1964:
1899:
1893:
1875:
1869:
1757:
1683:
1455:
1409:
591:
585:
476:
463:
234:
228:
1:
2729:
2794:Sven Gross, Arnold Reusken.
966:{\displaystyle k=1,2,\dots }
78:of the matrix is mandatory.
7:
1011:{\displaystyle x_{k},r_{k}}
672:Because of the symmetry of
62:In contrast to the popular
10:
2868:
1976:{\displaystyle \kappa (A)}
629:{\displaystyle \|\cdot \|}
255:{\displaystyle r(x):=b-Ax}
2798:. section 5.2: Springer.
1331:{\displaystyle \|r_{k}\|}
2852:Numerical linear algebra
2818:: CS1 maint: location (
2227:
2196:are maximal and minimal
973:in the following steps:
53:Christopher Conway Paige
2836:Minimal Residual Method
49:linear equation systems
37:Minimal Residual Method
2214:
2190:
2154:
2118:
2021:
2001:
1977:
1946:
1809:
1735:
1661:
1541:
1515:
1481:
1432:
1392:is calculated through
1386:
1359:
1332:
1299:
1219:
1139:
1012:
967:
927:
807:arbitrary and compute
801:
756:
729:
686:
663:
630:
604:
513:
483:
434:
392:
276:
256:
205:
166:
125:
45:Krylov subspace method
32:
29:boundary-value problem
2215:
2191:
2155:
2119:
2022:
2002:
1978:
1947:
1810:
1736:
1662:
1542:
1516:
1514:{\displaystyle l=1,2}
1482:
1433:
1387:
1385:{\displaystyle p_{k}}
1360:
1358:{\displaystyle x_{k}}
1333:
1300:
1220:
1140:
1013:
968:
928:
802:
757:
755:{\displaystyle s_{k}}
730:
728:{\displaystyle p_{k}}
687:
664:
631:
605:
514:
512:{\displaystyle x_{k}}
484:
435:
393:
277:
257:
206:
167:
126:
124:{\displaystyle Ax=b,}
57:Michael Alan Saunders
22:
2238:A, b, x0, maxit, tol
2204:
2164:
2128:
2031:
2011:
1991:
1958:
1831:
1744:
1670:
1551:
1525:
1493:
1442:
1396:
1369:
1342:
1309:
1228:
1148:
1023:
982:
939:
935:Then we iterate for
811:
769:
739:
712:
676:
644:
614:
523:
496:
444:
402:
289:
266:
222:
180:
135:
103:
16:Computational method
2027:is normal, we have
1540:{\displaystyle l=2}
398:is minimized. Here
2776:Nifa, M. Naoufal.
2210:
2186:
2150:
2114:
2017:
1997:
1973:
1942:
1805:
1731:
1657:
1537:
1511:
1477:
1428:
1382:
1355:
1328:
1295:
1215:
1135:
1008:
963:
923:
921:
797:
752:
725:
706:Conjugate Residual
682:
659:
626:
600:
509:
479:
430:
388:
272:
252:
201:
162:
121:
33:
2805:978-3-642-19685-0
2213:{\displaystyle A}
2174:
2138:
2109:
2092:
2063:
2020:{\displaystyle A}
2000:{\displaystyle A}
1911:
1902:
1878:
1655:
1133:
765:First you choose
685:{\displaystyle A}
275:{\displaystyle k}
72:positive definite
2859:
2824:
2823:
2817:
2809:
2791:
2785:
2784:
2782:
2773:
2767:
2766:
2742:
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:
2509:
2506:
2503:
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:
2247:
2244:
2241:
2237:
2234:
2231:
2220:, respectively.
2219:
2217:
2216:
2211:
2195:
2193:
2192:
2187:
2176:
2175:
2172:
2159:
2157:
2156:
2151:
2140:
2139:
2136:
2123:
2121:
2120:
2115:
2110:
2108:
2104:
2094:
2093:
2090:
2079:
2075:
2065:
2064:
2061:
2050:
2026:
2024:
2023:
2018:
2006:
2004:
2003:
1998:
1985:condition number
1982:
1980:
1979:
1974:
1951:
1949:
1948:
1943:
1935:
1934:
1922:
1921:
1916:
1912:
1910:
1903:
1889:
1886:
1879:
1865:
1862:
1846:
1845:
1814:
1812:
1811:
1806:
1804:
1803:
1788:
1787:
1769:
1768:
1756:
1755:
1740:
1738:
1737:
1732:
1730:
1729:
1714:
1713:
1695:
1694:
1682:
1681:
1666:
1664:
1663:
1658:
1656:
1654:
1650:
1649:
1631:
1630:
1611:
1607:
1606:
1588:
1587:
1574:
1569:
1568:
1546:
1544:
1543:
1538:
1520:
1518:
1517:
1512:
1486:
1484:
1483:
1478:
1476:
1475:
1454:
1453:
1437:
1435:
1434:
1429:
1427:
1426:
1408:
1407:
1391:
1389:
1388:
1383:
1381:
1380:
1364:
1362:
1361:
1356:
1354:
1353:
1337:
1335:
1334:
1329:
1324:
1323:
1304:
1302:
1301:
1296:
1294:
1293:
1278:
1277:
1259:
1258:
1240:
1239:
1224:
1222:
1221:
1216:
1214:
1213:
1198:
1197:
1179:
1178:
1160:
1159:
1144:
1142:
1141:
1136:
1134:
1132:
1128:
1127:
1109:
1108:
1089:
1085:
1084:
1066:
1065:
1046:
1041:
1040:
1017:
1015:
1014:
1009:
1007:
1006:
994:
993:
972:
970:
969:
964:
932:
930:
929:
924:
922:
918:
917:
898:
897:
884:
883:
867:
866:
853:
852:
827:
826:
806:
804:
803:
798:
796:
795:
790:
781:
780:
761:
759:
758:
753:
751:
750:
734:
732:
731:
726:
724:
723:
700:MINRES algorithm
692:, unlike in the
691:
689:
688:
683:
668:
666:
665:
660:
658:
657:
652:
636:is the standard
635:
633:
632:
627:
609:
607:
606:
601:
578:
577:
576:
575:
559:
535:
534:
518:
516:
515:
510:
508:
507:
488:
486:
485:
480:
475:
474:
456:
455:
439:
437:
436:
431:
429:
428:
423:
414:
413:
397:
395:
394:
389:
384:
383:
374:
373:
352:
351:
336:
335:
314:
313:
301:
300:
281:
279:
278:
273:
261:
259:
258:
253:
210:
208:
207:
202:
200:
199:
194:
174:symmetric matrix
171:
169:
168:
163:
161:
160:
149:
130:
128:
127:
122:
82:GMRES vs. MINRES
2867:
2866:
2862:
2861:
2860:
2858:
2857:
2856:
2842:
2841:
2832:
2827:
2811:
2810:
2806:
2792:
2788:
2780:
2774:
2770:
2763:10.1137/0712047
2743:
2736:
2732:
2727:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2235:
2232:
2229:
2226:
2205:
2202:
2201:
2171:
2167:
2165:
2162:
2161:
2135:
2131:
2129:
2126:
2125:
2089:
2085:
2084:
2080:
2060:
2056:
2055:
2051:
2049:
2032:
2029:
2028:
2012:
2009:
2008:
1992:
1989:
1988:
1959:
1956:
1955:
1930:
1926:
1917:
1888:
1887:
1864:
1863:
1861:
1857:
1856:
1841:
1837:
1832:
1829:
1828:
1822:
1817:
1793:
1789:
1777:
1773:
1764:
1760:
1751:
1747:
1745:
1742:
1741:
1719:
1715:
1703:
1699:
1690:
1686:
1677:
1673:
1671:
1668:
1667:
1639:
1635:
1620:
1616:
1612:
1596:
1592:
1583:
1579:
1575:
1573:
1558:
1554:
1552:
1549:
1548:
1526:
1523:
1522:
1494:
1491:
1490:
1465:
1461:
1449:
1445:
1443:
1440:
1439:
1416:
1412:
1403:
1399:
1397:
1394:
1393:
1376:
1372:
1370:
1367:
1366:
1349:
1345:
1343:
1340:
1339:
1319:
1315:
1310:
1307:
1306:
1283:
1279:
1267:
1263:
1248:
1244:
1235:
1231:
1229:
1226:
1225:
1203:
1199:
1187:
1183:
1168:
1164:
1155:
1151:
1149:
1146:
1145:
1117:
1113:
1098:
1094:
1090:
1074:
1070:
1055:
1051:
1047:
1045:
1030:
1026:
1024:
1021:
1020:
1002:
998:
989:
985:
983:
980:
979:
940:
937:
936:
920:
919:
913:
909:
899:
893:
889:
886:
885:
879:
875:
868:
862:
858:
855:
854:
848:
844:
828:
822:
818:
814:
812:
809:
808:
791:
786:
785:
776:
772:
770:
767:
766:
746:
742:
740:
737:
736:
719:
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2830:External links
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1953:
1827:
1823:
1019:
934:
764:
703:
694:GMRES method
671:
491:
217:
98:
85:
61:
40:
36:
34:
2198:eigenvalues
74:, only the
2730:References
2007:. Because
1987:of matrix
1521:(the step
2814:cite book
2169:λ
2133:λ
2087:λ
2058:λ
2035:κ
1962:κ
1937:‖
1924:‖
1891:κ
1881:−
1867:κ
1851:≤
1848:‖
1835:‖
1798:−
1775:β
1771:−
1758:←
1724:−
1701:β
1697:−
1684:←
1652:⟩
1644:−
1625:−
1614:⟨
1609:⟩
1601:−
1577:⟨
1556:β
1470:−
1456:←
1421:−
1410:←
1326:‖
1313:‖
1288:−
1272:−
1265:α
1261:−
1253:−
1208:−
1192:−
1185:α
1173:−
1130:⟩
1122:−
1103:−
1092:⟨
1087:⟩
1079:−
1060:−
1049:⟨
1035:−
1028:α
961:…
839:−
783:∈
624:‖
621:⋅
618:‖
595:‖
580:‖
565:∈
416:∈
368:−
354:…
322:
244:−
187:∈
155:×
142:∈
64:CG method
59:in 1975.
25:CG method
2846:Category
2233:= minres
2230:function
1018:through
978:Compute
519:through
76:symmetry
1983:is the
2802:
2661:'*
2646:'*
2568:'*
2553:'*
2489:'*
2423:'*
2408:'*
2124:where
1954:where
610:where
213:vector
131:where
68:matrix
41:MINRES
2781:(PDF)
2706:beta2
2682:beta2
2637:beta2
2613:beta1
2589:beta1
2544:beta1
2508:break
2468:alpha
2444:alpha
2399:alpha
2348:maxit
262:in a
172:is a
88:GMRES
43:is a
2820:link
2800:ISBN
2631:>
2628:iter
2495:<
2336:iter
2160:and
1489:for
319:span
176:and
86:The
55:and
35:The
2759:doi
2724:end
2721:end
2718:end
2511:end
2498:tol
2333:for
2200:of
2173:min
2137:max
2091:min
2062:max
1305:if
640:on
70:is
39:or
2848::
2816:}}
2812:{{
2755:12
2753:.
2749:.
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669:.
537::=
489:.
458::=
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215:.
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2808:.
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2258:=
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2184:)
2181:A
2178:(
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2106:|
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2041:A
2038:(
2015:A
1995:A
1971:)
1968:A
1965:(
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1932:0
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139:A
119:,
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113:=
110:x
107:A
31:.
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