Example of unaccelerated QR method: Matrix not tridiagonal a = 1 3 4 3 1 2 4 2 1 iteration = 1 q = -0.19611613513818 0.96836405227008 -0.15430334996209 -0.58834840541455 -0.24209101306752 -0.77151674981046 -0.78446454055274 -0.06052275326688 0.61721339984837 r = -5.09901951359279 -2.74562589193458 -2.74562589193458 0 2.54195563720897 3.32875142967841 0 0 -1.54303349962092 a = 4.76923076923077 -4.10684300668337 1.21045506533761 -4.10684300668337 -0.81684981684982 0.09338863578009 1.21045506533760 0.09338863578009 -0.95238095238095 iteration = 2 q = -0.74413097857936 -0.66743846032826 -0.02819553853306 0.64078029630091 -0.72506953269232 0.25233863087894 -0.18886423322336 0.16970584679374 0.96722811527218 r = -6.40912810582868 2.51497006467234 -0.66102481628175 0 3.34918648568970 -1.03724213560335 0 0 -0.93173350553742 a = 6.50561797752809 2.34199064927492 0.17597113409183 2.34199064927493 -2.60441913503119 -0.15812062354333 0.17597113409184 -0.15812062354333 -0.90119884249690 iteration = 3 q = -0.94058431373593 0.33953765992159 -0.00391487533378 -0.33860575201823 -0.93874191089251 -0.06410748344816 -0.02544196246519 -0.05897289401797 0.99793532070824 r = -6.91657077682730 -1.31694546886715 -0.08904686862815 0 3.24939625097152 0.26132958720756 0 0 -0.88989034570474 a = 6.95380881547013 -1.10691299871421 0.02264055677355 -1.10691299871421 -3.06575580793410 0.05247940904486 0.02264055677355 0.05247940904486 -0.88805300753603 iteration = 4 q = -0.98756141989399 -0.15723291597163 -0.00050206775744 0.15720083794616 -0.98741517232674 0.01729664730455 -0.00321535161345 0.01700257609931 0.99985027575132 r = -7.04139375575913 0.61103645101122 -0.01125373064709 0 3.20210924306362 -0.07047799432722 0 0 -0.88702369373125 a = 7.04990044228574 0.50360086773752 0.00285209306481 0.50360086773753 -3.16300955751059 -0.01508168785456 0.00285209306481 -0.01508168785456 -0.88689088477514 iteration = 5 q = -0.99745825109891 0.07125330389867 -0.00006323153615 -0.07125218928943 -0.99744688817426 -0.00477815790362 -0.00040352963615 -0.00476150764064 0.99998858253923 r = -7.06786518084220 -0.27694339916134 -0.00141235372654 0 3.19088907764887 0.01946934139711 0 0 -0.88680887638947 a = 7.06963383571183 -0.22736568901846 0.00035785366323 -0.22736568901846 -3.18283508442791 0.00422254724072 0.00035785366323 0.00422254724072 -0.88679875128391 iteration = 6 q = -0.99948323825965 -0.03214430548430 -0.00000793354744 0.03214426664948 -0.99948235831746 0.00132722784119 -0.00005059225788 0.00132628696256 0.99999911920127 r = -7.07328904086660 0.12493808184594 -0.00017707290259 0 3.18850162879336 -0.00540801405397 0 0 -0.88679236874949 a = 7.07364988768788 0.10249232017187 0.00004486482821 0.10249232017187 -3.18685830002398 -0.00117614115717 0.00004486482821 -0.00117614115717 -0.88679158766389 iteration = 7 q = -0.99989504642402 0.01448779264460 -0.00000099464325 -0.01448779129071 -0.99989497832263 -0.00036909172282 -0.00000634186314 -0.00036903857514 0.99999993188515 r = -7.07439237046502 -0.05631101785363 -0.00002219652099 0 3.18800893234355 0.00150392793314 0 0 -0.88679109320088 a = 7.07446571010267 -0.04618721758241 0.00000562390775 -0.04618721758241 -3.18767467730543 0.00032726012149 0.00000562390775 0.00032726012149 -0.88679103279724 iteration = 8 q = -0.99997868857765 -0.00652858258046 -0.00000012468077 0.00652858253326 -0.99997868330770 0.00010266602196 -0.00000079494171 0.00010266302001 0.99999999472984 r = -7.07461648024246 0.02537523578699 -0.00000278229600 0 3.18790829728683 -0.00041833050707 0 0 -0.88679099452591 a = 7.07463137442602 0.02081252275984 0.00000070494715 0.02081252275984 -3.18784038457364 -0.00009104064162 0.00000070494715 -0.00009104064162 -0.88679098985238 iteration = 9 q = -0.99999567277970 0.00294183987965 -0.00000001562852 -0.00294183987801 -0.99999567237190 -0.00002855882223 -0.00000009964393 -0.00002855865268 0.99999999959220 r = -7.07466198804700 -0.01143431672235 -0.00000034875377 0 3.18788781849546 0.00011636787734 0 0 -0.88679098689074 a = 7.07466501235497 -0.00937825552265 0.00000008836334 -0.00937825552265 -3.18787402582586 0.00002532555579 0.00000008836334 0.00002532555579 -0.88679098652910 iteration = 10 q = -0.99999912137857 -0.00132561008426 -0.00000000195900 0.00132561008420 -0.99999912134701 0.00000794434597 -0.00000001249010 0.00000794433639 0.99999999996844 r = -7.07467122831274 0.00515236932661 -0.00000004371535 0 3.18788365690210 -0.00003237061658 0 0 -0.88679098629992 a = 7.07467184238770 0.00422589072325 0.00000001107611 0.00422589072325 -3.18788085611576 -0.00000704496590 0.00000001107611 -0.00000704496590 -0.88679098627194 iteration = 11 q = -0.99999982160043 0.00059732664122 -0.00000000024555 -0.00059732664122 -0.99999982159798 -0.00000220992166 -0.00000000156560 -0.00000220992112 0.99999999999756 r = -7.07467310450637 -0.00232168380495 -0.00000000547960 0 3.18788281164407 0.00000900470939 0 0 -0.88679098625421 a = 7.07467322919129 -0.00190420733250 0.00000000138836 -0.00190420733250 -3.18788224293925 0.00000195973813 0.00000000138836 0.00000195973813 -0.88679098625204 iteration = 12 q = -0.99999996377690 -0.00026915833450 -0.00000000003078 0.00026915833450 -0.99999996377671 0.00000061474610 -0.00000000019624 0.00000061474607 0.99999999999981 r = -7.07467348545794 0.00104616218842 -0.00000000068685 0 3.18788263999813 -0.00000250488970 0 0 -0.88679098625067 a = 7.07467351077457 0.00085804518197 0.00000000017403 0.00085804518197 -3.18788252452406 -0.00000054515128 0.00000000017403 -0.00000054515128 -0.88679098625050 iteration = 13 q = -0.99999999264509 0.00012128406694 -0.00000000000386 -0.00012128406694 -0.99999999264507 -0.00000017100733 -0.00000000002460 -0.00000017100733 0.99999999999999 r = -7.07467356280817 -0.00047140581816 -0.00000000008609 0 3.18788260514472 0.00000069679905 0 0 -0.88679098625040 a = 7.07467356794858 -0.00038663936728 0.00000000002182 -0.00038663936728 -3.18788258169820 0.00000015164776 0.00000000002181 0.00000015164776 -0.88679098625038 iteration = 14 q = -0.99999999850662 -0.00005465119528 -0.00000000000048 0.00005465119528 -0.99999999850662 0.00000004757006 -0.00000000000308 0.00000004757006 1.00000000000000 r = -7.07467357851373 0.00021241777320 -0.00000000001079 0 3.18788259806779 -0.00000019383246 0 0 -0.88679098625037 a = 7.07467357955747 0.00017422159440 0.00000000000273 0.00017422159440 -3.18788259330709 -0.00000004218470 0.00000000000273 -0.00000004218470 -0.88679098625037 iteration = 15 q = -0.99999999969678 0.00002462609651 -0.00000000000006 -0.00002462609651 -0.99999999969678 -0.00000001323283 -0.00000000000039 -0.00000001323283 1.00000000000000 r = -7.07467358170266 -0.00009571648994 -0.00000000000135 0 3.18788259663085 0.00000005391945 0 0 -0.88679098625037 a = 7.07467358191459 -0.00007850510449 0.00000000000034 -0.00007850510449 -3.18788259566421 0.00000001173475 0.00000000000034 0.00000001173475 -0.88679098625037 iteration = 16 q = -0.99999999993843 -0.00001109663981 -0.00000000000001 0.00001109663981 -0.99999999993843 0.00000000368105 -0.00000000000005 0.00000000368105 1.00000000000000 r = -7.07467358235016 0.00004313031958 -0.00000000000017 0 3.18788259633909 -0.00000001499907 0 0 -0.88679098625037 a = 7.07467358239319 0.00003537478491 0.00000000000004 0.00003537478491 -3.18788259614282 -0.00000000326432 0.00000000000004 -0.00000000326432 -0.88679098625037 iteration = 17 q = -0.99999999998750 0.00000500020029 0.00000000000000 -0.00000500020029 -0.99999999998750 -0.00000000102398 -0.00000000000001 -0.00000000102398 1.00000000000000 r = -7.07467358248163 -0.00001943473342 -0.00000000000002 0 3.18788259627984 0.00000000417238 0 0 -0.88679098625037 a = 7.07467358249037 -0.00001594005149 0.00000000000001 -0.00001594005149 -3.18788259623999 0.00000000090805 0.00000000000001 0.00000000090805 -0.88679098625037 iteration = 18 q = -0.99999999999746 -0.00000225311476 0.00000000000000 0.00000225311476 -0.99999999999746 0.00000000028485 0.00000000000000 0.00000000028485 1.00000000000000 r = -7.07467358250833 0.00000875738615 0.00000000000000 0 3.18788259626782 -0.00000000116065 0 0 -0.88679098625037 a = 7.07467358251010 0.00000718266534 0.00000000000000 0.00000718266534 -3.18788259625972 -0.00000000025260 0.00000000000000 -0.00000000025260 -0.88679098625037 iteration = 19 q = -0.99999999999948 0.00000101526456 0.00000000000000 -0.00000101526456 -0.99999999999948 -0.00000000007924 0.00000000000000 -0.00000000007924 1.00000000000000 r = -7.07467358251374 -0.00000394612113 0.00000000000000 0 3.18788259626537 0.00000000032287 0 0 -0.88679098625037 a = 7.07467358251411 -0.00000323654421 0.00000000000000 -0.00000323654421 -3.18788259626373 0.00000000007027 0.00000000000000 0.00000000007027 -0.88679098625037 iteration = 20 q = -0.99999999999990 -0.00000045748319 0.00000000000000 0.00000045748319 -0.99999999999990 0.00000000002204 0.00000000000000 0.00000000002204 1.00000000000000 r = -7.07467358251485 0.00000177814153 0.00000000000000 0 3.18788259626488 -0.00000000008981 0 0 -0.88679098625037 a = 7.07467358251492 0.00000145840269 0.00000000000000 0.00000145840269 -3.18788259626454 -0.00000000001955 0.00000000000000 -0.00000000001955 -0.88679098625037