1、實驗七 矩陣特征值的計算 專業班級：信計131班 姓名：段雨博 學號：2013014907一、實驗目的1、熟悉matlab編程。2、學習QR算法原理及其計算機的實現。二、實驗題目1、已知矩陣(1)用MATLAB函數”eig”求矩陣全部特征值。（2）用基本QR算法求全部特征值（可用MATLAB函數“qr”實現矩陣的QR分解）2、用QR算法求矩陣的特征值 根據QR算法原理編制求（i）及（ii）中矩陣全部特征值的程序并輸出計算結果（要求誤差）三、實驗原理與理論基礎1、QR算法：目前QR算法主要用來計算：上海森伯格矩陣的全部特征值的問題；計算對稱三對角矩陣的全部特征值問題，且QR算法具有收斂性，算法穩

2、定等特點。對于一般矩陣 （或對稱矩陣），首先用豪斯霍爾德方法將A化為上海森伯格矩陣B（或對稱三對角矩陣），然后再用QR算法計算B的全部特征值。設，且對A進行QR分解，即A=QR，其中R為上三角矩陣，Q為正交矩陣，于是可得到一個新矩陣顯然，B是由A經過正交相似變換得到，因此B與A特征值相同，再對B進行QR分解，又可得一新的矩陣，重復這一過程可得到矩陣序列：設將進行QR分解作矩陣求得后將進行QR分解形成矩陣QR算法就是利用矩陣的QR分解，按上述地推法則構造矩陣序列的過程，只要A為非奇異矩陣，則由QR算法就完全確定。設，（1）如果A的特征值滿足： （2）A有標準形,其中且設有三角分解(L為單位下三角

3、矩陣，U為上三角矩陣)，則由QR算法產生的本質上收斂于上三角矩陣，即若記（1）（2）當時： 當時，極限不一定存在。如果對稱矩陣A滿足上述條件，則由QR算法產生的本質上收斂于對角矩陣四、實驗內容1、建立M文件：function =qr2(R,e)A=R;n=length(R);for i=1:n for j=1:n if i=j e(i,j)=1; I(i,j)=1; end end endfor i=1:(n-1)p=0; for j=i:i+1 p=p+R(j,i)*R(j,i); end q=sqrt(p); m=e(i:n,i); r=R(i:n,i); u=r+q*m; b=q*(q+

4、R(i,i); H=I(1:n-i+1,1:n-i+1)-(u*u')/b; if i>1 H=eye(i-1,n);zeros(n-i+1,i-1),H; end R=H*R; end R; Q=A*inv(R);A=R*Q z=0;for i=2:n for j=1:i-1 z=z+abs(A(i,j); endendzif z>e qr2(A);end求解過程：>> A=10 7 8 7;7 5 6 5;8 6 10 9;7 5 9 10A = 10 7 8 7 7 5 6 5 8 6 10 9 7 5 9 10>> eig(A)ans = 1

5、.015004839789237e-002 8.431071498550322e-001 3.858057455944950e+000 3.028868534580213e+001>> A=10 7 8 7;7 5 6 5;8 6 10 9;7 5 9 10;qr2(A)A = 3.056067457541445e+001 -8.985059828456483e-001 1.707247856786328e+000 3.705567965683247e-001 5.912156376155273e+000 4.053102159404461e-001 -7.972472470284

6、137e-001 -2.162017849026248e-001 -1.457218079687165e+000 -1.445873127400416e+000 3.332651555305440e+000 1.518103358310533e+000 1.060554876822455e+000 2.561200022419226e-001 -3.240384573153754e-002 7.013636533382514e-001z = 1.016432630803877e+001A = 3.036587661613971e+001 -4.854305363591701e+000 -8.7

7、35113626083455e-001 -5.092449286409494e+000 4.242056952198555e-001 3.535584533914850e+000 -2.002666116375007e+000 6.162314806185953e-001 5.740039408298876e-003 -3.393068368614086e-001 1.077923331289405e+000 -2.645149812037395e-003 8.625915256397850e-004 2.388243554946482e-002 7.066688455885042e-002

8、2.061551865460960e-002z = 8.646644831235181e-001用“eig”函數求得的矩陣特征值為： 1.015004839789237e-002 8.431071498550322e-001 3.858057455944950e+000 3.028868534580213e+001使用QR算法求得的特征值為： 3.036587661613971e+001 3.535584533914850e+000 1.077923331289405e+000 2.061551865460960e-002>> B=2 3 4 5 6;4 4 5 6 7;0 3 6

9、 7 8;0 0 2 8 9;0 0 0 1 0B = 2 3 4 5 6 4 4 5 6 7 0 3 6 7 8 0 0 2 8 9 0 0 0 1 0>> eig(B)ans = 1.317235139810320e+001 6.551878351915660e+000 1.595654573149936e+000 -3.907880454164885e-001 -9.290962777522975e-001>> B=2 3 4 5 6;4 4 5 6 7;0 3 6 7 8;0 0 2 8 9;0 0 0 1 0;qr2(B)A = 6.4000000000000

10、00e+000 6.514285714285715e+000 7.447257388345252e+000 6.450956186380818e+000 6.381818181818151e+000 2.800000000000000e+000 5.477551020408165e+000 7.707334510184234e+000 3.604625968036745e+000 7.633766233766214e+000 -6.415422057959204e-017 1.960140072683525e+000 7.829766052762572e+000 6.0183305685193

11、82e+000 6.551185076625914e+000 2.434357303797321e-016 -4.889059159370327e-016 1.679777895292521e+000 1.119129207820997e+000 9.449011522105670e-001 -1.742734613992503e-016 3.500033710581847e-016 2.692729173383974e-016 -5.916454984824948e-001 -8.264462809917336e-001z = 7.031563466458541e+000A = 9.6721

12、31147540988e+000 9.887091540254833e+000 7.449384462015936e+000 -8.938418294683963e+000 4.519004709455764e+000 1.244282813714488e+000 7.322835004168791e+000 5.932154406664927e+000 -6.054103365858815e+000 4.967937101618412e+000 -1.794041613703499e-016 2.450051419048138e+000 4.566668397697897e+000 -3.5

13、17759761031019e+000 4.917367677922591e-001 -6.973883493088713e-017 -1.144472277275882e-016 4.361831041747620e-001 -1.084281238976259e+000 -5.211495490169227e-002 -5.059111477194544e-017 -8.302422658817633e-017 4.132906177476206e-016 3.462896410405563e-001 -4.773533104314109e-001z = 4.476806977977944

14、e+000A = 1.104257886571281e+001 1.178929415751574e+001 2.169729834717949e+000 1.139125170160095e+001 6.881905101454944e-001 8.271053509292266e-001 8.330991374201149e+000 2.281361091988760e+000 7.498220163551565e+000 1.373611782080155e+000 -2.671537777365972e-017 9.083497128924405e-001 1.899197644406

15、293e+000 1.346118385047426e+000 -1.858659349586504e+000 -4.532640258557575e-018 -2.296776304096971e-017 1.596892484923356e-001 -9.289012384264184e-001 2.874626905070543e-002 1.940545753976591e-018 9.833119882687877e-018 -5.513737038834996e-017 -1.472186014241056e-001 -3.438666458938317e-001z = 2.042

16、362913738108e+000A = 1.196716591643307e+001 1.171510915026601e+001 2.077433375500999e+000 -1.175898184169138e+001 -9.752797465829078e-001 5.588840118252004e-001 7.688998321298111e+000 2.055908235242444e+000 -6.659602302989940e+000 9.106885491960837e-002 -1.272965888574176e-017 1.986607287185679e-001

17、 1.659398616069212e+000 2.224607027254660e-002 -2.011754825279250e+000 -1.234193415015871e-018 -1.314059359840290e-017 9.635192571322808e-002 -9.462778136313050e-001 -3.150911550044786e-001 -1.860963090300626e-019 -1.981387955384117e-018 1.244615177538549e-016 5.568218248401496e-002 -3.6928504016908

18、53e-001z = 9.095788487410115e-001用“eig”函數求得的矩陣特征值為：1.317235139810320e+001 6.551878351915660e+000 1.595654573149936e+000 -3.907880454164885e-001 -9.290962777522975e-001使用QR算法求得的特征值為：1.196716591643307e+001 7.688998321298111e+000 1.659398616069212e+000 -9.462778136313050e-001 -3.692850401690853e-001>

19、;> H6=1 1/2 1/3 1/4 1/5 1/6;1/2 1/3 1/4 1/5 1/6 1/7;1/3 1/4 1/5 1/6 1/7 1/8;1/4 1/5 1/6 1/7 1/8 1/9;1/5 1/6 1/7 1/8 1/9 1/10;1/6 1/7 1/8 1/9 1/10 1/11H6 = 1.000000000000000e+000 5.000000000000000e-001 3.333333333333333e-001 2.500000000000000e-001 2.000000000000000e-001 1.666666666666667e-001 5.00

20、0000000000000e-001 3.333333333333333e-001 2.500000000000000e-001 2.000000000000000e-001 1.666666666666667e-001 1.428571428571429e-001 3.333333333333333e-001 2.500000000000000e-001 2.000000000000000e-001 1.666666666666667e-001 1.428571428571429e-001 1.250000000000000e-001 2.500000000000000e-001 2.000

21、000000000000e-001 1.666666666666667e-001 1.428571428571429e-001 1.250000000000000e-001 1.111111111111111e-001 2.000000000000000e-001 1.666666666666667e-001 1.428571428571429e-001 1.250000000000000e-001 1.111111111111111e-001 1.000000000000000e-001 1.666666666666667e-001 1.428571428571429e-001 1.2500

22、00000000000e-001 1.111111111111111e-001 1.000000000000000e-001 9.090909090909091e-002>> eig(H6)ans = 1.082799484068105e-007 1.257075712262244e-005 6.157483541826516e-004 1.632152131987583e-002 2.423608705752096e-001 1.618899858924340e+000>> H6=1 1/2 1/3 1/4 1/5 1/6;1/2 1/3 1/4 1/5 1/6 1/

23、7;1/3 1/4 1/5 1/6 1/7 1/8;1/4 1/5 1/6 1/7 1/8 1/9;1/5 1/6 1/7 1/8 1/9 1/10;1/6 1/7 1/8 1/9 1/10 1/11;qr2(H6)A = 1.607262860669782e+000 1.021948842423596e-001 6.697063041856768e-003 3.727460607108758e-004 -5.353481023488704e-004 4.891768309901734e-004 1.493164216013435e-001 2.446385295405831e-001 1.3

24、46395600263824e-002 -6.681279862336608e-004 -4.793665675765238e-003 2.889083033165624e-003 8.333519252185923e-002 2.064144469083577e-001 1.573189501088843e-002 1.948498306742740e-003 1.483234431055164e-003 -2.337494211815255e-004 1.324054360170233e-001 3.935438661243592e-001 3.447419572554042e-002 7

25、.011113779915390e-003 1.072769230134642e-002 -4.430806105746643e-003 1.554849083140322e-002 5.684740486621217e-002 5.705231262604105e-003 1.642668194365771e-003 3.144659238821613e-003 -1.468218178669673e-003 1.169833964705411e-003 -6.305476776934365e-004 -3.578391080543362e-004 -3.080142443292248e-0

26、04 -7.987188975651961e-004 4.216199593536946e-004z = 1.082498307945416e+000A = 1.623258642417349e+000 -1.145752079935996e-001 7.290201450095246e-002 -2.887460370934568e-003 1.235200045304732e-003 8.810558953832148e-004 7.564686335741196e-002 -3.534858837245795e-002 5.644735183740964e-001 -9.86297741

27、5097146e-002 1.675863567642972e-002 5.651214271897440e-003 3.638789623275422e-002 -2.406996708580928e-002 2.838018093518541e-001 -4.833931306780879e-002 2.318961260601014e-003 1.212190561953645e-003 -3.731741457152312e-003 5.063232136687457e-003 -3.319310356339430e-002 5.736003105980065e-003 2.10959

28、7215139766e-003 4.267495583295251e-004 -2.947616344770366e-004 8.899207382117006e-004 -3.252227369881533e-003 8.893712249947085e-004 7.588281400470059e-004 1.258516183350374e-004 2.350123006057106e-005 -6.734184070903424e-006 1.690005713738185e-004 -3.823214127162336e-005 3.822054034075403e-006 3.98

29、3556522334020e-006z = 1.836603749815855e-0012、建立M文件function =qrfenjian2(R,e)A=R;n=length(R);for i=1:n for j=1:n if i=j e(i,j)=1; I(i,j)=1; end end endfor i=1:(n-1)p=0; for j=i:i+1 p=p+R(j,i)*R(j,i); end q=sqrt(p); m=e(i:n,i); r=R(i:n,i); u=r+q*m; b=q*(q+R(i,i); H=I(1:n-i+1,1:n-i+1)-(u*u')/b; if

30、i>1 H=eye(i-1,n);zeros(n-i+1,i-1),H; end R=H*R; end R; Q=A*inv(R); A=R*Q z=0;for i=2:n for j=1:i-1 z=z+abs(A(i,j); endendzif z>e qrfenjian2(A);end求解過程對于矩陣（i）>> R=6 2 1;2 3 1;1 1 1;qrfenjian2(R)%z為矩陣的精度 R = -6.48266920334518 -3.00416377715996 -1.42302494707577 0.02812324477671 -2.26712637

31、065064 -0.79608551413779 -0.00565878270107 0 0.61636025860349Q = -0.92399996486358 0.34221613518508 -0.06885858161768 -0.31222221246211 -0.90953612713914 -0.27316578318451 -0.15611110623105 -0.23422455683223 0.95948305607513A = 7.15010277589896 0.84722887430763 -0.09834616764998 0.80613912445676 2.2

32、5812034378243 -0.14646573813085 -0.09099196679049 -0.14630323516621 0.59177688031861z = 1.04343432641347R = -7.19655406385453 -1.09673421634796 0.12161985403843 0.00006499024505 -2.15305248081832 0.17058125588889 0.00000326090119 0 0.58089297645945Q = -0.99354432362905 0.11259603889567 0.00564953335

33、736 -0.11202630682034 -0.99173488755365 0.06254210164886 0.01264483223876 0.06151044443638 0.99802633582992A = 7.27449638630440 0.28484699471173 0.01213058220776 0.24329091849117 2.14575710646787 0.03558852582995 0.00734205438614 0.03573135231655 0.57974650722772z = 0.28636432519386R = -7.2785710106

34、4187 -0.35644725160942 -0.01389817791716 0.00000012385646 -2.13532984838098 -0.04476163520984 -0.00000000168285 0.00000000000000 0.57907075878044Q = -0.99944018840048 0.03343779118985 -0.00045432295230 -0.03342566032230 -0.99930351431684 -0.01658949130190 -0.00100872246897 -0.01656502670614 0.999862

35、28170131A = 7.28642488632161 0.11304987727588 -0.00467616343346 0.07141983846811 2.13458410353691 -0.00933143482345 -0.00058412000360 -0.00959232264021 0.57899101014148z = 0.08159628111192R = -7.28677494504435 -0.13396690686242 0.00481381185816 0.00000000022948 -2.13339505259620 0.01188544484657 0.0

36、0000000000085 0 0.57894308478582Q = -0.99995195971776 0.00980155726577 0.00003613705746 -0.00980129607902 -0.99994187744174 0.00449180735428 0.00008016166409 0.00449123739101 0.99998991112951A = 7.28773832152148 0.06255899842828 0.00394868714985 0.02091098909158 2.13332443457451 0.00230252534904 0.0

37、0004640904025 0.00260017082967 0.57893724390401z = 0.02355756896150R = -7.28776832210966 -0.06867996247290 -0.00395896430731 0.00000000000042 -2.13313773467168 -0.00299675619225 -0.00000000000000 0 0.57893399639881Q = -0.99999588343278 0.00286933244837 -0.00000287108779 -0.00286932681828 -0.99999514

38、079414 -0.00121875006167 -0.00000636807294 -0.00121873680654 0.99999925731975A = 7.28793541199065 0.04777342355549 -0.00385433383590 0.00612068839231 2.13313102157338 -0.00039699222094 -0.00000368669392 -0.00070556816997 0.57893356643597z = 0.00682994325619R = -7.28793798218680 -0.04956489188702 0.0

39、0385495874656 0.00000000000000 -2.13309026403375 0.00058524369513 0.00000000000000 0 0.57893340257865Q = -0.99999964733562 0.00083983827687 0.00000022807649 -0.00083983815549 -0.99999959263433 0.00033076132138 0.00000050586242 0.00033076101319 0.99999994529845A = 7.28797704042811 0.04344545748914 0.

40、00383690217923 0.00179145088890 2.13308958866180 -0.00012030009124 0.00000029286065 0.00019148859880 0.57893337091009z = 0.00198323234835R = -7.28797726060556 -0.04396978894538 -0.00383689575636 0.00000000000000 -2.13307885352647 0.00006927229061 -0.00000000000000 0 0.57893337930721Q = -0.9999999697

41、8895 0.00024580906936 -0.00000001811776 -0.00024580906675 -0.99999996575961 -0.00008977016808 -0.00000004018408 -0.00008977016091 0.99999999597066A = 7.28798784875508 0.04217868097061 -0.00383281652375 0.00052433011950 2.13307877427043 0.00026075913754 -0.00000002326390 -0.00005197094262 0.578933376

42、97449z = 5.763243260201910e-004R = -7.28798786761639 -0.04233214399423 0.00383279960167 0.00000000000000 -2.13307573486205 -0.00024692960443 -0.00000000000000 0.00000000000000 0.57893338315035Q = -0.99999999741200 0.00007194442821 0.00000000143922 -0.00007194442815 -0.99999999711519 0.00002436425814

43、 0.00000000319209 0.00002436425798 0.99999999970319A = 7.28799089432921 0.04180790713552 0.00383175772026 0.00015346291316 2.13307572269228 -0.00029890041220 0.00000000184801 0.00001410528230 0.57893338297852z = 1.675700434622167e-004R = -7.28799089594494 -0.04185282321068 -0.00383175157228 -0.00000

44、000000000 -2.13307484191870 0.00029515281499 0.00000000000000 0.00000000000000 0.57893338494194Q = -0.99999999977830 0.00002105695731 -0.00000000011433 -0.00002105695731 -0.99999999975644 -0.00000661264735 -0.00000000025357 -0.00000661264734 0.99999999997814A = 7.28799177562329 0.04169938562535 -0.0

45、0383147398103 0.00004491606581 2.13307483944743 0.00030925808668 -0.00000000014680 -0.00000382828231 0.57893338492928z = 4.874449491534070e-005R = -7.28799177576170 -0.04171253181528 0.00383147208665 0.00000000000000 -2.13307458241605 -0.00030824267414 -0.00000000000000 -0.00000000000000 0.578933385

46、48335Q = -0.99999999998101 0.00000616302367 0.00000000000908 -0.00000616302367 -0.99999999997940 0.00000179472462 0.00000000002014 0.00000179472462 0.99999999999839A = 7.28799203269869 0.04166762262505 0.00383139715795 0.00001314618913 2.13307458181890 -0.00031207095561 0.00000000001166 0.0000010390

47、2600 0.57893338548242z = 1.418522679238586e-005R = -7.28799203271054 -0.04167147029684 -0.00383139659595 -0.00000000000000 -2.13307450665500 0.00031179586684 -0.00000000000000 0.00000000000000 0.57893338563436Q = -0.99999999999837 0.00000180381497 -0.00000000000072 -0.00000180381497 -0.9999999999982

48、5 -0.00000048710250 -0.00000000000160 -0.00000048710250 0.99999999999988A = 7.28799210786632 0.04165832597392 -0.00383137629241 0.00000384767173 2.13307450649940 0.00031283489277 -0.00000000000093 -0.00000028199990 0.57893338563429z = 4.129672555200082e-006五、實驗結果1、（1）使用“eig”函數求得的矩陣特征值為： 1.0827994840

49、68105e-007 1.257075712262244e-005 6.157483541826516e-004 1.632152131987583e-002 2.423608705752096e-001 1.618899858924340e+000使用QR算法求得的特征值為： 1.623258642417349e+000 -3.534858837245795e-002 2.838018093518541e-001 5.736003105980065e-003 7.588281400470059e-004 3.983556522334020e-0062、該矩陣的特征值為：7.287992107

50、86632，2.13307450649940，0.57893338563429 對于矩陣（ii）： 由于輸出的結果太多，在這里只寫出了最后的一次迭代結果。>> a=2 3 4 5 6;4 4 5 6 7;0 3 6 7 8;0 0 2 8 9;0 0 0 1 0;qrfenjian2(a)R = Columns 1 through 4 -13.17234173641931 -11.22244182587705 -1.38322286385315 -12.28766033582139 0.00000000000000 -6.55188315760281 -1.45095128198600 -5.46616016847126 0.00000000000000 -0.00000000000000 -1.59565596180980 0