數(shù)值分析實驗報告7

1、實驗七、QR算法一、實驗?zāi)康?、熟悉matlab編程并學習QR算法原理及計算機實現(xiàn)；2、學習用matlab內(nèi)置函數(shù)eig和QR算法求矩陣的特征值，并比對二者差異。二、實驗題目1、課本第277頁第1題已知矩陣（1）用MATLAB函數(shù)“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(shù)“qr”實現(xiàn)矩陣的QR分解）。2、用QR算法求矩陣特征值： 根據(jù)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(jié)果（要求誤差10 -5）.三、實驗原理與理論基礎(chǔ)QR方法是一種變換方法，是計算一般矩陣（中小型矩陣）全部特征值問題的最有效方法之一。目前QR方法主要用來計算

2、上海森伯格矩陣和對稱三對角矩陣的全部特征值問題，且QR方法具有收斂快、算法穩(wěn)定等特點。對于一般矩陣（或?qū)ΨQ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?qū)ΨQ三對角矩陣），然后再用QR方法計算B的全部特征值。1、矩陣的QR分解設(shè)非奇異，則存在正交矩陣P，使PA=R，其中R為上三角矩陣。用Householder變換構(gòu)造正交矩陣P，記，它的第一列記為，不妨設(shè)，可按公式（3.2）（Th14,約化定理 設(shè)則存在初等反射矩陣H使，其中） 找到矩陣使于是其中一般地，設(shè)，其中為(j-1)階方陣，其對角線以下元素均為0，為(n-j+1)階方陣，設(shè)其第一列為，可選擇(n-j+1)的Householder

3、矩陣變換，使根據(jù)構(gòu)造n*n階的變換矩陣為于是有它和有類似的形式，只是為j階方陣，其對角線以下元素是0，這樣經(jīng)過n-1步運算得到其中為上三角矩陣，為正交矩陣，從而有PA=R。2、QR算法設(shè)，且對A進行QR分解，即，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(jīng)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(fù)這一過程可得到矩陣序列：設(shè)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，按上述遞推法則構(gòu)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。四、實驗內(nèi)容1、用matlab內(nèi)置函數(shù)eig求

4、矩陣的全部特征值；2、編寫求特征值的QR算法程序，并用之求矩陣特征值；3、比較兩種方法的結(jié)果差異。（1）QR算法的m文件function qrsf(A,r)Q,R=qr(A);t=A(1,1) %tempA=R*Q;for k=1:50 Q,R=qr(A); t=A(1,1); A=R*Q; if( abs( A(1,1)-t )r ) break; endendn=size(A,1);for i=1:n format long g disp( 特征值,num2str(i),=,num2str( A(i,i) ) );end%disp();for i=1:n disp(特征值); format

5、 long g,A(i,i)endformat long g,A,Q,R（2）改進后的QR算法的m文件function qrsf(A,r)Q,R=qr(A);%t=A(1,1) %tempt(1)=max(abs(diag(R);A=R*Q;for k=2:50 Q,R=qr(A); z=diag(A); t(k)=max(abs(diag(R); A=R*Q; if( abs( t(k) - t(k-1) ) A=10 7 8 7;7 5 6 5;8 6 10 9;7 5 9 10; 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; H

6、6=hilb(6);1、eig求矩陣特征值 eig(A),eig(B),eig(H6)ans = 0.0101500483978924 0.843107149855032 3.85805745594495 30.2886853458021ans = 13.1723513981032 6.55187835191566 1.59565457314994 -0.390788045416488 -0.929096277752298ans = 1.08279948406811e-007 1.25707571226224e-005 0.000615748354182652 0.01632152131987

7、58 0.24236087057521 1.618899858924342、QR算法求矩陣特征值 eig(A),qrsf(A,10-8)ans = 0.0101500483978924 0.843107149855032 3.85805745594495 30.2886853458021t = 10特征值1=30.2887特征值2=3.8581特征值3=0.84311特征值4=0.01015特征值ans = 30.2886853457915特征值ans = 3.85805737835431特征值ans = 0.843107227456257特征值ans = 0.0101500483978911

8、 eig(B),qrsf(B,10-8)ans = 13.1723513981032 6.55187835191566 1.59565457314994 -0.390788045416488 -0.929096277752298t = 2特征值1=13.1724特征值2=6.5519特征值3=1.5957特征值4=-0.9291特征值5=-0.39079特征值ans = 13.1723513891479特征值ans = 6.55187836087093特征值ans = 1.59565457937031特征值ans = -0.929096283974607特征值ans = -0.39078804

9、5414554 eig(H6),qrsf(H6,10-8)ans = 1.08279948406811e-007 1.25707571226224e-005 0.000615748354182652 0.0163215213198758 0.24236087057521 1.61889985892434t = 1特征值1=1.6189特征值2=0.24236特征值3=0.016322特征值4=0.00061575特征值5=1.2571e-005特征值6=1.0828e-007特征值ans = 1.6188998588068特征值ans = 0.24236087069274特征值ans = 0.

10、01632152131988特征值ans = 0.000615748354182639特征值ans = 1.25707571226506e-005特征值ans = 1.08279948456401e-0073、改進后的QR算法求特征值 eig(A),qrsf(A,10-8)ans = 0.0101500483978924 0.843107149855032 3.85805745594495 30.2886853458021特征值1=30.2887特征值2=3.8581特征值3=0.84311特征值4=0.01015特征值ans = 30.2886853458019特征值ans = 3.8580

11、5745223919特征值ans = 0.843107153560957特征值ans = 0.0101500483978911 eig(B),qrsf(B,10-8)ans = 13.1723513981032 6.55187835191566 1.59565457314994 -0.390788045416488 -0.929096277752298特征值1=13.1724特征值2=6.5519特征值3=1.5957特征值4=-0.9291特征值5=-0.39079特征值ans = 13.1723513936489特征值ans = 6.55187835636998特征值ans = 1.595

12、65456952802特征值ans = -0.929096274131193特征值ans = -0.390788045415675 eig(H6),qrsf(H6,10-8)ans = 1.08279948406811e-007 1.25707571226224e-005 0.000615748354182652 0.0163215213198758 0.24236087057521 1.61889985892434特征值1=1.6189特征值2=0.24236特征值3=0.016322特征值4=0.00061575特征值5=1.2571e-005特征值6=1.0828e-007特征值ans

13、= 1.61889985892171特征值ans = 0.242360870577844特征值ans = 0.0163215213198758特征值ans = 0.000615748354182638特征值ans = 1.25707571226506e-005特征值ans = 1.08279948456401e-007六、實驗結(jié)果分析與小結(jié) 從實驗結(jié)果可以看出，用MATLAB內(nèi)置函數(shù)eig求矩陣特征值與用QR算法求矩陣特征值的結(jié)果基本一致，數(shù)據(jù)只有微小差別。且單就QR算法而言，精度不同，計算出來特征值也存在一定的差異。不過這些微小差別對于計算來說影響不是很明顯，除了有特殊需要要求更精確的數(shù)值外

14、，這些結(jié)果已經(jīng)能夠滿足計算結(jié)果的要求。七、附錄3、eig與QR算法求矩陣特征值的結(jié)果比較 eig(A),qrsf(A,10-8)ans = 0.0101500483978924 0.843107149855032 3.85805745594495 30.2886853458021t = 10特征值1=30.2887特征值2=3.8581特征值3=0.84311特征值4=0.01015特征值ans = 30.2886853457915特征值ans = 3.85805737835431特征值ans = 0.843107227456257特征值ans = 0.0101500483978911A =

15、Columns 1 through 3 30.2886853457915 1.67263719509357e-005 2.4658715813052e-009 1.67263719456478e-005 3.85805737835431 0.000483698079436215 2.46587341708519e-009 0.000483698079436138 0.843107227456257 -5.08101348050736e-023 5.50511453327879e-018 -1.40098262801473e-013 Column 4 -1.07002288191827e-015

16、 8.90438591778815e-016 -1.38231041042597e-013 0.0101500483978911Q = Columns 1 through 3 -0.999999999990602 4.33543882155126e-006 4.37465097621799e-010 -4.33543785708671e-006 -0.999999835419719 0.000573708757424641 -2.92474424482639e-009 -0.000573708757417352 -0.999999835429117 -5.00590074187535e-021

17、 5.42373229907221e-016 -1.38027186974381e-011 Column 4 -1.31911901611722e-021 8.46111373361065e-015 -1.38027161147482e-011 1R = Columns 1 through 3 -30.2886853454344 -0.000148041080872945 -1.00648595392028e-007 0 -3.85805646596483 -0.00269709930446301 0 0 -0.84310736620718 0 0 0 Column 4 -1.07019946

18、786611e-015 -3.69340289178865e-015 -1.17754026710534e-011 0.0101500483978911 eig(B),qrsf(B,10-8)ans = 13.1723513981032 6.55187835191566 1.59565457314994 -0.390788045416488 -0.929096277752298t = 2特征值1=13.1724特征值2=6.5519特征值3=1.5957特征值4=-0.9291特征值5=-0.39079特征值ans = 13.1723513891479特征值ans = 6.5518783608

19、7093特征值ans = 1.59565457937031特征值ans = -0.929096283974607特征值ans = -0.390788045414554A = Columns 1 through 3 13.1723513891479 -11.2224332964075 1.38327278837818 -5.28299810482253e-009 6.55187836087093 -1.45097262375031 0 -4.57824183023911e-018 1.59565457937031 0 0 4.55829734055894e-008 0 0 0 Columns 4

20、 through 5 12.2876570415913 2.18439169839953 -5.46614907884632 -0.424112905513863 -0.344534349380365 2.00394783492873 -0.929096283974607 -0.294914932293193 3.5295461400519e-012 -0.390788045414554Q = Columns 1 through 3 -1 -8.06333361799512e-010 -2.31352650689045e-027 8.06333361799512e-010 -1 -2.8691

21、9358232593e-018 0 2.86919358232594e-018 -0.999999999999999 0 0 4.90616249270291e-008 0 0 0 Columns 4 through 5 -1.13505369739799e-034 -1.02516554521335e-045 -1.40767299379114e-025 -1.27139170196985e-036 -4.90616249270291e-008 -4.43118132496026e-019 -0.999999999999999 -9.03186825049292e-012 -9.031868

22、25049293e-012 1R = Columns 1 through 3 -13.1723513800989 11.2224333070288 -1.3832733912306 0 -6.55187836087093 1.45097289192847 0 0 -1.5956545624669 0 0 0 0 0 0 Columns 4 through 5 -12.2876569737454 2.18439169828855 5.46614900766307 -0.424112905464493 0.344534427647672 2.00394783493184 0.92909628397

23、7272 -0.294914932284802 0 -0.390788045414554 eig(H6),qrsf(H6,10-8)ans = 1.08279948406811e-007 1.25707571226224e-005 0.000615748354182652 0.0163215213198758 0.24236087057521 1.61889985892434t = 1特征值1=1.6189特征值2=0.24236特征值3=0.016322特征值4=0.00061575特征值5=1.2571e-005特征值6=1.0828e-007特征值ans = 1.618899858806

24、8特征值ans = 0.24236087069274特征值ans = 0.01632152131988特征值ans = 0.000615748354182639特征值ans = 1.25707571226506e-005特征值ans = 1.08279948456401e-007A = Columns 1 through 3 1.6188998588068 1.27197394498537e-005 1.19768529656603e-012 1.2719739449838e-005 0.24236087069274 3.08869246409526e-008 1.19788968013046

25、e-012 3.08869246092437e-008 0.01632152131988 -3.6564854489204e-021 -1.05134059527256e-016 -8.3901223728756e-011 -2.51874460802941e-031 -7.62887310800373e-027 -7.24698049380747e-021 7.33769428846597e-044 2.28755693138363e-039 2.38462685027147e-033 Columns 4 through 6 -4.65299860612185e-017 -7.8563087

26、5258594e-017 -4.99030380731949e-017 -4.69661140762161e-017 5.64100067650242e-017 -2.24039311154526e-017 -8.39012342417141e-011 -1.90672523361959e-017 9.70102034419473e-018 0.000615748354182639 8.55921938191856e-014 -5.5822503075125e-018 8.560823435926e-014 1.25707571226506e-005 -1.79372765726032e-01

27、7 -3.50196992607807e-026 -8.60412731528914e-018 1.08279948456401e-007Q = Columns 1 through 3 -0.999999998622786 5.24826445620704e-005 -2.59251450345072e-011 -5.24826445620255e-005 -0.999999998620995 1.89240475959173e-006 -7.33932613664668e-011 -1.8924047583461e-006 -0.9999999999982 5.93827888305781e

28、-018 1.70741925354901e-013 1.36258949226309e-007 2.00365386384803e-026 6.06874592641492e-022 5.76495148470373e-016 6.77659566066424e-037 2.11263208377376e-032 2.20227926247271e-026 Columns 4 through 6 -5.09824115576308e-019 3.4008272077694e-028 1.97571398705825e-039 8.71151586217316e-014 -1.09176212

29、19594e-022 -1.05044049606577e-033 -1.36258949226388e-007 3.51443236235137e-016 5.88051420890885e-027 -0.999999999999991 6.81010964765256e-009 2.17725918000353e-019 -6.81010964765255e-009 -1 -7.9461871176948e-011 -3.23418137522306e-019 -7.9461871176948e-011 1R = Columns 1 through 3 -1.61889985590967

30、-9.76838853035816e-005 -1.44084921219927e-010 0 -0.242360871026031 -4.89531790510286e-007 0 0 -0.0163215213199091 0 0 0 0 0 0 0 0 0 Columns 4 through 6 5.84784525152272e-017 7.8563087887546e-017 -4.99030380669521e-017 4.56367477791708e-014 -5.64098415545824e-017 -2.2403931119935e-017 2.3078545790686

31、3e-009 2.90479067967698e-017 9.70102034609643e-018 -0.000615748354182645 -4.2789060011646e-012 -5.58245625301427e-018 0 -1.25707571226506e-005 -1.01683315964937e-015 0 0 1.08279948456401e-007 4、eig與改進后的QR算法求矩陣特征值的結(jié)果比較 eig(A),qrsf(A,10-8)ans = 0.0101500483978924 0.843107149855032 3.85805745594495 30.

32、2886853458021特征值1=30.2887特征值2=3.8581特征值3=0.84311特征值4=0.01015特征值ans = 30.2886853458019特征值ans = 3.85805745223919特征值ans = 0.843107153560957特征值ans = 0.0101500483978911A = Columns 1 through 3 30.2886853458019 2.13054162245146e-006 6.86375089207106e-011 2.13054161716379e-006 3.85805745223919 0.00010570327

33、782691 6.86393453636055e-011 0.000105703277826833 0.843107153560957 1.70269961038915e-026 -1.44832419609733e-020 1.68662327113459e-015 Column 4 1.07002234228498e-015 -8.85182644520922e-016 -1.80487525890045e-016 0.0101500483978911Q = Columns 1 through 3 -0.999999999999848 5.52231697493189e-007 1.217

34、71530170591e-011 -5.52231691626367e-007 -0.999999992140593 0.000125373478981475 -8.14123620399204e-011 -0.000125373478981449 -0.999999992140745 1.6775285630588e-024 -1.42691358634137e-018 1.66168988069558e-013 Column 4 4.42050280607066e-025 -2.22600977082374e-017 1.66168986584696e-013 1R = Columns 1

35、 through 3 -30.2886853457962 -1.88569135427523e-005 -2.80162434078845e-009 0 -3.85805740866594 -0.000589401361893868 0 0 -0.843107160187151 0 0 0 Column 4 1.07002240146044e-015 -8.73123152404271e-016 1.3991777486471e-013 0.0101500483978911 eig(B),qrsf(B,10-8)ans = 13.1723513981032 6.55187835191566 1

36、.59565457314994 -0.390788045416488 -0.929096277752298特征值1=13.1724特征值2=6.5519特征值3=1.5957特征值4=-0.9291特征值5=-0.39079特征值ans = 13.1723513936489特征值ans = 6.55187835636998特征值ans = 1.59565456952802特征值ans = -0.929096274131193特征值ans = -0.390788045415675A = Columns 1 through 3 13.1723513936489 -11.2224332937522

37、1.38327313998092 -2.62774351496931e-009 6.55187835636998 -1.4509727793467 0 -1.11499208998337e-018 1.59565456952802 0 0 -2.6541440298262e-008 0 0 0 Columns 4 through 5 12.2876570042594 -2.18439169861631 -5.46614903246668 0.424112904658541 -0.344534421512392 -2.0039478265026 -0.929096274131193 0.2949

38-1.48456567492261e-012 -0.390788045415675Q = Columns 1 through 3 -1 -4.01067201196511e-010 -2.80252858398934e-028 4.01067201196511e-010 -1 -6.98767831333128e-019 0 6.98767831333129e-019 -0.999999999999999 0 0 -2.85669429931243e-008 0 0 0 Columns 4 through 5 8.00596742954247e-036 -3.04138

39、895236799e-047 1.99616608031226e-026 -7.58324027318754e-038 2.85669429931243e-008 -1.08523030585424e-019 -1 3.79890248009891e-012 3.79890248009891e-012 1R = Columns 1 through 3 -13.1723513891479 11.2224332990352 -1.38327278896012 0 -6.55187835636998 1.45097262319553 0 0 -1.59565457937031 0 0 0 0 0 0

40、 Columns 4 through 5 -12.2876570437836 -2.18439169856963 5.46614907391814 0.424112904637775 0.344534375921806 -2.00394782650391 0.929096274132314 0.294914989538372 0 -0.390788045415675 eig(H6),qrsf(H6,10-8)ans = 1.08279948406811e-007 1.25707571226224e-005 0.000615748354182652 0.0163215213198758 0.24

41、236087057521 1.61889985892434特征值1=1.6189特征值2=0.24236特征值3=0.016322特征值4=0.00061575特征值5=1.2571e-005特征值6=1.0828e-007特征值ans = 1.61889985892171特征值ans = 0.242360870577844特征值ans = 0.0163215213198758特征值ans = 0.000615748354182638特征值ans = 1.25707571226506e-005特征值ans = 1.08279948456401e-007A = Columns 1 through 3 1.61889985892171 1.90423583635592e-006 1.18725724292547e-014 1.90423583634025e-006 0.242360870577844 2.080045367086e-009 1.207695574630