大作业报告 2.docx
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大作业报告 2.docx
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大作业报告2
1、算法设计方案
(1)QR分解法
记A1=A,
令Q1=I。
对于r=1,2...n-1执行
(a)若a(r)ir(i=r+1,r+2....n)全为0,则令
Qr+1=Qr,Ar+1=Ar
转(e);否则转(b)
(b)计算
dr=
cr=-sgn(a(r)rr)dr
hr=cr2-cra(r)rr
(c)令ur=(0,0...0,a(r)rr-cr,....anr(r))
(d)计算
wr=Qrur
Qr+1=Qr-wrurT/hr
pr=ArTur/hr
Ar+1=Ar-urprT
(e)继续
(2)拟上三角化算法
记
A,并记
。
对于r=1,2,
,n-2执行
(a)若
全为零,则令
,转(e);否则转(b)。
(b)计算
,
(若
,则取
),
(c)令
(d)计算
,
,
,
,
(e)继续
(3)求特征值、特征向量算法
(a)使用矩阵的拟上三角化的算法把矩阵A
化为拟上三角矩阵
;给定精度水平
和迭代最大次数L。
(b)记
,令k=1,m=n。
(c)如果
,则得到A的一个特征值
,置m=n-1,转(d);否则转(e)。
(d)如果m=1,则得到A的一个特征值
,转(k);如果m=0,则直接转(k);如果m>1,则转(c)。
(e)求二阶子阵
的两个特征值
和
,即计算二次方程
的两个根
和
。
(f)如果m=2,则得到A的两个特征值
和
,转(k);否则转(g)。
(g)如果
,则得到A的两个特征值
和
,置m=m-2,转(d);否则转(h)。
(h)如果k=L,则计算终止,未得到A的全部特征值,否则转(i)。
(i)记
,计算
(
为m阶单位矩阵)
(对
作QR分解)
(j)置k=k+1,转(3)。
(k)A的全部特征值以计算完毕,停止计算。
(4)计算特征向量
计算迭代之后得到的分开上三角阵的特征向量,再通过原矩阵到迭代矩阵的变换矩阵,计算得到原矩阵的特征向量
2、作业结果
(1)矩阵A:
1-4列
-8.827516758830e-001+7.833269096275e-001+8.912073600614e-001+9.635581854172e-001
+9.320390859672e-001-4.609993049676e-001+9.995736030415e-001+9.738476308782e-001
+9.916648104525e-001+9.463000876874e-001+1.425348887938e+000+7.457052121767e-001
+8.084964038196e-001+6.754631805512e-001+5.155013718215e-001-1.216639521092e+000
+4.273798802338e-001+2.392493292140e-001+4.158066243329e-002-1.577456941432e-001
-5.837414342758e-002-2.555411020268e-001-4.425204432949e-001-6.118578909427e-001
-5.298361409085e-001-6.877661591840e-001-8.182771110644e-001-9.161659367495e-001
-8.715757724136e-001-9.516020738895e-001-9.936910036335e-001-9.961646088358e-001
-9.999232575641e-001-9.824526126243e-001-9.258146823277e-001-8.322674422239e-001
-8.834546557202e-001-7.727644875560e-001-6.312666378723e-001-4.646021794138e-001
4-8列
+9.974949866041e-001+9.916648104525e-001+9.463000876874e-001+8.632093666489e-001
+9.092974268257e-001+8.084964038196e-001+6.754631805512e-001+5.155013718215e-001
+5.984721441040e-001+4.273798802338e-001+2.392493292140e-001+4.158066243329e-002
+1.411200080599e-001-5.837414342758e-002-2.555411020268e-001-4.425204432949e-001
+6.638546982076e-003-5.298361409085e-001-6.877661591840e-001-8.182771110644e-001
-7.568024953079e-001+1.208825936461e+000-9.516020738895e-001-9.936910036335e-001
-9.775301176651e-001-9.999232575641e-001-1.429429375331e+000-9.258146823277e-001
-9.589242746631e-001-8.834546557202e-001-7.727644875560e-001+4.736156323789e-001
-7.055403255704e-001-5.506855425976e-001-3.738766648302e-001-1.821625042721e-001
-2.794154981989e-001-8.308940281750e-002+1.165492048505e-001+3.115413635134e-001
8-10列
+7.457052121767e-001+5.984721441040e-001
+3.349881501559e-001+1.411200080599e-001
-1.577456941432e-001-3.507832276896e-001
-6.118578909427e-001-7.568024953079e-001
-9.161659367495e-001-9.775301176651e-001
-9.961646088358e-001-9.589242746631e-001
-8.322674422239e-001-7.055403255704e-001
-4.646021794138e-001-2.794154981989e-001
+8.719827177217e-001+2.151199880878e-001
+4.941133511386e-001-1.499941239592e+000
(2)拟上三角阵:
1-4列
-8.827516758830e-001-9.933136491826e-002-1.103349285994e+000-7.600443585637e-001
-2.347878362416e+000+2.372370104937e+000+1.819290822208e+000+3.237804101546e-001
+0.000000000000e+000+1.728274599968e+000-1.171467642785e+000-1.243839262699e+000
+0.000000000000e+000+0.000000000000e+000-1.291669534130e+000-1.111603513396e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+1.560126298527e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
4-8列
+1.549101079914e-001-1.946591862872e+000-8.782436382927e-002-9.255889387184e-001
+2.205798440320e-001+2.102692662546e+000+1.816138086098e-001+1.278839089990e+000
-6.399758341743e-001-2.002833079037e+000+2.924947206124e-001-6.412830068395e-001
+1.171346824096e+000-1.307356030021e+000+1.803699177750e-001-4.246385358369e-001
+8.125049397515e-001+4.421756832923e-001-3.588616128137e-002+4.691742313671e-001
-7.707773755194e-001-1.583051425742e+000-3.042843176799e-001+2.528712446035e-001
+0.000000000000e+000-7.463453456938e-001-2.708365157020e-002-9.486521893682e-001
+0.000000000000e+000+0.000000000000e+000-7.701801374364e-001-4.697623990618e-001
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+7.013167092107e-001
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
8-10列
+6.032599440534e-001+1.518860956469e-001
-6.380578124405e-001-4.154075603804e-001
+9.783997621283e-002+2.557763574160e-001
+7.988955239303e-002+1.608819928069e-001
-2.736595050092e-001-7.359334657750e-002
-6.709925401449e-001+2.544619929082e-001
+1.195871081495e-001+1.929265617952e-002
+4.988259468008e-001+1.137691603776e-001
+1.582180688475e-001+3.862594614233e-001
+4.843807602783e-001+3.992777995177e-001
(3)拟上三角QR分解得到的Q:
1-4列
-3.519262579534e-001+4.427591982245e-001-6.955982513607e-001+6.486200753651e-002
-9.360277287361e-001-1.664679186545e-001+2.615299548560e-001-2.438671728934e-002
+0.000000000000e+000-8.810520554692e-001-3.989762796959e-001+3.720308728479e-002
+0.000000000000e+000+0.000000000000e+000-5.371806806439e-001-1.234945854205e-001
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+9.892235468621e-001
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
4-8列
+3.709718861896e-001+1.855847143605e-001-1.628942319628e-002-1.181053169648e-001
-1.394774360893e-001-6.977585391241e-002+6.124472142962e-003+4.440505443139e-002
+2.127794064090e-001+1.064463557221e-001-9.343171079756e-003-6.774200464527e-002
-7.063151608719e-001-3.533456368496e-001+3.101438948263e-002+2.248676491598e-001
-1.239411731211e-001-6.200358589825e-002+5.442272839460e-003+3.945881637235e-002
+5.323610690264e-001-6.733900344896e-001+5.910581205867e-002+4.285425323867e-001
+0.000000000000e+000-6.059761505746e-001-9.165783032817e-002-6.645586508974e-001
+0.000000000000e+000+0.000000000000e+000+9.933396625117e-001-9.690440311938e-002
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+5.410088006061e-001
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
8-10列
-5.255375383724e-002-5.486582943568e-002
+1.975907909728e-002+2.062836970533e-002
-3.014340698675e-002-3.146955080444e-002
+1.000601783527e-001+1.044622748702e-001
+1.755813350011e-002+1.833059462906e-002
+1.906901343193e-001+1.990794495295e-001
-2.957110877580e-001-3.087207462557e-001
-4.311990584470e-002-4.501694411183e-002
-5.817540838226e-001-6.073480580545e-001
-7.221591336735e-001+6.917269588876e-001
(4)拟上三角QR分解得到的R:
1-4列
+2.508342744917e+000-2.185646885493e+000-1.314609070786e+000-3.558787493835e-002
+0.000000000000e+000-1.961603277854e+000+2.407523727633e-001+7.054714572823e-001
+0.000000000000e+000+0.000000000000e+000+2.404534601993e+000+1.706758096328e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+1.577122080722e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
4-8列
-2.609857850388e-001-1.283121847090e+000-1.390878610606e-001-8.712897972161e-001
+5.957204318279e-001+5.526978774676e-001-3.268209924413e-001-5.769498668364e-002
-4.239566704091e-001+3.405332305815e+000-1.050017655853e-001+1.462257102734e+000
+6.399535133956e-001+3.468127872427e-001-5.701786649768e-002+4.014788054433e-001
-1.447846997770e+000-1.415724007744e+000-2.806139044665e-001-2.817910521892e-001
+0.000000000000e+000+1.231641451542e+000+1.619701003419e-001+1.962638275504e-001
+0.000000000000e+000+0.000000000000e+000-7.753441914209e-001-3.464514508821e-001
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+1.296312940612e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000+0.000000000000e+000
8-10列
+3.849367902971e-001+3.353802899665e-001
+2.871129330189e-001-8.895128754188e-002
-6.684487469283e-001-4.027646209664e-001
-2.222476176311e-001-6.317059236442e-002
-4.611434881852e-002+1.996629079956e-001
+5.350035621760e-001-1.509273424767e-001
+4.312226803504e-001+1.234643696237e-001
-4.288053318337e-001+2.737334158165e-001
-6.707396440648e-001-4.842320121884e-001
+0.000000000000e+000+7.168323926323e-002
(5)R、Q的乘积:
1-4列
+1.163074414164e+000+2.632670934508e+000-1.772796003272e+000-8.668899138521e-002
+1.836115060851e+000+1.144286420080e-001-9.880381403133e-001+5.589725694767e-001
+0.000000000000e+000-2.118520153533e+000-1.876189745783e+000-5.407071940597e-001
+0.000000000000e+000+0.000000000000e+000
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