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by1102130 刘晚春 电子信息工程学院 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数值分析上机作业2qr分解求特征值题目：双步位移qr分解法求a的全部特征值，对每一个实特征值求特征向量。已知1. 算法设计方案总体算法与老师的讲义和课本一致，在这里对自己的设计和改进做说明。算法的关键在于对于被运算的矩阵a的左下角理论为0的元素不再计算。1) a拟上三角化对于，记，则，这样去掉了系数，便于计算。设对a到的拟上三角化进行到到第k步，后半段不为0的部分是，则，只需要计算矩阵的右上角，右下角，和左下角即可，计算过程中按照行向量左乘矩阵以及列向量右乘矩阵的运算方法进行，左下角矩阵只有最右列有非零元素，可以简化运算。2) qr分解对于，记，则，这样去掉了系数，便于计算。设对a到的qr分解进行到到第k步，。后半段不为0的部分是，则，这样只需要计算矩阵的右上角和右下角矩阵。计算矩阵的右上角和右下角矩阵时，计算过程中按照行向量左乘矩阵以及列向量右乘矩阵的运算方法进行。3) 求特向量的方法对进行选主元的高斯消去法，化成上三角阵，进行带入求解，对于第k行，如果出现，并且的情况，直接令为1。特向量完成求解后再进行归一化。2. 拟上三角的a1) 12位有效数字的形式-8.82751675883e-001-9.93313649183e-002-1.10334928599e+000-7.60044358564e-0011.54910107991e-001-1.94659186287e+000-8.78243638293e-002-9.25588938718e-0016.03259944053e-0011.51886095647e-001-2.34787836242e+0002.37237010494e+0001.81929082221e+0003.23780410155e-0012.20579844032e-0012.10269266255e+0001.81613808610e-0011.27883908999e+000-6.38057812440e-001-4-0010.00000000000e+0001.72827459997e+000-1+000-1.24383926270e+000-6.39975834174e-001-2.00283307904e+0002.92494720612e-001-6.41283006840e-0019.78399762128e-0022.55776357416e-0010.00000000000e+0000.00000000000e+000-1.29166953413e+000-1.11160351340e+0001+000-1.30735603002e+0001.80369917775e-001-4.24638535837e-0017.98895523930e-0021.60881992807e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0001.56012629853e+0008.12504939751e-0014.42175683292e-001-3.58861612814e-0024.69174231367e-001-2.73659505009e-001-7.35933465775e-0020.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.70777375519e-001-1.58305142574e+000-3.04284317680e-0012.52871244603e-001-6.70992540145e-0012.54461992908e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.46345345694e-001-2.70836515702e-002-9.48652189368e-0011.19587108150e-0011.92926561795e-0020.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.70180137436e-001-4.69762399062e-0014.98825946801e-0011-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0007.01316709211e-0011.58218068848e-0013.86259461423e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0004.84380760278e-0013.99277799518e-0012) 3位小数位的形式-0.883-0.099-1.103-0.7600.155-1.947-0.088-0.9260.6030.152-2.3482.3721.8190.3240.2212.1030.1821.279-0.638-0.4150.0001.728-1.171-1.244-0.640-2.0030.292-0.6410.0980.2560.0000.000-1.292-1.1121.171-1.3070.180-0.4250.0800.1610.0000.0000.0001.5600.8130.442-0.0360.469-0.274-0.0740.0000.0000.0000.000-0.771-1.583-0.3040.253-0.6710.2540.0000.0000.0000.0000.000-0.746-0.027-0.9490.1200.0190.0000.0000.0000.0000.0000.000-0.770-0.4700.4990.1140.0000.0000.0000.0000.0000.0000.0000.7010.1580.3860.0000.0000.0000.0000.0000.0000.0000.0000.4840.3993. a(n-1)进行qr分解得到q，r，以及rq1) 12位有效数字的e型输出：拟上三角a,qr之后的q：-3.51926257953e-001-9.36027728736e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.39595535350e-001-3.31120858752e+000-8.81052055469e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0002.40254545121e-001-4.48034907538e+000-5+000-5.37180680644e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-1.43125202505e-0012.66904781503e+0006.54274505882e+0001.19520089248e+0009.89223546862e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-1.96077644897e-0013.65652310377e+0008.96338324661e+0003.85510755662e-0012.70771861608e+0005.32361069026e-001-0.00000000000e+000-0.00000000000e+000-0.00000000000e+000-0.00000000000e+000-5.63148539010e-0011.05017869060e+0012.57434557752e+0011.10721351706e+0004.11588263550e+0009.26346777492e-001-6.05976150575e-0010.00000000000e+0000.00000000000e+0000.00000000000e+000-3-0015.87640573131e+0001.44050714811e+0016.19555121015e-0012.30309341878e+0003.78744079911e-002-7.93763324139e-0019.93339662512e-0010.00000000000e+0000.00000000000e+000-9-0021.71224940381e+0004.19730634392e+0001.80524105226e-0016.71068424056e-0011.10357309328e-002-1.60433215213e+0002.40658785801e+0005.41008800606e-0010.00000000000e+000-1.56690148242e-0012.92201157088e+0007.16284536362e+0003.08070496692e-0011+0001.88328484197e-002-2.73784712766e+0002.07736354786e+0002.51193105972e-001-7.22159133673e-001-8.85680693195e-0021.65164770259e+0004.04875093812e+0001.74134809446e-0016.47317276562e-0011.06451429335e-002-1.54754996986e+0001+000-2.51782156703e-0016.91726958888e-001拟上三角a,qr之后的r：2.50834274492e+000-2+000-1.31460907079e+000-3.55878749384e-002-2.60985785039e-001-1.28312184709e+000-1.39087861061e-001-8.71289797216e-0013.84936790297e-0013.35380289967e-0010.00000000000e+000-1.96160327785e+0002.40752372763e-0017.05471457282e-0015.95720431828e-0015.52697877468e-001-3.26820992441e-001-5.76949866836e-0022.87112933019e-001-8.89512875419e-0020.00000000000e+0000.00000000000e+0002.40453460199e+0001.70675809633e+000-4.23956670409e-0013.40533230581e+000-1.05001765585e-0011.46225710273e+000-6.68448746928e-001-4.02764620966e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0001.57712208072e+0006.39953513396e-0013.46812787243e-001-5.70178664977e-0024.01478805443e-001-2.22247617631e-001-6.31705923644e-0020.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-1.44784699777e+000-1.41572400774e+000-2.80613904467e-001-2.81791052189e-001-4.61143488185e-0021.99662907996e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0001.23164145154e+0001.61970100342e-0011.96263827550e-0015.35003562176e-001-1.50927342477e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.75344191421e-001-3.46451450882e-0014.31222680350e-0011.23464369624e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0001.29631294061e+000-4.28805331834e-0012.73733415816e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-6.70739644065e-001-4.84232012188e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0007.16832392632e-002拟上三角a,qr之后的rq：1.16307441416e+0006.54021266522e+0001.71698760912e+001-1.50944630535e+001-2.16705228879e+001-6.04262699693e+001-3.39980110592e+001-1+001-1.81045579997e+001-1.00161647810e+0011.83611506085e+0006.28316224606e+0007+000-2.22795075154e+000-2.83544695057e+000-1.04594830181e+001-6.02700244190e+000-1.27420738622e+000-2+000-1.44657481541e+0000.00000000000e+000-2.11852015353e+000-1.32735235180e+0011.73527877647e+0012.28756497487e+0016.52639670964e+0013.63833511946e+0011.34796269561e+0012.07757966695e+0011+0010.00000000000e+0000.00000000000e+000-8.47199512781e-0012.51803480278e+0002.52544119293e+0004.73600481425e+0002.90818562795e+0001.65542017861e+0002.20518743642e+0001.26449870642e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-1.43224434245e+000-4.67403861531e+000-7.10057435575e+000-3.44531980370e+000-1.24013323626e+000-1.65761440713e+000-6.99183495559e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0006.55677959800e-0011.04277707175e+0001-0015-0012.30840029271e-001-2.46194560930e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0004.69840088488e-0012.71295815438e-0016.43439025273e-0011.42222749450e+0007.69904500551e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0001.28767905894e+0002.88770352479e+0002.38752121986e+0001.81946570709e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-3.62876050354e-0011.81207395928e-001-1.66076063019e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-5-0024.95852290988e-0022) 3位小数形式输出拟上三角a,qr之后的q：-0.352-0.9360.0000.0000.0000.0000.0