北航数值分析上机作业 二 报告+所有程序大全Word文档格式.docx

北航数值分析上机作业 二 报告+所有程序大全Word文档格式.docx

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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

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A

1）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-001-4.15407560380e-001

0.00000000000e+0001.72827459997e+000-1.17146764279e+000-1.24383926270e+000-6.39975834174e-001-2.00283307904e+0002.92494720612e-001-6.41283006840e-0019.78399762128e-0022.55776357416e-001

0.00000000000e+0000.00000000000e+000-1.29166953413e+000-1.11160351340e+0001.17134682410e+000-1.30735603002e+0001.80369917775e-001-4.24638535837e-0017.98895523930e-0021.60881992807e-001

0.00000000000e+0000.00000000000e+0000.00000000000e+0001.56012629853e+0008.12504939751e-0014.42175683292e-001-3.58861612814e-0024.69174231367e-001-2.73659505009e-001-7.35933465775e-002

0.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.70777375519e-001-1.58305142574e+000-3.04284317680e-0012.52871244603e-001-6.70992540145e-0012.54461992908e-001

0.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.46345345694e-001-2.70836515702e-002-9.48652189368e-0011.19587108150e-0011.92926561795e-002

0.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.70180137436e-001-4.69762399062e-0014.98825946801e-0011.13769160378e-001

0.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0007.01316709211e-0011.58218068848e-0013.86259461423e-001

0.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0004.84380760278e-0013.99277799518e-001

2）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

-2.3482.3721.8190.3240.2212.1030.1821.279-0.638-0.415

0.0001.728-1.171-1.244-0.640-2.0030.292-0.6410.0980.256

0.0000.000-1.292-1.1121.171-1.3070.180-0.4250.0800.161

0.0000.0000.0001.5600.8130.442-0.0360.469-0.274-0.074

0.0000.0000.0000.000-0.771-1.583-0.3040.253-0.6710.254

0.0000.0000.0000.0000.000-0.746-0.027-0.9490.1200.019

0.0000.0000.0000.0000.0000.000-0.770-0.4700.4990.114

0.0000.0000.0000.0000.0000.0000.0000.7010.1580.386

0.0000.0000.0000.0000.0000.0000.0000.0000.4840.399

3.A（n-1）进行QR分解得到Q，R，以及RQ

1）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+000

2.40254545121e-001-4.48034907538e+000-5.13890963843e+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

-3.15116782680e-0015.87640573131e+0001.44050714811e+0016.19555121015e-0012.30309341878e+0003.78744079911e-002-7.93763324139e-0019.93339662512e-0010.00000000000e+0000.00000000000e+000

-9.18177790891e-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-0011.14520098274e+0001.88328484197e-002-2.73784712766e+0002.07736354786e+0002.51193105972e-001-7.22159133673e-001

-8.85680693195e-002