1、矩阵与数值分析实验报告姓名学号院系机械工程学院班级003班老师2014年12月20日第1页，共29页1方程在X30附近有根，试写出其三种不同的等价形式以构成两种不同的迭代格式，再用这两种迭代求根，并绘制误差下降曲线，观察这两种迭代是否收敛及收敛的快慢。解三种不同的等价形式32242195XXX5421923XXX19425223XXX两种不同的迭代方法方法一简单迭代法选择19425223XXX作为简单迭代法K1KXX的迭代格式，K1K|XX时，迭代终止，这里取1E5，简单迭代法的程序如下程序11简单迭代法第一题简单迭代法程序CLCCLEARDEL1E5X3E1E10N0WHILEEDELNN1
2、TSQRT2X319X42/5EABSTXENEXTENDX,NPLOTE,G,LINEWIDTH,3TITLE误差下降曲线图XLABEL迭代次数YLABEL迭,代,误,差,ROTATION,0计算结果11X20000N1202468101200050101502025误差下降曲线图迭代次数迭代误差第2页，共29页结果分析11这里的迭代结果是收敛的的，但是我也试过第3个迭代方程，发现是发散的，由计算结果可知，迭代法的收敛性与构造的迭代函数是直接相关的，我们应该根据定理45来构造迭代函数，这样可以得到收敛的结果。方法2NEWTON迭代法1042195223XXXXF2191062XXXF3191
3、064219522231XXXXXXXFXFXXKKKK当K1K|XX时，迭代终止，X03,1E5程序12NEWTON迭代法第一题NEWTON迭代法程序CLCCLEARNEWTON迭代法DEL1E5X3E1E10条件判断值，足够大即可N0WHILEEDELNN1TX2X35X219X42/6X210X19EABSTXENEXTENDXNPLOTE,G,LINEWIDTH,3TITLE误差下降曲线图XLABEL迭代次数YLABEL迭,代,误,差,ROTATION,0计算结果12X35000N6115225335445555600204060811214误差下降曲线图迭代次数迭代误差第3页，共29
4、页结果分析12NEWTON迭代法是二阶收敛，弦截法是在前者基础上变化来的，收敛阶1618，在该题中使用简单迭代法的迭代步数为12，NEWTON迭代法为6，很明显，NEWTON迭代法收敛比较快。2用复化梯形公式、复化辛普森公式、龙贝格公式求下列定积分，要求绝对误差为，并将计算结果与精确解进行比较（1）（2）解程序和结果如下程序211使用复化梯形公式解第一个等式FUNCTIONSTIXING1A,B,NFORMATLONGA1B2N150000EROR100HBA/2NINDEX1AH2HBHINDEX2A2H2HB2HS1SUMSUBSFUN2,INDEX1S2SUMSUBSFUN2,INDEX
5、2SHSUBSFUN2,ASUBSFUN2,B2S2CEXP4ERRORABSCS计算结果211S54598150039042402C54598150033144236ERROR5898165511553088E009程序212使用复化SIMPSON公式解第一个等式FUNCTIONSSIMPONA,B,NFORMATLONGA1B2N370EROR100HBA/2NINDEX1AH2HBHINDEX2A2H2HB2HS1SUMSUBSFUN2,INDEX1S2SUMSUBSFUN2,INDEX2SHSUBSFUN2,ASUBSFUN2,B4S12S2/3CEXP4ERRORABSCS第4页，共
6、29页计算结果212S54598150034153207C54598150033144236ERROR1008970684779342E009程序213使用复化ROMBERG公式解第一个等式FUNCTIONTROMBERGF,A,B,ETZEROS10,4T1,1BA/2FAFBFORK24SUM0FORI12K2SUMSUMFA2I1BA/2K1ENDTK,105TK1,1BA/2K1SUMFORI2KTK,I4I1TK,I1TK1,I1/4I11ENDENDFORK515SUM0FORI12K2SUMSUMFA2I1BA/2K1ENDTK,105TK1,1BA/2K1SUMFORI24TK
7、,I4I1TK,I1TK1,I1/4I11ENDIFK6IFABSTK,4TK1,415DISP溢出END计算结果213第5页，共29页程序221使用复化梯形公式解第2个等式FUNCTIONSTIXING2F,A,B,TERROR05108N2S0WHILEABSSTERRORHBA/NS10FORK1N1XAHKS1S1FEVALF,XENDNN1SHFEVALF,AFEVALF,B2S1/2ENDNN1END计算结果221在MATLAB命令框内输入TIXING2X2X/X23,2,3,LOG6得到积分（2）的梯形公式计算量TIXING2X2X/X23,2,3,LOG6N14908ANS17
8、918程序222使用复化SIMPSON公式解第2个等式FUNCTIONSFUHUASIMPSONF,A,B,TEP05108N2S0WHILEABSSTEPHBA/2NS10S20FORK1NXAH2K1第6页，共29页S1S1FEVALF,XENDFORK1N1XAH2KS2S2FEVALF,XENDNN1SHFEVALF,AFEVALF,B4S12S2/3ENDNN1END计算结果222N95ANS1791759474175664程序223使用复化ROMBERG公式解第2个等式FUNCTIONTROMBERGF,A,B,ETZEROS15,4T1,1BA/2FAFBFORK24SUM0FO
9、RI12K2SUMSUMFA2I1BA/2K1ENDTK,105TK1,1BA/2K1SUMFORI2KTK,I4I1TK,I1TK1,I1/4I11ENDENDFORK515SUM0FORI12K2SUMSUMFA2I1BA/2K1ENDTK,105TK1,1BA/2K1SUMFORI24TK,I4I1TK,I1TK1,I1/4I11ENDIFK6IFABSTK,4TK1,415DISP溢出END第7页，共29页计算结果223结果分析2从运行结果可看出，给予充分多的迭代次数，三种迭代方法均能得出足够精确的结果；但迭代速度方面，龙贝格公式最快（远高于另两种），复化梯形公式速度最慢（慢很多），这
10、与预想结果是相符的。3使用带选主元的分解法求解线性方程组,其中，当时对于的情况分别求解精确解为对得到的结果与精确解的差异进行解释解程序及结果如下程序3列组元消去法GAUSS列组元消去法CLCN3N7,N11,下次运行时，替换掉3就可以，程序不变B1,1NFORI1NFORJ1NAI,JIJ1ENDIFI1BI,1IN1/I1ENDENDABNLENGTHAXZEROSN,1AABFORK1N1MAXKFORIK1NIFAI,KAMAX,KMAXIENDENDTEMPAK,KN1AK,KN1AMAX,KN1第8页，共29页AMAX,KN1TEMPFORIK1NAI,KAI,K/AK,KAI,K1
11、N1AI,K1N1AI,KAK,K1N1ENDENDXN,1AN,N1/AN,NFORIN111SUM0FORJI1NSUMSUMXJ,1AI,JENDXI,1AI,N1SUM/AI,IENDX计算结果31列组元消去法N3时计算结果32列组元消去法N7时第9页，共29页计算结果33列组元消去法N11时结果分析3在A方阵阶数较小时即3阶和7阶时结果与精确解相符；但当阶数达到11阶时，结果有了一些误差，经计算机计算，A条件数是11645E014，远大于3阶的709231和7阶的24459E007，可见，方阵的稳定性对运行结果的影响，在稳定性差的情况下，即使是较好的计算方法也会影响结果的精确性。4用
12、4阶RUNGEKUTTA法求解微分方程TTTTEETUUUUU222101,1010,21令10H，使用上述程序执行20步，然后令050H，使用上述程序执行40步2比较两个近似解与精确解3当H减半时，1中的最终全局误差是否和预期相符4在同一坐标系上画出两个近似解与精确解（提示输出矩阵R包含近似解的X和Y坐标，用命令PLOTR,1,R,2画出相应图形）解程序及结果如下第10页，共29页程序4FUNCTIONFRUNGEKX,TUOXTNTH01,计算20步，并将计算结果储存在U1中U1ZEROS20,1H101I1WHILEIRUNGEK01,0U10163744036772803第11页，共2
13、9页0201092763722473021952092245022802246606912768670220724075006086021083270630627401972747464109720181704477615480016529692258991601488672366782470132962563748465011793241554476601039823423374110091214639845090007965902873249500692956044337170060071851516817005191512031959600447416518870090038462992
14、616638U20163746031471215020109583086042702195244446711900224664269349821022072746367585902108357672042110197277413772783018170673356637701652987795370120148868724949069013296372307314001179332893298460103982973946124009121507030136800796592954192980069295740471532006007188559469800519150768155920044
15、7415511626650038462851405422U0163746150615596020109601381069202195246544376110224664482058611022072766470286502108359483385410197277571153285018170686619519001652988882215870148868811560274013296379003480101179333392762360103983009500067009121509393782700796593093885820069295746763223006007188592858
16、700519150726498560044741543712331第12页，共29页0038462841666342R110E005021138427938050003250088218470500373198738237712037907817433124903589696779798810324203226748088028247423131955202388579709966530196563167076391015748820267613701226286335753720092373147053682006671626564952500454092736731910028065608
17、73425100142329505195550003441176943397000476697403772900108174677562740015095029667900R210E00601191443817494250182950264993043020976642051140402127087897207950201027005930499018113433014610401573805020194110132628812654634010868457450685700866112054576100066961660771803004994639000999200355539431090
18、570023636458845733001396928478447600062916901549800000333889339788000416573571471200074503333155620009739080236715R310E0050199469841205557030671379534774603522220961865720357807295359169033886697738683103060897937334770266736181117611第13页，共29页022559508973119001856947096257060148827082130376011593246
19、74981920087378508052682006316087133861900430456277886180026668680255804001360378150405700034077880094190004350400466258001007243442471800141211216442280246810121416182000050101502025结果分析4近似解与精确解的差距随着T值得增大也同时增大，当H减半时，误差并未改变，可见减小H，增加步数并不能提高精度。5设为阶的三对角方阵，是一个阶的对称正定矩阵其中为阶单位矩阵。设为线性方程组的真解，右边的向量由这个真解给出。1用CH
20、OLESKY分解法求解该方程2用JACOBI和GAUSSSEIDEL迭代法解该方程组，误差设为其中取值为4，5，6解程序及结果如下第14页，共29页程序51CHOLESKY分解方程组CHOLESKY分解法解方程组程序CLCFORMATLONGN4FORI1NFORJ1NIFIJTI,J2CONTINUEENDIFIJ1|JI1TI,J1CONTINUEENDTI,J0ENDENDTAN2,N20FORI1NFORJ1NIFIJFORK1NANI1K,NJ1K4IFNI1K1AZEROS16,1JACOBI5A,4JACOBI迭代法收敛X09999999948557700999999991676
21、46109999999916764610999999994855770099999999167646109999999865322320999999986532232099999999167646109999999916764610999999986532232099999998653223209999999916764610999999994855770099999999167646109999999916764610999999994855770I87计算结果522JACOBI迭代法N5时CLEARAZEROS25,1JACOBI5A,5JACOBI迭代法收敛X09999999947677
22、900999999991030498099999998953558109999999910304980999999994767790099999999103049809999999843033710999999982060996第21页，共29页099999998430337109999999910304980999999989535581099999998206099609999999790711620999999982060996099999998953558109999999910304980999999984303371099999998206099609999999843033710
23、99999999103049809999999947677900999999991030498099999998953558109999999910304980999999994767790I126计算结果523JACOBI迭代法N6时CLEARAZEROS36,1JACOBI5A,6JACOBI迭代法收敛X0999999995214836099999999137743109999999892478330999999989247833099999999137743109999999952148360999999991377432099999998446266809999999806252640
24、999999980625264099999998446266809999999913774310999999989247833099999998062526409999999758401000999999975840100099999998062526409999999892478330999999989247833099999998062526409999999758401000999999975840100099999998062526409999999892478330999999991377431099999998446266809999999806252640999999980625
25、2640999999984462668099999999137743109999999952148360999999991377431099999998924783309999999892478330999999991377431第22页，共29页0999999995214836I172程序531GAUSSSEIDEL迭代法FUNCTIONFGAUSSXO,NFORMATLONG生成TN矩阵TZEROSNFORI1NTI,I2ENDFORI1N1TI,I11ENDFORI2NTI,I11END生成系数矩阵AIEYENBT2ICIAZEROSN2FORI1NANI11IN,NI11INBENDF
26、ORI1N1AIN1I1N,I1N1INCAI1N1IN,IN1I1NCEND生成右边项BYONESN2,1BAY用GAUSSSEIDEL法求解CALZEROSN2UZEROSN2DZEROSN2FORI1N2DI,IAI,IENDFORI2N2LI,1I1CI,1I1ENDFORI1N21UI,I1N2CI,I1N2ENDBINVDLUFINVDLBI1WHILEIAZEROS16,1GAUSSA,4迭代法收敛X0999999995998180099999999476155009999999957620050999999997881002099999999476155009999999931
27、427800999999994452392099999999722619609999999957620050999999994452392099999999551189109999999977559460999999997881002099999999722619609999999977559460999999998877973I46计算结果532GAUSSSEIDEL迭代法N5时CLEARAZEROS25,1GAUSSA,5迭代法收敛X0999999996678773099999999501816009999999950181600999999996263620099999999813181
28、009999999950181600999999992527240099999999252724009999999943954300999999997197715099999999501816009999999925272400999999992527240099999999439543009999999971977150999999996263620099999999439543009999999943954300999999995796572099999999789828609999999981318100999999997197715099999999719771509999999978
29、982860999999998949143第24页，共29页计算结果532GAUSSSEIDEL迭代法N6时CLEARAZEROS36,1GAUSSA,6迭代法收敛X0999999996872701099999999492286309999999942958880999999994860773099999999628680109999999981434000999999994922863099999999175731909999999907394320999999991656517099999999397165909999999969858290999999994295888099999999
30、073943209999999895958470999999990626182099999999322722009999999966136100999999994860773099999999165651709999999906261820999999991554481099999999389793609999999969489680999999996286801099999999397165909999999932272200999999993897936099999999559113109999999977955660999999998143400099999999698582909999
31、99996613610099999999694896809999999977955660999999998897783I91结果分析5三种方法均可得出精确解（对称正定的方阵有较好的性质），在跌代法方面，GAUSSSEIDEL法的迭代速度快于JACOBI法。6设考虑空间的一个等距划分，分点为设为插值于这些等分点上的LAGRANGE插值多项式。1选择不断增大的分点数目画出原函数与插值多项式在的图像，并比较分析实验结果。第25页，共29页（2）选择，重复上述的实验看其结果如何解程序及结果如下程序61LAGRANGE插值法LAGRANGE插值函数FUNCTIONYLAGRANGE11X0,Y0,XML
32、ENGTHX/区间长度/NLENGTHX0FORI1NLI1ENDFORI1MFORJ1NFORK1NIFJKCONTINUEENDLJXIX0K/X0JX0KLJENDENDENDY0FORI1NYY0ILIYENDTEXTMX100011Y125X21PPOLYFITX,Y,NPYVPAPOLY2SYMP,10PLOT_X100011F1POLYVALP,PLOT_XFIGUREPLOTX,Y,R,PLOT_X,F1XLABELXYLABELY,ROTATION,0TITLELAGRANGE插值计算结果611LAGRANGE插值法N6时N6PY45859499191015X54982698
33、138X68666417318X437444553241015X34572379867X2572561311016X0777412545108060402002040608102002040608112XYLAGRANGE插值真实值近似值第26页，共29页计算结果611LAGRANGE插值法N15时N15PY00000000007283856714X152575155632X140000000002666711901X139647387451X120000000003893888888X111459327005X100000000002885657039X91143148595X8000000
34、0001140457452X74952249448X600000000002301690577X51180067335X4203367621011X31478427248X251337841441013X09544785302108060402002040608102002040608112XYLAGRANGE插值程序621LAGRANGE插值法解LAGRANGE插值函数FUNCTIONYLAGRANGE11X0,Y0,XMLENGTHX/区间长度/NLENGTHX0FORI1NLI1ENDFORI1MFORJ1NFORK1NIFJKCONTINUEENDLJXIX0K/X0JX0KLJEND
35、ENDENDY0FORI1NYY0ILIYENDTEXTMX5015YX1X41PPOLYFITX,Y,NPYVPAPOLY2SYMP,10PLOT_X5015F1POLYVALP,PLOT_XFIGUREPLOTX,Y,R,PLOT_X,F1XLABELXYLABELY,ROTATION,0第27页，共29页TITLELAGRANGE插值计算结果621第二问第一个方程N10时N10PY000001753800946X9175021441019X101104745111017X80001077005207X724358788321016X6002316106611X52190593511015
36、X402033800473X368894232481015X206089155935X3395128792101554321012345080604020020406XYLAGRANGE插值真实值近似值计算结果622第二问第一个方程N20时N20PY70612004191021X2000000000001797649761X1982291542651019X18000000002333397096X1740598925031017X160000001300145831X1511044620971015X14000004065752637X1318082665491014X12000078315
37、82068X1118247248011013X100009598399158X911162678511012X8007465055542X739056353521012X603558348923X569099347331012X409566390032X347218619541012X2116411936X5302103039101354321012345080604020020406XYLAGRANGE插值第28页，共29页程序623第二问第二个方程LAGRANGE插值函数FUNCTIONYLAGRANGE11X0,Y0,XMLENGTHX/区间长度/NLENGTHX0FORI1NLI1EN
38、DFORI1MFORJ1NFORK1NIFJKCONTINUEENDLJXIX0K/X0JX0KLJENDENDENDY0FORI1NYY0ILIYENDTEXTMX5015YATANXPPOLYFITX,Y,NPYVPAPOLY2SYMP,10PLOT_X5015F1POLYVALP,PLOT_XFIGUREPLOTX,Y,R,PLOT_X,F1XLABELXYLABELY,ROTATION,0TITLELAGRANGE插值计算结果6231第二问第二个方程N5时N5PY00008996130265X55660064711018X4003928426983X310267658911016X20
39、7160939679X183873274710165432101234515105005115XYLAGRANGE插值真实值近似值第29页，共29页计算结果6232第二问第二个方程N10时N10PY00000057621837X938993660041020X1025136109091018X800003795239723X756896513271017X60009269421815X553074391981016X401082281259X317647640921015X208716535438X913074079210165432101234515105005115XYLAGRANGE插值结果分析6对于第一个小题，可以看出，在N值增大的情况，与原函数值接近的点增多了，但同时插值多项式的振荡度也越来越大，这说明，LAGRANGE插值在多点插值情况下，并不是点越多与原函数越接近；另外两个的情况与第一个有些出入，这说明对于某些函数，在一定的点数范围内，LAGRANGE插值多项式与原函数的拟合度较好。
