2014年1月计算方法期末考试题A

第1页共6页上海海事大学试卷2013—2014学年第一学期期末考试《计算方法》（A卷）班级学号姓名总分1(18’)(a)Findallfixedpointsof2x39.0g(x).(b)Towhichofthefixed-pointsisFixed-PointIterationlocallyconvergent,why?(c)Compute3stepsfortheconvergentfixed-pointstartingwiththeinitialvalue4.0x0,andfindtheconvergentrateS.2(18’)(a)ThezeroofthefunctionAxxf3)(isthecubicrootofanumberA.ShowthattheNewton’sMethodtothisfunctionproducetheiteration21k332xkkxAx(b)Apply3stepsofNewtoniterationmethodtofindthecubicrootof2to9decimalplacesstartingwith10x.(c)Assumetheerroris32iixe,ifafter3stepstheerror6310e,estimate4e.3(20’)(a)FindtheLUfactorizationofthegivenmatrices.Checkbymatrixmultiplication.1-12044310120211-1A(b)UsetheresultofLandUtocomputethesecondcolumnof1-A,thenjustifytheresultisright.(c)Computetheapproximationnumberofoperationsinvolvingthewholesolutionprocess.题目得分阅卷人--------------------------------------------------------------------------------------装订线------------------------------------------------------------------------------------第2页共6页4(14’)(a)Rearrangetheequationstoformastrictlydiagonallydominantsystem.Apply2stepsoftheJacobiandGauss-SeidelMethodsfromstartingvector[0,0,0].2345128wvuwvuwvu(b)Theexactsolutionofthesystemis]4957,74,4961[X,andtheerror2-normbetweentheexactsolutionandtheJacobiseconditerationis3760878601.022YX,estimatethesameonesabouttheGauss-Seideliteration.IsGauss-SeidelmethodfasterthanJacobimethod?5(20’)(a)Findthedegree2interpolatingpolynomial)(2xPthroughthepoints)0,(),1,2(),0,0(.(b)Calculateanapproximationfor)4sin(.(c)Usingtheinterpolatingerrorformulatogiveanerrorboundfortheapproximationinpart(b).(d)Comparetheactualerrortoyourerrorbound.6(10)Let)(xPisanpolynomialanddxxP)(istheintegralof)(xP.WriteouttheprogramwithMATLABtoevaluatedthepolynomialdxxP)(.(assumethattheintegralconstantC=1)第3页共6页参考答案1(18’)(a)since2x39.0g(x),so2x39.0x,3.1,3.0x21x(b)-2x(x)g,convergent,16.03.02(0.3)gdivergent,16.2)3.1(2(-1.3)g(c),27636359.03371.039.0,3371.023.039.0,23.04.039.04.02322210xxsimilarlyxx(d)S=0.62(18’)(a)2123232,3232(x)ff(x)-xg(x)kkkxxxsoxx(b),259933493.1263888889,1,34323213210xxsimilarlyxx(c)1223433107937.07937.0)2(f2)2(f(r)f2(r)fMMee3(20’)(a)1010012100100001L,100021000120211-1U,bychecking,LU=A.第4页共6页(b),1210,,00101010012100100001,Ly432143212yyyythenyyyye,142529,,12101000210001202111,Ux43214321xxxxthenxxxxy(c)Triangularfactorizationportion:623223nnnTheforwardandbacksubstitutionportion:nn22Replacenby4andsumthetworesults,wegetthesummation:624(14’)(a)Therearrangedsystemis4512823wvuwvuwvuJacobi:5482132111kkkkkkkkkvuwwuvwvu548132111wvu2423120494039222wvuGauss-Seidel:5482132111111kkkkkkkkkvuwwuvwvu403924532111wvu8089720361180191222wvu(b)TheerrornormregardingtotheGauss-Seideliteration:2031262636.02YXGauss-SeidelmethodisfasterthanJacobimethod.第5页共6页5(20’)(a)2232023044)2)(02())(0())(())((1)(xxxxxxxxxxxxxP.(b)43)4(P(c)242.0128!3)4)(24)(04()4(32E(d)043.043226(8’)functionhornerIntegral(a,x0)N=length(a);P=a(n)/n;Fork=n-1:-1:1P=p*x0+a(k)/k;EndP=p*x0+1;第6页共6页计算方法复习提纲1（18）（a）给定迭代函数，求出不动点。（b）判断不动点迭代的收敛性。（c）对给定的初始值x0,计算3步迭代并确定收敛常数S。2（18）（a）求出n次方根的牛顿迭代公式。（b）对给定的初始值计算某方根数的3步迭代。（c）给定误差ei，估计误差ei+1。3（20）（a）计算矩阵A的LU分解，并检验分解结果是正确的。（b）用已得到的L和U计算1-A的某一列并验证其正确性。（c）计算整个求解过程的运算总数。4（12）将某个任意的线性方程组调整为对角线严格占优的形式，对给定的初始值用Jacobi和Gauss-Seidel方法各迭代2步。5（20）（a）用Lagrange插值法求出经过3个已知点的2阶多项式。（b）计算某个点的内插值。（c）用插值误差公式求出(b)中近似计算的误差上界。（d）比较实际误差和误差上界。6（12）写一段简单的小程序。