數(shù)值計(jì)算方法復(fù)習(xí)

Review復(fù)習(xí)1PreliminariesTheProgramforSolvingEngineeringProblemEngineeringProblem

AcademicModelPracticalModelResolvableModelComputerProgramResultsNumericalMethodAnalysisMethodComputerLanguageTheoryforEngineeringTheConceptof

Numericalanalysis

Numericalanalysisisthestudyofalgorithmsfortheproblemsofcontinuousmathematics----LloydN.Trefethen“計(jì)算數(shù)學(xué)”就是研究在計(jì)算機(jī)上解決數(shù)學(xué)問題的理論和數(shù)值方法。

數(shù)值計(jì)算方法是一門根據(jù)計(jì)算機(jī)特點(diǎn)，研究通過計(jì)算機(jī)求工程問題滿足精度要求的近似解的學(xué)科。TheWayofNumericalMethodsDispersing

(離散化)

只計(jì)算定義域上有限個(gè)變量的值，而不是所有函數(shù)變量的值

Approach(逼近)

用簡(jiǎn)單函數(shù)y(x)近似替代函數(shù)f(x),但誤差E(x)=f(x)-y(x)要滿足精度要求。Deducebydegrees(遞推)

遞推是將一個(gè)復(fù)雜的計(jì)算過程轉(zhuǎn)化為簡(jiǎn)單過程的多次重復(fù)的數(shù)學(xué)方法。TheAbsoluteError&.RelativeErrorTheAbsoluteError:Ep=|p-p’|

絕對(duì)誤差TheRelativeError:Rp

=|p-p’|/|p|相對(duì)誤差SignificantDigits（有效數(shù)字）|p-p’|/|p|<10-d/2Thenumberp’issaidtoapproximateptodsignificantdigitsifdisthelargestpositiveinteger1、選用數(shù)值穩(wěn)定性好的算法2、相近兩數(shù)避免相減3、絕對(duì)值相對(duì)太小的數(shù)不宜作除數(shù)4、警惕大數(shù)吃小數(shù)5、簡(jiǎn)化計(jì)算方法，盡量減少過程誤差LossofSignificance(精度損失）2TheSolutionofNonlinearEquationsf(x)=0Themethods(1)Findtheareafortheroot(2)Approach2.1IterationforSolvingx=g(x)迭代法迭代函數(shù)迭代序列迭代收斂Theorem2.1

convergencedivergenceTheorem2.2Theorem2.3

P:AttractivefixedPointP:RepellingfixedPoint2.2BracketingMethodsforLocatingaRoot所謂二分法，是使用對(duì)分區(qū)間的方法，保留有根區(qū)間，舍去無根區(qū)間，并且如此不斷地對(duì)分下去，以逐步逼近方程根的方程求解方法y=f(x)xy0x*baf(a)f(b)x1=(a+b)/2f((a+b)/2)=f(x1)xkTheBisectionMethodofBolzanoTheorem2.4BisectionTheoremConvergenceoftheFalsePositionMethody=f(x)xy0x*baf(a)f(b)xk2.3Newton-RaphsonandSecantMethods

(牛頓迭代法和弦割法)是否可以將它轉(zhuǎn)換成線性方程進(jìn)行求解？如何轉(zhuǎn)化？TheGeometricConstructionforNewton-RaphsonCorollary2.2(Newton’sIterationforfindingSquareRoots)(求平方根的牛頓迭代法）牛頓迭代法的收斂性判斷定理3證明=？TheConvergenceofNewton-RaphsonIterationTheDivision-by-ZeroError0Definition2.4(OrderofaRoot)SpeedofConvergenceDefinition2.5(OrderofConvergence)Theorem2.6(ConvergenceRateforNewtonRaphsonIteration)TheSecantMethodsExample2.16(SecantMethodataSimpleRoot)AcceleratedConvergenceComparisonoftheSpeedofConvergence3TheSolutionofLinearSystemsAX=B主要內(nèi)容高斯消去法三角分解法追趕法平方根法

雅可比迭代法高斯—賽德爾迭代法向量與矩陣范數(shù)譜半徑定義迭代法收斂判斷===平方根法的解法=步驟雅可比迭代法高斯—賽德爾迭代法§4向量與矩陣范數(shù)常用向量范數(shù)矩陣范數(shù)常見矩陣范數(shù)舉例譜半徑定義迭代法收斂判斷收斂條件雅可比迭代法高斯—賽德爾迭代法超松弛迭代法收斂收斂收斂收斂A陣為對(duì)角占優(yōu)矩陣收斂收斂A陣為對(duì)稱正定矩陣收斂當(dāng)時(shí)收斂A陣為正定矩陣的充分必要條件是A陣的順序主子式大于零?；贛ATLAB的線性方程組矩陣除法矩陣除法舉例a=[-0.0022210.7812503.9965.56254]b=[0.41.38167.4178]’X=a\bx=1.9273-0.69850.9004矩陣除法舉例a=[-0.0022210.7812503.9965.56254]b=[0.41.38167.4178]’X=a\bx=1.9273-0.69850.9004平方根法a=[42-2;22-3;-2-314]L=chol(a)L=21-101-2003MATLAB實(shí)現(xiàn)：Jacobi.mfunctiony=jacobi(a,b,x0)D=diag(diag(a));U=-triu(a,1);L=-tril(a,-1);B=D\(L+U);f=D\b;y=B*x0+f;n=1;whilenorm(y-x0)>=1.0e-6x0=y;y=B*x0+f;n=n+1;endyn10*x1-x2=9-x1+10*x2-2*x3=7-x2+10*x3=6>>a=[10-10;-110-2;0-210];>>b=[9;7;6];>>jacobi(a,b,[0;0;0])y=0.99580.95790.7916n=11ans=0.99580.95790.7916MATLAB實(shí)現(xiàn)：Jacobi.mfunctiony=jacobi(a,b,x0)D=diag(diag(a));U=-triu(a,1);L=-tril(a,-1);B=D\(L+U);f=D\b;y=B*x0+f;n=1;whilenorm(y-x0)>=1.0e-6x0=y;y=B*x0+f;n=n+1;endyn10*x1-x2=9-x1+10*x2-2*x3=7-x2+10*x3=6>>a=[10-10;-110-2;0-210];>>b=[9;7;6];>>jacobi(a,b,[0;0;0])y=0.99580.95790.7916n=11ans=0.99580.95790.7916seidel.mfunctiony=seidel(a,b,x0)D=diag(diag(a));U=-triu(a,1);L=-tril(a,-1);G=(D-L)\U;f=(D-L)\b;y=G*x0+f;n=1;whilenorm(y-x0)>=1.0e-6x0=y;y=G*x0+f;n=n+1;endyn10*x1-x2=9-x1+10*x2-2*x3=7-x2+10*x3=6>>a=[10-10;-110-2;0-210];>>b=[9;7;6];>>seidel(a,b,[0;0;0])y=0.99580.95790.7916n=7ans=0.99580.95790.79164InterpolationandPolynomialApproximation插值法Interpolation插值概念與基礎(chǔ)理論Introduction插值多項(xiàng)式的求法插值概念與基礎(chǔ)理論概念在工程實(shí)踐和科學(xué)實(shí)驗(yàn)中，常常需要從一組實(shí)驗(yàn)觀測(cè)數(shù)據(jù)揭示自變量x與因變量y之間的關(guān)系，一般可以用一個(gè)近似的函數(shù)關(guān)系式y(tǒng)＝f(x)來表示．如何確定插值多項(xiàng)式？

插值余項(xiàng)對(duì)t求導(dǎo)，k(x)看成常數(shù)4.3LagrangeApproximation當(dāng)n=1時(shí)稱線性插值當(dāng)n=2時(shí)拋物線插值MATALAB實(shí)現(xiàn)Lagrange插值%lagrangeinsertfunctiony=lagrange(x0,y0,x)n=length(x0);m=length(x);fori=1:mz=x(i)s=0.0fork=1:np=1.0;forj=1:nifj~=kp=p*(z-x0(j))/(x0(k)-x0(j));endends=p*y0(k)+s;endy(i)=s;endx=[0.4:0.1:0.8];y=[-0.916291-0.693147-0.510826-0.356675-0.223144];lagrange(x,y,0.54)ans=-0.61612．2差商與牛頓基本括值多項(xiàng)式

前面構(gòu)造的拉格朗日插值多項(xiàng)式，其形式具有對(duì)稱性，既便于記憶，又便于應(yīng)用與編制程序．但是，由于公式中的都依賴于全部插值節(jié)點(diǎn)，在增加或減少節(jié)點(diǎn)時(shí)，必須全部重新計(jì)算．為克服這個(gè)缺點(diǎn)，插值多項(xiàng)式可以如何構(gòu)造？這種形式的插值多項(xiàng)式稱為n次牛頓插值多項(xiàng)式NewtonPolynomials利用MATALAB進(jìn)行插值計(jì)算一維插值分段線性插值分段拋物插值分段低次插值分段低次插值Runge現(xiàn)象產(chǎn)生x=[-5:1:5];y=1./(1+x.^2);x0=[-5:0.1:5];y0=lagrange(x,y,x0);y1=1./(1+x0.^2);plot(x,y)plot(x0,y0,'--r')分段線性插值分段拋物插值3.1分段線性插值與分段拋物插值MATALAB實(shí)現(xiàn)Lagrange插值%lagrangeinsertfunctiony=lagrange(x0,y0,x)n=length(x0);m=length(x);fori=1:mz=x(i)s=0.0fork=1:np=1.0;forj=1:nifj~=kp=p*(z-x0(j))/(x0(k)-x0(j));endends=p*y0(k)+s;endy(i)=s;endx=[0.4:0.1:0.8];y=[-0.916291-0.693147-0.510826-0.356675-0.223144];lagrange(x,y,0.54)ans=-0.6161利用MATALAB進(jìn)行插值計(jì)算一維插值利用MATLAB軟件進(jìn)行插值高維插值氣旋變化情況可視化5CurveFitting曲線擬合的基本概念5.1Least-squaresLine最小二乘擬合曲線實(shí)驗(yàn)數(shù)據(jù)帶有誤差實(shí)驗(yàn)數(shù)據(jù)很多最小二乘法問題提法:在曲線擬合時(shí)，如果所選擇的擬合函數(shù)在所有數(shù)據(jù)點(diǎn)處的偏差的平方和最小，則稱這種擬合方法為最小二乘法。Theorem定理5.1(Least-squaresLine)ThePowerFity=AxMy=AxM5.2CurveFittingDataLinearizationMethodfory=CeAxNonlinearLeast-squaresMethodfory=CeAxInterpolationbySplineFunction分段低次插值Runge現(xiàn)象產(chǎn)生x=[-5:1:5];y=1./(1+x.^2);x0=[-5:0.1:5];y0=lagrange(x,y,x0);y1=1./(1+x0.^2);plot(x,y)plot(x0,y0,'--r')分段線性插值分段拋物插值3.1分段線性插值與分段拋物插值三次樣條插值

三次樣條插值函數(shù)求法邊界條件：三次樣條插值函數(shù)簡(jiǎn)化計(jì)算方法由確定兩個(gè)積分常數(shù)6NumericalDifferentiation6.1導(dǎo)數(shù)的近似值6.1.2中心差分公式6.1.4Richardson外推法6.2數(shù)值差分6.2.1更多的中心差分公式6.2.2誤差分析利用插值多項(xiàng)式構(gòu)造微分公式6.2.3拉格朗日多項(xiàng)式微分6.2.3牛頓多項(xiàng)式微分6.2.4利用三次樣條插值函數(shù)構(gòu)造微分公式RelatedExampleExample6.2(P315);Exercises4(P325);AlgrorithmsandPrograms2(P328);Example6.5(P332);Exercises3,8(P339,340);AlgrorithmsandPrograms1(P341);7NumericalIntegration構(gòu)造數(shù)值積分公式的基本方法與有關(guān)概念7.1IntroductiontoQuadrature數(shù)值積分余項(xiàng)E(Pi)=Ri[f]數(shù)值積分公式的精度例題Thetrapezoidalrulen=1Simpson’srule(n=2)Simpson’s3/8rulen=3Boole’srulen=4牛頓科茨公式余項(xiàng)梯形公式余項(xiàng)：辛普森公式余項(xiàng)：科茨公式余項(xiàng)：代數(shù)精度？當(dāng)積分區(qū)間比較大時(shí)，精度？7.2CompositeTrapezoidalandSimpson’sRuleab=CompositeTrapezoidalRule=abCompositeSimpsonRuleabCompositeBool’sRuleab復(fù)合牛頓科茨公式余項(xiàng)復(fù)合梯形公式余項(xiàng)復(fù)合辛普森公式余項(xiàng)復(fù)合布爾公式余項(xiàng)例題7.3RecursiveRulesandRombergIntegrationRecursiveTrapezoidalRuleabRombergIntegration數(shù)值積分在MATLAB中的應(yīng)用quad采用遞推自適應(yīng)Simpson法計(jì)算積分;精度較高，較常用quadl采用遞推自適應(yīng)Lobatto法計(jì)算積分;精度高，最常用trapz

采用梯形法計(jì)算積分;速度快，精度差cumtrapz

采用梯形法計(jì)算一個(gè)區(qū)間上的積分曲線;速度快，精度差Fnint

利用樣條函數(shù)求不定積分；與spline、ppval配合使用；主要對(duì)付“表格”函數(shù)的積分【例題】（1）符號(hào)解析法symsx;IS=int('exp(-x*x)','x',0,1)vpa(IS)

IS=1/2*erf(1)*pi^(1/2)ans=.74682413281242702539946743613185（2）MATLAB指令quad和quadl求積fun=inline('exp(-x.*x)','x');Isim=quad(fun,0,1),IL=quadl(fun,0,1)Isim=0.7468IL=0.7468

（3）10參數(shù)Gauss法Ig=gauss10(fun,0,1)