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1、 MATLAB作业二参考答案1、试求下面线性代数方程的解析解与数值解，并检验解的正确性。【求解】求出A, A;B 两个矩阵的秩，可见二者相同，所以方程不是矛盾方程，应该有无穷多解。 A=2,-9,3,-2,-1; 10,-1,10,5,0; 8,-2,-4,-6,3; -5,-6,-6,-8,-4;B=-1,-4,0; -3,-8,-4; 0,3,3; 9,-5,3;rank(A), rank(A B)ans =4 4用下面的语句可以求出方程的解析解，并可以验证该解没有误差。 x0=null(sym(A);x_analytical=sym(A)B; syms a;x=a*x0 x0 x0+x_
2、analyticalx = a+967/1535, a-943/1535, a-159/1535 -1535/1524*a, -1535/1524*a, -1535/1524*a -3659/1524*a-1807/1535,-3659/1524*a-257/1535,-3659/1524*a-141/1535 1321/508*a+759/1535, 1321/508*a-56/1535, 1321/508*a-628/1535 -170/127*a-694/307, -170/127*a+719/307, -170/127*a+103/307 A*x-Bans = 0, 0, 0 0, 0
3、, 0 0, 0, 0 0, 0, 0用数值解方法也可以求出方程的解，但会存在误差，且与任意常数a 的值有关。 x0=null(A); x_numerical=AB; syms a;x=a*x0 x0 x0+x_numerical; vpa(x,10)ans = .2474402553*a+.1396556436, .2474402553*a-.6840666849, .2474402553*a-.1418420333-.2492262414*a+.4938507789,-.2492262414*a+.7023776988e-1,-.2492262414*a+.3853511888e-1 -.
4、5940839201*a, -.5940839201*a, -.5940839201*a .6434420813*a-.7805411315, .6434420813*a-.2178190763,.6434420813*a-.5086089095-.3312192394*a-1.604263460, -.3312192394*a+2.435364854, -.3312192394*a+.3867176824 A*x-Bans = 1/18014398509481984*a, 1/18014398509481984*a, 1/18014398509481984*a -5/450359962737
5、0496*a, -5/4503599627370496*a, -5/4503599627370496*a -25/18014398509481984*a, -25/18014398509481984*a, -25/18014398509481984*a 13/18014398509481984*a, 13/18014398509481984*a, 13/18014398509481984*a2、求解下面的联立方程，并检验得出的高精度数值解（准解析解）的精度。【求解】给出的方程可以由下面的语句直接求解，经检验可见，精度是相当高的。 x1,x2=solve(x12-x2-1=0,(x1-2)2+(
6、x2-0.5)2-1=0,x1,x2)x1 =-1.3068444845633173592407431426632-1.2136904451605911320167045558746*sqrt(-1)-1.3068444845633173592407431426632+1.2136904451605911320167045558746*sqrt(-1)1.06734608580668971340859731280701.5463428833199450050728889725194x2 =-.76520198984055124633885565606586+3.1722093284506318
7、231178646481143*sqrt(-1)-.76520198984055124633885565606586-3.1722093284506318231178646481143*sqrt(-1).139227666886861440483624988051411.3911763127942410521940863240803 norm(double(x1.2-x2-1 (x1-2).2+(x2-0.5).2-1)ans =6.058152713871457e-0313、用Jacobi、Gauss-Seidel迭代法求解方程组 ，给定初值为。【求解】：编写Jacobi、Gauss-Sei
8、del函数计算，function y=jacobi(a,b,x0)D=diag(diag(a); U=-triu(a,1); L=-tril(a,-1);B=D(L+U); f=Db;y=B*x0+f;n=1;while norm(y-x0)=1.0e-6 x0=y; y=B*x0+f; n=n+1;endn a=10,-1,-2;-1,10,-2;-1,-1,5;b=72;83;42; jacobi(a,b,0;0;0)n = 17ans = 11.0000 12.0000 13.0000function y=seidel(a,b,x0)D=diag(diag(a);U=-triu(a,1)
9、;L=-tril(a,-1);G=(D-L)U ;f=(D-L)b;y=G*x0+f; n=1;while norm(y-x0)=1.0e-6 x0=y; y=G*x0+f; n=n+1;endn seidel(a,b,0;0;0)n = 10ans = 11.0000 12.0000 13.00004555、假设已知一组数据，试用插值方法绘制出区间内的光滑函数曲线，比较各种插值算法的优劣。-2-1.7-1.4-1.1-0.8-0.5-0.20.10.40.711.3.10289.11741.13158.14483.15656.16622.17332.1775.17853.17635.1710
10、9.163021.61.92.22.52.83.13.43.744.34.64.9.15255.1402.12655.11219.09768.08353.07015.05786.04687.03729.02914.02236【求解】用下面的语句可以立即得出给定样本点数据的三次插值与样条插值，得出的结果如，可见，用两种插值方法对此例得出的结果几乎一致，效果均很理想。 x=-2,-1.7,-1.4,-1.1,-0.8,-0.5,-0.2,0.1,0.4,0.7,1,1.3,.1.6,1.9,2.2,2.5,2.8,3.1,3.4,3.7,4,4.3,4.6,4.9;y=0.10289,0.1174
11、1,0.13158,0.14483,0.15656,0.16622,0.17332,.0.1775,0.17853,0.17635,0.17109,0.16302,0.15255,0.1402,.0.12655,0.11219,0.09768,0.08353,0.07019,0.05786,0.04687,.0.03729,0.02914,0.02236;x0=-2:0.02:4.9;y1=interp1(x,y,x0,cubic);y2=interp1(x,y,x0,spline);plot(x0,y1,:,x0,y2,x,y,o)5、 假设已知实测数据由下表给出，试对在(0.1,0.1)(
12、1.1,1.1)区域内的点进行插值，并用三维曲面的方式绘制出插值结果。00.10.20.30.40.50.60.70.80.911.10.1.83041.82727.82406.82098.81824.8161.81481.81463.81579.81853.823040.2.83172.83249.83584.84201.85125.86376.87975.89935.92263.94959.98010.3.83587.84345.85631.87466.89867.9284.963771.00451.05021.11.15290.4.84286.86013.88537.91865.9598
13、51.00861.06421.12531.19041.2571.32220.5.85268.88251.92286.973461.03361.10191.17641.2541.33081.40171.46050.6.86532.91049.968471.03831.1181.20461.29371.37931.45391.50861.53350.7.88078.943961.02171.11181.21021.3111.40631.48591.53771.54841.50520.8.89904.982761.0821.19221.30611.41381.50211.55551.55731.49151.3460.9.920061.02661.14821.27681.40051.50341.56611.56781.48891.31561.04541.943811.07521.21911.36241.48661.56841.58211.50321.3151.0155.624771.1.970231.12791.29291.44481.55641.59641.53411.34731.0321.61268.14763【求解】直接采用插值方法可以解决该问题，得出的插值曲面。 x,y=meshgrid(0.1
