西交大计算方法上机报告.doc

计算方法（B）实验报告姓名：学号：学院：专业：实验一 三对角方程组的求解一、 实验目的掌握三对角方程组求解的方法。二、 实验内容求三对角方程组的解，其中： ， 三、 算法组织设系数矩阵为三对角矩阵则方程组称为三对角方程组。设矩阵T非奇异，T可分解为T=LU，其中L为下三角矩阵，U为单位上三角矩阵，记可先依次求出中的元素后，令，先求解下三角方程组得出y，再求解上三角方程组。追赶法的算法组织如下：1.输入三对角矩阵和右端向量；2.将压缩为四个一维数组，是T的三对角线性方程组的三个对角，是右端向量。将分解矩阵压缩为三个一维数组。3.对做分解（也可以用分解）导出追赶法的计算步骤如下： 4.回代求解 5. 停止，输出结果。四、 MATLAB 程序MATLAB程序见附件1.五、 结果及分析实验结果为：实验二 Jacobi迭代和Gauss-Seidel迭代解线性方程组一、 实验目的掌握Jacobi迭代和Gauss-Seidel迭代解线性方程组的方法。二、 实验内容用Jacobi迭代和Gauss-Seidel迭代解电路电流方程组，使各部分电流误差均小于。三、 算法组织形如的方程组，用Jacobi迭代求解，算法组织如下：1. 将系数矩阵分解成对角元、下三角部分元和上三角部分元,于是.2. 由。3. 从而构成形如迭代格式：其中4. 选取初始向量进行迭代计算。5. 当迭代后的解满足题中的约束条件时迭代停止。形如的方程组，用GaussSeidel迭代求解，算法组织如下：1. 将系数矩阵分解成对角元、下三角部分元和上三角部分元,于是.2. 由。3. 从而构成形如迭代格式：其中4. 选取初始向量进行迭代计算。5. 当迭代后的解满足题中的约束条件时迭代停止。四、 MATLAB 程序MATLAB程序见附件2，其中1为Jacobi迭代，2为GaussSeidel迭代。五、 结果及分析Jacobi迭代结果:方程组的解为迭代次数GaussSeidel迭代结果：方程组的解为迭代次数由以上结果可知,达到相同的计算精度，GaussSeidel迭代比Jacobi迭代的速度快，GaussSeidel迭代比Jacobi迭代次数少。实验三多项式插值及误差计算一、实验目的掌握多项式插值的原理和基本方法。二、实验内容已知，对a. 计算函数在点处的值;b. 求插值数据点的插值多项式和三次样条插值多项式;c. 对，计算和相应的函数值;d. 计算，解释所得到结果。三、 算法组织（一）本题第一问是简单的用matlab程序可以计算，算法很简单。（二） 本题在算法上第二问中的Newton插值多项式和三次样条插值多项式。计算两种插值多项式的算法如下：1. 求Newton插值多项式，算法组织如下：Newton插值多项式的表达式如下：其中每一项的系数ci的表达式如下：根据上述公式，为了得到系数需计算：1) 一阶差商2) 二阶差商 3) n阶差商4) n+1阶差商2. 求三次样条插值多项式，算法组织如下：所谓三次样条插值多项式是一种对区间进行分段的分段函数，然后在每一段上进行分析，即它在节点分成的每个小区间上是3次多项式，其在此区间上的表达式如下：因此，只要确定了的值，就确定了整个表达式，的计算方法如下：令：则满足如下n-1个方程：方程中有n+1个未知量，则令和分别为零，则由上面的方程组可得到的值，可得到整个区间上的三次样条插值多项式。（三） 第三问和第四问的算法与第二问的算法类似，不再赘述。四、 MATLAB 程序MATLAB程序见附件3。五、 结果及分析第一问当n=5时,各节点及f(x)值为：x(0)=-1,y(0)=3.846154e-02x(1)=-6.000000e-01,y(1)=1.000000e-01x(2)=-2.000000e-01,y(2)=5.000000e-01x(3)=2.000000e-01,y(3)=5.000000e-01x(4)=6.000000e-01,y(4)=1.000000e-01x(5)=1,y(5)=3.846154e-02当n=10时,各节点及f(x)值为：x(0)=-1,y(0)=3.846154e-02x(1)=-8.000000e-01,y(1)=5.882353e-02x(2)=-6.000000e-01,y(2)=1.000000e-01x(3)=-4.000000e-01,y(3)=2.000000e-01x(4)=-2.000000e-01,y(4)=5.000000e-01x(5)=0,y(5)=1x(6)=2.000000e-01,y(6)=5.000000e-01x(7)=4.000000e-01,y(7)=2.000000e-01x(8)=6.000000e-01,y(8)=1.000000e-01x(9)=8.000000e-01,y(9)=5.882353e-02x(10)=1,y(10)=3.846154e-02当n=20时,各节点及f(x)值为：x(0)=-1,y(0)=3.846154e-02x(1)=-9.000000e-01,y(1)=4.705882e-02x(2)=-8.000000e-01,y(2)=5.882353e-02x(3)=-7.000000e-01,y(3)=7.547170e-02x(4)=-6.000000e-01,y(4)=1.000000e-01x(5)=-5.000000e-01,y(5)=1.379310e-01x(6)=-4.000000e-01,y(6)=2.000000e-01x(7)=-3.000000e-01,y(7)=3.076923e-01x(8)=-2.000000e-01,y(8)=5.000000e-01x(9)=-1.000000e-01,y(9)=8.000000e-01x(10)=0,y(10)=1x(11)=1.000000e-01,y(11)=8.000000e-01x(12)=2.000000e-01,y(12)=5.000000e-01x(13)=3.000000e-01,y(13)=3.076923e-01x(14)=4.000000e-01,y(14)=2.000000e-01x(15)=5.000000e-01,y(15)=1.379310e-01x(16)=6.000000e-01,y(16)=1.000000e-01x(17)=7.000000e-01,y(17)=7.547170e-02x(18)=8.000000e-01,y(18)=5.882353e-02x(19)=9.000000e-01,y(19)=4.705882e-02x(20)=1,y(20)=3.846154e-02第二问牛顿插值算法当n=5时Newton插值多项式的系数分别为：c0=3.846154e-02c1=1.538462e-01c2=1.057692e+00c3=-1.923077e+00c4=1.201923e+00c5=-5.551115e-16当n=10时Newton插值多项式的系数分别为：c0=3.846154e-02c1=1.018100e-01c2=2.601810e-01c3=7.918552e-01c4=2.686652e+00c5=-6.363122e+00c6=-1.767534e+01c7=8.484163e+01c8=-1.679157e+02c9=2.209417e+02c10=-2.209417e+02当n=20时Newton插值多项式的系数分别为：c0=3.846154e-02c1=8.597285e-02c2=1.583710e-01c3=2.860070e-01c4=5.335952e-01c5=1.037751e+00c6=2.001902e+00c7=2.796775e+00c8=-7.543931e+00c9=-1.011991e+02c10=-6.439941e+01c11=2.152780e+03c12=-7.267934e+03c13=1.139374e+04c14=-3.538429e+03c15=-2.830744e+04c16=8.669152e+04c17=-1.592293e+05c18=2.237536e+05c19=-2.601786e+05c20=2.601786e+05三次样条插值算法当n=5时，取边界条件为自然样条的三次样条插值多项式的系数分别为：m = 0 4.1296 -3.8259 -3.8259 4.1296 0当n=10时，取边界条件为自然样条的三次样条插值多项式的系数分别为：m = 0 0.4101 1.4820 2.4856 18.5755 -46.7878 18.5755 2.4856 1.4820 0.4101 0当n=20时，取边界条件为自然样条的三次样条插值多项式的系数分别为：m = 0 0.3615 0.4543 0.7514 1.2681 2.2177 4.3438 7.7810 15.3016 -4.3719 -57.8141 -4.3719 15.3016 7.7810 4.3438 2.2177 1.2681 0.7514 0.4543 0.3615 0第三问当n=5时，给定点xi的f(xi),N(xi),S(xi),分别为:x(1)=-9.800000e-01,f=3.998401e-02,N=1.369250e-02,S=3.604615e-02x(2)=-9.600000e-01,f=4.159734e-02,N=-6.920000e-03,S=3.371336e-02x(3)=-9.400000e-01,f=4.330879e-02,N=-2.359981e-02,S=3.154575e-02x(4)=-9.200000e-01,f=4.512635e-02,N=-3.656615e-02,S=2.962591e-02x(5)=-9.000000e-01,f=4.705882e-02,N=-4.603365e-02,S=2.803644e-02x(6)=-8.800000e-01,f=4.911591e-02,N=-5.221231e-02,S=2.685992e-02x(7)=-8.600000e-01,f=5.130836e-02,N=-5.530750e-02,S=2.617895e-02x(8)=-8.400000e-01,f=5.364807e-02,N=-5.552000e-02,S=2.607611e-02x(9)=-8.200000e-01,f=5.614823e-02,N=-5.304596e-02,S=2.663401e-02x(10)=-8.000000e-01,f=5.882353e-02,N=-4.807692e-02,S=2.793522e-02x(11)=-7.800000e-01,f=6.169031e-02,N=-4.079981e-02,S=3.006235e-02x(12)=-7.600000e-01,f=6.476684e-02,N=-3.139692e-02,S=3.309798e-02x(13)=-7.400000e-01,f=6.807352e-02,N=-2.004596e-02,S=3.712470e-02x(14)=-7.200000e-01,f=7.163324e-02,N=-6.920000e-03,S=4.222510e-02x(15)=-7.000000e-01,f=7.547170e-02,N=7.812500e-03,S=4.848178e-02x(16)=-6.800000e-01,f=7.961783e-02,N=2.398769e-02,S=5.597733e-02x(17)=-6.600000e-01,f=8.410429e-02,N=4.144635e-02,S=6.479433e-02x(18)=-6.400000e-01,f=8.896797e-02,N=6.003385e-02,S=7.501538e-02x(19)=-6.200000e-01,f=9.425071e-02,N=7.960019e-02,S=8.672308e-02x(20)=-6.000000e-01,f=1.000000e-01,N=1.000000e-01,S=1.000000e-01x(21)=-5.800000e-01,f=1.062699e-01,N=1.210925e-01,S=1.148885e-01x(22)=-5.600000e-01,f=1.131222e-01,N=1.427415e-01,S=1.312696e-01x(23)=-5.400000e-01,f=1.206273e-01,N=1.648156e-01,S=1.489844e-01x(24)=-5.200000e-01,f=1.288660e-01,N=1.871877e-01,S=1.678737e-01x(25)=-5.000000e-01,f=1.379310e-01,N=2.097356e-01,S=1.877783e-01x(26)=-4.800000e-01,f=1.479290e-01,N=2.323415e-01,S=2.085393e-01x(27)=-4.600000e-01,f=1.589825e-01,N=2.548925e-01,S=2.299974e-01x(28)=-4.400000e-01,f=1.712329e-01,N=2.772800e-01,S=2.519935e-01x(29)=-4.200000e-01,f=1.848429e-01,N=2.994002e-01,S=2.743686e-01x(30)=-4.000000e-01,f=2.000000e-01,N=3.211538e-01,S=2.969636e-01x(31)=-3.800000e-01,f=2.169197e-01,N=3.424463e-01,S=3.196192e-01x(32)=-3.600000e-01,f=2.358491e-01,N=3.631877e-01,S=3.421765e-01x(33)=-3.400000e-01,f=2.570694e-01,N=3.832925e-01,S=3.644763e-01x(34)=-3.200000e-01,f=2.808989e-01,N=4.026800e-01,S=3.863595e-01x(35)=-3.000000e-01,f=3.076923e-01,N=4.212740e-01,S=4.076670e-01x(36)=-2.800000e-01,f=3.378378e-01,N=4.390031e-01,S=4.282397e-01x(37)=-2.600000e-01,f=3.717472e-01,N=4.558002e-01,S=4.479184e-01x(38)=-2.400000e-01,f=4.098361e-01,N=4.716031e-01,S=4.665441e-01x(39)=-2.200000e-01,f=4.524887e-01,N=4.863540e-01,S=4.839577e-01x(40)=-2.000000e-01,f=5.000000e-01,N=5.000000e-01,S=5.000000e-01x(41)=-1.800000e-01,f=5.524862e-01,N=5.124925e-01,S=5.145385e-01x(42)=-1.600000e-01,f=6.097561e-01,N=5.237877e-01,S=5.275466e-01x(43)=-1.400000e-01,f=6.711409e-01,N=5.338463e-01,S=5.390243e-01x(44)=-1.200000e-01,f=7.352941e-01,N=5.426338e-01,S=5.489717e-01x(45)=-1.000000e-01,f=8.000000e-01,N=5.501202e-01,S=5.573887e-01x(46)=-8.000000e-02,f=8.620690e-01,N=5.562800e-01,S=5.642753e-01x(47)=-6.000000e-02,f=9.174312e-01,N=5.610925e-01,S=5.696316e-01x(48)=-4.000000e-02,f=9.615385e-01,N=5.645415e-01,S=5.734575e-01x(49)=-2.000000e-02,f=9.900990e-01,N=5.666156e-01,S=5.757530e-01x(50)=0,f=1,N=5.673077e-01,S=5.765182e-01x(51)=2.000000e-02,f=9.900990e-01,N=5.666156e-01,S=5.757530e-01x(52)=4.000000e-02,f=9.615385e-01,N=5.645415e-01,S=5.734575e-01x(53)=6.000000e-02,f=9.174312e-01,N=5.610925e-01,S=5.696316e-01x(54)=8.000000e-02,f=8.620690e-01,N=5.562800e-01,S=5.642753e-01x(55)=1.000000e-01,f=8.000000e-01,N=5.501202e-01,S=5.573887e-01x(56)=1.200000e-01,f=7.352941e-01,N=5.426338e-01,S=5.489717e-01x(57)=1.400000e-01,f=6.711409e-01,N=5.338463e-01,S=5.390243e-01x(58)=1.600000e-01,f=6.097561e-01,N=5.237877e-01,S=5.275466e-01x(59)=1.800000e-01,f=5.524862e-01,N=5.124925e-01,S=5.145385e-01x(60)=2.000000e-01,f=5.000000e-01,N=5.000000e-01,S=5.000000e-01x(61)=2.200000e-01,f=4.524887e-01,N=4.863540e-01,S=4.839577e-01x(62)=2.400000e-01,f=4.098361e-01,N=4.716031e-01,S=4.665441e-01x(63)=2.600000e-01,f=3.717472e-01,N=4.558002e-01,S=4.479184e-01x(64)=2.800000e-01,f=3.378378e-01,N=4.390031e-01,S=4.282397e-01x(65)=3.000000e-01,f=3.076923e-01,N=4.212740e-01,S=4.076670e-01x(66)=3.200000e-01,f=2.808989e-01,N=4.026800e-01,S=3.863595e-01x(67)=3.400000e-01,f=2.570694e-01,N=3.832925e-01,S=3.644763e-01x(68)=3.600000e-01,f=2.358491e-01,N=3.631877e-01,S=3.421765e-01x(69)=3.800000e-01,f=2.169197e-01,N=3.424463e-01,S=3.196192e-01x(70)=4.000000e-01,f=2.000000e-01,N=3.211538e-01,S=2.969636e-01x(71)=4.200000e-01,f=1.848429e-01,N=2.994002e-01,S=2.743686e-01x(72)=4.400000e-01,f=1.712329e-01,N=2.772800e-01,S=2.519935e-01x(73)=4.600000e-01,f=1.589825e-01,N=2.548925e-01,S=2.299974e-01x(74)=4.800000e-01,f=1.479290e-01,N=2.323415e-01,S=2.085393e-01x(75)=5.000000e-01,f=1.379310e-01,N=2.097356e-01,S=1.877783e-01x(76)=5.200000e-01,f=1.288660e-01,N=1.871877e-01,S=1.678737e-01x(77)=5.400000e-01,f=1.206273e-01,N=1.648156e-01,S=1.489844e-01x(78)=5.600000e-01,f=1.131222e-01,N=1.427415e-01,S=1.312696e-01x(79)=5.800000e-01,f=1.062699e-01,N=1.210925e-01,S=1.148885e-01x(80)=6.000000e-01,f=1.000000e-01,N=1.000000e-01,S=1.000000e-01x(81)=6.200000e-01,f=9.425071e-02,N=7.960019e-02,S=8.672308e-02x(82)=6.400000e-01,f=8.896797e-02,N=6.003385e-02,S=7.501538e-02x(83)=6.600000e-01,f=8.410429e-02,N=4.144635e-02,S=6.479433e-02x(84)=6.800000e-01,f=7.961783e-02,N=2.398769e-02,S=5.597733e-02x(85)=7.000000e-01,f=7.547170e-02,N=7.812500e-03,S=4.848178e-02x(86)=7.200000e-01,f=7.163324e-02,N=-6.920000e-03,S=4.222510e-02x(87)=7.400000e-01,f=6.807352e-02,N=-2.004596e-02,S=3.712470e-02x(88)=7.600000e-01,f=6.476684e-02,N=-3.139692e-02,S=3.309798e-02x(89)=7.800000e-01,f=6.169031e-02,N=-4.079981e-02,S=3.006235e-02x(90)=8.000000e-01,f=5.882353e-02,N=-4.807692e-02,S=2.793522e-02x(91)=8.200000e-01,f=5.614823e-02,N=-5.304596e-02,S=2.663401e-02x(92)=8.400000e-01,f=5.364807e-02,N=-5.552000e-02,S=2.607611e-02x(93)=8.600000e-01,f=5.130836e-02,N=-5.530750e-02,S=2.617895e-02x(94)=8.800000e-01,f=4.911591e-02,N=-5.221231e-02,S=2.685992e-02x(95)=9.000000e-01,f=4.705882e-02,N=-4.603365e-02,S=2.803644e-02x(96)=9.200000e-01,f=4.512635e-02,N=-3.656615e-02,S=2.962591e-02x(97)=9.400000e-01,f=4.330879e-02,N=-2.359981e-02,S=3.154575e-02x(98)=9.600000e-01,f=4.159734e-02,N=-6.920000e-03,S=3.371336e-02x(99)=9.800000e-01,f=3.998401e-02,N=1.369250e-02,S=3.604615e-02当n=10时，给定点xi的f(xi),N(xi),S(xi),分别为:x(1)=-9.800000e-01,f=3.998401e-02,N=1.230317e+00,S=4.022710e-02x(2)=-9.600000e-01,f=4.159734e-02,N=1.804385e+00,S=4.200907e-02x(3)=-9.400000e-01,f=4.330879e-02,N=1.958952e+00,S=4.382384e-02x(4)=-9.200000e-01,f=4.512635e-02,N=1.845845e+00,S=4.568782e-02x(5)=-9.000000e-01,f=4.705882e-02,N=1.578721e+00,S=4.761740e-02x(6)=-8.800000e-01,f=4.911591e-02,N=1.240213e+00,S=4.962900e-02x(7)=-8.600000e-01,f=5.130836e-02,N=8.880811e-01,S=5.173901e-02x(8)=-8.400000e-01,f=5.364807e-02,N=5.604444e-01,S=5.396383e-02x(9)=-8.200000e-01,f=5.614823e-02,N=2.802176e-01,S=5.631987e-02x(10)=-8.000000e-01,f=5.882353e-02,N=5.882353e-02,S=5.882353e-02x(11)=-7.800000e-01,f=6.169031e-02,N=-1.007243e-01,S=6.149562e-02x(12)=-7.600000e-01,f=6.476684e-02,N=-2.012964e-01,S=6.437461e-02x(13)=-7.400000e-01,f=6.807352e-02,N=-2.496082e-01,S=6.750337e-02x(14)=-7.200000e-01,f=7.163324e-02,N=-2.546027e-01,S=7.092479e-02x(15)=-7.000000e-01,f=7.547170e-02,N=-2.261963e-01,S=7.468173e-02x(16)=-6.800000e-01,f=7.961783e-02,N=-1.743355e-01,S=7.881707e-02x(17)=-6.600000e-01,f=8.410429e-02,N=-1.083152e-01,S=8.337369e-02x(18)=-6.400000e-01,f=8.896797e-02,N=-3.631685e-02,S=8.839447e-02x(19)=-6.200000e-01,f=9.425071e-02,N=3.487332e-02,S=9.392228e-02x(20)=-6.000000e-01,f=1.000000e-01,N=1.000000e-01,S=1.000000e-01x(21)=-5.800000e-01,f=1.062699e-01,N=1.553586e-01,S=1.066700e-01x(22)=-5.600000e-01,f=1.131222e-01,N=1.987262e-01,S=1.139730e-01x(23)=-5.400000e-01,f=1.206273e-01,N=2.292273e-01,S=1.219491e-01x(24)=-5.200000e-01,f=1.288660e-01,N=2.471535e-01,S=1.306384e-01x(25)=-5.000000e-01,f=1.379310e-01,N=2.537555e-01,S=1.400810e-01x(26)=-4.800000e-01,f=1.479290e-01,N=2.510206e-01,S=1.503172e-01x(27)=-4.600000e-01,f=1.589825e-01,N=2.414494e-01,S=1.613870e-01x(28)=-4.400000e-01,f=1.712329e-01,N=2.278397e-01,S=1.733307e-01x(29)=-4.200000e-01,f=1.848429e-01,N=2.130858e-01,S=1.861883e-01x(30)=-4.000000e-01,f=2.000000e-01,N=2.000000e-01,S=2.000000e-01x(31)=-3.800000e-01,f=2.169197e-01,N=1.911589e-01,S=2.149065e-01x(32)=-3.600000e-01,f=2.358491e-01,N=1.887784e-01,S=2.314509e-01x(33)=-3.400000e-01,f=2.570694e-01,N=1.946178e-01,S=2.502767e-01x(34)=-3.200000e-01,f=2.808989e-01,N=2.099145e-01,S=2.720276e-01x(35)=-3.000000e-01,f=3.076923e-01,N=2.353466e-01,S=2.973471e-01x(36)=-2.800000e-01,f=3.378378e-01,N=2.710241e-01,S=3.268788e-01x(37)=-2.600000e-01,f=3.717472e-01,N=3.165048e-01,S=3.612664e-01x(38)=-2.400000e-01,f=4.098361e-01,N=3.708328e-01,S=4.011534e-01x(39)=-2.200000e-01,f=4.524887e-01,N=4.325960e-01,S=4.471834e-01x(40)=-2.000000e-01,f=5.000000e-01,N=5.000000e-01,S=5.000000e-01x(41)=-1.800000e-01,f=5.524862e-01,N=5.709536e-01,S=5.597038e-01x(42)=-1.600000e-01,f=6.097561e-01,N=6.431626e-01,S=6.242233e-01x(43)=-1.400000e-01,f=6.711409e-01,N=7.142283e-01,S=6.909440e-01x(44)=-1.200000e-01,f=7.352941e-01,N=7.817471e-01,S=7.572512e-01x(45)=-1.000000e-01,f=8.000000e-01,N=8.434074e-01,S=8.205306e-01x(46)=-8.000000e-02,f=8.620690e-01,N=8.970803e-01,S=8.781675e-01x(47)=-6.000000e-02,f=9.174312e-01,N=9.409023e-01,S=9.275474e-01x(48)=-4.000000e-02,f=9.615385e-01,N=9.733459e-01,S=9.660558e-01x(49)=-2.000000e-02,f=9.900990e-01,N=9.932776e-01,S=9.910782e-01x(50)=0,f=1,N=1.000000e+00,S=1x(51)=2.000000e-02,f=9.