数值实验三 LU分解法的优点

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1、数值实验三LU分解法的优点实验目的给定矩阵A与向量bn - 1A= :21对称nn 1 n-i0b=:0L。(1)求A的LU分解(2)利用A的LU分解解下列方程：A*x=bA2 * x = b A3 * x = b对第题分析一下，如果先求M=A3,再解M*相比有何缺点?(3)利用A的LU分解法求A-七其中n由自己选择，例如取n=5二：实验原理输入方程阶数n,系数矩阵A,右端向量bK=L/n (分解 A二L*U)k-lUk j = ak j 2 Iks * Us j(j = k, n)S = 1输出失败信息，停Lkk = (aikEyUsUsk) /Ukk(i=k+L，n)yk=bk-Ss=i1

2、ksys)(k=l,2,!)(解方程组 L*y=b)Xk=(yk-?=k+iUksXs)/Ukk(k=n,nl,，1)输出XpX2,.,Xn,结束三：实验过程实验代码:Option Base 1Dim a() As Single, u() As Single, 1() As SinglePrivate Sub Commandl_Click()Dim m As Integer; p As Integer; n As Integer; k As Integer; i As Integer; j As Integer; s As Integer;t As Singlen = Val(Textl.Te

3、xt)ReDim a(n, n), u(n, n), l(nz n)For i = lTo nFor j = iTo na(ij) = n + i-ja(j, i) = n + ijNextNextt = 0For k = 1 To nFor j = kTo n t = 0For s = 1 To k -1t = t+l(k, s) * u(sj)Nextu(k,j) = a(k,j)-tNextIf k on ThenFori = k + 1 To nt = 0For s = 1 To k -1t = t+ l(i, s) * u(s, k)Nextl(i, k) = (a(i, k)-t)

4、/u(k, k)NextEnd IfNextForm = lTonl(mz m) = 1NextFor i = lTo nFor j = ITo nText2.Text = Text2.Text & a(i, j) & vbCrLfNextNextFor i = lTo nForj = ITo nTextS.Text = TextB.Text & l(i, j) & vbCrLfNextNextFor i = lTo nForj = ITo nText4.Text = Text4.Text & u(i, j) & vbCrLfNextNextEnd SubPrivate Sub Command

5、2_Click()Dim y() As Single, x() As Single, b() As SingleDim n As Integer; k As Integer; i As Integer; j As Integer; s As Integer, t As Singlen = Val(Textl.Text)ReDim y(n)/x(n)/ b(n)b(l) = lFor i = 2To nb(i) = 0Next iFor k = 1 To nt = 0For s = ITo k -1t = t+ l(k, s) * y(s)Nexty(k) = b(k)-1NextFor k =

6、 n To 1 Step -1t = 0For s = k + 1 To nt = t + u(k, s) * x(s)Nextx(k) = (y(k) -1) / u(k, k)NextFor i = 1 To nText5.Text = Text5.Text & x(i) & vbCrLfNextEnd SubPrivate Sub Command3_Click()Dim y() As Single, x() As Single, b() As SingleDim n As Integer; k As Integer; i As Integer; j As Integer; s As In

7、teger; t As Single n = Val(Textl.Text)ReDim y(n)/x(n)/ b(n)b(l) = lFor i = 2To nb(i) = 0Next iFori = lTo2For k = ITo nt = 0Fors = lTo(k-l)t = t+l(k, s) * y(s)NextV(k) = b(k)-tNextFor k = n To 1 Step -1t = 0Fors = k + lTo nt = t+ u(k, s) * x(s)Nextx(k) = (y(k)-t)/u(k, k)NextFor j = ITo nb(j)=x(j)Next

8、NextFor i = lTo nText6.Text = Text6.Text & b(i) & vbCrLfNextEnd SubPrivate Sub Command4_Click()Dim y() As Single, x() As Single, b() As Single, v() As SingleDim n As Integer; k As Integer; i As Integer; j As Integer; s As Integer, t As Single n = Val(Textl.Text)ReDimy(n)zx(n)z b(n)b(l) = lFor i = 2T

9、o nb(i) = 0Next iFor i = 1 To 3For k = ITo nt = 0Fors = lTo(k-l)t = t+l(k, s) * y(s)NextV(k) = b(k)-tNextFor k = n To 1 Step -1t = 0Fors = k + lTo nt = t+ u(k, s) * x(s)Nextx(k) = (y(k)-t)/u(k, k)NextForj = ITo nb(j)=x(j)NextNextFor i = lTo nText7.Text = Text7.Text & b(i) & vbCrLfNextEnd SubPrivate

10、Sub Command5_Click()EndEnd SubPrivate Sub Command6_Click()Dim y() As Single, x() As Single, b() As Single, v() As SingleDim n As Integer; k As Integer, i As Integer; j As Integer; s As Integer, t As Singlen = Val(Textl.Text)ReDim v(l To n, 1 To n)ReDim y(l To n)For i = lTo nReDim b(l To n)b(i) = 1Fo

11、r k = ITo nt = 0For s = 1 To k -1t = t+l(k, s) * y(s)NextV(k) = b(k)-tNextFor k = n To 1 Step -1t = 0Fors = k + lTo nt = t+ u(k, s) * v(s, i)Nextv(k, i) = (y(k) -1)/ u(k, k)NextNextFor i = 1 To nForj = ITo nText8.Text = Text8.Text & v(iz j) & vbCrLfNextNextEnd Sub四：实验结果当n=10时A*x=b 的解 xl= 0 .5454544,

12、-0.4999996,3.971573E07, 2.5836E-07, -1.444715E-07,8.344654E-08, -5.188799E-08, 2.407111E-08, -3.632159E 08, 4.545457E-02 一】A2 * x = b 的解为 x2=【0. 5495862, -0. 7727261, 0. 2499987, 8. 174827E-07, -4. 631004E-07,3. 457071E-07, -2. 65427E-07, 9. 026667E-08, -2. 272733EP2, 4. 958682EP2】一】A3 * x = b的解为 x3=【0. 6883911, T. 172518, 0.6363608, -0. 1249981, -1. 1815E-069. 874508E-07, -7. 90294IE-07, 1. 136398E-02, -4. 752079EP2, 6. 339223E-02五：实验分析LU分解法比较简便迅速，当解多个系数矩阵为A的线性方程做时，LU分解法就显得特别优 越，只要对系数矩阵做一次LU分解，以后只要解三角形方程即可。也可以更具系数矩阵的 形状来设计算法