理学院 数值计算方法 实验二

1、实验二实验名称线性方程组的数值解法姓 名张见学 号08119054班 级信计（2）指导教师张昆实验日期2010-11-12成 绩一、 实验目的1、 掌握Gauss消去法及Gauss列主元消去法，能用这两种方法求解方程组；2、 掌握Jacobi和G-S迭代法，能应用Jacobi和G-S迭代法求解方程组；1、 掌握相应数值算法的程序编写；2、 理解迭代法收敛的充要条件，会判断迭代法的收敛性。二、 实验题目1、 分别用列主元消去法与不选主元消去法求解，分析算法对结果的影响：a) A=0.310-1559.14315.291-6.130-1211.29521211,b=59.1746.7812b) A

2、=10-701-32.099999625-15-10102,b=85.909901512、 给定矩阵A=hilb(n)与向量bA=11215121316151619,b=12345, a) 求的LU分解；b) 利用的LU分解求解方程组：Ax=b；A2x=b；A3x=b；c) 利用的LU分解求A-1，n值自己给定。三、 实验原理1、 当系数矩阵A的秩增广矩阵B的秩时，方程组无解向量； 当系数矩阵A的秩=增广矩阵B的秩 d Then d = Abs(a(i, k) Ni = i End If Next i For j = k To n t = a(Ni, j) a(Ni, j) = a(k, j)

3、 a(k, j) = t Next j e = b(Ni) b(Ni) = b(k) b(k) = e For i = k + 1 To n g = a(i, k) / a(k, k) For j = k To n a(i, j) = a(i, j) - a(k, j) * g Next j b(i) = b(i) - b(k) * g Next iNext kx(n) = b(n) / a(n, n)For i = n - 1 To 1 Step -1 t = 0 For j = i + 1 To n t = t + a(i, j) * x(j) Next j x(i) = (b(i) -

4、t) / a(i, i)Next iPicture2.Print 解得结果为：For i = 1 To nPicture2.Print x(i)Next iEnd Sub(b) 不选主元消元法Private Sub Command1_Click()Dim a() As Double, b() As Double, p As Double, t As Double, d As Double, Ni As Integer, x() As DoubleDim n As Integer, m As Integerm = 0n = Val(InputBox(输入线性方程组的阶数n)输入系数矩阵ReDim

5、 Preserve a(n, n)Picture1.Print 系数矩阵：For v = 1 To n For w = 1 To n a(v, w) = Val(InputBox(a( & v & , & w & ) m = m + 1 Picture1.Print a(v, w); If m Mod n = 0 Then Picture1.Print Next wNext vPrintPicture1.Print 常数向量：ReDim Preserve x(n)ReDim Preserve b(n)For v = 1 To n b(v) = Val(InputBox(输入常数向量 & b(

6、& v & )Picture1.Print b(v);Next v For k = 1 To n For i = k + 1 To n g = a(i, k) / a(k, k) For j = k To n a(i, j) = a(i, j) - a(k, j) * g Next j b(i) = b(i) - b(k) * g Next iNext kx(n) = b(n) / a(n, n)For i = n - 1 To 1 Step -1 t = 0For j = i + 1 To nt = t + a(i, j) * x(j) Next jx(i) = (b(i) - t) / a

7、(i, i)Next iPicture2.Print 解得结果为：For i = 1 To nPicture2.Print x(i)Next iEnd Sub2) Private Sub Command1_Click()Dim a() As Double, b() As Double, t As Double, d As Double, Ni As Integer, U() As Double, L() As Double, m As DoubleDim n As IntegerDim x() As Double, y() As Doublem = 0n = Val(InputBox(输入线性

8、方程组的阶数n)输入系数矩阵ReDim Preserve a(n, n)ReDim Preserve U(n, n)ReDim Preserve L(n, n)Picture1.Print 系数矩阵：For v = 1 To n For w = 1 To n a(v, w) = Val(InputBox(a( & v & , & w & ) m = m + 1 Picture1.Print a(v, w); If m Mod n = 0 Then Picture1.Print Next wNext vReDim Preserve b(n)Picture1.Print 常数向量：For v =

9、1 To n b(v) = Val(InputBox(输入常数向量 & b( & v & )Picture1.Print b(v);Next vFor k = 1 To n For i = k + 1 To n m = a(i, k) / a(k, k) L(i, k) = m For j = k To n a(i, j) = a(i, j) - a(k, j) * m Next j b(i) = b(i) - b(k) * m Next iNext kFor i = 1 To n For j = i To n U(i, j) = a(i, j) Next j Next iPicture2.P

10、rint L为： For i = 1 To n For j = 1 To n L(i, i) = 1 Picture2.Print L(i, j); Space(5); Next j Picture2.Print Next i Picture2.Print * Picture2.Print U为： For i = 1 To n For j = 1 To n Picture2.Print U(i, j); Space(5); Next j Picture2.Print Next i Picture2.Print *ReDim Preserve x(n)ReDim Preserve y(n)x(n) = b(n) / a(n, n)For i = n - 1 To 1