将方阵化为上Hessenberg矩阵

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上Hessenberg矩阵

形如把

b =

[ 54.0000000000000 40.0000000000000 10.0000000000000 76.0000000000000

47.0000000000000 20.0000000000000 94.0000000000000 49.0000000000000

26.0000000000000 80.0000000000000 94.0000000000000 70.0000000000000

3.00000000000000 92.0000000000000 83.0000000000000 45.0000000000000 ]

化为:

b1 =

[ 54.0000000000000 50.3829787234043 82.9789551473672 10.0000000000000

47.0000000000000 75.1276595744681 114.602178385252 94.0000000000000

0.00000000000000 135.991851516523 95.6102950602596 77.0000000000000

0.00000000000000 0.00000000000000 5.47923255500914 -11.737954634727 ]

的这种矩阵。b1的特点是在矩阵的下三角部分的次对焦线元素不为0外,其它元素均为0,则b1就叫做b的上H(Hessenberg)矩阵。

注意:Hessenberg矩阵与原矩阵的行列式、特征值均相同。

//*********************************************************************************************************************************

下面是代码的实现。

Math_Matrix_Hessenberg(A,n,ret)

A:我们要转化的n阶方阵

n:方阵A的阶数

ret:转化成功的n阶方阵(其就是我们要的值)

函数返回值为n

Public Function Math_Matrix_Hessenberg(ByVal A(,) As Double, ByVal n As Int16, ByRef ret(,) As Double) As Int16

Dim i As Int16

Dim j As Int16

Dim k As Int16

Dim temp As Double

Dim MaxNumber As Int16

n -= 1

ReDim ret(n, n)

For k = 1 To n - 1

i = k - 1

MaxNumber = k

temp = Math.Abs(A(k, i))

For j = k + 1 To n

If Math.Abs(A(j, i)) > temp Then

MaxNumber = j

End If

ret(0, 0) = A(MaxNumber, i) '储存最大值

i = MaxNumber

If ret(0, 0) <> 0 Then

If i <> k Then

For j = k - 1 To n

temp = A(i, j)

A(i, j) = A(k, j)

A(k, j) = temp

For j = 0 To n

temp = A(j, i)

A(j, i) = A(j, k)

A(j, k) = temp

End If

For i = k + 1 To n

temp = A(i, k - 1) / ret(0, 0)

A(i, k - 1) = 0

For j = k To n

A(i, j) -= temp * A(k, j)

For j = 0 To n

A(j, k) += temp * A(j, i)

End If

For i = 0 To n

For j = 0 To n

ret(i, j) = A(i, j)

Return n + 1

End Function

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