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請教MATLAB兩個問題?

問題一:

求下列系統的 ||A||e‵||A||1以及||A||∞

8 2 -10

[A]=[-9 1 3]

15 -1 6

在決定範數之前, 先調整矩陣的大小使得每一列中最大元素為1

問題二:

除了希伯特矩陣,還有其他的矩陣天生就是這樣病態條件,例如凡德芒矩陣,如下:

2

X1 X1 1

2

[ X2 X2 1 ]

2

X3 X3 1

2

P.S:X1意思就是X1的平方,以此類推

Update:

題目來自"應用數值方法"這本書的第10章 P10-11 10.6 10.9題

譯者:王晉中

1 Answer

Rating
  • 老師
    Lv 7
    1 decade ago
    Favorite Answer

    1.

    將A矩陣每ㄧ列除上該列絕對值最大的元素 則A矩陣變成

    >> A=[8/(-10),2/(10),1;1,1/(-9),3/(-9);1,-1/15,6/15]

    A =

    -0.8000 0.2000 1.0000

    1.0000 -0.1111 -0.3333

    1.0000 -0.0667 0.4000

    然後利用MATLAB來求範數(norm)

    >> norm(A,'fro')

    ans =

    1.9920

    >> norm(A,1)

    ans =

    2.8000

    >> norm(A,inf)

    ans =

    2

    --------------------------------------------------------------

    2.

    (a).

    已知 x1=4,x2=2,x3=7 代入凡德芒矩陣(Vandermonde matrix) 所以 Vandermonde matrix 為

    >> A=[16,4,1;4,2,1;49,7,1]

    A =

    16 4 1

    4 2 1

    49 7 1

    此矩陣的列和範數(row-sum norm)為 49+7+1=57

    A的反矩陣為

    >> inv(A)

    ans =

    -0.1667 0.1000 0.0667

    1.5000 -1.1000 -0.4000

    -2.3333 2.8000 0.5333

    此矩陣的列和範數為 abs(-2.3333)+2.8+0.5333=5.6667

    所以條件數(condition number)為

    57*(5.6667)=323

    若是使用MATLAB計算

    >> cond(A,inf)

    ans =

    323

    (b).

    光譜條件數(Spectral norm)

    >> cond(A)

    ans =

    216.1294

    夫拉班條件數(Frobenius norm)

    >> cond(A,'fro')

    ans =

    217.4843

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