函數norm

 

X為向量,
n = norm(X)       % 即求歐幾理德範數(Euclidean norm)

n = norm(X, inf)  % 即求無窮範數(maximum norm)

n = norm(X, 1)    % 即求1-範數

n = norm(X, -inf) % 即求向量X的元素的絕對值的最小值

n = norm(X, p)    % 即求p-範數,所以norm(X, 2) = norm(X)

 

矩陣的範數

A為矩陣,

n = norm(A)       % 求歐幾理德範數(Euclidean norm),等於A矩陣的最大奇異值(singular value)

n = norm(A, 1)   % 求A的列向量的1-範數中的最大值

n = norm(A, 2)   % 求A的歐幾理德範數(Euclidean norm),等同於norm(A)

n = norm(A, inf)  % 求行範數,等同於A的行向量的1-範數中的最大值,即: max(sum(abs(A')))

n = norm(A, 'fro') % 求A矩陣的Frobenius範數

矩陣的p階範數估計需要自己寫程式求得,

計算公式舉例如下,

a = magic(3) 

sum(sum(abs(a)^4))^(1/4)

a =

     8   1   6

     3   5   7

     4   9   2

ans =

   19.7411

 

註一:

p階範數公式:

p-norm.png  

註二:

Matlab的magic square函數

M = magic(n)

returns an n-by-n matrix constructed from the integers 1 through n^2 with equal row and column sums.

The order n must be a scalar greater than or equal to 3.

 

Tips:

A magic square, scaled by its magic sum, is doubly stochastic.

 

Examples:

The magic square of order 3 is

M = magic(3)

M = 

    8    1    6
    3    5    7
    4    9    2

This is called a magic square because the sum of the elements in each column is the same.

 sum(M) =

     15    15    15

And the sum of the elements in each row, obtained by transposing twice, is the same.

 sum(M')' = 

     15
     15
     15

This is also a special magic square because the diagonal elements have the same sum.

sum(diag(M)) =

     15

The value of the characteristic sum for a magic square of order n is

sum(1:n^2)/n

which, when n = 3, is 15.

 

Algorithm:

There are three different algorithms:

  • n odd

  • n even but not divisible by four

  • n divisible by four

 

To make this apparent, type

for n = 3:20
    A = magic(n);
    r(n) = rank(A);
end

For n odd, the rank of the magic square is n. For n divisible by 4, the rank is 3. For n even but not divisible by 4, the rank is n/2 + 2.

[(3:20)',r(3:20)']
ans =
     3     3
     4     3
     5     5
     6     5
     7     7
     8     3
     9     9
    10     7
    11    11
    12     3
    13    13
    14     9
    15    15
    16     3
    17    17
    18    11
    19    19
    20     3

Plotting A for n = 18, 19, 20 shows the characteristic plot for each category.

magic_18.jpg    magic_19.jpg    magic_20.jpg    

Limitations:

If you supply n less than 3magic returns either a nonmagic square, or else the degenerate magic squares 1 and [].

創作者介紹

Totui

Totui 發表在 痞客邦 PIXNET 留言(0) 人氣()