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pagenorm

Page-wise matrix or vector norm

Description

example

N = pagenorm(X) returns the 2-norm of each matrix page of N-D array X. Each page of the output array N is given by N(1,1,i) = norm(X(:,:,i)).

If X has more than three dimensions, then pagenorm returns an N-D array with the same dimensions, as in N(1,1,i,j,k) = norm(X(:,:,i,j,k)).

example

N = pagenorm(X,p) returns the p-norm of each page of X, where p is 1, 2, or Inf.

  • If p = 1, then N contains the maximum absolute column sum of each page of X.

  • If p = 2, then N contains the maximum singular value of each page of X.

  • If p = Inf, then N contains the maximum absolute row sum of each page of X.

example

N = pagenorm(X,"fro") returns the Frobenius norm of each page of X.

example

N = pagenorm(V), where V is an array with a vector on each page, returns the 2-norm of each page of N-D array V. Each page of the output array N is given by N(1,1,i) = norm(V(:,1,i)) or N(1,1,i) = norm(V(1,:,i)), depending on the orientation of the vectors.

example

N = pagenorm(V,p) returns the generalized vector p-norm of each page of V, where p is any positive real value or Inf.

Examples

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Create a 3-D array with a matrix on each of two pages.

a = magic(3);
b = pascal(3);
A = cat(3,a,b);

Calculate the 2-norm of each page. This command returns the maximum singular value of each matrix.

N = pagenorm(A)
N = 
N(:,:,1) =

    15


N(:,:,2) =

    7.8730

Now calculate the infinity norm of each page. This command returns the maximum absolute row sum of each matrix.

Ninf = pagenorm(A,Inf)
Ninf = 
Ninf(:,:,1) =

    15


Ninf(:,:,2) =

    10

When each page of a multidimensional array contains a vector, you can use pagenorm to calculate vector norms of each page.

Create a 3-D array with a vector on each of three pages.

a = 1:10;
b = 11:20;
c = 21:30;
V = cat(3,a,b,c);

Calculate the 2-norm of each page in the array. This command returns the magnitude of each vector.

N = pagenorm(V)
N = 
N(:,:,1) =

   19.6214


N(:,:,2) =

   49.8498


N(:,:,3) =

   81.1480

Now calculate the 1-norm of each page. This command returns the sum of the element magnitudes for each vector.

N1 = pagenorm(V,1)
N1 = 
N1(:,:,1) =

    55


N1(:,:,2) =

   155


N1(:,:,3) =

   255

Create a 3-D array with a matrix on each of three pages.

a = randi(10,3,3);
b = hilb(3);
c = pascal(3);
A = cat(3,a,b,c);

Calculate the Frobenius norm of each page.

N = pagenorm(A,"fro")
N = 
N(:,:,1) =

   21.9089


N(:,:,2) =

    1.4136


N(:,:,3) =

    7.9373

Input Arguments

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Input array with matrix pages, specified as a matrix or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

Input array with vector pages, specified as a vector or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

Norm type, specified as 2 (default), a positive real scalar, or Inf. The valid values of p and what they return depend on whether each page of the input array is a matrix or vector, as shown in the table. See norm for definitions of each norm type.

Note

This table does not reflect the actual algorithms used in calculations.

pMatrix pagesVector pages
1max(sum(abs(A)))sum(abs(u))
2 max(svd(A))sum(abs(u).^2)^(1/2)
Positive, real-valued numeric scalarsum(abs(u).^p)^(1/p)
Infmax(sum(abs(A')))max(abs(u))

Example: pagenorm(X,Inf) returns the maximum absolute row sum of each matrix page in X.

Example: pagenorm(V,2) returns the magnitude of each vector page in V.

Output Arguments

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Norm value, returned as a scalar or multidimensional array. The norm gives a measure of the magnitude of the elements. By convention, pagenorm returns NaN if the input contains any NaN values on a page.

More About

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Array Pages

Page-wise functions like pagenorm operate on 2-D matrices that have been arranged into a multidimensional array. For example, with a 3-D array the elements in the third dimension of the array are commonly called pages because they stack on top of each other like pages in a book. Each page is a matrix that gets operated on by the function.

3-D array with several matrices stacked on top of each other as pages in the third dimension

You can also assemble a collection of 2-D matrices into a higher dimensional array, like a 4-D or 5-D array, and in these cases pagenorm still treats the fundamental unit of the array as a 2-D matrix that gets operated on, such as X(:,:,i,j,k,l).

The cat function is useful to assemble a collection of matrices into a multidimensional array, and the zeros function is useful to preallocate a multidimensional array.

Tips

  • The squeeze function is useful to reshape results from pagenorm as a column vector.

  • When pagenorm operates on a single matrix or vector, the results might differ from norm slightly due to floating-point round-off error.

Extended Capabilities

Version History

Introduced in R2022b

See Also

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