pagenorm

Page-wise matrix or vector norm

Syntax

N = pagenorm(X)

N = pagenorm(X,p)

N = pagenorm(X,"fro")

N = pagenorm(V)

N = pagenorm(V,p)

Description

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)).

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.

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

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.

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

Norm of Matrix Pages

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

Norm of Vector Pages

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

Frobenius Norm of Matrix Pages

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

X — Input array with matrix pages

matrix | multidimensional array

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

Data Types: single | double
Complex Number Support: Yes

V — Input array with vector pages

vector | multidimensional array

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

Data Types: single | double
Complex Number Support: Yes

p — Norm type

2 (default) | positive real scalar | Inf

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.

p	Matrix pages	Vector pages
1	max(sum(abs(A)))	sum(abs(u))
2	max(svd(A))	sum(abs(u).^2)^(1/2)
Positive, real-valued numeric scalar	—	sum(abs(u).^p)^(1/p)
Inf	max(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

N — Norm value

scalar | multidimensional array

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

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.

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

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• Pages larger than 32x32 are not supported when calculating the 2-norm of each matrix page using pagenorm(X) or pagenorm(X,2).

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced in R2022b