pagepinv

Page-wise Moore-Penrose pseudoinverse

Since R2024a

Syntax

B = pagepinv(A)

B = pagepinv(A,tol)

Description

B = pagepinv(A) computes the Moore-Penrose pseudoinverse of each page of N-D array A. Each page of the output array B is given by B(:,:,i) = pinv(A(:,:,i)).

If A has more than three dimensions, then pagepinv returns an N-D array with the same dimensions, as in B(:,:,i,j,k) = pinv(A(:,:,i,j,k)).

B = pagepinv(A,tol) specifies a value for one or more tolerances. pagepinv treats singular values of the pages of A that are less than or equal to the corresponding tolerance as zero.

Examples

Pseudoinverse of 3-D Array Pages

Create a 3-by-3-by-3 array.

A = cat(3,magic(3),hilb(3),pascal(3))

A = 
A(:,:,1) =

     8     1     6
     3     5     7
     4     9     2


A(:,:,2) =

    1.0000    0.5000    0.3333
    0.5000    0.3333    0.2500
    0.3333    0.2500    0.2000


A(:,:,3) =

     1     1     1
     1     2     3
     1     3     6

Calculate the matrix pseudoinverse of each array page.

B = pagepinv(A)

B = 
B(:,:,1) =

    0.1472   -0.1444    0.0639
   -0.0611    0.0222    0.1056
   -0.0194    0.1889   -0.1028


B(:,:,2) =

    9.0000  -36.0000   30.0000
  -36.0000  192.0000 -180.0000
   30.0000 -180.0000  180.0000


B(:,:,3) =

    3.0000   -3.0000    1.0000
   -3.0000    5.0000   -2.0000
    1.0000   -2.0000    1.0000

Set Tolerances of 3-D Array Pages

Create a 3-by-3-by-3 array.

A = cat(3,magic(3),hilb(3),pascal(3))

A = 
A(:,:,1) =

     8     1     6
     3     5     7
     4     9     2


A(:,:,2) =

    1.0000    0.5000    0.3333
    0.5000    0.3333    0.2500
    0.3333    0.2500    0.2000


A(:,:,3) =

     1     1     1
     1     2     3
     1     3     6

Create a 1-by-1-by-3 array of tolerances.

tol = cat(3,2,0.2,1)

tol = 
tol(:,:,1) =

     2


tol(:,:,2) =

    0.2000


tol(:,:,3) =

     1

Calculate the matrix pseudoinverse of each page with the corresponding tolerance value.

B = pagepinv(A,tol)

B = 
B(:,:,1) =

    0.1472   -0.1444    0.0639
   -0.0611    0.0222    0.1056
   -0.0194    0.1889   -0.1028


B(:,:,2) =

    0.4857    0.2701    0.1899
    0.2701    0.1502    0.1056
    0.1899    0.1056    0.0742


B(:,:,3) =

    0.0048    0.0116    0.0212
    0.0116    0.0283    0.0516
    0.0212    0.0516    0.0939

Input Arguments

A — Input array

matrix | multidimensional array

Input array, specified as a matrix or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

tol — Singular value tolerance

scalar | multidimensional array

Singular value tolerance, specified as a scalar or a multidimensional array of size [1 1 size(A, 3:ndims(A))]. Specify a scalar tolerance to use the same tolerance for all array pages. Specify an array of tolerances to use a different tolerance for each page. pagepinv treats singular values that are less than or equal to tol as zeros during the computation of the pseudoinverse.

The default tolerance is max(size(A, [1 2])) * eps(pagenorm(A)).

Example: pagepinv(A,1e-4)

More About

Array Pages

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

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 pagepinv still treats the fundamental unit of the array as a 2-D matrix that the function operates on, such as X(:,:,i,j,k,l).

The cat function is useful for assembling a collection of matrices into a multidimensional array, and the zeros function is useful for preallocating a multidimensional array.

Tips

• Results obtained using pagepinv are numerically equivalent to computing the pseudoinverses of each of the same matrices in a for-loop. However, the two results might differ slightly due to floating-point round-off error.

Version History

Introduced in R2024a