pageinv

Page-wise matrix inverse

Since R2022a

Syntax

Y = pageinv(X)

[Y,RC] = pageinv(X)

Description

Y = pageinv(X) computes the inverse of each page of N-D array X. Each page of the output array Y is given by Y(:,:,i) = inv(X(:,:,i)).

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

[Y,RC] = pageinv(X) also returns the reciprocal condition number of each page of X. The value of RC(1,1,i) is the reciprocal of the condition number of X(:,:,i). If RC(1,1,i) < eps, then the call inv(X(:,:,i)) returns a warning that X(:,:,i) is ill conditioned. However, pageinv does not issue a warning for ill-conditioned inputs.

Examples

Inverse of 3-D Array Pages

Create a 3-by-3-by-3 array by concatenating several 3-by-3 matrices.

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

X = 
X(:,:,1) =

     8     1     6
     3     5     7
     4     9     2


X(:,:,2) =

    1.0000    0.5000    0.3333
    0.5000    0.3333    0.2500
    0.3333    0.2500    0.2000


X(:,:,3) =

     1     1     1
     1     2     3
     1     3     6

Calculate the matrix inverse of each array page.

Y = pageinv(X)

Y = 
Y(:,:,1) =

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


Y(:,:,2) =

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


Y(:,:,3) =

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

The ith page of the output, Y(:,:,i), is found by calculating the matrix inverse Y(:,:,i) = inv(X(:,:,i)).

Check Reciprocal Condition Number of Array Pages

Create a 10-by-10-by-2 array with a magic square matrix on the first page and a Hilbert matrix on the second page.

X = cat(3,magic(10),hilb(10))

X = 
X(:,:,1) =

    92    99     1     8    15    67    74    51    58    40
    98    80     7    14    16    73    55    57    64    41
     4    81    88    20    22    54    56    63    70    47
    85    87    19    21     3    60    62    69    71    28
    86    93    25     2     9    61    68    75    52    34
    17    24    76    83    90    42    49    26    33    65
    23     5    82    89    91    48    30    32    39    66
    79     6    13    95    97    29    31    38    45    72
    10    12    94    96    78    35    37    44    46    53
    11    18   100    77    84    36    43    50    27    59


X(:,:,2) =

    1.0000    0.5000    0.3333    0.2500    0.2000    0.1667    0.1429    0.1250    0.1111    0.1000
    0.5000    0.3333    0.2500    0.2000    0.1667    0.1429    0.1250    0.1111    0.1000    0.0909
    0.3333    0.2500    0.2000    0.1667    0.1429    0.1250    0.1111    0.1000    0.0909    0.0833
    0.2500    0.2000    0.1667    0.1429    0.1250    0.1111    0.1000    0.0909    0.0833    0.0769
    0.2000    0.1667    0.1429    0.1250    0.1111    0.1000    0.0909    0.0833    0.0769    0.0714
    0.1667    0.1429    0.1250    0.1111    0.1000    0.0909    0.0833    0.0769    0.0714    0.0667
    0.1429    0.1250    0.1111    0.1000    0.0909    0.0833    0.0769    0.0714    0.0667    0.0625
    0.1250    0.1111    0.1000    0.0909    0.0833    0.0769    0.0714    0.0667    0.0625    0.0588
    0.1111    0.1000    0.0909    0.0833    0.0769    0.0714    0.0667    0.0625    0.0588    0.0556
    0.1000    0.0909    0.0833    0.0769    0.0714    0.0667    0.0625    0.0588    0.0556    0.0526

Calculate the matrix inverse of each array page. Specify two outputs to also return the reciprocal condition number of each matrix that is being inverted.

[Y,RC] = pageinv(X);
RC

RC = 
RC(:,:,1) =

   3.0812e-18


RC(:,:,2) =

   2.8285e-14

Compare the reciprocal condition numbers to eps. The results indicate that the magic square matrix is ill conditioned, so the results of the inversion operation for that page are not reliable.

RC < eps

ans = 1x1x2 logical array
ans(:,:,1) =

   1


ans(:,:,2) =

   0

Check the results by calculating the residual norm between X(:,:,i)*Y(:,:,i) and the identity matrix. The large residual norm for the first page confirms that the matrix inverse for that page is not accurate.

norm(X(:,:,1)*Y(:,:,1) - eye(10))

ans = 23.2199

norm(X(:,:,2)*Y(:,:,2) - eye(10))

ans = 1.5594e-04

Input Arguments

X — Input array

matrix | multidimensional array

Input array, specified as a matrix or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

Y — Page-wise inverses

multidimensional array

Page-wise inverses, returned as a multidimensional array. Y has the same size and data type as X. Each page Y(:,:,i,...) is the matrix inverse of X(:,:,i,...).

RC — Reciprocal condition numbers

multidimensional array

Reciprocal condition numbers, returned as a multidimensional array with the same number of pages as X. The value of RC(1,1,i) is the reciprocal of the condition number of X(:,:,i). If RC(1,1,i) < eps, then the call inv(X(:,:,i)) returns a warning that X(:,:,i) is ill conditioned. However, pageinv does not issue a warning for ill-conditioned inputs.

More About

Array Pages

Page-wise functions like pageinv 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 pageinv 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 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 pageinv are numerically equivalent to computing the inverses of each of the same matrices in a for-loop. However, the two results might differ 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™.

This function fully supports GPU arrays. 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 R2022a