pagelsqminnorm

Page-wise minimum-norm least-squares solution to linear equation

Since R2024a

Syntax

X = pagelsqminnorm(A,B)

X = pagelsqminnorm(A,B,tol)

X = pagelsqminnorm(___,RegularizationFactor=alpha)

Description

X = pagelsqminnorm(A,B) computes the minimum-norm solution of the least-squares system AX = B for each page of the N-D arrays A and B. Each page of the output array X is given by X(:,:,i) = lsqminnorm(A(:,:,i),B(:,:,i)).

If A and B have more than three dimensions, then pagelsqminnorm implicitly expands the additional dimensions to solve the least-squares systems of all page combinations, as in X(:,:,i,j,k) = lsqminnorm(A(:,:,i,j,k),B(:,:,i,j,k)).

X = pagelsqminnorm(A,B,tol) specifies one or more tolerances that pagelsqminnorm uses to determine the rank of each page of A.

X = pagelsqminnorm(___,RegularizationFactor=alpha) specifies the Tikhonov regularization factors to apply to each page of the solution X. (since R2025a)

Examples

Least-Squares Solution of 3-D Array Pages

Create a 3-by-3-by-2 array A and a 3-by-1-by-2 array B.

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

B = ones(3,1,2)

B = 
B(:,:,1) =

     1
     1
     1


B(:,:,2) =

     1
     1
     1

Solve the linear system A(:,:,i)*X(:,:,i) = B(:,:,i) for each corresponding set of pages in A and B using pagelsqminnorm.

X = pagelsqminnorm(A,B)

X = 
X(:,:,1) =

    0.0667
    0.0667
    0.0667


X(:,:,2) =

    3.0000
  -24.0000
   30.0000

Input Arguments

A — Coefficient array

matrix | multidimensional array

Coefficient array, specified as a matrix or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

B — Input array

matrix | multidimensional array

Input array, specified as a matrix or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

tol — Rank tolerance

nonnegative scalar | multidimensional array

Rank tolerance, specified as a nonnegative scalar or multidimensional array of size [1 1 size(A,3:ndims(A))]. Specifying the tolerance can help the solutions be resilient against random noise in the coefficient matrix. 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.

By default, pagelsqminnorm computes the tolerance based on the QR decomposition of each page of A.

Example: X = pagelsqminnorm(A,B,1e-2)

Data Types: double

alpha — Tikhonov regularization factors

0 (default) | real number | multidimensional array

Since R2025a

Tikhonov regularization factors for the least-squares solution, specified as a real number or multidimensional array of size [1 1 size(X,3:ndims(X))]. Specifying the regularization factors as alpha returns a solution X that minimizes pagenorm(A.*X-B).^2 + alpha.^2.*pagenorm(X).^2 for each column of each page of X. Specify a real number to use the same regularization factor for all pages. Specify an array to use a different regularization factor for each page.

For ill-conditioned problems, specifying the regularization factors gives preference to solutions with smaller norms.

Data Types: double | single

More About

Array Pages

Page-wise functions like pagelsqminnorm 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 pagelsqminnorm 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

• pagelsqminnorm(A,B,tol) is typically more efficient than pagemtimes(pagepinv(A,tol),B) for computing minimum-norm least-squares solutions to linear systems. pagelsqminnorm uses the complete orthogonal decomposition (COD) to find a low-rank approximation of each page of A, while pagepinv uses the singular value decomposition (SVD). Therefore, the results of pagepinv and pagelsqminnorm do not match exactly.

• Results obtained using pagelsqminnorm are numerically equivalent to using lsqminnorm to compute the minimum-norm solution of each of the same least-squares systems in a for-loop. However, the two results might differ slightly due to floating-point round-off error.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™. (since R2025a)

Usage notes and limitations:

• Code generation does not support Tikhonov regularization for this function.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™. (since R2024b)

The pagelsqminnorm function supports GPU array input with these usage notes and limitations:

• Coefficient array A must be nonsparse.

• Specifying Tikhonov regularization factors is not supported.

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

Version History

Introduced in R2024a

R2025a: Apply Tikhonov regularization to least-squares solution

You can specify the Tikhonov regularization factors alpha to return a solution X that minimizes pagenorm(A.*X-B).^2 + alpha.^2.*pagenorm(X).^2 for each column of each page of X.

R2025a: Support for code generation

You can generate C and C++ code for the pagelsqminnorm function. However, code generation does not support Tikhonov regularization for this function.