linalg::mtx_pinverse Interface Reference

Computes the Moore-Penrose pseudo-inverse of a M-by-N matrix using the singular value decomposition of the matrix. More...

Public Member Functions
	mtx_pinverse_dbl
	mtx_pinverse_cmplx

Detailed Description

Computes the Moore-Penrose pseudo-inverse of a M-by-N matrix using the singular value decomposition of the matrix.

Syntax

subroutine mtx_pinverse(real(real64) a(:,:), real(real64) ainv(:,:), optional real(real64) tol, optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)
subroutine mtx_pinverse(complex(real64) a(:,:), complex(real64) ainv(:,:), optional real(real64) tol, optional complex(real64) work(:), optional integer(int32) olwork, optional real(real64) rwork(:), optional class(errors) err)

Parameters

[in,out]	a	On input, the M-by-N matrix to invert. The matrix is overwritten on output.
[out]	ainv	The N-by-M matrix where the pseudo-inverse of a will be written.
[in]	tol	An optional input, that if supplied, overrides the default tolerance on singular values such that singular values less than this tolerance are forced to have a reciprocal of zero, as opposed to 1/S(I). The default tolerance is: MAX(M, N) * EPS * MAX(S). If the supplied value is less than a value that causes an overflow, the tolerance reverts back to its default value, and the operation continues; however, a warning message is issued.
[out]	work	An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.
[out]	olwork	An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.
[out]	rwork	An optional input, that if provided, prevents any local memory allocation for real-valued workspaces. If not provided, the memory required is allocated within. If provided, the length of the array must be at least 6 * MIN(M, N).
[in,out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. LA_ARRAY_SIZE_ERROR: Occurs if any of the input arrays are not sized appropriately. LA_OUT_OF_MEMORY_ERROR: Occurs if local memory must be allocated, and there is insufficient memory available. LA_CONVERGENCE_ERROR: Occurs as a warning if the QR iteration process could not converge to a zero value.

See Also

• Wikipedia
• Wolfram MathWorld
• MathWorks

Usage

The following example illustrates how to compute the Moore-Penrose pseudo-inverse of a matrix.

program example
    use iso_fortran_env, only : int32, real64
    use linalg
    implicit none
 
    ! Variables
    real(real64) :: a(3,2), ai(2,3), ao(3,2), c(2,2)
    integer(int32) :: i
 
    ! Create the 3-by-2 matrix A
    !     | 1   0 |
    ! A = | 0   1 |
    !     | 0   1 |
    a = reshape([1.0d0, 0.0d0, 0.0d0, 0.0d0, 1.0d0, 1.0d0], [3, 2])
    ao = a  ! Just making a copy for later as mtx_pinverse will destroy the
            ! contents of the original matrix
 
    ! The Moore-Penrose pseudo-inverse of this matrix is:
    !         | 1   0    0  |
    ! A**-1 = |             |
    !         | 0  1/2  1/2 |
    call mtx_pinverse(a, ai)
 
    ! Notice, A**-1 * A is an identity matrix.
    c = matmul(ai, ao)
 
    ! Display the inverse
    print '(A)', "Inverse:"
    do i = 1, size(ai, 1)
        print *, ai(i,:)
    end do
 
    ! Display the result of inv(A) * A
    print '(A)', "A**-1 * A:"
    do i = 1, size(c, 1)
        print *, c(i,:)
    end do
end program

The above program produces the following output.

Inverse:
 1.0000000000000000        0.0000000000000000        0.0000000000000000
 0.0000000000000000       0.49999999999999978       0.49999999999999989
A**-1 * A:
 1.0000000000000000        0.0000000000000000
 0.0000000000000000       0.99999999999999967

Definition at line 3157 of file linalg.f90.

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The documentation for this interface was generated from the following file:

• D:/Code/linalg/src/linalg.f90