linalg::solve_least_squares Interface Reference

Solves the overdetermined or underdetermined system \( A X = B \) of M equations of N unknowns. Notice, it is assumed that matrix A has full rank. More...

Public Member Functions
	solve_least_squares_mtx
	solve_least_squares_mtx_cmplx
	solve_least_squares_vec
	solve_least_squares_vec_cmplx

Detailed Description

Solves the overdetermined or underdetermined system \( A X = B \) of M equations of N unknowns. Notice, it is assumed that matrix A has full rank.

Syntax

subroutine solve_least_squares(real(real64) a(:,:), real(real64) b(:,:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)
subroutine solve_least_squares(complex(real64) a(:,:), complex(real64) b(:,:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)
subroutine solve_least_squares(real(real64) a(:,:), real(real64) b(:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)
subroutine solve_least_squares(complex(real64) a(:,:), complex(real64) b(:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

Parameters

[in,out]	a	On input, the M-by-N matrix A. On output, if M >= N, the QR factorization of A in the form as output by qr_factor; else, if M < N, the LQ factorization of A.
[in,out]	b	If M >= N, the M-by-NRHS matrix B. On output, the first N rows contain the N-by-NRHS solution matrix X. If M < N, an N-by-NRHS matrix with the first M rows containing the matrix B. On output, the N-by-NRHS solution matrix X.
[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.
[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_INVALID_OPERATION_ERROR: Occurs if a is not of full rank.

Notes

This routine utilizes the LAPACK routine DGELS (ZGELS in the complex case).

Usage

The following example illustrates the least squares solution of an overdetermined system of linear equations.

program example
    use iso_fortran_env, only : real64, int32
    use linalg
    implicit none
 
    ! Local Variables
    real(real64) :: a(3,2), b(3)
    integer(int32) :: i
 
    ! Build the 3-by-2 matrix A
    !     | 2   1 |
    ! A = |-3   1 |
    !     |-1   1 |
    a = reshape([2.0d0, -3.0d0, -1.0d0, 1.0d0, 1.0d0, 1.0d0], [3, 2])
 
    ! Build the right-hand-side vector B.
    !     |-1 |
    ! b = |-2 |
    !     | 1 |
    b = [-1.0d0, -2.0d0, 1.0d0]
 
    ! The solution is:
    ! x = [0.13158, -0.57895]**T
 
    ! Compute the solution via a least-squares approach.  The results overwrite
    ! the first 2 elements in b.
    call solve_least_squares(a, b)
 
    ! Display the results
    print '(A)', "Least Squares Solution: X = "
    print '(F9.5)', (b(i), i = 1, size(a, 2))
end program

The above program produces the following output.

Least Squares Solution: X =
 0.13158
-0.57895

Definition at line 2753 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