linalg::form_lq Interface Reference

Forms the orthogonal matrix Q from the elementary reflectors returned by the LQ factorization algorithm. More...

Public Member Functions
	form_lq_no_pivot
	form_lq_no_pivot_cmplx

Detailed Description

Forms the orthogonal matrix Q from the elementary reflectors returned by the LQ factorization algorithm.

Syntax

subroutine form_lq(real(real64) l(:,:), real(real64) tau(:), real(real64) q(:,:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)
subroutine form_lq(complex(real64) l(:,:), complex(real64) tau(:), complex(real64) q(:,:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

Parameters

[in,out]	l	On input, an M-by-N matrix where the elements above the diagonal contain the elementary reflectors generated from the LQ factorization performed by lq_factor. On and below the diagonal the matrix contains the matrix L. On output, the elements above the diagonal are zeroed sucht hat the remaining matrix is the M-by-N lower trapezoidal matrix L where only the M-by-M submatrix is the lower triangular matrix L. Notice, M must be less than or equal to N for this routine.
[in]	tau	A MIN(M, N)-element array containing the scalar factors of each elementary reflector defined in l.
[out]	q	An N-by-N matrix where the orthogonal matrix Q will be written.
[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.

Notes

This routine utilizes the LAPACK routine DORGLQ (ZUNGLQ in the complex case).

Usage

The folowing example illustrates the solution of a system of equations using LQ factorization.

program example
    use iso_fortran_env, only : real64, int32
    use linalg
    implicit none
 
    ! Variables
    real(real64) :: a(3,3), b(3), q(3,3), tau(3), x(3)
    integer(int32) :: i
 
    ! Build the 3-by-3 matrix A.
    !     | 1   2   3 |
    ! A = | 4   5   6 |
    !     | 7   8   0 |
    a = reshape( &
        [1.0d0, 4.0d0, 7.0d0, 2.0d0, 5.0d0, 8.0d0, 3.0d0, 6.0d0, 0.0d0], &
        [3, 3])
 
    ! Build the right-hand-side vector B.
    !     | -1 |
    ! b = | -2 |
    !     | -3 |
    b = [-1.0d0, -2.0d0, -3.0d0]
 
    ! The solution is:
    !     |  1/3 |
    ! x = | -2/3 |
    !     |   0  |
 
    ! Compute the LQ factorization
    call lq_factor(a, tau)
 
    ! Build L and Q.  A is overwritten with L
    call form_lq(a, tau, q)
 
    ! Solve the lower triangular problem and store the solution in B.
    !
    ! A few notes about this solution noting we've factored A = L * Q.
    !
    ! We then have to solve: L * Q * X = B for X.  If we let Y = Q * X, then
    ! we solve the lower triangular system L * Y = B for Y.
    call solve_triangular_system(.false., .false., .true., a, b)
 
    ! Now we've solved the lower triangular system L * Y = B for Y.  At
    ! this point we solve the problem: Q * X = Y.  Q is an orthogonal matrix;
    ! therefore, inv(Q) = Q**T.  We can solve this by multiplying both
    ! sides by Q**T:
    !
    ! Compute Q**T * B = X
    call mtx_mult(.true., 1.0d0, q, b, 0.0d0, x)
 
    ! Display the results
    print '(A)', "LQ Solution: X = "
    print '(F8.4)', (x(i), i = 1, size(x))
end program

The above program produces the following output.

LQ Solution: X = 
   0.3333
  -0.6667
   0.0000

See Also

• LAPACK Users Manual

Definition at line 3684 of file linalg.f90.

---

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

• D:/Code/linalg/src/linalg.f90