linalg::mult_lq Interface Reference

Multiplies a general matrix by the orthogonal matrix Q from a LQ factorization. More...

Public Member Functions
	mult_lq_mtx
	mult_lq_mtx_cmplx
	mult_lq_vec
	mult_lq_vec_cmplx

Detailed Description

Multiplies a general matrix by the orthogonal matrix Q from a LQ factorization.

Syntax 1

Multiplies a general matrix by the orthogonal matrix \( Q \) from a LQ factorization such that: \( C = op(Q) C \), or \( C = C op(Q) \).

subroutine mult_qr(logical lside, logical trans, real(real64) a(:,:), real(real64) tau(:), real(real64) c(:,:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)
subroutine mult_qr(logical lside, logical trans, complex(real64) a(:,:), complex(real64) tau(:), complex(real64) c(:,:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

Parameters

[in]	lside	Set to true to apply \( Q \) or \( Q^T \) from the left; else, set to false to apply \( Q \) or \( Q^T \) from the right.
[in]	trans	Set to true to apply \( Q^T \); else, set to false. In the event \( Q \) is complex-valued, \( Q^H \) is computed instead of \( Q^T \).
[in]	a	On input, an K-by-P matrix containing the elementary reflectors output from the LQ factorization. If lside is set to true, P = M; else, if lside is set to false, P = N.
[in]	tau	A K-element array containing the scalar factors of each elementary reflector defined in a.
[in,out]	c	On input, the M-by-N matrix C. On output, the product of the orthogonal matrix Q and the original matrix C.
[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.
[in,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.

Syntax 2

Multiplies a vector by the orthogonal matrix \( Q \) from a QR factorization such that: \( C = op(Q) C\).

subroutine mult_qr(logical trans, real(real64) a(:,:), real(real64) tau(:), real(real64) c(:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)
subroutine mult_qr(logical trans, complex(real64) a(:,:), complex(real64) tau(:), complex(real64) c(:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

Parameters

[in]	trans	Set to true to apply \( Q^T \); else, set to false. In the event \( Q \) is complex-valued, \( Q^H \) is computed instead of \( Q^T \).
[in]	a	On input, an K-by-M matrix containing the elementary reflectors output from the LQ factorization. Notice, the contents of this matrix are restored on exit.
[in]	tau	A K-element array containing the scalar factors of each elementary reflector defined in a.
[in,out]	c	On input, the M-element vector C. On output, the product of the orthogonal matrix Q and the original vector C.
[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 DORMLQ (ZUNMLQ 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
 
    ! Local Variables
    real(real64) :: a(3,3), tau(3), b(3)
    integer(int32) :: i, pvt(3)
 
    ! 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)
 
    ! Solve the lower triangular problem and store the solution in B.
    !
    ! A comment 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 mult_lq(.true., a, tau, b)
 
    ! Display the results
    print '(A)', "LQ Solution: X = "
    print '(F8.4)', (b(i), i = 1, size(b))
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 3833 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