qrfac Subroutine

public subroutine qrfac(m, n, a, lda, pivot, ipvt, lipvt, rdiag, acnorm, wa)

this subroutine uses householder transformations with column pivoting (optional) to compute a qr factorization of the m by n matrix a. that is, qrfac determines an orthogonal matrix q, a permutation matrix p, and an upper trapezoidal matrix r with diagonal elements of nonincreasing magnitude, such that ap = qr. the householder transformation for column k, k = 1,2,...,min(m,n), is of the form

                  t
  i - (1/u(k))*u*u

where u has zeros in the first k-1 positions. the form of this transformation and the method of pivoting first appeared in the corresponding linpack subroutine.

the subroutine statement is

subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa)

where

m is a positive integer input variable set to the number of rows of a.

n is a positive integer input variable set to the number of columns of a.

a is an m by n array. on input a contains the matrix for which the qr factorization is to be computed. on output the strict upper trapezoidal part of a contains the strict upper trapezoidal part of r, and the lower trapezoidal part of a contains a factored form of q (the non-trivial elements of the u vectors described above).

lda is a positive integer input variable not less than m which specifies the leading dimension of the array a.

pivot is a logical input variable. if pivot is set true, then column pivoting is enforced. if pivot is set false, then no column pivoting is done.

ipvt is an integer output array of length lipvt. ipvt defines the permutation matrix p such that ap = qr. column j of p is column ipvt(j) of the identity matrix. if pivot is false, ipvt is not referenced.

lipvt is a positive integer input variable. if pivot is false, then lipvt may be as small as 1. if pivot is true, then lipvt must be at least n.

rdiag is an output array of length n which contains the diagonal elements of r.

acnorm is an output array of length n which contains the norms of the corresponding columns of the input matrix a. if this information is not needed, then acnorm can coincide with rdiag.

wa is a work array of length n. if pivot is false, then wa can coincide with rdiag.

Arguments

Type IntentOptional Attributes Name
integer :: m
integer :: n
real(kind=wp) :: a(lda,n)
integer :: lda
logical :: pivot
integer :: ipvt(lipvt)
integer :: lipvt
real(kind=wp) :: rdiag(n)
real(kind=wp) :: acnorm(n)
real(kind=wp) :: wa(n)

Calls

proc~~qrfac~~CallsGraph proc~qrfac minpack_module::qrfac proc~dpmpar minpack_module::dpmpar proc~qrfac->proc~dpmpar proc~enorm minpack_module::enorm proc~qrfac->proc~enorm

Called by

proc~~qrfac~~CalledByGraph proc~qrfac minpack_module::qrfac proc~hybrd minpack_module::hybrd proc~hybrd->proc~qrfac proc~hybrj minpack_module::hybrj proc~hybrj->proc~qrfac proc~hybrd1 minpack_module::hybrd1 proc~hybrd1->proc~hybrd proc~hybrj1 minpack_module::hybrj1 proc~hybrj1->proc~hybrj proc~halo_to_rv_diffcorr halo_orbit_module::halo_to_rv_diffcorr proc~halo_to_rv_diffcorr->proc~hybrd1

Source Code

    subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa)

    implicit none

    integer m , n , lda , lipvt
    integer ipvt(lipvt)
    logical pivot
    real(wp) a(lda,n) , rdiag(n) , acnorm(n) , wa(n)

      integer i , j , jp1 , k , kmax , minmn
      real(wp) ajnorm , epsmch , sum , temp

      real(wp),parameter :: p05 = 5.0e-2_wp

      epsmch = dpmpar(1)  ! the machine precision
!
!     compute the initial column norms and initialize several arrays.
!
      do j = 1 , n
         acnorm(j) = enorm(m,a(1,j))
         rdiag(j) = acnorm(j)
         wa(j) = rdiag(j)
         if ( pivot ) ipvt(j) = j
      enddo
!
!     reduce a to r with householder transformations.
!
      minmn = min(m,n)
      do j = 1 , minmn
         if ( pivot ) then
!
!        bring the column of largest norm into the pivot position.
!
            kmax = j
            do k = j , n
               if ( rdiag(k)>rdiag(kmax) ) kmax = k
            enddo
            if ( kmax/=j ) then
               do i = 1 , m
                  temp = a(i,j)
                  a(i,j) = a(i,kmax)
                  a(i,kmax) = temp
               enddo
               rdiag(kmax) = rdiag(j)
               wa(kmax) = wa(j)
               k = ipvt(j)
               ipvt(j) = ipvt(kmax)
               ipvt(kmax) = k
            endif
         endif
!
!        compute the householder transformation to reduce the
!        j-th column of a to a multiple of the j-th unit vector.
!
         ajnorm = enorm(m-j+1,a(j,j))
         if ( ajnorm/=zero ) then
            if ( a(j,j)<zero ) ajnorm = -ajnorm
            do i = j , m
               a(i,j) = a(i,j)/ajnorm
            enddo
            a(j,j) = a(j,j) + one
!
!        apply the transformation to the remaining columns
!        and update the norms.
!
            jp1 = j + 1
            if ( n>=jp1 ) then
               do k = jp1 , n
                  sum = zero
                  do i = j , m
                     sum = sum + a(i,j)*a(i,k)
                  enddo
                  temp = sum/a(j,j)
                  do i = j , m
                     a(i,k) = a(i,k) - temp*a(i,j)
                  enddo
                  if ( .not.(.not.pivot .or. rdiag(k)==zero) ) then
                     temp = a(j,k)/rdiag(k)
                     rdiag(k) = rdiag(k)*sqrt(max(zero,one-temp**2))
                     if ( p05*(rdiag(k)/wa(k))**2<=epsmch ) then
                        rdiag(k) = enorm(m-j,a(jp1,k))
                        wa(k) = rdiag(k)
                     endif
                  endif
               enddo
            endif
         endif
         rdiag(j) = -ajnorm
      enddo

      end subroutine qrfac