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subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa)
integer m,n,lda,lipvt
integer ipvt(lipvt)
logical pivot
double precision a(lda,n),rdiag(n),acnorm(n),wa(n)
c **********
c
c subroutine qrfac
c
c this subroutine uses householder transformations with column
c pivoting (optional) to compute a qr factorization of the
c m by n matrix a. that is, qrfac determines an orthogonal
c matrix q, a permutation matrix p, and an upper trapezoidal
c matrix r with diagonal elements of nonincreasing magnitude,
c such that a*p = q*r. the householder transformation for
c column k, k = 1,2,...,min(m,n), is of the form
c
c t
c i - (1/u(k))*u*u
c
c where u has zeros in the first k-1 positions. the form of
c this transformation and the method of pivoting first
c appeared in the corresponding linpack subroutine.
c
c the subroutine statement is
c
c subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa)
c
c where
c
c m is a positive integer input variable set to the number
c of rows of a.
c
c n is a positive integer input variable set to the number
c of columns of a.
c
c a is an m by n array. on input a contains the matrix for
c which the qr factorization is to be computed. on output
c the strict upper trapezoidal part of a contains the strict
c upper trapezoidal part of r, and the lower trapezoidal
c part of a contains a factored form of q (the non-trivial
c elements of the u vectors described above).
c
c lda is a positive integer input variable not less than m
c which specifies the leading dimension of the array a.
c
c pivot is a logical input variable. if pivot is set true,
c then column pivoting is enforced. if pivot is set false,
c then no column pivoting is done.
c
c ipvt is an integer output array of length lipvt. ipvt
c defines the permutation matrix p such that a*p = q*r.
c column j of p is column ipvt(j) of the identity matrix.
c if pivot is false, ipvt is not referenced.
c
c lipvt is a positive integer input variable. if pivot is false,
c then lipvt may be as small as 1. if pivot is true, then
c lipvt must be at least n.
c
c rdiag is an output array of length n which contains the
c diagonal elements of r.
c
c acnorm is an output array of length n which contains the
c norms of the corresponding columns of the input matrix a.
c if this information is not needed, then acnorm can coincide
c with rdiag.
c
c wa is a work array of length n. if pivot is false, then wa
c can coincide with rdiag.
c
c subprograms called
c
c minpack-supplied ... dpmpar,enorm
c
c fortran-supplied ... dmax1,dsqrt,min0
c
c argonne national laboratory. minpack project. march 1980.
c burton s. garbow, kenneth e. hillstrom, jorge j. more
c
c **********
integer i,j,jp1,k,kmax,minmn
double precision ajnorm,epsmch,one,p05,sum,temp,zero
double precision dpmpar,enorm
data one,p05,zero /1.0d0,5.0d-2,0.0d0/
c
c epsmch is the machine precision.
c
epsmch = dpmpar(1)
c
c compute the initial column norms and initialize several arrays.
c
do 10 j = 1, n
acnorm(j) = enorm(m,a(1,j))
rdiag(j) = acnorm(j)
wa(j) = rdiag(j)
if (pivot) ipvt(j) = j
10 continue
c
c reduce a to r with householder transformations.
c
minmn = min0(m,n)
do 110 j = 1, minmn
if (.not.pivot) go to 40
c
c bring the column of largest norm into the pivot position.
c
kmax = j
do 20 k = j, n
if (rdiag(k) .gt. rdiag(kmax)) kmax = k
20 continue
if (kmax .eq. j) go to 40
do 30 i = 1, m
temp = a(i,j)
a(i,j) = a(i,kmax)
a(i,kmax) = temp
30 continue
rdiag(kmax) = rdiag(j)
wa(kmax) = wa(j)
k = ipvt(j)
ipvt(j) = ipvt(kmax)
ipvt(kmax) = k
40 continue
c
c compute the householder transformation to reduce the
c j-th column of a to a multiple of the j-th unit vector.
c
ajnorm = enorm(m-j+1,a(j,j))
if (ajnorm .eq. zero) go to 100
if (a(j,j) .lt. zero) ajnorm = -ajnorm
do 50 i = j, m
a(i,j) = a(i,j)/ajnorm
50 continue
a(j,j) = a(j,j) + one
c
c apply the transformation to the remaining columns
c and update the norms.
c
jp1 = j + 1
if (n .lt. jp1) go to 100
do 90 k = jp1, n
sum = zero
do 60 i = j, m
sum = sum + a(i,j)*a(i,k)
60 continue
temp = sum/a(j,j)
do 70 i = j, m
a(i,k) = a(i,k) - temp*a(i,j)
70 continue
if (.not.pivot .or. rdiag(k) .eq. zero) go to 80
temp = a(j,k)/rdiag(k)
rdiag(k) = rdiag(k)*dsqrt(dmax1(zero,one-temp**2))
if (p05*(rdiag(k)/wa(k))**2 .gt. epsmch) go to 80
rdiag(k) = enorm(m-j,a(jp1,k))
wa(k) = rdiag(k)
80 continue
90 continue
100 continue
rdiag(j) = -ajnorm
110 continue
return
c
c last card of subroutine qrfac.
c
end
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