subroutine dqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info)
integer ldx,n,k,job,info
double precision x(ldx,1),qraux(1),y(1),qy(1),qty(1),b(1),rsd(1),
* xb(1)
c
c dqrsl applies the output of dqrdc to compute coordinate
c transformations, projections, and least squares solutions.
c for k .le. min(n,p), let xk be the matrix
c
c xk = (x(jpvt(1)),x(jpvt(2)), ... ,x(jpvt(k)))
c
c formed from columnns jpvt(1), ... ,jpvt(k) of the original
c n x p matrix x that was input to dqrdc (if no pivoting was
c done, xk consists of the first k columns of x in their
c original order). dqrdc produces a factored orthogonal matrix q
c and an upper triangular matrix r such that
c
c xk = q * (r)
c (0)
c
c this information is contained in coded form in the arrays
c x and qraux.
c
c on entry
c
c x double precision(ldx,p).
c x contains the output of dqrdc.
c
c ldx integer.
c ldx is the leading dimension of the array x.
c
c n integer.
c n is the number of rows of the matrix xk. it must
c have the same value as n in dqrdc.
c
c k integer.
c k is the number of columns of the matrix xk. k
c must nnot be greater than min(n,p), where p is the
c same as in the calling sequence to dqrdc.
c
c qraux double precision(p).
c qraux contains the auxiliary output from dqrdc.
c
c y double precision(n)
c y contains an n-vector that is to be manipulated
c by dqrsl.
c
c job integer.
c job specifies what is to be computed. job has
c the decimal expansion abcde, with the following
c meaning.
c
c if a.ne.0, compute qy.
c if b,c,d, or e .ne. 0, compute qty.
c if c.ne.0, compute b.
c if d.ne.0, compute rsd.
c if e.ne.0, compute xb.
c
c note that a request to compute b, rsd, or xb
c automatically triggers the computation of qty, for
c which an array must be provided in the calling
c sequence.
c
c on return
c
c qy double precision(n).
c qy conntains q*y, if its computation has been
c requested.
c
c qty double precision(n).
c qty contains trans(q)*y, if its computation has
c been requested. here trans(q) is the
c transpose of the matrix q.
c
c b double precision(k)
c b contains the solution of the least squares problem
c
c minimize norm2(y - xk*b),
c
c if its computation has been requested. (note that
c if pivoting was requested in dqrdc, the j-th
c component of b will be associated with column jpvt(j)
c of the original matrix x that was input into dqrdc.)
c
c rsd double precision(n).
c rsd contains the least squares residual y - xk*b,
c if its computation has been requested. rsd is
c also the orthogonal projection of y onto the
c orthogonal complement of the column space of xk.
c
c xb double precision(n).
c xb contains the least squares approximation xk*b,
c if its computation has been requested. xb is also
c the orthogonal projection of y onto the column space
c of x.
c
c info integer.
c info is zero unless the computation of b has
c been requested and r is exactly singular. in
c this case, info is the index of the first zero
c diagonal element of r and b is left unaltered.
c
c the parameters qy, qty, b, rsd, and xb are not referenced
c if their computation is not requested and in this case
c can be replaced by dummy variables in the calling program.
c to save storage, the user may in some cases use the same
c array for different parameters in the calling sequence. a
c frequently occuring example is when one wishes to compute
c any of b, rsd, or xb and does not need y or qty. in this
c case one may identify y, qty, and one of b, rsd, or xb, while
c providing separate arrays for anything else that is to be
c computed. thus the calling sequence
c
c call dqrsl(x,ldx,n,k,qraux,y,dum,y,b,y,dum,110,info)
c
c will result in the computation of b and rsd, with rsd
c overwriting y. more generally, each item in the following
c list contains groups of permissible identifications for
c a single callinng sequence.
c
c 1. (y,qty,b) (rsd) (xb) (qy)
c
c 2. (y,qty,rsd) (b) (xb) (qy)
c
c 3. (y,qty,xb) (b) (rsd) (qy)
c
c 4. (y,qy) (qty,b) (rsd) (xb)
c
c 5. (y,qy) (qty,rsd) (b) (xb)
c
c 6. (y,qy) (qty,xb) (b) (rsd)
c
c in any group the value returned in the array allocated to
c the group corresponds to the last member of the group.
c
c linpack. this version dated 08/14/78 .
c g.w. stewart, university of maryland, argonne national lab.
c
c dqrsl uses the following functions and subprograms.
c
c blas daxpy,dcopy,ddot
c fortran dabs,min0,mod
c
c internal variables
c
integer i,j,jj,ju,kp1
double precision ddot,t,temp
logical cb,cqy,cqty,cr,cxb
c
c
c set info flag.
c
info = 0
c
c determine what is to be computed.
c
cqy = job/10000 .ne. 0
cqty = mod(job,10000) .ne. 0
cb = mod(job,1000)/100 .ne. 0
cr = mod(job,100)/10 .ne. 0
cxb = mod(job,10) .ne. 0
ju = min0(k,n-1)
c
c special action when n=1.
c
if (ju .ne. 0) go to 40
if (cqy) qy(1) = y(1)
if (cqty) qty(1) = y(1)
if (cxb) xb(1) = y(1)
if (.not.cb) go to 30
if (x(1,1) .ne. 0.0d0) go to 10
info = 1
go to 20
10 continue
b(1) = y(1)/x(1,1)
20 continue
30 continue
if (cr) rsd(1) = 0.0d0
go to 250
40 continue
c
c set up to compute qy or qty.
c
if (cqy) call dcopy(n,y,1,qy,1)
if (cqty) call dcopy(n,y,1,qty,1)
if (.not.cqy) go to 70
c
c compute qy.
c
do 60 jj = 1, ju
j = ju - jj + 1
if (qraux(j) .eq. 0.0d0) go to 50
temp = x(j,j)
x(j,j) = qraux(j)
t = -ddot(n-j+1,x(j,j),1,qy(j),1)/x(j,j)
call daxpy(n-j+1,t,x(j,j),1,qy(j),1)
x(j,j) = temp
50 continue
60 continue
70 continue
if (.not.cqty) go to 100
c
c compute trans(q)*y.
c
do 90 j = 1, ju
if (qraux(j) .eq. 0.0d0) go to 80
temp = x(j,j)
x(j,j) = qraux(j)
t = -ddot(n-j+1,x(j,j),1,qty(j),1)/x(j,j)
call daxpy(n-j+1,t,x(j,j),1,qty(j),1)
x(j,j) = temp
80 continue
90 continue
100 continue
c
c set up to compute b, rsd, or xb.
c
if (cb) call dcopy(k,qty,1,b,1)
kp1 = k + 1
if (cxb) call dcopy(k,qty,1,xb,1)
if (cr .and. k .lt. n) call dcopy(n-k,qty(kp1),1,rsd(kp1),1)
if (.not.cxb .or. kp1 .gt. n) go to 120
do 110 i = kp1, n
xb(i) = 0.0d0
110 continue
120 continue
if (.not.cr) go to 140
do 130 i = 1, k
rsd(i) = 0.0d0
130 continue
140 continue
if (.not.cb) go to 190
c
c compute b.
c
do 170 jj = 1, k
j = k - jj + 1
if (x(j,j) .ne. 0.0d0) go to 150
info = j
c ......exit
go to 180
150 continue
b(j) = b(j)/x(j,j)
if (j .eq. 1) go to 160
t = -b(j)
call daxpy(j-1,t,x(1,j),1,b,1)
160 continue
170 continue
180 continue
190 continue
if (.not.cr .and. .not.cxb) go to 240
c
c compute rsd or xb as required.
c
do 230 jj = 1, ju
j = ju - jj + 1
if (qraux(j) .eq. 0.0d0) go to 220
temp = x(j,j)
x(j,j) = qraux(j)
if (.not.cr) go to 200
t = -ddot(n-j+1,x(j,j),1,rsd(j),1)/x(j,j)
call daxpy(n-j+1,t,x(j,j),1,rsd(j),1)
200 continue
if (.not.cxb) go to 210
t = -ddot(n-j+1,x(j,j),1,xb(j),1)/x(j,j)
call daxpy(n-j+1,t,x(j,j),1,xb(j),1)
210 continue
x(j,j) = temp
220 continue
230 continue
240 continue
250 continue
return
end