Revision **616fe292ea4c66f8db35d04c9fa045fea56eea02** authored by Roger Koenker on **04 September 2016, 13:02:15 UTC**, committed by cran-robot on **04 September 2016, 13:02:15 UTC**

dsel05.f

```
subroutine dsel05( k, n, x)
integer k, n
double precision x(n)
c
c Selects the smallest k elements of the array x[1:n].
c The input array is permuted so that the smallest k elements of
c x are x(i), i = 1,...,k, (in arbitrary order) and x(k) is the
c kth smallest element.
c
c This is a Fortran 77 version of the Algol 68 procedure from
c
c R.W. Floyd and R.L. Rivest: "Algorithm 489: The Algorithm
c SELECT---for Finding the $i$th Smallest of $n$ Elements",
c Comm. ACM 18, 3 (1975) 173,
c
c including some modifications suggested in
c
c T. Brown: "Remark on Algorithm 489", ACM Trans. Math.
c Software 3, 2 (1976), 301-304.
c
c Array stack(2,nstack) permits up to nstack levels of recursion.
c For standard parameters cs <= 1 and cutoff >= 600,
c nstack = 5 suffices for n up to 2**31-1 (maximum integer*4).
integer nstack
parameter (nstack=10)
integer stack(2,nstack)
c
c Parameters cutoff, cs and csd are as originally proposed.
integer cutoff
parameter (cutoff=600)
double precision cs, csd
parameter (cs=0.5d0, csd=0.5d0)
c Brown's version
c parameter (cs=0.5d0, csd=0.1d0)
c
c Subprograms called:
intrinsic dble, exp, log, max, min, sign
c
c Written by K.C. Kiwiel, 8 March 2006, kiwiel@ibspan.waw.pl
c
c Local variables:
integer i, j, jstack, l, m, r, s, sd
double precision dm, swap, v, z
l=1
r=n
jstack=0
c entry to SELECT( x, n, l, r, k)
c SELECT will rearrange the values of the array segment x[l:r] so
c that x(k) (for some given k; l <= k <= r) will contain the
c (k-l+1)-th smallest value, l <= i <= k will imply x(i) <= x(k),
c and k <= i <= r will imply x(k) <= x(i).
c while r > l do
1 continue
if (l.ge.r) goto 6
c The additional test below prevents stack overflow.
if (r-l.gt.cutoff .and. jstack.lt.nstack) then
c Use SELECT recursively on a sample of size s to get an
c estimate for the (k-l+1)-th smallest element into x(k),
c biased slightly so that the (k-l+1)-th element is
c expected to lie in the smaller set after partitioning.
m=r-l+1
i=k-l+1
dm=m
z=log(dm)
s=cs*exp(2*z/3)+0.5d0
sd=csd*sqrt(z*s*(1-s/dm))*sign(1d0,i-dm/2)+0.5d0
if (i.eq.m/2) sd=0
c Brown's modification
c sd=csd*sqrt(z*s*(1-s/dm))*(2*i/dm-1)+0.5d0
c Push the current l and r on the stack.
jstack=jstack+1
stack(1,jstack)=l
stack(2,jstack)=r
c Find new l and r for the next recursion.
l=max(dble(l),k-i*(s/dm)+sd)+0.5d0
r=min(dble(r),k-i*(s/dm)+sd+s)+0.5d0
c call SELECT( x, n, l, r, k)
goto 1
endif
2 continue
c Partition x[l:r] about the pivot v := x(k).
v=x(k)
c Initialize pointers for partitioning.
i=l
j=r
c Swap x(l) and x(k).
x(k)=x(l)
x(l)=v
if (v.lt.x(r)) then
c Swap x(l) and x(r).
x(l)=x(r)
x(r)=v
endif
c while i < j do
3 continue
if (i.lt.j) then
c Swap x(i) and x(j).
swap=x(j)
x(j)=x(i)
x(i)=swap
i=i+1
j=j-1
c Scan up to find element >= v.
4 continue
if (x(i).lt.v) then
i=i+1
goto 4
endif
c Scan down to find element <= v.
5 continue
if (x(j).gt.v) then
j=j-1
goto 5
endif
goto 3
c end of while i < j do
endif
if (x(l).eq.v) then
c Swap x(l) and x(j).
swap=x(j)
x(j)=v
x(l)=swap
else
j=j+1
c Swap x(j) and x(r).
swap=x(j)
x(j)=x(r)
x(r)=swap
endif
c Now adjust l, r so that they surround the subset containing
c the (k-l+1)-th smallest element.
if (j.le.k) l=j+1
if (k.le.j) r=j-1
goto 1
c end of while r > l do
6 continue
c Exit if the stack is empty.
if (jstack.eq.0) return
c Pop l and r from the stack.
l=stack(1,jstack)
r=stack(2,jstack)
jstack=jstack-1
c Continue as if after a return from a recursive call.
goto 2
end
```

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