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subroutine dogleg(n,r,lr,diag,qtb,delta,x,wa1,wa2)
integer n,lr
double precision delta
double precision r(lr),diag(n),qtb(n),x(n),wa1(n),wa2(n)
c **********
c
c subroutine dogleg
c
c given an m by n matrix a, an n by n nonsingular diagonal
c matrix d, an m-vector b, and a positive number delta, the
c problem is to determine the convex combination x of the
c gauss-newton and scaled gradient directions that minimizes
c (a*x - b) in the least squares sense, subject to the
c restriction that the euclidean norm of d*x be at most delta.
c
c this subroutine completes the solution of the problem
c if it is provided with the necessary information from the
c qr factorization of a. that is, if a = q*r, where q has
c orthogonal columns and r is an upper triangular matrix,
c then dogleg expects the full upper triangle of r and
c the first n components of (q transpose)*b.
c
c the subroutine statement is
c
c subroutine dogleg(n,r,lr,diag,qtb,delta,x,wa1,wa2)
c
c where
c
c n is a positive integer input variable set to the order of r.
c
c r is an input array of length lr which must contain the upper
c triangular matrix r stored by rows.
c
c lr is a positive integer input variable not less than
c (n*(n+1))/2.
c
c diag is an input array of length n which must contain the
c diagonal elements of the matrix d.
c
c qtb is an input array of length n which must contain the first
c n elements of the vector (q transpose)*b.
c
c delta is a positive input variable which specifies an upper
c bound on the euclidean norm of d*x.
c
c x is an output array of length n which contains the desired
c convex combination of the gauss-newton direction and the
c scaled gradient direction.
c
c wa1 and wa2 are work arrays of length n.
c
c subprograms called
c
c minpack-supplied ... dpmpar,enorm
c
c fortran-supplied ... dabs,dmax1,dmin1,dsqrt
c
c argonne national laboratory. minpack project. march 1980.
c burton s. garbow, kenneth e. hillstrom, jorge j. more
c
c **********
integer i,j,jj,jp1,k,l
double precision alpha,bnorm,epsmch,gnorm,one,qnorm,sgnorm,sum,
* temp,zero
double precision dpmpar,enorm
data one,zero /1.0d0,0.0d0/
c
c epsmch is the machine precision.
c
epsmch = dpmpar(1)
c
c first, calculate the gauss-newton direction.
c
jj = (n*(n + 1))/2 + 1
do 50 k = 1, n
j = n - k + 1
jp1 = j + 1
jj = jj - k
l = jj + 1
sum = zero
if (n .lt. jp1) go to 20
do 10 i = jp1, n
sum = sum + r(l)*x(i)
l = l + 1
10 continue
20 continue
temp = r(jj)
if (temp .ne. zero) go to 40
l = j
do 30 i = 1, j
temp = dmax1(temp,dabs(r(l)))
l = l + n - i
30 continue
temp = epsmch*temp
if (temp .eq. zero) temp = epsmch
40 continue
x(j) = (qtb(j) - sum)/temp
50 continue
c
c test whether the gauss-newton direction is acceptable.
c
do 60 j = 1, n
wa1(j) = zero
wa2(j) = diag(j)*x(j)
60 continue
qnorm = enorm(n,wa2)
if (qnorm .le. delta) go to 140
c
c the gauss-newton direction is not acceptable.
c next, calculate the scaled gradient direction.
c
l = 1
do 80 j = 1, n
temp = qtb(j)
do 70 i = j, n
wa1(i) = wa1(i) + r(l)*temp
l = l + 1
70 continue
wa1(j) = wa1(j)/diag(j)
80 continue
c
c calculate the norm of the scaled gradient and test for
c the special case in which the scaled gradient is zero.
c
gnorm = enorm(n,wa1)
sgnorm = zero
alpha = delta/qnorm
if (gnorm .eq. zero) go to 120
c
c calculate the point along the scaled gradient
c at which the quadratic is minimized.
c
do 90 j = 1, n
wa1(j) = (wa1(j)/gnorm)/diag(j)
90 continue
l = 1
do 110 j = 1, n
sum = zero
do 100 i = j, n
sum = sum + r(l)*wa1(i)
l = l + 1
100 continue
wa2(j) = sum
110 continue
temp = enorm(n,wa2)
sgnorm = (gnorm/temp)/temp
c
c test whether the scaled gradient direction is acceptable.
c
alpha = zero
if (sgnorm .ge. delta) go to 120
c
c the scaled gradient direction is not acceptable.
c finally, calculate the point along the dogleg
c at which the quadratic is minimized.
c
bnorm = enorm(n,qtb)
temp = (bnorm/gnorm)*(bnorm/qnorm)*(sgnorm/delta)
temp = temp - (delta/qnorm)*(sgnorm/delta)**2
* + dsqrt((temp-(delta/qnorm))**2
* +(one-(delta/qnorm)**2)*(one-(sgnorm/delta)**2))
alpha = ((delta/qnorm)*(one - (sgnorm/delta)**2))/temp
120 continue
c
c form appropriate convex combination of the gauss-newton
c direction and the scaled gradient direction.
c
temp = (one - alpha)*dmin1(sgnorm,delta)
do 130 j = 1, n
x(j) = temp*wa1(j) + alpha*x(j)
130 continue
140 continue
return
c
c last card of subroutine dogleg.
c
end
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