1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177
|
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
|