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subroutine lmpar(n,r,ldr,ipvt,diag,qtb,delta,par,x,sdiag,wa1,
* wa2)
integer n,ldr
integer ipvt(n)
double precision delta,par
double precision r(ldr,n),diag(n),qtb(n),x(n),sdiag(n),wa1(n),
* wa2(n)
c **********
c
c subroutine lmpar
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,
c the problem is to determine a value for the parameter
c par such that if x solves the system
c
c a*x = b , sqrt(par)*d*x = 0 ,
c
c in the least squares sense, and dxnorm is the euclidean
c norm of d*x, then either par is zero and
c
c (dxnorm-delta) .le. 0.1*delta ,
c
c or par is positive and
c
c abs(dxnorm-delta) .le. 0.1*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, with column pivoting, of a. that is, if
c a*p = q*r, where p is a permutation matrix, q has orthogonal
c columns, and r is an upper triangular matrix with diagonal
c elements of nonincreasing magnitude, then lmpar expects
c the full upper triangle of r, the permutation matrix p,
c and the first n components of (q transpose)*b. on output
c lmpar also provides an upper triangular matrix s such that
c
c t t t
c p *(a *a + par*d*d)*p = s *s .
c
c s is employed within lmpar and may be of separate interest.
c
c only a few iterations are generally needed for convergence
c of the algorithm. if, however, the limit of 10 iterations
c is reached, then the output par will contain the best
c value obtained so far.
c
c the subroutine statement is
c
c subroutine lmpar(n,r,ldr,ipvt,diag,qtb,delta,par,x,sdiag,
c 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 n by n array. on input the full upper triangle
c must contain the full upper triangle of the matrix r.
c on output the full upper triangle is unaltered, and the
c strict lower triangle contains the strict upper triangle
c (transposed) of the upper triangular matrix s.
c
c ldr is a positive integer input variable not less than n
c which specifies the leading dimension of the array r.
c
c ipvt is an integer input array of length n which defines the
c permutation matrix p such that a*p = q*r. column j of p
c is column ipvt(j) of the identity matrix.
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 par is a nonnegative variable. on input par contains an
c initial estimate of the levenberg-marquardt parameter.
c on output par contains the final estimate.
c
c x is an output array of length n which contains the least
c squares solution of the system a*x = b, sqrt(par)*d*x = 0,
c for the output par.
c
c sdiag is an output array of length n which contains the
c diagonal elements of the upper triangular matrix s.
c
c wa1 and wa2 are work arrays of length n.
c
c subprograms called
c
c minpack-supplied ... dpmpar,enorm,qrsolv
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,iter,j,jm1,jp1,k,l,nsing
double precision dxnorm,dwarf,fp,gnorm,parc,parl,paru,p1,p001,
* sum,temp,zero
double precision dpmpar,enorm
data p1,p001,zero /1.0d-1,1.0d-3,0.0d0/
c
c dwarf is the smallest positive magnitude.
c
dwarf = dpmpar(2)
c
c compute and store in x the gauss-newton direction. if the
c jacobian is rank-deficient, obtain a least squares solution.
c
nsing = n
do 10 j = 1, n
wa1(j) = qtb(j)
if (r(j,j) .eq. zero .and. nsing .eq. n) nsing = j - 1
if (nsing .lt. n) wa1(j) = zero
10 continue
if (nsing .lt. 1) go to 50
do 40 k = 1, nsing
j = nsing - k + 1
wa1(j) = wa1(j)/r(j,j)
temp = wa1(j)
jm1 = j - 1
if (jm1 .lt. 1) go to 30
do 20 i = 1, jm1
wa1(i) = wa1(i) - r(i,j)*temp
20 continue
30 continue
40 continue
50 continue
do 60 j = 1, n
l = ipvt(j)
x(l) = wa1(j)
60 continue
c
c initialize the iteration counter.
c evaluate the function at the origin, and test
c for acceptance of the gauss-newton direction.
c
iter = 0
do 70 j = 1, n
wa2(j) = diag(j)*x(j)
70 continue
dxnorm = enorm(n,wa2)
fp = dxnorm - delta
if (fp .le. p1*delta) go to 220
c
c if the jacobian is not rank deficient, the newton
c step provides a lower bound, parl, for the zero of
c the function. otherwise set this bound to zero.
c
parl = zero
if (nsing .lt. n) go to 120
do 80 j = 1, n
l = ipvt(j)
wa1(j) = diag(l)*(wa2(l)/dxnorm)
80 continue
do 110 j = 1, n
sum = zero
jm1 = j - 1
if (jm1 .lt. 1) go to 100
do 90 i = 1, jm1
sum = sum + r(i,j)*wa1(i)
90 continue
100 continue
wa1(j) = (wa1(j) - sum)/r(j,j)
110 continue
temp = enorm(n,wa1)
parl = ((fp/delta)/temp)/temp
120 continue
c
c calculate an upper bound, paru, for the zero of the function.
c
do 140 j = 1, n
sum = zero
do 130 i = 1, j
sum = sum + r(i,j)*qtb(i)
130 continue
l = ipvt(j)
wa1(j) = sum/diag(l)
140 continue
gnorm = enorm(n,wa1)
paru = gnorm/delta
if (paru .eq. zero) paru = dwarf/dmin1(delta,p1)
c
c if the input par lies outside of the interval (parl,paru),
c set par to the closer endpoint.
c
par = dmax1(par,parl)
par = dmin1(par,paru)
if (par .eq. zero) par = gnorm/dxnorm
c
c beginning of an iteration.
c
150 continue
iter = iter + 1
c
c evaluate the function at the current value of par.
c
if (par .eq. zero) par = dmax1(dwarf,p001*paru)
temp = dsqrt(par)
do 160 j = 1, n
wa1(j) = temp*diag(j)
160 continue
call qrsolv(n,r,ldr,ipvt,wa1,qtb,x,sdiag,wa2)
do 170 j = 1, n
wa2(j) = diag(j)*x(j)
170 continue
dxnorm = enorm(n,wa2)
temp = fp
fp = dxnorm - delta
c
c if the function is small enough, accept the current value
c of par. also test for the exceptional cases where parl
c is zero or the number of iterations has reached 10.
c
if (dabs(fp) .le. p1*delta
* .or. parl .eq. zero .and. fp .le. temp
* .and. temp .lt. zero .or. iter .eq. 10) go to 220
c
c compute the newton correction.
c
do 180 j = 1, n
l = ipvt(j)
wa1(j) = diag(l)*(wa2(l)/dxnorm)
180 continue
do 210 j = 1, n
wa1(j) = wa1(j)/sdiag(j)
temp = wa1(j)
jp1 = j + 1
if (n .lt. jp1) go to 200
do 190 i = jp1, n
wa1(i) = wa1(i) - r(i,j)*temp
190 continue
200 continue
210 continue
temp = enorm(n,wa1)
parc = ((fp/delta)/temp)/temp
c
c depending on the sign of the function, update parl or paru.
c
if (fp .gt. zero) parl = dmax1(parl,par)
if (fp .lt. zero) paru = dmin1(paru,par)
c
c compute an improved estimate for par.
c
par = dmax1(parl,par+parc)
c
c end of an iteration.
c
go to 150
220 continue
c
c termination.
c
if (iter .eq. 0) par = zero
return
c
c last card of subroutine lmpar.
c
end