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
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
|
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
|
