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
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
|
C/MEMBR ADD NAME=REMEZ,SSI=0
subroutine remez(ngr,nfc,iext,ad,x,y,des,grid,wt,a,p,q,alpha)
c!purpose
c this subroutine implements the remez exchange algorithm
c for the weighted chebyshev approximation of a continuous
c function with a sum of cosines. inputs to the subroutine
c are a dense grid which replaces the frequency axis, the
c desired function on this grid, the weight function on the
c grid, the number of cosines, and an initial guess of the
c extremal frequencies. the program minimizes the chebyshev
c error by determining the best location of the extremal
c frequencies (points of maximum error) and then calculates
c the coefficients of the best approximation.
c!calling sequence
c subroutine remez(ngr,nfc,iext,ad,x,y,des,grid,wt,a,p,q,alpha)
c dimensioning of arrays in calling program is as follows:
c working vectors ad,x,y,a,p,q: nfc+2
c input vectors des,grid,wt: ngr
c input vector iext: nfc+2
c output vector alpha: nfc+2
c!
dimension iext(*),ad(*),alpha(*),x(*),y(*)
dimension des(*),grid(*),wt(*)
dimension a(*),p(*),q(*)
double precision pi2,dnum,dden,dtemp,a,p,q
double precision dk,dak
double precision ad,dev,x,y
double precision gee,dgee01
double precision alpha
common /rem001/ pi2,dev,nfcns,ngrid
common /oops/niter,iout
c
c the program allows a maximum number of iterations of 25
c
itrmax=25
ngrid=ngr
nfcns=nfc
pi=4.0d+0*atan(1.0d+0)
pi2=2.0d+0*pi
devl=-1.0d+0
nz=nfcns+1
nzz=nfcns+2
niter=0
100 continue
iext(nzz)=ngrid+1
niter=niter+1
if(niter.gt.itrmax) go to 400
do 110 j=1,nz
jxt=iext(j)
dtemp=grid(jxt)
dtemp=cos(dtemp*pi2)
110 x(j)=dtemp
jet=(nfcns-1)/15+1
do 120 j=1,nz
120 ad(j)=dgee01(j,nz,jet,x)
dnum=0.0d+0
dden=0.0d+0
k=1
do 130 j=1,nz
l=iext(j)
dtemp=ad(j)*des(l)
dnum=dnum+dtemp
dtemp=dble(k)*ad(j)/wt(l)
dden=dden+dtemp
130 k=-k
dev=dnum/dden
c write(6,131) dev
c 131 format(1x,12hdeviation = ,f12.9)
nu=1
if(dev.gt.0.0d+0) nu=-1
dev=-dble(nu)*dev
k=nu
do 140 j=1,nz
l=iext(j)
dtemp=dble(k)*dev/wt(l)
y(j)=des(l)+dtemp
140 k=-k
if(dev.gt.devl) go to 150
call ouch
go to 400
150 devl=dev
jchnge=0
k1=iext(1)
knz=iext(nz)
klow=0
nut=-nu
j=1
c
c search for the extremal frequencies of the best
c approximation
c
200 if(j.eq.nzz) ynz=comp
if(j.ge.nzz) go to 300
kup=iext(j+1)
l=iext(j)+1
nut=-nut
if(j.eq.2) y1=comp
comp=dev
if(l.ge.kup) go to 220
err=gee(l,nz,ad,x,y,grid)
err=(err-des(l))*wt(l)
dtemp=dble(nut)*err-comp
if(dtemp.le.0.0d+0) go to 220
comp=dble(nut)*err
210 l=l+1
if(l.ge.kup) go to 215
err=gee(l,nz,ad,x,y,grid)
err=(err-des(l))*wt(l)
dtemp=dble(nut)*err-comp
if(dtemp.le.0.0d+0) go to 215
comp=dble(nut)*err
go to 210
215 iext(j)=l-1
j=j+1
klow=l-1
jchnge=jchnge+1
go to 200
220 l=l-1
225 l=l-1
if(l.le.klow) go to 250
err=gee(l,nz,ad,x,y,grid)
err=(err-des(l))*wt(l)
dtemp=dble(nut)*err-comp
if(dtemp.gt.0.0d+0) go to 230
if(jchnge.le.0) go to 225
go to 260
230 comp=dble(nut)*err
235 l=l-1
if(l.le.klow) go to 240
err=gee(l,nz,ad,x,y,grid)
err=(err-des(l))*wt(l)
dtemp=dble(nut)*err-comp
if(dtemp.le.0.0d+0) go to 240
comp=dble(nut)*err
go to 235
240 klow=iext(j)
iext(j)=l+1
j=j+1
jchnge=jchnge+1
go to 200
250 l=iext(j)+1
if(jchnge.gt.0) go to 215
255 l=l+1
if(l.ge.kup) go to 260
err=gee(l,nz,ad,x,y,grid)
err=(err-des(l))*wt(l)
dtemp=dble(nut)*err-comp
if(dtemp.le.0.0d+0) go to 255
comp=dble(nut)*err
go to 210
260 klow=iext(j)
j=j+1
go to 200
300 if(j.gt.nzz) go to 320
if(k1.gt.iext(1)) k1=iext(1)
if(knz.lt.iext(nz)) knz=iext(nz)
nut1=nut
nut=-nu
l=0
kup=k1
comp=ynz*(1.000010d+0)
luck=1
310 l=l+1
if(l.ge.kup) go to 315
err=gee(l,nz,ad,x,y,grid)
err=(err-des(l))*wt(l)
dtemp=dble(nut)*err-comp
if(dtemp.le.0.0d+0) go to 310
comp=dble(nut)*err
j=nzz
go to 210
315 luck=6
go to 325
320 if(luck.gt.9) go to 350
if(comp.gt.y1) y1=comp
k1=iext(nzz)
325 l=ngrid+1
klow=knz
nut=-nut1
comp=y1*(1.000010d+0)
330 l=l-1
if(l.le.klow) go to 340
err=gee(l,nz,ad,x,y,grid)
err=(err-des(l))*wt(l)
dtemp=dble(nut)*err-comp
if(dtemp.le.0.0d+0) go to 330
j=nzz
comp=dble(nut)*err
luck=luck+10
go to 235
340 if(luck.eq.6) go to 370
do 345 j=1,nfcns
nzzmj=nzz-j
nzmj=nz-j
345 iext(nzzmj)=iext(nzmj)
iext(1)=k1
go to 100
350 kn=iext(nzz)
do 360 j=1,nfcns
360 iext(j)=iext(j+1)
iext(nz)=kn
go to 100
370 if(jchnge.gt.0) go to 100
c
c calculation of the coefficients of the best approximation
c using the inverse discrete fourier transform
c
400 continue
nm1=nfcns-1
fsh=1.0e-06
gtemp=grid(1)
x(nzz)=-2.0d+0
cn=2*nfcns-1
delf=1.0d+0/cn
l=1
kkk=0
if(grid(1).lt.0.010d+0.and.grid(ngrid).gt.0.490d+0) kkk=1
if(nfcns.le.3) kkk=1
if(kkk.eq.1) go to 405
dtemp=cos(pi2*grid(1))
dnum=cos(pi2*grid(ngrid))
aa=2.0d+0/(dtemp-dnum)
bb=-(dtemp+dnum)/(dtemp-dnum)
405 continue
do 430 j=1,nfcns
ft=j-1
ft=ft*delf
xt=cos(pi2*ft)
if(kkk.eq.1) go to 410
xt=(xt-bb)/aa
xt1=sqrt(1.0d+0-xt*xt)
ft=atan2(xt1,xt)/pi2
410 xe=x(l)
if(xt.gt.xe) go to 420
if((xe-xt).lt.fsh) go to 415
l=l+1
go to 410
415 a(j)=y(l)
go to 425
420 if((xt-xe).lt.fsh) go to 415
grid(1)=ft
a(j)=gee(1,nz,ad,x,y,grid)
425 continue
if(l.gt.1) l=l-1
430 continue
grid(1)=gtemp
dden=pi2/cn
do 510 j=1,nfcns
dtemp=0.0d+0
dnum=j-1
dnum=dnum*dden
if(nm1.lt.1) go to 505
do 500 k=1,nm1
dak=a(k+1)
dk=k
500 dtemp=dtemp+dak*cos(dnum*dk)
505 dtemp=2.0d+0*dtemp+a(1)
510 alpha(j)=dtemp
do 550 j=2,nfcns
550 alpha(j)=2.0d+0*alpha(j)/cn
alpha(1)=alpha(1)/cn
if(kkk.eq.1) go to 545
p(1)=2.0d+0*alpha(nfcns)*bb+alpha(nm1)
p(2)=2.0d+0*aa*alpha(nfcns)
q(1)=alpha(nfcns-2)-alpha(nfcns)
do 540 j=2,nm1
if(j.lt.nm1) go to 515
aa=0.5*aa
bb=0.5*bb
515 continue
p(j+1)=0.0d+0
do 520 k=1,j
a(k)=p(k)
520 p(k)=2.0d+0*bb*a(k)
p(2)=p(2)+a(1)*2.0d+0*aa
jm1=j-1
do 525 k=1,jm1
525 p(k)=p(k)+q(k)+aa*a(k+1)
jp1=j+1
do 530 k=3,jp1
530 p(k)=p(k)+aa*a(k-1)
if(j.eq.nm1) go to 540
do 535 k=1,j
535 q(k)=-a(k)
nf1j=nfcns-1-j
q(1)=q(1)+alpha(nf1j)
540 continue
do 543 j=1,nfcns
543 alpha(j)=p(j)
545 continue
if(nfcns.gt.3)return
alpha(nfcns+1)=0.0d+0
alpha(nfcns+2)=0.0d+0
return
end
|
