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 | #include <cfloat>
#include <Rcpp.h>
//[[Rcpp::export]]
Rcpp::List vunirootRcpp(Rcpp::Function f,
Rcpp::NumericVector lower,
Rcpp::NumericVector upper,
Rcpp::NumericVector fa,
Rcpp::NumericVector fb,
int numiter,
double tol) {
using namespace Rcpp;
int size = lower.size();
NumericVector a(clone(lower)), b(clone(upper)), c(clone(a)), Tol(size,0.0);
NumericVector fc(clone(fa));
LogicalVector converged(size,false);
IntegerVector ns(size,-1);
int i;
/* First test if we have found a root at an endpoint */
for(i=0; i<size; i++) {
if (fa[i]==0.0) {
converged[i]=true;
b[i]=a[i];
}
if (fb[i]==0.0) {
converged[i]=true;
}
}
for (int n = 1; n<=numiter; n++) {
R_CheckUserInterrupt(); /* be polite -- did the user hit ctrl-C? */
for(i=0; i<size; i++) {
if (!converged[i]) {
double prev_step = b[i]-a[i]; /* Distance from the last but one
to the last approximation */
double tol_act; /* Actual tolerance */
double p; /* Interpolation step is calcu- */
double q; /* lated in the form p/q; divi-
* sion operations is delayed
* until the last moment */
double new_step; /* Step at this iteration */
if( fabs(fc[i]) < fabs(fb[i]) )
{ /* Swap data for b to be the */
a[i] = b[i]; b[i] = c[i]; c[i] = a[i]; /* best approximation */
fa[i]=fb[i]; fb[i]=fc[i]; fc[i]=fa[i];
}
tol_act = 2*DBL_EPSILON*fabs(b[i]) + tol/2;
new_step = (c[i]-b[i])/2.0;
if( fabs(new_step) <= tol_act || fb[i] == (double)0 )
{
// *Maxit -= maxit;
Tol[i] = fabs(c[i]-b[i]);
converged[i] = true; /* Acceptable approx. is found */
ns[i] = n;
} else {
/* Decide if the interpolation can be tried */
if( fabs(prev_step) >= tol_act /* If prev_step was large enough*/
&& fabs(fa[i]) > fabs(fb[i]) ) { /* and was in true direction,
* Interpolation may be tried */
double t1,cb,t2;
cb = c[i]-b[i];
if( a[i]==c[i] ) { /* If we have only two distinct */
/* points linear interpolation */
t1 = fb[i]/fa[i]; /* can only be applied */
p = cb*t1;
q = 1.0 - t1;
}
else { /* Quadric inverse interpolation*/
q = fa[i]/fc[i]; t1 = fb[i]/fc[i]; t2 = fb[i]/fa[i];
p = t2 * ( cb*q*(q-t1) - (b[i]-a[i])*(t1-1.0) );
q = (q-1.0) * (t1-1.0) * (t2-1.0);
}
if( p>(double)0 ) /* p was calculated with the */
q = -q; /* opposite sign; make p positive */
else /* and assign possible minus to */
p = -p; /* q */
if( p < (0.75*cb*q-fabs(tol_act*q)/2) /* If b+p/q falls in [b,c]*/
&& p < fabs(prev_step*q/2) ) /* and isnt too large */
new_step = p/q; /* it is accepted
* If p/q is too large then the
* bisection procedure can
* reduce [b,c] range to more
* extent */
}
if( fabs(new_step) < tol_act) { /* Adjust the step to be not less*/
if( new_step > (double)0 ) /* than tolerance */
new_step = tol_act;
else
new_step = -tol_act;
}
a[i] = b[i]; fa[i] = fb[i]; /* Save the previous approx. */
b[i] += new_step; // fb = (*f)(b, info); /* Do step to a new approxim. */
}
}
}
if (is_true(all(converged))) break;
fb = f(b);
for(i=0; i<size; i++) {
if( (fb[i] > 0 && fc[i] > 0) || (fb[i] < 0 && fc[i] < 0) ) {
/* Adjust c for it to have a sign opposite to that of b */
c[i] = a[i]; fc[i] = fa[i];
}
}
}
if (is_false(all(converged)))
for (i=0; i<size; i++)
if (!converged[i]) {
Tol[i]=fabs(c[i]-b[i]);
ns[i] = -1;
}
return Rcpp::List::create(_("root") = b, _("iter") = ns, _("tol")=Tol);
}
//[[Rcpp::export]]
Rcpp::NumericVector voptimizeRcpp(Rcpp::Function f,
Rcpp::NumericVector ax,
Rcpp::NumericVector bx,
double tol) {
using Rcpp::NumericVector;
using Rcpp::clone;
using Rcpp::as;
const int size = ax.size();
/* c is the squared inverse of the golden ratio */
const double c = (3. - sqrt(5.)) * .5;
/* Local variables */
NumericVector a = clone(ax), b=clone(bx);
NumericVector w, x, fx, fu, fv, fw;
double p, q, r;
double eps, tol3;
NumericVector d(size, 0.0), e(size, 0.0), u(size, 0.0), xm(size,0.0),
t2(size,0.0), tol1(size,0.0), v(size,0.0);
/* eps is approximately the square root of the relative machine precision. */
eps = DBL_EPSILON;
for (int i=0; i<size; ++i)
tol1[i] = eps + 1.;/* the smallest 1.000... > 1 */
eps = sqrt(eps);
for (int i=0; i<size; ++i)
v[i] = a[i] + c * (b[i] - a[i]);
w = clone(v);
x = clone(v);
fx = f(x);
fv = clone(fx);
fw = clone(fx);
tol3 = tol / 3.;
/* main loop starts here ----------------------------------- */
for(;;) {
R_CheckUserInterrupt(); /* be polite -- did the user hit ctrl-C? */
for (int i=0; i<size; ++i) {
xm[i] = (a[i] + b[i]) * .5;
tol1[i] = eps * fabs(x[i]) + tol3;
t2[i] = tol1[i] * 2.;
}
/* check stopping criterion -- do any fail? */
bool any_failures = false;
for (int i=0; i<size; ++i) {
if (fabs(x[i] - xm[i]) > t2[i] - (b[i] - a[i]) * .5) {
any_failures = true;
break;
}
}
if (!any_failures) break;
for (int i=0; i<size; ++i) {
p = 0.;
q = 0.;
r = 0.;
if (fabs(e[i]) > tol1[i]) { /* fit parabola */
r = (x[i] - w[i]) * (fx[i] - fv[i]);
q = (x[i] - v[i]) * (fx[i] - fw[i]);
p = (x[i] - v[i]) * q - (x[i] - w[i]) * r;
q = (q - r) * 2.;
if (q > 0.) p = -p; else q = -q;
r = e[i];
e[i] = d[i];
}
if (fabs(p) >= fabs(q * .5 * r) ||
p <= q * (a[i] - x[i]) || p >= q * (b[i] - x[i])) { /* a golden-section step */
if (x[i] < xm[i]) e[i] = b[i] - x[i]; else e[i] = a[i] - x[i];
d[i] = c * e[i];
}
else { /* a parabolic-interpolation step */
d[i] = p / q;
u[i] = x[i] + d[i];
/* f must not be evaluated too close to ax or bx */
if (u[i] - a[i] < t2[i] || b[i] - u[i] < t2[i]) {
d[i] = tol1[i];
if (x[i] >= xm[i]) d[i] = -d[i];
}
}
/* f must not be evaluated too close to x */
if (fabs(d[i]) >= tol1[i])
u[i] = x[i] + d[i];
else if (d[i] > 0.)
u[i] = x[i] + tol1[i];
else
u[i] = x[i] - tol1[i];
}
fu = f(u);
/* update a, b, v, w, and x */
for (int i=0; i<size; ++i) {
if (fu[i] <= fx[i]) {
if (u[i] < x[i]) b[i] = x[i]; else a[i] = x[i];
v[i] = w[i]; w[i] = x[i]; x[i] = u[i];
fv[i] = fw[i]; fw[i] = fx[i]; fx[i] = fu[i];
} else {
if (u[i] < x[i]) a[i] = u[i]; else b[i] = u[i];
if (fu[i] <= fw[i] || w[i] == x[i]) {
v[i] = w[i]; fv[i] = fw[i];
w[i] = u[i]; fw[i] = fu[i];
} else if (fu[i] <= fv[i] || v[i] == x[i] || v[i] == w[i]) {
v[i] = u[i]; fv[i] = fu[i];
}
}
}
}
/* end of main loop */
return x;
}
|