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
|
/* Calculate coefficients of finite difference formulas using
algorithm by Fornberg, 1998.
Reference: http://epubs.siam.org/sam-bin/dbq/article/32250
Author: Pearu Peterson, September 2002
Compilation:
c++ genpfdf.cc -o genpfdf -lcln -lginac
or
c++ `ginac-config --cppflags --libs` genpfdf.cc -o genpfdf
Usage:
./genpfdf n [m] > output_file.c
where
n -- max. number of grid intervals.
m -- max. order of derivative [default n].
*/
#include <iostream>
#include <string>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
static int gen_apply_periodic_fd(void);
static int gen_apply_periodic_fd2(void);
static int gen_fd_coeffs(const numeric &,const numeric &,const numeric &);
int main(int argc, const char *const *argv)
{
numeric n,m;
switch (argc) {
case 2:
n = numeric(argv[1]);
m = n;
break;
case 3:
n = numeric(argv[1]);
m = numeric(argv[2]);
break;
default:
cout << "Usage:" << endl;
cout << " " << *argv << " n [m [s]]" << endl;
cout << " where" << endl;
cout << " n -- max. number of grid intervals" << endl;
cout << " m -- max. order of derivative [default n]" << endl;
return 1;
}
cout <<"/*****************************************************/"<<endl;
cout <<"/******** Periodic Finite Difference Formulae ********/"<<endl;
cout <<"/*****************************************************/"<<endl;
cout <<"/* Author: Pearu Peterson, September 2002 */"<<endl;
cout <<"/* */"<<endl;
cout <<"/* This file is automatically generated using */"<<endl;
cout <<"/* genpfdf.cc utility from fdf package and it */"<<endl;
cout <<"/* contains the following C function */"<<endl;
cout <<"/* */"<<endl;
cout <<"/* void periodic_finite_difference(int n, */"<<endl;
cout <<"/* double *x, double *y, double h, int k, int m); */"<<endl;
cout <<"/* where */"<<endl;
cout <<"/* n -- length of arrays x and y */"<<endl;
cout <<"/* x -- input array */"<<endl;
cout <<"/* y -- output array */"<<endl;
cout <<"/* h -- discretization step of arrays x and y */"<<endl;
cout <<"/* k -- the order of derivative */"<<endl;
cout <<"/* m -- number of grid intervals used to evaluate */"<<endl;
cout <<"/* finite differences at the center point. The */"<<endl;
cout <<"/* error is O(h^(2*(m-1))) within numerical */"<<endl;
cout <<"/* accuracy. */"<<endl;
cout <<"/*****************************************************/"<<endl;
cout <<"/* Reference: */"<<endl;
cout <<"/* http://epubs.siam.org/sam-bin/dbq/article/32250 */"<<endl;
cout <<"/*****************************************************/"<<endl;
cout <<"/* Parameters to genpfdf n="<<n<<" m="<<m<<" */"<<endl;
cout <<"#include <stdio.h>"<<endl;
cout <<"#include <math.h>"<<endl;
for (int i=1;i<=n;++i)
for (int j=1;j<=m && j<=i;++j)
gen_fd_coeffs(numeric(i),numeric(j),numeric(i)/2);
gen_apply_periodic_fd();
gen_apply_periodic_fd2();
cout << "extern void periodic_finite_difference("<<endl
<<"int n, double *x, double *y, double h, int k, int m) {"
<< endl <<"double coeff = pow(1.0/h,k), *c=NULL;"
<< endl <<" switch (k*1000+m) {"<<endl;
for (int i=1;i<=n;++i)
for (int j=1;j<=m&&j<=i;++j)
cout <<" case "<<j*1000+i
<<": c=fd_coeffs_"<<i<<"_"<<j<<"; break;"<<endl;
cout << " default:"<<endl
<<" fprintf(stderr,"<<endl
<<" \"Unsupported derivative/order combination: k,m=%d,%d\\n\","
<<endl<<" k,m);"<<endl<<" }"
<<endl<<" if (c!=NULL) {"
<<endl<<" if (k%2)"
<<endl<<" apply_periodic_fd(n,x,y,coeff,m,c);"
<<endl<<" else"
<<endl<<" apply_periodic_fd2(n,x,y,coeff,m,c);"
<<endl<<" }"
<<endl<<"}"<<endl;
return 0;
}
static
int gen_fd_coeffs(const numeric &n,const numeric &m,const numeric &s) {
symbol x("x");
ex poly = pow(x,s)*pow(log(x),m);
if (n.is_odd())
poly = poly * pow(numeric(2),-m);
//else
// poly = poly;
poly = expand(series_to_poly(poly.series(x==1,n.to_int()+1)));
Digits = 16;
//cout << " /* poly=" << poly << " */"<<endl;
cout <<"static double fd_coeffs_"<<n.to_int()<<"_"<<m.to_int()<<"[] = {"<<endl;
if (n.is_even() && m.is_even())
cout <<" "<< ex_to<numeric>(poly.coeff(x,n.to_int()/2).evalf())
<<","
//<< " /* " << poly.coeff(x,n.to_int()/2) << " */"
<<endl;
for (int i=n.to_int()/2+1,j=1; i<=n; ++i,++j) {
cout << " "
<< ex_to<numeric>(poly.coeff(x,i).evalf());
if (i<n)
cout <<",";
//cout<< " /* " << poly.coeff(x,i) << " */";
cout<<endl;
}
cout<<"};"<<endl;
}
static int gen_apply_periodic_fd(void) {
cout <<"static void apply_periodic_fd"
<<"(int n,double *x,double *y,double coeff,int m,double *c) {"<<endl
<<" int i,k,l=(m+1)/2;"<<endl
<<" if (m%2) {"<<endl
<<" for(i=m;i<n-m;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=m;k>0;k-=2)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]-x[i-k]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" for(i=0;i<m;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=m;k>0;k-=2)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]-x[(i+n-k)%n]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" for(i=n-m;i<n;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=m;k>0;k-=2)"<<endl
<<" d += (*(--c_ptr)) * (x[(i+k)%n]-x[i-k]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" } else {"<<endl
<<" for(i=l;i<n-l;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=l;k>0;--k)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]-x[i-k]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" for(i=0;i<l;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=l;k>0;--k)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]-x[(i+n-k)%n]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" for(i=n-l;i<n;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=l;k>0;--k)"<<endl
<<" d += (*(--c_ptr)) * (x[(i+k)%n]-x[i-k]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" }"<<endl
<<"}"<<endl;
}
static int gen_apply_periodic_fd2(void) {
cout <<"static void apply_periodic_fd2"
<<"(int n,double *x,double *y,double coeff,int m,double *c) {"<<endl
<<" int i,k,l=(m+1)/2;"<<endl
<<" if (m%2) {"<<endl
<<" for(i=m;i<n-m;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=m;k>0;k-=2)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]+x[i-k]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" for(i=0;i<m;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=m;k>0;k-=2)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]+x[(i+n-k)%n]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" for(i=n-m;i<n;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l;"<<endl
<<" for(k=m;k>0;k-=2)"<<endl
<<" d += (*(--c_ptr)) * (x[(i+k)%n]+x[i-k]);"<<endl
<<" y[i] = coeff * d;"<<endl
<<" }"<<endl
<<" } else {"<<endl
<<" for(i=l;i<n-l;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l + 1;"<<endl
<<" for(k=l;k>0;--k)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]+x[i-k]);"<<endl
<<" y[i] = coeff * (d+(*c)*x[i]);"<<endl
<<" }"<<endl
<<" for(i=0;i<l;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l + 1;"<<endl
<<" for(k=l;k>0;--k)"<<endl
<<" d += (*(--c_ptr)) * (x[i+k]+x[(i+n-k)%n]);"<<endl
<<" y[i] = coeff * (d+(*c)*x[i]);"<<endl
<<" }"<<endl
<<" for(i=n-l;i<n;++i) {"<<endl
<<" double d = 0.0, *c_ptr = c + l + 1;"<<endl
<<" for(k=l;k>0;--k)"<<endl
<<" d += (*(--c_ptr)) * (x[(i+k)%n]+x[i-k]);"<<endl
<<" y[i] = coeff * (d+(*c)*x[i]);"<<endl
<<" }"<<endl
<<" }"<<endl
<<"}"<<endl;
}
|
