1 #include "poly-laguerre-solve.h"
2 #include <iterator>
4 typedef std::complex<double> cdouble;
6 cdouble laguerre_internal_complex(Poly const & p,
7 double x0,
8 double tol,
9 bool & quad_root) {
10 cdouble a = 2*tol;
11 cdouble xk = x0;
12 double n = p.degree();
13 quad_root = false;
14 const unsigned shuffle_rate = 10;
15 static double shuffle[] = {0, 0.5, 0.25, 0.75, 0.125, 0.375, 0.625, 0.875, 1.0};
16 unsigned shuffle_counter = 0;
17 while(std::norm(a) > (tol*tol)) {
18 //std::cout << "xk = " << xk << std::endl;
19 cdouble b = p.back();
20 cdouble d = 0, f = 0;
21 double err = abs(b);
22 double abx = abs(xk);
23 for(int j = p.size()-2; j >= 0; j--) {
24 f = xk*f + d;
25 d = xk*d + b;
26 b = xk*b + p[j];
27 err = abs(b) + abx*err;
28 }
30 err *= 1e-7; // magic epsilon for convergence, should be computed from tol
32 cdouble px = b;
33 if(abs(b) < err)
34 return xk;
35 //if(std::norm(px) < tol*tol)
36 // return xk;
37 cdouble G = d / px;
38 cdouble H = G*G - f / px;
40 //std::cout << "G = " << G << "H = " << H;
41 cdouble radicand = (n - 1)*(n*H-G*G);
42 //assert(radicand.real() > 0);
43 if(radicand.real() < 0)
44 quad_root = true;
45 //std::cout << "radicand = " << radicand << std::endl;
46 if(G.real() < 0) // here we try to maximise the denominator avoiding cancellation
47 a = - sqrt(radicand);
48 else
49 a = sqrt(radicand);
50 //std::cout << "a = " << a << std::endl;
51 a = n / (a + G);
52 //std::cout << "a = " << a << std::endl;
53 if(shuffle_counter % shuffle_rate == 0)
54 ;//a *= shuffle[shuffle_counter / shuffle_rate];
55 xk -= a;
56 shuffle_counter++;
57 if(shuffle_counter >= 90)
58 break;
59 }
60 //std::cout << "xk = " << xk << std::endl;
61 return xk;
62 }
64 double laguerre_internal(Poly const & p,
65 Poly const & pp,
66 Poly const & ppp,
67 double x0,
68 double tol,
69 bool & quad_root) {
70 double a = 2*tol;
71 double xk = x0;
72 double n = p.degree();
73 quad_root = false;
74 while(a*a > (tol*tol)) {
75 //std::cout << "xk = " << xk << std::endl;
76 double px = p(xk);
77 if(px*px < tol*tol)
78 return xk;
79 double G = pp(xk) / px;
80 double H = G*G - ppp(xk) / px;
82 //std::cout << "G = " << G << "H = " << H;
83 double radicand = (n - 1)*(n*H-G*G);
84 assert(radicand > 0);
85 //std::cout << "radicand = " << radicand << std::endl;
86 if(G < 0) // here we try to maximise the denominator avoiding cancellation
87 a = - sqrt(radicand);
88 else
89 a = sqrt(radicand);
90 //std::cout << "a = " << a << std::endl;
91 a = n / (a + G);
92 //std::cout << "a = " << a << std::endl;
93 xk -= a;
94 }
95 //std::cout << "xk = " << xk << std::endl;
96 return xk;
97 }
100 std::vector<cdouble >
101 laguerre(Poly p, const double tol) {
102 std::vector<cdouble > solutions;
103 //std::cout << "p = " << p << " = ";
104 while(p.size() > 1)
105 {
106 double x0 = 0;
107 bool quad_root = false;
108 cdouble sol = laguerre_internal_complex(p, x0, tol, quad_root);
109 //if(abs(sol) > 1) break;
110 Poly dvs;
111 if(quad_root) {
112 dvs.push_back((sol*conj(sol)).real());
113 dvs.push_back(-(sol + conj(sol)).real());
114 dvs.push_back(1.0);
115 //std::cout << "(" << dvs << ")";
116 //solutions.push_back(sol);
117 //solutions.push_back(conj(sol));
118 } else {
119 //std::cout << sol << std::endl;
120 dvs.push_back(-sol.real());
121 dvs.push_back(1.0);
122 solutions.push_back(sol);
123 //std::cout << "(" << dvs << ")";
124 }
125 Poly r;
126 p = divide(p, dvs, r);
127 //std::cout << r << std::endl;
128 }
129 return solutions;
130 }
132 std::vector<double>
133 laguerre_real_interval(Poly const & ply,
134 const double lo, const double hi,
135 const double tol) {
136 }
138 /*
139 Local Variables:
140 mode:c++
141 c-file-style:"stroustrup"
142 c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
143 indent-tabs-mode:nil
144 fill-column:99
145 End:
146 */
147 // vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :