std::vector<std::complex<double> > solve(Poly const & pp) {
Poly p(pp);
p.normalize();
gsl_poly_complex_workspace * w
= gsl_poly_complex_workspace_alloc (p.size());
gsl_complex_packed_ptr z = new double[p.degree()*2];
double* a = new double[p.size()];
for(int i = 0; i < p.size(); i++)
a[i] = p[i];
std::vector<std::complex<double> > roots;
//roots.resize(p.degree());
gsl_poly_complex_solve (a, p.size(), w, z);
delete[]a;
gsl_poly_complex_workspace_free (w);
for (int i = 0; i < p.degree(); i++) {
roots.push_back(std::complex<double> (z[2*i] ,z[2*i+1]));
//printf ("z%d = %+.18f %+.18f\n", i, z[2*i], z[2*i+1]);
}
delete[] z;
return roots;
}
int main (int argc, char **argv) {
int i;
const gsl_rng_type *rngType;
gsl_rng *rng;
gsl_rng_env_setup();
rngType = gsl_rng_default;
rng = gsl_rng_alloc(rngType);
double a[16];
for (i = 0; i < 16; i++) {
a[i] = gsl_ran_ugaussian(rng);
printf("%e\n", a[i]);
}
double z[30];
gsl_poly_complex_workspace *w = gsl_poly_complex_workspace_alloc(16);
gsl_poly_complex_solve(a, 16, w, z);
gsl_poly_complex_workspace_free(w);
for (i = 0; i < 30; i++) {
printf("z%d = %+.18f %+.18f\n", i, z[2*i], z[2*i+1]);
}
gsl_rng_free(rng);
return 0;
}
std::vector<std::complex<double> >
PolyBase::calcRoots(const double epsilon)
/**
Calculate all the roots of the polynominal.
Uses the GSL which uses a Hessian reduction of the
characteristic compainion matrix.
\f[ A= \left( -a_{m-1}/a_m -a_{m-2}/a_m ... -a_0/a_m \right) \f]
where the matrix component below A is the Indenty.
However, GSL requires that the input coefficient is a_m == 1,
hence the call to this->compress().
@param epsilon :: tolerance factor (-ve to use default)
@return roots (not sorted/uniqued)
*/
{
compress(epsilon);
std::vector<std::complex<double> > Out(iDegree);
// Zero State:
if (iDegree==0)
return Out;
// x+a_0 =0
if (iDegree==1)
{
Out[0]=std::complex<double>(-afCoeff[0]);
return Out;
}
// x^2+a_1 x+c = 0
if (iDegree==2)
{
solveQuadratic(Out[0],Out[1]);
return Out;
}
// x^3+a_2 x^2+ a_1 x+c=0
if (iDegree==2)
{
solveCubic(Out[0],Out[1],Out[2]);
return Out;
}
// THERE IS A QUARTIC / QUINTIC Solution availiable but...
// WS contains the the hessian matrix if required (eigenvalues/vectors)
//
gsl_poly_complex_workspace* WS
(gsl_poly_complex_workspace_alloc(iDegree+1));
double* RZ=new double[2*(iDegree+1)];
gsl_poly_complex_solve(&afCoeff.front(),iDegree+1, WS, RZ);
for(int i=0;i<iDegree;i++)
Out[i]=std::complex<double>(RZ[2*i],RZ[2*i+1]);
gsl_poly_complex_workspace_free (WS);
delete [] RZ;
return Out;
}
int main(int argc, char *argv[])
{
/* 0 + 2x - 3x^2 + 1x^3 */
double p[] = {0, 2, -3, 1};
double z[6];
gsl_poly_complex_workspace *w = gsl_poly_complex_workspace_alloc(4);
gsl_poly_complex_solve(p, 4, w, z);
gsl_poly_complex_workspace_free(w);
for(int i = 0; i < 3; ++i)
printf("%.12f\n", z[2 * i]);
return 0;
}
int
main ()
{
int i;
double a[6] = { -1, 0, 0, 0, 0, 1 }; /* P(x) = x^5 - 1 */
double z[10];
gsl_poly_complex_workspace * w = gsl_poly_complex_workspace_alloc (6) ;
gsl_poly_complex_solve (a, 6, w, z) ;
gsl_poly_complex_workspace_free (w) ;
for (i = 0; i < 5 ; i++)
{
printf("z%d = %+.18f %+.18f\n", i, z[2*i], z[2*i+1]) ;
}
}
double gp_max_fixed(double m, double s)
{
double coeff[5];
coeff[4] = rt_powd_snf(m, 4.0) / (4.0 * rt_powd_snf(s, 4.0));
coeff[3] = 3.0 * (m * m) / (2.0 * (s * s));
coeff[2] = 3.75 - m * m / (2.0 * rt_powd_snf(s, 4.0));
coeff[1] = -5.0 / (2.0 * (s * s));
coeff[0] = 1.0 / (4.0 * rt_powd_snf(s, 4.0));
double foundRoots[10];
gsl_poly_complex_workspace * workspace = gsl_poly_complex_workspace_alloc(5);
gsl_poly_complex_solve(coeff, 5, workspace, foundRoots);
gsl_poly_complex_workspace_free(workspace);
int numRealRoots = 0;
double realRoots[10];
double re;
double im;
for (int i = 0; i < 4; i++)
{
re = foundRoots[2 * i];
im = foundRoots[2 * i + 1];
if (im <= 0.000000000000001) {
if (re >= 0) {
realRoots[numRealRoots] = re;
numRealRoots++;
}
}
}
double y[10];
onestagepdf_prime_fixed(realRoots, numRealRoots, m, s, y);
double largest = 0;
for (int i = 0; i < numRealRoots; i++) {
if (y[i] > largest) {
largest = y[i];
}
}
return largest;
}
pure_expr* wrap_gsl_poly_complex_solve(double* a, size_t n)
{
double tz[2*(n-1)];
pure_expr *z[n-1];
double t[2];
int i, r;
gsl_poly_complex_workspace* w = gsl_poly_complex_workspace_alloc(n);
r = gsl_poly_complex_solve(a, n, w, tz);
gsl_poly_complex_workspace_free(w);
if (r == GSL_SUCCESS) {
for (i = 0; i < n-1; ++i) {
t[0] = tz[2*i];
t[1] = tz[2*i+1];
z[i] = pure_complex(t);
}
return pure_listv(n-1, z);
} else
return pure_listl(0);
}
int
main (void)
{
int i;
/* coefficient of P(x) = -1 + x^5 */
double a[6] = { -1, 0, 0, 0, 0, 1 };
double z[10];
gsl_poly_complex_workspace * w
= gsl_poly_complex_workspace_alloc (6);
gsl_poly_complex_solve (a, 6, w, z);
gsl_poly_complex_workspace_free (w);
for (i = 0; i < 5; i++)
{
printf ("z%d = %+.18f %+.18f\n",
i, z[2*i], z[2*i+1]);
}
return 0;
}
int
main (void)
{
const double eps = 100.0 * GSL_DBL_EPSILON;
gsl_ieee_env_setup ();
/* Polynomial evaluation */
{
double x, y;
double c[3] = { 1.0, 0.5, 0.3 };
x = 0.5;
y = gsl_poly_eval (c, 3, x);
gsl_test_rel (y, 1 + 0.5 * x + 0.3 * x * x, eps,
"gsl_poly_eval({1, 0.5, 0.3}, 0.5)");
}
{
double x, y;
double d[11] = { 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1 };
x = 1.0;
y = gsl_poly_eval (d, 11, x);
gsl_test_rel (y, 1.0, eps,
"gsl_poly_eval({1,-1, 1, -1, 1, -1, 1, -1, 1, -1, 1}, 1.0)");
}
{
gsl_complex x, y;
double c[1] = {0.3};
GSL_SET_REAL (&x, 0.75);
GSL_SET_IMAG (&x, 1.2);
y = gsl_poly_complex_eval (c, 1, x);
gsl_test_rel (GSL_REAL (y), 0.3, eps, "y.real, gsl_poly_complex_eval ({0.3}, 0.75 + 1.2i)");
gsl_test_rel (GSL_IMAG (y), 0.0, eps, "y.imag, gsl_poly_complex_eval ({0.3}, 0.75 + 1.2i)");
}
{
gsl_complex x, y;
double c[4] = {2.1, -1.34, 0.76, 0.45};
GSL_SET_REAL (&x, 0.49);
GSL_SET_IMAG (&x, 0.95);
y = gsl_poly_complex_eval (c, 4, x);
gsl_test_rel (GSL_REAL (y), 0.3959143, eps, "y.real, gsl_poly_complex_eval ({2.1, -1.34, 0.76, 0.45}, 0.49 + 0.95i)");
gsl_test_rel (GSL_IMAG (y), -0.6433305, eps, "y.imag, gsl_poly_complex_eval ({2.1, -1.34, 0.76, 0.45}, 0.49 + 0.95i)");
}
{
gsl_complex x, y;
gsl_complex c[1];
GSL_SET_REAL (&c[0], 0.674);
GSL_SET_IMAG (&c[0], -1.423);
GSL_SET_REAL (&x, -1.44);
GSL_SET_IMAG (&x, 9.55);
y = gsl_complex_poly_complex_eval (c, 1, x);
gsl_test_rel (GSL_REAL (y), 0.674, eps, "y.real, gsl_complex_poly_complex_eval ({0.674 - 1.423i}, -1.44 + 9.55i)");
gsl_test_rel (GSL_IMAG (y), -1.423, eps, "y.imag, gsl_complex_poly_complex_eval ({0.674 - 1.423i}, -1.44 + 9.55i)");
}
{
gsl_complex x, y;
gsl_complex c[4];
GSL_SET_REAL (&c[0], -2.31);
GSL_SET_IMAG (&c[0], 0.44);
GSL_SET_REAL (&c[1], 4.21);
GSL_SET_IMAG (&c[1], -3.19);
GSL_SET_REAL (&c[2], 0.93);
GSL_SET_IMAG (&c[2], 1.04);
GSL_SET_REAL (&c[3], -0.42);
GSL_SET_IMAG (&c[3], 0.68);
GSL_SET_REAL (&x, 0.49);
GSL_SET_IMAG (&x, 0.95);
y = gsl_complex_poly_complex_eval (c, 4, x);
gsl_test_rel (GSL_REAL (y), 1.82462012, eps, "y.real, gsl_complex_poly_complex_eval ({-2.31 + 0.44i, 4.21 - 3.19i, 0.93 + 1.04i, -0.42 + 0.68i}, 0.49 + 0.95i)");
gsl_test_rel (GSL_IMAG (y), 2.30389412, eps, "y.imag, gsl_complex_poly_complex_eval ({-2.31 + 0.44i, 4.21 - 3.19i, 0.93 + 1.04i, -0.42 + 0.68i}, 0.49 + 0.95i)");
}
/* Quadratic */
{
double x0, x1;
int n = gsl_poly_solve_quadratic (4.0, -20.0, 26.0, &x0, &x1);
gsl_test (n != 0, "gsl_poly_solve_quadratic, no roots, (2x - 5)^2 = -1");
}
{
double x0, x1;
int n = gsl_poly_solve_quadratic (4.0, -20.0, 25.0, &x0, &x1);
gsl_test (n != 2, "gsl_poly_solve_quadratic, one root, (2x - 5)^2 = 0");
gsl_test_rel (x0, 2.5, 1e-9, "x0, (2x - 5)^2 = 0");
gsl_test_rel (x1, 2.5, 1e-9, "x1, (2x - 5)^2 = 0");
gsl_test (x0 != x1, "x0 == x1, (2x - 5)^2 = 0");
}
{
double x0, x1;
int n = gsl_poly_solve_quadratic (4.0, -20.0, 21.0, &x0, &x1);
gsl_test (n != 2, "gsl_poly_solve_quadratic, two roots, (2x - 5)^2 = 4");
gsl_test_rel (x0, 1.5, 1e-9, "x0, (2x - 5)^2 = 4");
gsl_test_rel (x1, 3.5, 1e-9, "x1, (2x - 5)^2 = 4");
}
{
double x0, x1;
int n = gsl_poly_solve_quadratic (4.0, 7.0, 0.0, &x0, &x1);
gsl_test (n != 2, "gsl_poly_solve_quadratic, two roots, x(4x + 7) = 0");
gsl_test_rel (x0, -1.75, 1e-9, "x0, x(4x + 7) = 0");
gsl_test_rel (x1, 0.0, 1e-9, "x1, x(4x + 7) = 0");
}
{
double x0, x1;
int n = gsl_poly_solve_quadratic (5.0, 0.0, -20.0, &x0, &x1);
gsl_test (n != 2,
"gsl_poly_solve_quadratic, two roots b = 0, 5 x^2 = 20");
gsl_test_rel (x0, -2.0, 1e-9, "x0, 5 x^2 = 20");
gsl_test_rel (x1, 2.0, 1e-9, "x1, 5 x^2 = 20");
}
{
double x0, x1;
int n = gsl_poly_solve_quadratic (0.0, 3.0, -21.0, &x0, &x1);
gsl_test (n != 1,
"gsl_poly_solve_quadratic, one root (linear) 3 x - 21 = 0");
gsl_test_rel (x0, 7.0, 1e-9, "x0, 3x - 21 = 0");
}
{
double x0, x1;
int n = gsl_poly_solve_quadratic (0.0, 0.0, 1.0, &x0, &x1);
gsl_test (n != 0,
"gsl_poly_solve_quadratic, no roots 1 = 0");
}
/* Cubic */
{
double x0, x1, x2;
int n = gsl_poly_solve_cubic (0.0, 0.0, -27.0, &x0, &x1, &x2);
gsl_test (n != 1, "gsl_poly_solve_cubic, one root, x^3 = 27");
gsl_test_rel (x0, 3.0, 1e-9, "x0, x^3 = 27");
}
{
double x0, x1, x2;
int n = gsl_poly_solve_cubic (-51.0, 867.0, -4913.0, &x0, &x1, &x2);
gsl_test (n != 3, "gsl_poly_solve_cubic, three roots, (x-17)^3=0");
gsl_test_rel (x0, 17.0, 1e-9, "x0, (x-17)^3=0");
gsl_test_rel (x1, 17.0, 1e-9, "x1, (x-17)^3=0");
gsl_test_rel (x2, 17.0, 1e-9, "x2, (x-17)^3=0");
}
{
double x0, x1, x2;
int n = gsl_poly_solve_cubic (-57.0, 1071.0, -6647.0, &x0, &x1, &x2);
gsl_test (n != 3,
"gsl_poly_solve_cubic, three roots, (x-17)(x-17)(x-23)=0");
gsl_test_rel (x0, 17.0, 1e-9, "x0, (x-17)(x-17)(x-23)=0");
gsl_test_rel (x1, 17.0, 1e-9, "x1, (x-17)(x-17)(x-23)=0");
gsl_test_rel (x2, 23.0, 1e-9, "x2, (x-17)(x-17)(x-23)=0");
}
{
double x0, x1, x2;
int n = gsl_poly_solve_cubic (-11.0, -493.0, +6647.0, &x0, &x1, &x2);
gsl_test (n != 3,
"gsl_poly_solve_cubic, three roots, (x+23)(x-17)(x-17)=0");
gsl_test_rel (x0, -23.0, 1e-9, "x0, (x+23)(x-17)(x-17)=0");
gsl_test_rel (x1, 17.0, 1e-9, "x1, (x+23)(x-17)(x-17)=0");
gsl_test_rel (x2, 17.0, 1e-9, "x2, (x+23)(x-17)(x-17)=0");
}
{
double x0, x1, x2;
int n = gsl_poly_solve_cubic (-143.0, 5087.0, -50065.0, &x0, &x1, &x2);
gsl_test (n != 3,
"gsl_poly_solve_cubic, three roots, (x-17)(x-31)(x-95)=0");
gsl_test_rel (x0, 17.0, 1e-9, "x0, (x-17)(x-31)(x-95)=0");
gsl_test_rel (x1, 31.0, 1e-9, "x1, (x-17)(x-31)(x-95)=0");
gsl_test_rel (x2, 95.0, 1e-9, "x2, (x-17)(x-31)(x-95)=0");
}
{
double x0, x1, x2;
int n = gsl_poly_solve_cubic (-109.0, 803.0, 50065.0, &x0, &x1, &x2);
gsl_test (n != 3,
"gsl_poly_solve_cubic, three roots, (x+17)(x-31)(x-95)=0");
gsl_test_rel (x0, -17.0, 1e-9, "x0, (x+17)(x-31)(x-95)=0");
gsl_test_rel (x1, 31.0, 1e-9, "x1, (x+17)(x-31)(x-95)=0");
gsl_test_rel (x2, 95.0, 1e-9, "x2, (x+17)(x-31)(x-95)=0");
}
/* Quadratic with complex roots */
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (4.0, -20.0, 26.0, &z0, &z1);
gsl_test (n != 2,
"gsl_poly_complex_solve_quadratic, 2 roots (2x - 5)^2 = -1");
gsl_test_rel (GSL_REAL (z0), 2.5, 1e-9, "z0.real, (2x - 5)^2 = -1");
gsl_test_rel (GSL_IMAG (z0), -0.5, 1e-9, "z0.imag, (2x - 5)^2 = -1");
gsl_test_rel (GSL_REAL (z1), 2.5, 1e-9, "z1.real, (2x - 5)^2 = -1");
gsl_test_rel (GSL_IMAG (z1), 0.5, 1e-9, "z1.imag, (2x - 5)^2 = -1");
}
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (4.0, -20.0, 25.0, &z0, &z1);
gsl_test (n != 2,
"gsl_poly_complex_solve_quadratic, one root, (2x - 5)^2 = 0");
gsl_test_rel (GSL_REAL (z0), 2.5, 1e-9, "z0.real, (2x - 5)^2 = 0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag (2x - 5)^2 = 0");
gsl_test_rel (GSL_REAL (z1), 2.5, 1e-9, "z1.real, (2x - 5)^2 = 0");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag (2x - 5)^2 = 0");
gsl_test (GSL_REAL (z0) != GSL_REAL (z1),
"z0.real == z1.real, (2x - 5)^2 = 0");
gsl_test (GSL_IMAG (z0) != GSL_IMAG (z1),
"z0.imag == z1.imag, (2x - 5)^2 = 0");
}
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (4.0, -20.0, 21.0, &z0, &z1);
gsl_test (n != 2,
"gsl_poly_complex_solve_quadratic, two roots, (2x - 5)^2 = 4");
gsl_test_rel (GSL_REAL (z0), 1.5, 1e-9, "z0.real, (2x - 5)^2 = 4");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, (2x - 5)^2 = 4");
gsl_test_rel (GSL_REAL (z1), 3.5, 1e-9, "z1.real, (2x - 5)^2 = 4");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag, (2x - 5)^2 = 4");
}
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (4.0, 7.0, 0.0, &z0, &z1);
gsl_test (n != 2,
"gsl_poly_complex_solve_quadratic, two roots, x(4x + 7) = 0");
gsl_test_rel (GSL_REAL (z0), -1.75, 1e-9, "z0.real, x(4x + 7) = 0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, x(4x + 7) = 0");
gsl_test_rel (GSL_REAL (z1), 0.0, 1e-9, "z1.real, x(4x + 7) = 0");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag, x(4x + 7) = 0");
}
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (5.0, 0.0, -20.0, &z0, &z1);
gsl_test (n != 2,
"gsl_poly_complex_solve_quadratic, two roots b = 0, 5 x^2 = 20");
gsl_test_rel (GSL_REAL (z0), -2.0, 1e-9, "z0.real, 5 x^2 = 20");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, 5 x^2 = 20");
gsl_test_rel (GSL_REAL (z1), 2.0, 1e-9, "z1.real, 5 x^2 = 20");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag, 5 x^2 = 20");
}
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (5.0, 0.0, 20.0, &z0, &z1);
gsl_test (n != 2,
"gsl_poly_complex_solve_quadratic, two roots b = 0, 5 x^2 = -20");
gsl_test_rel (GSL_REAL (z0), 0.0, 1e-9, "z0.real, 5 x^2 = -20");
gsl_test_rel (GSL_IMAG (z0), -2.0, 1e-9, "z0.imag, 5 x^2 = -20");
gsl_test_rel (GSL_REAL (z1), 0.0, 1e-9, "z1.real, 5 x^2 = -20");
gsl_test_rel (GSL_IMAG (z1), 2.0, 1e-9, "z1.imag, 5 x^2 = -20");
}
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (0.0, 3.0, -21.0, &z0, &z1);
gsl_test (n != 1,
"gsl_poly_complex_solve_quadratic, one root (linear) 3 x - 21 = 0");
gsl_test_rel (GSL_REAL (z0), 7.0, 1e-9, "z0.real, 3x - 21 = 0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, 3x - 21 = 0");
}
{
gsl_complex z0, z1;
int n = gsl_poly_complex_solve_quadratic (0.0, 0.0, 1.0, &z0, &z1);
gsl_test (n != 0,
"gsl_poly_complex_solve_quadratic, no roots 1 = 0");
}
/* Cubic with complex roots */
{
gsl_complex z0, z1, z2;
int n = gsl_poly_complex_solve_cubic (0.0, 0.0, -27.0, &z0, &z1, &z2);
gsl_test (n != 3, "gsl_poly_complex_solve_cubic, three root, x^3 = 27");
gsl_test_rel (GSL_REAL (z0), -1.5, 1e-9, "z0.real, x^3 = 27");
gsl_test_rel (GSL_IMAG (z0), -1.5 * sqrt (3.0), 1e-9,
"z0.imag, x^3 = 27");
gsl_test_rel (GSL_REAL (z1), -1.5, 1e-9, "z1.real, x^3 = 27");
gsl_test_rel (GSL_IMAG (z1), 1.5 * sqrt (3.0), 1e-9, "z1.imag, x^3 = 27");
gsl_test_rel (GSL_REAL (z2), 3.0, 1e-9, "z2.real, x^3 = 27");
gsl_test_rel (GSL_IMAG (z2), 0.0, 1e-9, "z2.imag, x^3 = 27");
}
{
gsl_complex z0, z1, z2;
int n = gsl_poly_complex_solve_cubic (-1.0, 1.0, 39.0, &z0, &z1, &z2);
gsl_test (n != 3,
"gsl_poly_complex_solve_cubic, three root, (x+3)(x^2-4x+13) = 0");
gsl_test_rel (GSL_REAL (z0), -3.0, 1e-9, "z0.real, (x+3)(x^2+1) = 0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, (x+3)(x^2+1) = 0");
gsl_test_rel (GSL_REAL (z1), 2.0, 1e-9, "z1.real, (x+3)(x^2+1) = 0");
gsl_test_rel (GSL_IMAG (z1), -3.0, 1e-9, "z1.imag, (x+3)(x^2+1) = 0");
gsl_test_rel (GSL_REAL (z2), 2.0, 1e-9, "z2.real, (x+3)(x^2+1) = 0");
gsl_test_rel (GSL_IMAG (z2), 3.0, 1e-9, "z2.imag, (x+3)(x^2+1) = 0");
}
{
gsl_complex z0, z1, z2;
int n =
gsl_poly_complex_solve_cubic (-51.0, 867.0, -4913.0, &z0, &z1, &z2);
gsl_test (n != 3,
"gsl_poly_complex_solve_cubic, three roots, (x-17)^3=0");
gsl_test_rel (GSL_REAL (z0), 17.0, 1e-9, "z0.real, (x-17)^3=0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, (x-17)^3=0");
gsl_test_rel (GSL_REAL (z1), 17.0, 1e-9, "z1.real, (x-17)^3=0");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag, (x-17)^3=0");
gsl_test_rel (GSL_REAL (z2), 17.0, 1e-9, "z2.real, (x-17)^3=0");
gsl_test_rel (GSL_IMAG (z2), 0.0, 1e-9, "z2.imag, (x-17)^3=0");
}
{
gsl_complex z0, z1, z2;
int n =
gsl_poly_complex_solve_cubic (-57.0, 1071.0, -6647.0, &z0, &z1, &z2);
gsl_test (n != 3,
"gsl_poly_complex_solve_cubic, three roots, (x-17)(x-17)(x-23)=0");
gsl_test_rel (GSL_REAL (z0), 17.0, 1e-9, "z0.real, (x-17)(x-17)(x-23)=0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, (x-17)(x-17)(x-23)=0");
gsl_test_rel (GSL_REAL (z1), 17.0, 1e-9, "z1.real, (x-17)(x-17)(x-23)=0");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag, (x-17)(x-17)(x-23)=0");
gsl_test_rel (GSL_REAL (z2), 23.0, 1e-9, "z2.real, (x-17)(x-17)(x-23)=0");
gsl_test_rel (GSL_IMAG (z2), 0.0, 1e-9, "z2.imag, (x-17)(x-17)(x-23)=0");
}
{
gsl_complex z0, z1, z2;
int n =
gsl_poly_complex_solve_cubic (-11.0, -493.0, +6647.0, &z0, &z1, &z2);
gsl_test (n != 3,
"gsl_poly_complex_solve_cubic, three roots, (x+23)(x-17)(x-17)=0");
gsl_test_rel (GSL_REAL (z0), -23.0, 1e-9,
"z0.real, (x+23)(x-17)(x-17)=0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, (x+23)(x-17)(x-17)=0");
gsl_test_rel (GSL_REAL (z1), 17.0, 1e-9, "z1.real, (x+23)(x-17)(x-17)=0");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag, (x+23)(x-17)(x-17)=0");
gsl_test_rel (GSL_REAL (z2), 17.0, 1e-9, "z2.real, (x+23)(x-17)(x-17)=0");
gsl_test_rel (GSL_IMAG (z2), 0.0, 1e-9, "z2.imag, (x+23)(x-17)(x-17)=0");
}
{
gsl_complex z0, z1, z2;
int n =
gsl_poly_complex_solve_cubic (-143.0, 5087.0, -50065.0, &z0, &z1, &z2);
gsl_test (n != 3,
"gsl_poly_complex_solve_cubic, three roots, (x-17)(x-31)(x-95)=0");
gsl_test_rel (GSL_REAL (z0), 17.0, 1e-9, "z0.real, (x-17)(x-31)(x-95)=0");
gsl_test_rel (GSL_IMAG (z0), 0.0, 1e-9, "z0.imag, (x-17)(x-31)(x-95)=0");
gsl_test_rel (GSL_REAL (z1), 31.0, 1e-9, "z1.real, (x-17)(x-31)(x-95)=0");
gsl_test_rel (GSL_IMAG (z1), 0.0, 1e-9, "z1.imag, (x-17)(x-31)(x-95)=0");
gsl_test_rel (GSL_REAL (z2), 95.0, 1e-9, "z2.real, (x-17)(x-31)(x-95)=0");
gsl_test_rel (GSL_IMAG (z2), 0.0, 1e-9, "z2.imag, (x-17)(x-31)(x-95)=0");
}
{
/* Wilkinson polynomial: y = (x-1)(x-2)(x-3)(x-4)(x-5) */
double a[6] = { -120, 274, -225, 85, -15, 1 };
double z[6*2];
gsl_poly_complex_workspace *w = gsl_poly_complex_workspace_alloc (6);
int status = gsl_poly_complex_solve (a, 6, w, z);
gsl_poly_complex_workspace_free (w);
gsl_test (status,
"gsl_poly_complex_solve, 5th-order Wilkinson polynomial");
gsl_test_rel (z[0], 1.0, 1e-9, "z0.real, 5th-order polynomial");
gsl_test_rel (z[1], 0.0, 1e-9, "z0.imag, 5th-order polynomial");
gsl_test_rel (z[2], 2.0, 1e-9, "z1.real, 5th-order polynomial");
gsl_test_rel (z[3], 0.0, 1e-9, "z1.imag, 5th-order polynomial");
gsl_test_rel (z[4], 3.0, 1e-9, "z2.real, 5th-order polynomial");
gsl_test_rel (z[5], 0.0, 1e-9, "z2.imag, 5th-order polynomial");
gsl_test_rel (z[6], 4.0, 1e-9, "z3.real, 5th-order polynomial");
gsl_test_rel (z[7], 0.0, 1e-9, "z3.imag, 5th-order polynomial");
gsl_test_rel (z[8], 5.0, 1e-9, "z4.real, 5th-order polynomial");
gsl_test_rel (z[9], 0.0, 1e-9, "z4.imag, 5th-order polynomial");
}
{
/* : 8-th order polynomial y = x^8 + x^4 + 1 */
double a[9] = { 1, 0, 0, 0, 1, 0, 0, 0, 1 };
double z[8*2];
double C = 0.5;
double S = sqrt (3.0) / 2.0;
gsl_poly_complex_workspace *w = gsl_poly_complex_workspace_alloc (9);
int status = gsl_poly_complex_solve (a, 9, w, z);
gsl_poly_complex_workspace_free (w);
gsl_test (status, "gsl_poly_complex_solve, 8th-order polynomial");
gsl_test_rel (z[0], -S, 1e-9, "z0.real, 8th-order polynomial");
gsl_test_rel (z[1], C, 1e-9, "z0.imag, 8th-order polynomial");
gsl_test_rel (z[2], -S, 1e-9, "z1.real, 8th-order polynomial");
gsl_test_rel (z[3], -C, 1e-9, "z1.imag, 8th-order polynomial");
gsl_test_rel (z[4], -C, 1e-9, "z2.real, 8th-order polynomial");
gsl_test_rel (z[5], S, 1e-9, "z2.imag, 8th-order polynomial");
gsl_test_rel (z[6], -C, 1e-9, "z3.real, 8th-order polynomial");
gsl_test_rel (z[7], -S, 1e-9, "z3.imag, 8th-order polynomial");
gsl_test_rel (z[8], C, 1e-9, "z4.real, 8th-order polynomial");
gsl_test_rel (z[9], S, 1e-9, "z4.imag, 8th-order polynomial");
gsl_test_rel (z[10], C, 1e-9, "z5.real, 8th-order polynomial");
gsl_test_rel (z[11], -S, 1e-9, "z5.imag, 8th-order polynomial");
gsl_test_rel (z[12], S, 1e-9, "z6.real, 8th-order polynomial");
gsl_test_rel (z[13], C, 1e-9, "z6.imag, 8th-order polynomial");
gsl_test_rel (z[14], S, 1e-9, "z7.real, 8th-order polynomial");
gsl_test_rel (z[15], -C, 1e-9, "z7.imag, 8th-order polynomial");
}
{
int i;
double xa[7] = {0.16, 0.97, 1.94, 2.74, 3.58, 3.73, 4.70 };
double ya[7] = {0.73, 1.11, 1.49, 1.84, 2.30, 2.41, 3.07 };
double dd_expected[7] = { 7.30000000000000e-01,
4.69135802469136e-01,
-4.34737219941284e-02,
2.68681098870099e-02,
-3.22937056934996e-03,
6.12763259971375e-03,
-6.45402453527083e-03 };
double dd[7], coeff[7], work[7];
gsl_poly_dd_init (dd, xa, ya, 7);
for (i = 0; i < 7; i++)
{
gsl_test_rel (dd[i], dd_expected[i], 1e-10, "divided difference dd[%d]", i);
}
for (i = 0; i < 7; i++)
{
double y = gsl_poly_dd_eval(dd, xa, 7, xa[i]);
gsl_test_rel (y, ya[i], 1e-10, "divided difference y[%d]", i);
}
gsl_poly_dd_taylor (coeff, 1.5, dd, xa, 7, work);
for (i = 0; i < 7; i++)
{
double y = gsl_poly_eval(coeff, 7, xa[i] - 1.5);
gsl_test_rel (y, ya[i], 1e-10, "taylor expansion about 1.5 y[%d]", i);
}
}
{
double c[6] = { +1.0, -2.0, +3.0, -4.0, +5.0, -6.0 };
double dc[6];
double x;
x = -0.5;
gsl_poly_eval_derivs(c, 6, x, dc, 6);
gsl_test_rel (dc[0], c[0] + c[1]*x + c[2]*x*x + c[3]*x*x*x + c[4]*x*x*x*x + c[5]*x*x*x*x*x , eps, "gsl_poly_eval_dp({+1, -2, +3, -4, +5, -6}, 3.75)");
gsl_test_rel (dc[1], c[1] + 2.0*c[2]*x + 3.0*c[3]*x*x + 4.0*c[4]*x*x*x + 5.0*c[5]*x*x*x*x , eps, "gsl_poly_eval_dp({+1, -2, +3, -4, +5, -6} deriv 1, -12.375)");
gsl_test_rel (dc[2], 2.0*c[2] + 3.0*2.0*c[3]*x + 4.0*3.0*c[4]*x*x + 5.0*4.0*c[5]*x*x*x , eps, "gsl_poly_eval_dp({+1, -2, +3, -4, +5, -6} deriv 2, +48.0)");
gsl_test_rel (dc[3], 3.0*2.0*c[3] + 4.0*3.0*2.0*c[4]*x + 5.0*4.0*3.0*c[5]*x*x , eps,"gsl_poly_eval_dp({+1, -2, +3, -4, +5, -6} deriv 3, -174.0)");
gsl_test_rel (dc[4], 4.0*3.0*2.0*c[4] + 5.0*4.0*3.0*2.0*c[5]*x, eps, "gsl_poly_eval_dp({+1, -2, +3, -4, +5, -6} deriv 4, +480.0)");
gsl_test_rel (dc[5], 5.0*4.0*3.0*2.0*c[5] , eps, "gsl_poly_eval_dp({+1, -2, +3, -4, +5, -6} deriv 5, -720.0)");
}
/* now summarize the results */
exit (gsl_test_summary ());
}
double K_E_n(const gsl_matrix* alle, double n, double sigma) {
double m_1 = MSUN_TO_MJUP(MGET(alle, 0, MASS));
double m_2 = MGET(alle, 1, MASS);
double m_3 = MGET(alle, 2, MASS);
double m_12 = m_1 + m_2;
double a_i = MGET(alle, 1, SMA);
double a_o = MGET(alle, 2, SMA);
double e_i = MGET(alle, 1, ECC);
double e_o = MGET(alle, 2, ECC);
double xi = acosh(1/e_o) * sqrt(1-e_o*e_o);
double E_22 = 4.*SQRT_TWOPI/3. * pow(1-e_o*e_o, 0.75)/(e_o * e_o) * pow(sigma, 5./2.) * exp(-sigma*xi);
double I_22 = 9./4. * m_3 / m_12 * POW_3(a_i/a_o) * E_22;
double e_i_ind = sqrt(e_i * e_i + I_22 * I_22);
double eps_o = sqrt(1-e_i*e_i);
double e_eq;
if (e_i < 0.) {
e_eq = (5./4.) * e_o * m_3 * (m_1 - m_2) * SQR(a_i/a_o) * sigma /
(eps_o * fabs(m_1*m_2 - m_12*m_3 * (a_i/a_o) * eps_o * sigma));
} else {
double A = 0.75 * (m_3 / m_12) * POW_3(a_i/a_o) / POW_3(eps_o);
double B = 15./64. * (m_3 / m_12) * (m_1-m_2)/m_12 * POW_4(a_i/a_o) / POW_5(eps_o);
double C = 3./4. * (m_1 * m_2 / SQR(m_12)) * SQR(a_i/a_o) / POW_4(eps_o);
double D = 15./64. * (m_1 * m_2 / SQR(m_12)) * (m_1-m_2)/m_12 * POW_3(a_i/a_o) * (1+4*e_o * e_o)/(e_o * POW_6(eps_o));
double a[9] = {-B*B, 2*A*B, B*B + C*C - A*A,
-2*(A*B + 4*C*D), A*A + 3*C*C + 16*D*D,
-18*C*D, 9./4. * C*C + 24*D*D, -9*C*D, 9*D*D};
gsl_poly_complex_workspace * w = gsl_poly_complex_workspace_alloc(9);
double z[16];
gsl_poly_complex_solve(a, 9, w, z);
for (int i = 0; i < 16; i+=2)
if (z[i] > 0 && z[i] < 1) {
e_eq = z[i];
break;
}
gsl_poly_complex_workspace_free(w);
}
double alpha = fabs(1-e_i/e_eq);
double e_i_oct = (alpha < 1 ? (1+alpha) * e_eq : e_i + 2*e_eq);
e_i = MAX(e_i_ind, e_i_oct);
double s = -3*e_i + 13./8. * POW_3(e_i) + 5./192. * POW_5(e_i) - 227./3072. * POW_7(e_i);
double F = E_22/(TWOPI * n);
double M_i = m_3 / (m_12 + m_3);
double M_o = (m_1 * m_2 / POW_2(m_12)) * pow(m_12 / (m_12+m_3), 2./3.);
double A_n = -9./2. * s * F * (M_i + M_o * pow(n, 2./3.));
double E_n = 0.5 * POW_2(sigma-n) - 2.*A_n;
/*
PRINTNUM(A_n);
PRINTNUM(E_n);
PRINTNUM(I_22);
PRINTNUM(n);
PRINTNUM(sigma);
PRINTNUM(s);
PRINTNUM(F);
PRINTNUM(e_i);
PRINTNUM(e_i_oct);
PRINTNUM(e_i_ind);
PRINTNUM(e_eq);
*/
return E_n;
}
int
main(int argc, char **argv)
{
int i, highest_power;
char *cp, *arg;
double *a, *z;
gsl_poly_complex_workspace *w;
if (argc <= 1) {
fprintf(stderr, "Coefficients must be specified on the command line.\n");
printf("\npolyroots version 1.0 - numerical polynomial equation solver\n");
printf("\nSolves polynomial = 0 when given all real coefficients of the polynomial.\n");
printf("Double precision floating point arithmetic is used.\n");
printf("\nUsage: %s highest-power-coefficient ... constant-term\n", argv[0]);
printf("\nThe coefficients must be decimal, floating point numbers.\n");
exit(2);
}
highest_power = argc - 2;
a = calloc(highest_power + 1, sizeof(double)); /* allocate real double input array */
z = calloc(2 * highest_power, sizeof(double)); /* allocate complex double output array */
/* parse the command line into the coefficient array a[] */
for (i = 0; i < argc - 1; i++) {
arg = argv[argc-i-1];
errno = 0;
a[i] = strtod(arg, &cp);
if (arg == cp || *cp) {
fprintf(stderr, "\"%s\" is not a floating point number.\n", arg);
exit(2);
}
if (errno) {
fprintf(stderr, "\"%s\" is out of range.\n", arg);
exit(2);
}
}
/* nicely display the actual polynomial equation we are solving */
printf("The %d approximate floating point solutions of:\n", highest_power);
for (i = highest_power; i >= 0; i--) {
if (a[i]) {
if (i) {
if (a[i] == 1.0) {
printf("+x");
} else if (a[i] == -1.0) {
printf("-x");
} else {
printf("%+.14g*x", a[i]);
}
if (i > 1) {
printf("^%d", i);
}
} else {
printf("%+.14g", a[i]);
}
printf(" ");
}
}
printf("= 0\nare:\n");
/* solve the polynomial equation */
w = gsl_poly_complex_workspace_alloc(highest_power + 1);
if (gsl_poly_complex_solve(a, highest_power + 1, w, z) != GSL_SUCCESS) {
fprintf(stderr, "Approximation failed.\n");
exit(1);
}
gsl_poly_complex_workspace_free(w);
/* display all solutions */
for (i = 0; i < highest_power; i++) {
#if true /* zero out relatively very small values (which are floating point error) */
if (fabs(z[2*i] * epsilon) > fabs(z[2*i+1]))
z[2*i+1] = 0.0;
else if (fabs(z[2*i+1] * epsilon) > fabs(z[2*i]))
z[2*i] = 0.0;
#endif
printf("x = %+.14g", z[2*i]); /* output real part */
if (z[2*i+1])
printf(" %+.14g*i", z[2*i+1]); /* output imaginary part */
printf("\n");
}
exit(0);
}
//20100224追加
Integer DAESolver::get_status_code()
{
Integer status_code(0);
Real minimum_occurrence_time(INF);
const RealMatrix::size_type a_taylor_size(3);
RealVector tmp_min;
tmp_min.resize(0);
for (StatusEventArray::size_type i(0); i < the_status_event_.size(); ++i)
{
Integer index = the_status_event_[i].variable_index;
Integer flag = the_status_event_[i].variable_flag;
Real threshold = the_status_event_[i].threshold;
Integer code = the_status_event_[i].status_code;
Real a[a_taylor_size + 1];
if (flag == 0)
{
a[0] = the_value_differential_[index] - threshold;
a[1] = the_taylor_series_[0][index];
Real power = 1.0;
for (Integer j(0); j < a_taylor_size - 1; ++j)
{
power = power * the_tolerable_step_interval_;
a[j + 2] = the_taylor_series_[j + 1][index] / power;
}
}
else
{
a[0] = the_value_algebraic_[index] - threshold;
a[1] = the_taylor_series_[0][
index + the_function_differential_size_];
Real power = 1.0;
for (Integer j(0); j < a_taylor_size - 1; ++j)
{
power = power * the_tolerable_step_interval_;
a[j + 2] = the_taylor_series_[j + 1][
index + the_function_differential_size_] / power;
}
}
Real z[2 * a_taylor_size];
gsl_poly_complex_workspace * w
= gsl_poly_complex_workspace_alloc(a_taylor_size + 1);
gsl_poly_complex_solve(a, a_taylor_size + 1, w, z);
gsl_poly_complex_workspace_free(w);
for (Integer k(0); k < a_taylor_size; k++)
{
const Real solution(z[2 * k]);
if (z[2 * k + 1] == 0.0 && solution >= 0. &&
solution < the_next_time_ - the_current_time_)
{
if (solution < minimum_occurrence_time)
{
minimum_occurrence_time = solution;
status_code = code;
tmp_min.push_back(minimum_occurrence_time);
}
}
}
}
Integer count = 0;
for (Integer n(0); n < tmp_min.size(); n++)
{
if (tmp_min[n] == minimum_occurrence_time) count++;
}
if (count > 1)
{
throw "Some status events occurred at the same time.";
}
return status_code;
}
