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#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_poly.h>
/* Find a minimum in x=[0,1] of the interpolating quadratic through
* (0,f0) (1,f1) with derivative fp0 at x=0. The interpolating
* polynomial is q(x) = f0 + fp0 * z + (f1-f0-fp0) * z^2
*/
static double
interp_quad (double f0, double fp0, double f1, double zl, double zh)
{
double fl = f0 + zl*(fp0 + zl*(f1 - f0 -fp0));
double fh = f0 + zh*(fp0 + zh*(f1 - f0 -fp0));
double c = 2 * (f1 - f0 - fp0); /* curvature */
double zmin = zl, fmin = fl;
if (fh < fmin) { zmin = zh; fmin = fh; }
if (c > 0) /* positive curvature required for a minimum */
{
double z = -fp0 / c; /* location of minimum */
if (z > zl && z < zh) {
double f = f0 + z*(fp0 + z*(f1 - f0 -fp0));
if (f < fmin) { zmin = z; fmin = f; };
}
}
return zmin;
}
/* Find a minimum in x=[0,1] of the interpolating cubic through
* (0,f0) (1,f1) with derivatives fp0 at x=0 and fp1 at x=1.
*
* The interpolating polynomial is:
*
* c(x) = f0 + fp0 * z + eta * z^2 + xi * z^3
*
* where eta=3*(f1-f0)-2*fp0-fp1, xi=fp0+fp1-2*(f1-f0).
*/
static double
cubic (double c0, double c1, double c2, double c3, double z)
{
return c0 + z * (c1 + z * (c2 + z * c3));
}
static void
check_extremum (double c0, double c1, double c2, double c3, double z,
double *zmin, double *fmin)
{
/* could make an early return by testing curvature >0 for minimum */
double y = cubic (c0, c1, c2, c3, z);
if (y < *fmin)
{
*zmin = z; /* accepted new point*/
*fmin = y;
}
}
static double
interp_cubic (double f0, double fp0, double f1, double fp1, double zl, double zh)
{
double eta = 3 * (f1 - f0) - 2 * fp0 - fp1;
double xi = fp0 + fp1 - 2 * (f1 - f0);
double c0 = f0, c1 = fp0, c2 = eta, c3 = xi;
double zmin, fmin;
double z0, z1;
zmin = zl; fmin = cubic(c0, c1, c2, c3, zl);
check_extremum (c0, c1, c2, c3, zh, &zmin, &fmin);
{
int n = gsl_poly_solve_quadratic (3 * c3, 2 * c2, c1, &z0, &z1);
if (n == 2) /* found 2 roots */
{
if (z0 > zl && z0 < zh)
check_extremum (c0, c1, c2, c3, z0, &zmin, &fmin);
if (z1 > zl && z1 < zh)
check_extremum (c0, c1, c2, c3, z1, &zmin, &fmin);
}
else if (n == 1) /* found 1 root */
{
if (z0 > zl && z0 < zh)
check_extremum (c0, c1, c2, c3, z0, &zmin, &fmin);
}
}
return zmin;
}
static double
interpolate (double a, double fa, double fpa,
double b, double fb, double fpb, double xmin, double xmax,
int order)
{
/* Map [a,b] to [0,1] */
double z, alpha, zmin, zmax;
zmin = (xmin - a) / (b - a);
zmax = (xmax - a) / (b - a);
if (zmin > zmax)
{
double tmp = zmin;
zmin = zmax;
zmax = tmp;
};
if (order > 2 && GSL_IS_REAL(fpb)) {
z = interp_cubic (fa, fpa * (b - a), fb, fpb * (b - a), zmin, zmax);
} else {
z = interp_quad (fa, fpa * (b - a), fb, zmin, zmax);
}
alpha = a + z * (b - a);
return alpha;
}
/* recommended values from Fletcher are
rho = 0.01, sigma = 0.1, tau1 = 9, tau2 = 0.05, tau3 = 0.5 */
static int
minimize (gsl_function_fdf * fn, double rho, double sigma,
double tau1, double tau2, double tau3,
int order, double alpha1, double *alpha_new)
{
double f0, fp0, falpha, falpha_prev, fpalpha, fpalpha_prev, delta,
alpha_next;
double alpha = alpha1, alpha_prev = 0.0;
double a, b, fa, fb, fpa, fpb;
const size_t bracket_iters = 100, section_iters = 100;
size_t i = 0;
GSL_FN_FDF_EVAL_F_DF (fn, 0.0, &f0, &fp0);
falpha_prev = f0;
fpalpha_prev = fp0;
/* Avoid uninitialized variables morning */
a = 0.0; b = alpha;
fa = f0; fb = 0.0;
fpa = fp0; fpb = 0.0;
/* Begin bracketing */
while (i++ < bracket_iters)
{
falpha = GSL_FN_FDF_EVAL_F (fn, alpha);
/* Fletcher's rho test */
if (falpha > f0 + alpha * rho * fp0 || falpha >= falpha_prev)
{
a = alpha_prev; fa = falpha_prev; fpa = fpalpha_prev;
b = alpha; fb = falpha; fpb = GSL_NAN;
break; /* goto sectioning */
}
fpalpha = GSL_FN_FDF_EVAL_DF (fn, alpha);
/* Fletcher's sigma test */
if (fabs (fpalpha) <= -sigma * fp0)
{
*alpha_new = alpha;
return GSL_SUCCESS;
}
if (fpalpha >= 0)
{
a = alpha; fa = falpha; fpa = fpalpha;
b = alpha_prev; fb = falpha_prev; fpb = fpalpha_prev;
break; /* goto sectioning */
}
delta = alpha - alpha_prev;
{
double lower = alpha + delta;
double upper = alpha + tau1 * delta;
alpha_next = interpolate (alpha_prev, falpha_prev, fpalpha_prev,
alpha, falpha, fpalpha, lower, upper, order);
}
alpha_prev = alpha;
falpha_prev = falpha;
fpalpha_prev = fpalpha;
alpha = alpha_next;
}
/* Sectioning of bracket [a,b] */
while (i++ < section_iters)
{
delta = b - a;
{
double lower = a + tau2 * delta;
double upper = b - tau3 * delta;
alpha = interpolate (a, fa, fpa, b, fb, fpb, lower, upper, order);
}
falpha = GSL_FN_FDF_EVAL_F (fn, alpha);
if ((a-alpha)*fpa <= GSL_DBL_EPSILON) {
/* roundoff prevents progress */
return GSL_ENOPROG;
};
if (falpha > f0 + rho * alpha * fp0 || falpha >= fa)
{
/* a_next = a; */
b = alpha; fb = falpha; fpb = GSL_NAN;
}
else
{
fpalpha = GSL_FN_FDF_EVAL_DF (fn, alpha);
if (fabs(fpalpha) <= -sigma * fp0)
{
*alpha_new = alpha;
return GSL_SUCCESS; /* terminate */
}
if ( ((b-a) >= 0 && fpalpha >= 0) || ((b-a) <=0 && fpalpha <= 0))
{
b = a; fb = fa; fpb = fpa;
a = alpha; fa = falpha; fpa = fpalpha;
}
else
{
a = alpha; fa = falpha; fpa = fpalpha;
}
}
}
return GSL_SUCCESS;
}
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