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/*
Theseus - maximum likelihood superpositioning of macromolecular structures
Copyright (C) 2004-2015 Douglas L. Theobald
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the:
Free Software Foundation, Inc.,
59 Temple Place, Suite 330,
Boston, MA 02111-1307 USA
-/_|:|_|_\-
*/
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
#include "integrate.h"
/* WARNING: this integration is not very precise (see EPS below) */
#define EPS 1.0e-7
#define JMAX 20
double
trapzd(double (*func)(double, double, double),
double param1, double param2, double a, double b, int n)
{
double x, tnm, sum, del, val;
static double s;
int it, j;
if (n == 1)
{
return (s = 0.5 * (b - a) * (func(a, param1, param2) + func(b, param1, param2)));
}
else
{
for (it = 1, j = 1; j < n-1; ++j)
it <<= 1;
/* SCREAMD(it); */
tnm = it;
del = (b - a) / tnm;
x = a + 0.5 * del;
sum = 0.0;
for (j = 1; j <= it; j++, x += del)
{
val = func(x, param1, param2);
/* printf("\n--> %12.6f ", val); */
sum += val;
}
s = 0.5 * (s + (b - a) * sum / tnm);
return (s);
}
}
double
integrate_qsimp(double (*func)(double a, double param1, double param2),
double param1, double param2, double a, double b)
{
int j;
double s, st, ost = 0.0, os = 0.0;
for (j = 1; j <= JMAX; ++j)
{
st = trapzd(func, param1, param2, a, b, j);
s = (4.0 * st - ost) / 3.0;
/* if (j > 5) */
/* { */
if ((fabs(s - os) < EPS * fabs(os)) ||
(s == 0.0 && os == 0.0))
{
/* SCREAMD(j); */
return (s);
}
/* } */
os = s;
ost = st;
}
fprintf(stderr, "\n ERROR303: Too many iterations in routine qsimp ");
/* exit (EXIT_FAILURE); */
return (s);
}
#undef EPS
#undef JMAX
/*////////////////////////////////////////////////////////////////////////////////
// File: rombergs_method.c //
// Routines: //
// Rombergs_Integration_Method //
////////////////////////////////////////////////////////////////////////////////*/
#include <math.h> /* // required for fabs() */
static const double richardson[] = {
3.333333333333333333e-01, 6.666666666666666667e-02, 1.587301587301587302e-02,
3.921568627450980392e-03, 9.775171065493646139e-04, 2.442002442002442002e-04,
6.103888176768601599e-05, 1.525902189669642176e-05, 3.814711817595739730e-06,
9.536752259018191355e-07, 2.384186359449949133e-07, 5.960464832810451556e-08,
1.490116141589226448e-08, 3.725290312339701922e-09, 9.313225754828402544e-10,
2.328306437080797376e-10, 5.820766091685553902e-11, 1.455191522857861004e-11,
3.637978807104947841e-12, 9.094947017737554185e-13, 2.273736754432837583e-13,
5.684341886081124604e-14, 1.421085471520220567e-14, 3.552713678800513551e-15,
8.881784197001260212e-16, 2.220446049250313574e-16
};
#define MAX_COLUMNS 1+sizeof(richardson)/sizeof(richardson[0])
#define max(x,y) ( (x) < (y) ? (y) : (x) )
#define min(x,y) ( (x) < (y) ? (x) : (y) )
/* //////////////////////////////////////////////////////////////////////////////// */
/* // double Rombergs_Integration_Method( double a, double h, double tolerance, // */
/* // int max_cols, double (*f)(double), int *err ); // */
/* // // */
/* // Description: // */
/* // If T(f,h,a,b) is the result of applying the trapezoidal rule to approx- // */
/* // imating the integral of f(x) on [a,b] using subintervals of length h, // */
/* // then if I(f,a,b) is the integral of f(x) on [a,b], then // */
/* // I(f,a,b) = lim T(f,h,a,b) // */
/* // where the limit is taken as h approaches 0. // */
/* // The classical Romberg method applies Richardson Extrapolation to the // */
/* // limit of the sequence T(f,h,a,b), T(f,h/2,a,b), T(f,h/4,a,b), ... , // */
/* // in which the limit is approached by successively deleting error terms // */
/* // in the Euler-MacLaurin summation formula. // */
/* // // */
/* // Arguments: // */
/* // double a The lower limit of the integration interval. // */
/* // double h The length of the interval of integration, h > 0. // */
/* // The upper limit of integration is a + h. // */
/* // double tolerance The acceptable error estimate of the integral. // */
/* // Iteration stops when the magnitude of the change of // */
/* // the extrapolated estimate falls below the tolerance. // */
/* // int max_cols The maximum number of columns to be used in the // */
/* // Romberg method. This corresponds to a minimum // */
/* // integration subinterval of length 1/2^max_cols * h. // */
/* // double *f Pointer to the integrand, a function of a single // */
/* // variable of type double. // */
/* // int *err 0 if the extrapolated error estimate falls below the // */
/* // tolerance; -1 if the extrapolated error estimate is // */
/* // greater than the tolerance and the number of columns // */
/* // is max_cols. // */
/* // // */
/* // Return Values: // */
/* // The integral of f(x) from a to a + h. // */
/* // // */
/* //////////////////////////////////////////////////////////////////////////////// */
/* // // */
double
romberg_int(double a, double h, double tolerance,
int max_cols, double (*f) (double), int *err)
{
double upper_limit = a + h;
//upper limit of integration
double dt[MAX_COLUMNS];
//dt[i] is the last element in column i.
double integral = 0.5 * ((*f) (a) + (*f) (a + h));
double x, old_h, delta = 0.0;
int j, k;
/* // Initialize err and the first column, dt[0], to the numerical estimate // */
/* // of the integral using the trapezoidal rule with a step size of h. // */
*err = 0;
dt[0] = 0.5 * h * ((*f) (a) + (*f) (a + h));
/* // For each possible succeeding column, halve the step size, calculate // */
/* // the composite trapezoidal rule using the new step size, and up date // */
/* // preceeding columns using Richardson extrapolation. // */
max_cols = min(max(max_cols, 0), MAX_COLUMNS);
for (k = 1; k < max_cols; k++)
{
old_h = h;
/* // Calculate T(f,h/2,a,b) using T(f,h,a,b) // */
h *= 0.5;
integral = 0.0;
for (x = a + h; x < upper_limit; x += old_h)
integral += (*f) (x);
integral = h * integral + 0.5 * dt[0];
/* // Calculate the Richardson Extrapolation to the limit // */
for (j = 0; j < k; j++)
{
delta = integral - dt[j];
dt[j] = integral;
integral += richardson[j] * delta;
}
/* // If the magnitude of the change in the extrapolated estimate // */
/* // for the integral is less than the preassigned tolerance, // */
/* // return the estimate with err = 0. // */
if (fabs(delta) < tolerance)
return (integral);
/* // Store the current esimate in the kth column. // */
dt[k] = integral;
}
/* // The process didn't converge within the preassigned tolerance // */
/* // using the maximum number of columns designated. // */
/* // Return the current estimate of integral and set err = -1. // */
*err = -1;
return (integral);
}
double
integrate_romberg(double (*f)(double a, double p1, double p2),
double p1, double p2, double a, double b)
{
double h = b - a;
double upper_limit = a + h;
//upper limit of integration
double dt[MAX_COLUMNS];
//dt[i] is the last element in column i.
double integral = 0.5 * ((*f)(a, p1, p2) + (*f)(a+h, p1, p2));
double x, old_h, delta = 0.0;
int j, k;
int max_cols = 5;
double tolerance = 1e-8;
/* // Initialize err and the first column, dt[0], to the numerical estimate // */
/* // of the integral using the trapezoidal rule with a step size of h. // */
dt[0] = 0.5 * h * ((*f)(a, p1, p2) + (*f)(a+h, p1, p2));
/* // For each possible succeeding column, halve the step size, calculate // */
/* // the composite trapezoidal rule using the new step size, and up date // */
/* // preceeding columns using Richardson extrapolation. // */
max_cols = min(max(max_cols, 0), MAX_COLUMNS);
for (k = 1; k < max_cols; k++)
{
old_h = h;
/* // Calculate T(f,h/2,a,b) using T(f,h,a,b) // */
h *= 0.5;
integral = 0.0;
for (x = a + h; x < upper_limit; x += old_h)
integral += (*f)(x, p1, p2);
integral = h * integral + 0.5 * dt[0];
/* // Calculate the Richardson Extrapolation to the limit // */
for (j = 0; j < k; j++)
{
delta = integral - dt[j];
dt[j] = integral;
integral += richardson[j] * delta;
}
/* // If the magnitude of the change in the extrapolated estimate // */
/* // for the integral is less than the preassigned tolerance, // */
/* // return the estimate with err = 0. // */
if (fabs(delta) < tolerance)
return (integral);
/* // Store the current esimate in the kth column. // */
dt[k] = integral;
}
/* // The process didn't converge within the preassigned tolerance // */
/* // using the maximum number of columns designated. // */
/* // Return the current estimate of integral and set err = -1. // */
return (integral);
}
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