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integration.cpp

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00001 
00030 #include <itpp/base/math/integration.h>
00031 #include <itpp/base/math/elem_math.h>
00032 #include <itpp/base/help_functions.h>
00033 #include <itpp/base/matfunc.h>
00034 #include <itpp/base/specmat.h>
00035 
00036 
00037 namespace itpp
00038 {
00039 
00041 double quadstep(double(*f)(double), double a, double b,
00042                 double fa, double fm, double fb, double is)
00043 {
00044   double Q, m, h, fml, fmr, i1, i2;
00045   vec x(2), y(2);
00046 
00047   m = (a + b) / 2;
00048   h = (b - a) / 4;
00049   x = vec_2(a + h, b - h);
00050   y = apply_function<double>(f, x);
00051   fml = y(0);
00052   fmr = y(1);
00053 
00054   i1 = h / 1.5 * (fa + 4 * fm + fb);
00055   i2 = h / 3 * (fa + 4 * (fml + fmr) + 2 * fm + fb);
00056   i1 = (16 * i2 - i1) / 15;
00057 
00058   if ((is + (i1 - i2) == is) || (m <= a) || (b <= m)) {
00059     if ((m <= a) || (b <= m)) {
00060       it_warning("Interval contains no more machine number. Required tolerance may not be met");
00061     }
00062     Q = i1;
00063     return Q;
00064   }
00065   else {
00066     Q = quadstep(f, a, m, fa, fml, fm, is) + quadstep(f, m, b, fm, fmr, fb, is);
00067   }
00068   return Q;
00069 }
00070 
00071 
00072 double quad(double(*f)(double), double a, double b, double tol)
00073 {
00074   vec x(3), y(3), yy(5);
00075   double Q, fa, fm, fb, is;
00076 
00077   x = vec_3(a, (a + b) / 2, b);
00078   y = apply_function<double>(f, x);
00079   fa = y(0);
00080   fm = y(1);
00081   fb = y(2);
00082   yy = apply_function<double>(f, a + vec(".9501 .2311 .6068 .4860 .8913")
00083                               * (b - a));
00084   is = (b - a) / 8 * (sum(y) + sum(yy));
00085 
00086   if (is == 0.0)
00087     is = b - a;
00088 
00089   is = is * tol / std::numeric_limits<double>::epsilon();
00090   Q = quadstep(f, a, b, fa, fm, fb, is);
00091 
00092   return Q;
00093 }
00094 
00095 
00096 //--------------------- quadl() ----------------------------------------
00097 
00099 double quadlstep(double(*f)(double), double a, double b,
00100                  double fa, double fb, double is)
00101 {
00102   double Q, h, m, alpha, beta, mll, ml, mr, mrr, fmll, fml, fm, fmr, fmrr,
00103   i1, i2;
00104   vec x(5), y(5);
00105 
00106   h = (b - a) / 2;
00107   m = (a + b) / 2;
00108   alpha = std::sqrt(2.0 / 3);
00109   beta = 1.0 / std::sqrt(5.0);
00110   mll = m - alpha * h;
00111   ml = m - beta * h;
00112   mr = m + beta * h;
00113   mrr = m + alpha * h;
00114   x(0) = mll;
00115   x(1) = ml;
00116   x(2) = m;
00117   x(3) = mr;
00118   x(4) = mrr;
00119 
00120   y = apply_function<double>(f, x);
00121 
00122   fmll = y(0);
00123   fml = y(1);
00124   fm = y(2);
00125   fmr = y(3);
00126   fmrr = y(4);
00127 
00128   i2 = (h / 6) * (fa + fb + 5 * (fml + fmr));
00129   i1 = (h / 1470) * (77 * (fa + fb) + 432 * (fmll + fmrr) + 625 * (fml + fmr) + 672 * fm);
00130 
00131   if ((is + (i1 - i2) == is) || (mll <= a) || (b <= mrr)) {
00132     if ((m <= a) || (b <= m)) {
00133       it_warning("Interval contains no more machine number. Required tolerance may not be met");
00134     }
00135     Q = i1;
00136     return Q;
00137   }
00138   else {
00139     Q = quadlstep(f, a, mll, fa, fmll, is) + quadlstep(f, mll, ml, fmll, fml, is) + quadlstep(f, ml, m, fml, fm, is) +
00140         quadlstep(f, m, mr, fm, fmr, is) + quadlstep(f, mr, mrr, fmr, fmrr, is) + quadlstep(f, mrr, b, fmrr, fb, is);
00141   }
00142   return Q;
00143 }
00144 
00145 double quadl(double(*f)(double), double a, double b, double tol)
00146 {
00147   double Q, m, h, alpha, beta, x1, x2, x3, fa, fb, i1, i2, is, s, erri1, erri2, R;
00148   vec x(13), y(13);
00149   double tol2 = tol;
00150 
00151   m = (a + b) / 2;
00152   h = (b - a) / 2;
00153 
00154   alpha = std::sqrt(2.0 / 3);
00155   beta = 1.0 / std::sqrt(5.0);
00156 
00157   x1 = .942882415695480;
00158   x2 = .641853342345781;
00159   x3 = .236383199662150;
00160   x(0) = a;
00161   x(1) = m - x1 * h;
00162   x(2) = m - alpha * h;
00163   x(3) = m - x2 * h;
00164   x(4) = m - beta * h;
00165   x(5) = m - x3 * h;
00166   x(6) = m;
00167   x(7) = m + x3 * h;
00168   x(8) = m + beta * h;
00169   x(9) = m + x2 * h;
00170   x(10) = m + alpha * h;
00171   x(11) = m + x1 * h;
00172   x(12) = b;
00173 
00174   y = apply_function<double>(f, x);
00175 
00176   fa = y(0);
00177   fb = y(12);
00178   i2 = (h / 6) * (y(0) + y(12) + 5 * (y(4) + y(8)));
00179   i1 = (h / 1470) * (77 * (y(0) + y(12)) + 432 * (y(2) + y(10)) + 625 * (y(4) + y(8)) + 672 * y(6));
00180 
00181   is = h * (.0158271919734802 * (y(0) + y(12)) + .0942738402188500 * (y(1) + y(11)) + .155071987336585 * (y(2) + y(10)) +
00182             .188821573960182 * (y(3) + y(9)) + .199773405226859 * (y(4) + y(8)) + .224926465333340 * (y(5) + y(7)) + .242611071901408 * y(6));
00183 
00184   s = sign(is);
00185   if (s == 0.0)
00186     s = 1;
00187 
00188   erri1 = std::abs(i1 - is);
00189   erri2 = std::abs(i2 - is);
00190 
00191   R = 1;
00192   if (erri2 != 0.0)
00193     R = erri1 / erri2;
00194 
00195   if (R > 0 && R < 1)
00196     tol2 = tol2 / R;
00197 
00198   is = s * std::abs(is) * tol2 / std::numeric_limits<double>::epsilon();
00199   if (is == 0.0)
00200     is = b - a;
00201 
00202   Q = quadlstep(f, a, b, fa, fb, is);
00203 
00204   return Q;
00205 }
00206 
00207 
00208 } // namespace itpp
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