1 | /** |
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2 | * \file |
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3 | * \brief Provides code for the general c2_function algebra which supports |
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4 | * fast, flexible operations on piecewise-twice-differentiable functions |
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5 | * |
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6 | * \author Created by R. A. Weller and Marcus H. Mendenhall on 7/9/05. |
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7 | * \author Copyright 2005 __Vanderbilt University__. All rights reserved. |
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8 | * |
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9 | * \version c2_function.cc,v 1.43 2007/11/12 20:22:54 marcus Exp |
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10 | */ |
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11 | |
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12 | #include <iostream> |
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13 | #include <vector> |
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14 | #include <algorithm> |
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15 | #include <cstdlib> |
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16 | #include <numeric> |
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17 | #include <functional> |
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18 | #include <iterator> |
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19 | #include <cmath> |
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20 | #include <limits> |
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21 | #include <sstream> |
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22 | |
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23 | template <typename float_type> const std::string c2_function<float_type>::cvs_file_vers() const |
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24 | { return "c2_function.cc,v 1.43 2007/11/12 20:22:54 marcus Exp"; } |
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25 | |
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26 | // find a pre-bracketed root of a c2_function, which is a MUCH easier job than general root finding |
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27 | // since the derivatives are known exactly, and smoothness is guaranteed. |
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28 | // this searches for f(x)=value, to make life a little easier than always searching for f(x)=0 |
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29 | template <typename float_type> float_type c2_function<float_type>::find_root(float_type lower_bracket, float_type upper_bracket, |
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30 | float_type start, float_type value, int *error, |
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31 | float_type *final_yprime, float_type *final_yprime2) const throw(c2_exception) |
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32 | { |
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33 | // find f(x)=value within the brackets, using the guarantees of smoothness associated with a c2_function |
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34 | // can use local f(x)=a*x**2 + b*x + c and solve quadratic to find root, then iterate |
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35 | reset_evaluations(); |
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36 | |
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37 | float_type yp, yp2; // we will make unused pointers point here, to save null checks later |
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38 | if (!final_yprime) final_yprime=&yp; |
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39 | if (!final_yprime2) final_yprime2=&yp2; |
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40 | |
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41 | float_type ftol=5*(std::numeric_limits<float_type>::epsilon()*std::abs(value)+std::numeric_limits<float_type>::min()); |
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42 | float_type xtol=5*(std::numeric_limits<float_type>::epsilon()*(std::abs(upper_bracket)+std::abs(lower_bracket))+std::numeric_limits<float_type>::min()); |
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43 | |
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44 | float_type root=start; // start looking in the middle |
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45 | if(error) *error=0; // start out with error flag set to OK, if it is expected |
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46 | |
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47 | float_type c, b; |
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48 | |
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49 | // this new logic is to keep track of where we were before, and lower the number of |
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50 | // function evaluations if we are searching inside the same bracket as before. |
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51 | // Since this root finder has, very often, the bracket of the entire domain of the function, |
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52 | // this makes a big difference, especially to c2_inverse_function |
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53 | if(!rootInitialized || upper_bracket != lastRootUpperX || lower_bracket != lastRootLowerX) { |
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54 | lastRootUpperY=value_with_derivatives(upper_bracket, final_yprime, final_yprime2); |
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55 | increment_evaluations(); |
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56 | lastRootUpperX=upper_bracket; |
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57 | |
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58 | lastRootLowerY=value_with_derivatives(lower_bracket, final_yprime, final_yprime2); |
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59 | increment_evaluations(); |
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60 | lastRootLowerX=lower_bracket; |
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61 | rootInitialized=true; |
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62 | } |
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63 | |
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64 | float_type clower=lastRootLowerY-value; |
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65 | float_type cupper=lastRootUpperY-value; |
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66 | if(clower*cupper >0) { |
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67 | // argh, no sign change in here! |
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68 | if(error) { *error=1; return 0.0; } |
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69 | else { |
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70 | std::ostringstream outstr; |
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71 | outstr << "unbracketed root in find_root at xlower= " << lower_bracket << ", xupper= " << upper_bracket; |
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72 | outstr << ", value= " << value << ": bailing"; |
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73 | throw c2_exception(outstr.str().c_str()); |
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74 | } |
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75 | } |
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76 | |
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77 | float_type delta=upper_bracket-lower_bracket; // first error step |
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78 | c=value_with_derivatives(root, final_yprime, final_yprime2)-value; // compute initial values |
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79 | b=*final_yprime; // make a local copy for readability |
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80 | increment_evaluations(); |
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81 | |
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82 | while( |
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83 | std::abs(delta) > xtol && // absolute x step check |
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84 | std::abs(c) > ftol && // absolute y tolerance |
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85 | std::abs(c) > xtol*std::abs(b) // comparison to smallest possible Y step from derivative |
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86 | ) |
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87 | { |
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88 | float_type a=(*final_yprime2)/2; // second derivative is 2*a |
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89 | float_type disc=b*b-4*a*c; |
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90 | // std::cout << std::endl << "find_root_debug a,b,c,d " << a << " " << b << " " << c << " " << disc << std::endl; |
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91 | |
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92 | if(disc >= 0) { |
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93 | float_type q=-0.5*((b>=0)?(b+std::sqrt(disc)):(b-std::sqrt(disc))); |
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94 | if(q*q > std::abs(a*c)) delta=c/q; // since x1=q/a, x2=c/q, x1/x2=q^2/ac, this picks smaller step |
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95 | else delta=q/a; |
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96 | root+=delta; |
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97 | } |
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98 | |
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99 | if(disc < 0 || root<lower_bracket || root>upper_bracket || |
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100 | std::abs(delta) >= 0.5*(upper_bracket-lower_bracket)) { |
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101 | // if we jump out of the bracket, or aren't converging well, bisect |
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102 | root=0.5*(lower_bracket+upper_bracket); |
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103 | delta=upper_bracket-lower_bracket; |
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104 | } |
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105 | c=value_with_derivatives(root, final_yprime, final_yprime2)-value; // compute initial values |
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106 | b=*final_yprime; // make a local copy for readability |
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107 | increment_evaluations(); |
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108 | |
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109 | // now, close in bracket on whichever side this still brackets |
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110 | if(c*clower < 0.0) { |
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111 | cupper=c; |
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112 | upper_bracket=root; |
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113 | } else { |
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114 | clower=c; |
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115 | lower_bracket=root; |
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116 | } |
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117 | // std::cout << "find_root_debug x, y, dx " << root << " " << c << " " << delta << std::endl; |
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118 | } |
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119 | return root; |
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120 | } |
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121 | |
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122 | /* def partial_integrals(self, xgrid): |
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123 | Return the integrals of a function between the sampling points xgrid. The sum is the definite integral. |
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124 | This method uses an exact integration of the polynomial which matches the values and derivatives at the |
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125 | endpoints of a segment. Its error scales as h**6, if the input functions really are smooth. |
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126 | This could very well be used as a stepper for adaptive Romberg integration. |
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127 | For InterpolatingFunctions, it is likely that the Simpson's rule integrator is sufficient. |
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128 | #the weights come from an exact mathematica solution to the 5th order polynomial with the given values & derivatives |
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129 | #yint = (y0+y1)*dx/2 + dx^2*(yp0-yp1)/10 + dx^3 * (ypp0+ypp1)/120 ) |
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130 | */ |
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131 | |
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132 | // the recursive part of the integrator is agressively designed to minimize copying of data... lots of pointers |
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133 | template <typename float_type> float_type c2_function<float_type>::integrate_step(c2_integrate_recur &rb) const |
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134 | { |
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135 | struct c2_integrate_fblock *fbl[3]={rb.f0, rb.f1, rb.f2}; |
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136 | struct c2_integrate_fblock f1; // will hold new middle values |
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137 | float_type retvals[2]={0.0,0.0}; |
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138 | float_type lr[2]; |
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139 | |
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140 | // std::cout << "entering with " << rb.f0->x << " " << rb.f1->x << " " << rb.f2->x << std::endl; |
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141 | |
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142 | int depth=rb.depth; // save this from the recursion block |
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143 | float_type abs_tol=rb.abs_tol; // this is the value we will pass down |
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144 | float_type *rblr=rb.lr; // save pointer to our parent's lr[2] array since it will get trampled in recursion |
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145 | |
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146 | if(!depth) { |
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147 | switch(rb.derivs) { |
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148 | case 0: |
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149 | rb.eps_scale=0.1; rb.extrap_coef=16; break; |
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150 | case 1: |
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151 | rb.eps_scale=0.1; rb.extrap_coef=64; break; |
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152 | case 2: |
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153 | rb.eps_scale=0.02; rb.extrap_coef=1024; break; |
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154 | default: |
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155 | throw c2_exception("derivs must be 0, 1 or 2 in partial_integrals"); |
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156 | } |
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157 | |
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158 | rb.extrap2=1.0/(rb.extrap_coef-1.0); |
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159 | } |
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160 | |
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161 | for (int i=0; i<(depth==0?1:2); i++) { // handle left and right intervals, but only left one for depth=0 |
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162 | struct c2_integrate_fblock *f0=fbl[i], *f2=fbl[i+1]; |
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163 | f1.x=0.5*(f0->x + f2->x); // center of interval |
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164 | float_type dx=f2->x - f0->x; |
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165 | float_type dx2 = 0.5*dx; |
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166 | float_type total; |
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167 | |
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168 | f1.y=value_with_derivatives(f1.x, &(f1.yp), &(f1.ypp)); |
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169 | increment_evaluations(); |
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170 | |
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171 | // check for underflow on step size, which prevents us from achieving specified accuracy. |
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172 | if(std::abs(dx) < std::abs(f1.x)*rb.rel_tol) { |
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173 | std::ostringstream outstr; |
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174 | outstr << "Step size underflow in adaptive_partial_integrals at depth=" << depth << ", x= " << f1.x; |
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175 | throw c2_exception(outstr.str().c_str()); |
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176 | } |
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177 | |
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178 | if(!depth) { // top level, total has not been initialized yet |
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179 | switch(rb.derivs) { // create estimate of next lower order for first try |
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180 | case 0: |
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181 | total=0.5*(f0->y+f2->y)*dx; break; |
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182 | case 1: |
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183 | total=(f0->y+4.0*f1.y+f2->y)*dx/6.0; break; |
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184 | case 2: |
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185 | total=( (14*f0->y + 32*f1.y + 14*f2->y) + dx * (f0->yp - f2->yp) ) * dx /60.; break; |
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186 | default: |
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187 | total=0.0; // just to suppress missing default warnings |
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188 | } |
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189 | } else total=rblr[i]; // otherwise, get it from previous level |
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190 | |
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191 | float_type left, right; |
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192 | |
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193 | switch(rb.derivs) { |
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194 | case 2: |
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195 | // use ninth-order estimates for each side, from full set of all values (!) (Thanks, Mathematica!) |
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196 | left= ( ( (169*f0->ypp + 1024*f1.ypp - 41*f2->ypp)*dx2 + |
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197 | (2727*f0->yp - 5040*f1.yp + 423*f2->yp) )*dx2 + |
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198 | (17007*f0->y + 24576*f1.y - 1263*f2->y) )* (dx2/40320.0); |
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199 | right= ( ( (169*f2->ypp + 1024*f1.ypp - 41*f0->ypp)*dx2 - |
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200 | (2727*f2->yp - 5040*f1.yp + 423*f0->yp) )*dx2 + |
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201 | (17007*f2->y + 24576*f1.y - 1263*f0->y) )* (dx2/40320.0); |
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202 | // std::cout << f0->x << " " << f1.x << " " << f2->x << std::endl ; |
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203 | // std::cout << f0->y << " " << f1.y << " " << f2->y << " " << left << " " << right << " " << total << std::endl ; |
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204 | break; |
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205 | case 1: |
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206 | left= ( (202*f0->y + 256*f1.y + 22*f2->y) + dx*(13*f0->yp - 40*f1.yp - 3*f2->yp) ) * dx /960.; |
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207 | right= ( (202*f2->y + 256*f1.y + 22*f0->y) - dx*(13*f2->yp - 40*f1.yp - 3*f0->yp) ) * dx /960.; |
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208 | break; |
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209 | case 0: |
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210 | left= (5*f0->y + 8*f1.y - f2->y)*dx/24.; |
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211 | right= (5*f2->y + 8*f1.y - f0->y)*dx/24.; |
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212 | break; |
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213 | default: |
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214 | left=right=0.0; // suppress warnings about missing default |
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215 | break; |
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216 | } |
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217 | |
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218 | lr[0]= left; // left interval |
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219 | lr[1]= right; // right interval |
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220 | float_type lrsum=left+right; |
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221 | |
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222 | float_type eps=std::abs(total-lrsum)*rb.eps_scale; |
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223 | if(rb.extrapolate) eps*=rb.eps_scale; |
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224 | |
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225 | if(!rb.adapt || eps < abs_tol || eps < std::abs(total)*rb.rel_tol) { |
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226 | if(depth==0 || !rb.extrapolate) retvals[i]=lrsum; |
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227 | else { |
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228 | retvals[i]=(rb.extrap_coef*lrsum - total)*rb.extrap2; |
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229 | // std::cout << "extrapolating " << lrsum << " " << total << " " << retvals[i] << std::endl; |
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230 | |
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231 | } |
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232 | } else { |
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233 | rb.depth=depth+1; // increment depth counter |
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234 | rb.lr=lr; // point to our left-right values array for recursion |
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235 | rb.abs_tol=abs_tol*0.5; // each half has half the error budget |
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236 | rb.f0=f0; rb.f1=&f1; rb.f2=f2; // insert pointers to data into our recursion block |
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237 | // std::cout << "recurring with " << f0->x << " " << f1.x << " " << f2->x << std::endl ; |
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238 | retvals[i]=integrate_step(rb); // and recur |
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239 | } |
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240 | } |
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241 | return retvals[0]+retvals[1]; |
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242 | } |
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243 | |
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244 | template <typename float_type> bool c2_function<float_type>::check_monotonicity( |
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245 | const std::vector<float_type> &data, const char message[]) throw(c2_exception) |
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246 | { |
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247 | size_t np=data.size(); |
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248 | if(np < 2) return false; // one point has no direction! |
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249 | |
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250 | bool rev=(data[1] < data[0]); // which way do data point? |
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251 | size_t i; |
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252 | |
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253 | if(!rev) for(i = 2; i < np && (data[i-1] < data[i]) ; i++); |
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254 | else for(i = 2; i < np &&(data[i-1] > data[i]) ; i++); |
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255 | |
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256 | if(i != np) throw c2_exception(message); |
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257 | |
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258 | return rev; |
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259 | } |
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260 | |
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261 | template <typename float_type> void c2_function<float_type>::set_sampling_grid(const std::vector<float_type> &grid) throw(c2_exception) |
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262 | { |
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263 | bool rev=check_monotonicity(grid, "set_sampling_grid: sampling grid not monotonic"); |
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264 | |
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265 | if(!sampling_grid || no_overwrite_grid) sampling_grid=new std::vector<float_type>; |
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266 | (*sampling_grid)=grid; no_overwrite_grid=0; |
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267 | |
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268 | if(rev) std::reverse(sampling_grid->begin(), sampling_grid->end()); // make it increasing |
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269 | } |
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270 | |
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271 | template <typename float_type> std::vector<float_type> &c2_function<float_type>::get_sampling_grid(float_type xmin, float_type xmax) const |
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272 | { |
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273 | std::vector<float_type> *result=new std::vector<float_type>; |
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274 | |
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275 | if( !(sampling_grid) || !(sampling_grid->size()) || (xmax <= sampling_grid->front()) || (xmin >= sampling_grid->back()) ) { |
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276 | // nothing is known about the function in this region, return xmin and xmax |
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277 | result->push_back(xmin); |
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278 | result->push_back(xmax); |
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279 | } else { |
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280 | std::vector<float_type> &sg=*sampling_grid; // just a shortcut |
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281 | int np=sg.size(); |
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282 | int klo=0, khi=np-1, firstindex=0, lastindex=np-1; |
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283 | |
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284 | result->push_back(xmin); |
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285 | |
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286 | if(xmin > sg.front() ) { |
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287 | // hunt through table for position bracketing starting point |
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288 | while(khi-klo > 1) { |
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289 | int km=(khi+klo)/2; |
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290 | if(sg[km] > xmin) khi=km; |
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291 | else klo=km; |
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292 | } |
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293 | khi=klo+1; |
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294 | // khi now points to first point definitively beyond our first point, or last point of array |
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295 | firstindex=khi; |
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296 | khi=np-1; // restart upper end of search |
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297 | } |
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298 | |
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299 | if(xmax < sg.back()) { |
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300 | // hunt through table for position bracketing starting point |
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301 | while(khi-klo > 1) { |
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302 | int km=(khi+klo)/2; |
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303 | if(sg[km] > xmax) khi=km; |
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304 | else klo=km; |
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305 | } |
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306 | khi=klo+1; |
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307 | // khi now points to first point definitively beyond our last point, or last point of array |
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308 | lastindex=klo; |
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309 | } |
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310 | |
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311 | int initsize=result->size(); |
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312 | result->resize(initsize+(lastindex-firstindex+2)); |
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313 | std::copy(sg.begin()+firstindex, sg.begin()+lastindex+1, result->begin()+initsize); |
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314 | result->back()=xmax; |
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315 | |
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316 | // this is the unrefined sampling grid... now check for very close points on front & back and fix if needed. |
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317 | preen_sampling_grid(result); |
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318 | } |
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319 | return *result; |
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320 | } |
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321 | |
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322 | template <typename float_type> void c2_function<float_type>::preen_sampling_grid(std::vector<float_type> *result) const |
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323 | { |
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324 | // this is the unrefined sampling grid... now check for very close points on front & back and fix if needed. |
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325 | if(result->size() > 2) { // may be able to prune dangerously close points near the ends if there are at least 3 points |
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326 | bool deleteit=false; |
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327 | float_type x0=(*result)[0], x1=(*result)[1]; |
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328 | float_type dx1=x1-x0; |
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329 | |
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330 | float_type ftol=10.0*(std::numeric_limits<float_type>::epsilon()*(std::abs(x0)+std::abs(x1))+std::numeric_limits<float_type>::min()); |
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331 | if(dx1 < ftol) deleteit=true; |
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332 | float_type dx2=(*result)[2]-x0; |
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333 | if(dx1/dx2 < 0.1) deleteit=true; // endpoint is very close to internal interesting point |
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334 | |
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335 | if(deleteit) result->erase(result->begin()+1); // delete redundant interesting point |
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336 | } |
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337 | |
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338 | if(result->size() > 2) { // may be able to prune dangerously close points near the ends if there are at least 3 points |
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339 | bool deleteit=false; |
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340 | int pos=result->size()-3; |
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341 | float_type x0=(*result)[pos+1], x1=(*result)[pos+2]; |
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342 | float_type dx1=x1-x0; |
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343 | |
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344 | float_type ftol=10.0*(std::numeric_limits<float_type>::epsilon()*(std::abs(x0)+std::abs(x1))+std::numeric_limits<float_type>::min()); |
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345 | if(dx1 < ftol) deleteit=true; |
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346 | float_type dx2=x1-(*result)[pos]; |
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347 | if(dx1/dx2 < 0.1) deleteit=true; // endpoint is very close to internal interesting point |
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348 | |
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349 | if(deleteit) result->erase(result->end()-2); // delete redundant interesting point |
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350 | } |
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351 | } |
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352 | |
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353 | template <typename float_type> std::vector<float_type> &c2_function<float_type>:: |
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354 | refine_sampling_grid(const std::vector<float_type> &grid, size_t refinement) const |
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355 | { |
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356 | size_t np=grid.size(); |
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357 | size_t count=(np-1)*refinement + 1; |
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358 | float_type dxscale=1.0/refinement; |
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359 | |
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360 | std::vector<float_type> *result=new std::vector<float_type>(count); |
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361 | |
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362 | for(size_t i=0; i<(np-1); i++) { |
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363 | float_type x=grid[i]; |
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364 | float_type dx=(grid[i+1]-x)*dxscale; |
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365 | for(size_t j=0; j<refinement; j++, x+=dx) (*result)[i*refinement+j]=x; |
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366 | } |
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367 | (*result)[count-1]=grid.back(); |
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368 | return *result; |
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369 | } |
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370 | |
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371 | template <typename float_type> float_type c2_function<float_type>::integral(float_type xmin, float_type xmax, std::vector<float_type> *partials, |
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372 | float_type abs_tol, float_type rel_tol, int derivs, bool adapt, bool extrapolate) const |
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373 | { |
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374 | std::vector<float_type> &grid=get_sampling_grid(xmin, xmax); |
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375 | float_type intg=partial_integrals(grid, partials, abs_tol, rel_tol, adapt, extrapolate); |
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376 | delete &grid; |
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377 | return intg; |
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378 | } |
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379 | |
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380 | template <typename float_type> c2_function<float_type> &c2_function<float_type>::normalized_function(float_type xmin, float_type xmax, float_type norm) |
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381 | { |
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382 | float_type intg=integral(xmin, xmax); |
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383 | return *new c2_scaled_function<float_type>(*this, norm/intg); |
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384 | } |
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385 | |
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386 | template <typename float_type> c2_function<float_type> &c2_function<float_type>::square_normalized_function(float_type xmin, float_type xmax, float_type norm) |
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387 | { |
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388 | c2_quadratic<float_type> q(0., 0., 0., 1.); |
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389 | c2_composed_function<float_type> mesquared(q,*this); |
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390 | |
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391 | std::vector<float_type> grid(get_sampling_grid(xmin, xmax)); |
---|
392 | float_type intg=mesquared.partial_integrals(grid); |
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393 | |
---|
394 | return *new c2_scaled_function<float_type>(*this, std::sqrt(norm/intg)); |
---|
395 | } |
---|
396 | |
---|
397 | template <typename float_type> c2_function<float_type> &c2_function<float_type>::square_normalized_function( |
---|
398 | float_type xmin, float_type xmax, const c2_function<float_type> &weight, float_type norm) |
---|
399 | { |
---|
400 | c2_quadratic<float_type> q(0., 0., 0., 1.); |
---|
401 | c2_composed_function<float_type> mesquared(q,*this); |
---|
402 | c2_product<float_type> weighted(mesquared, weight); |
---|
403 | |
---|
404 | std::vector<float_type> grid(get_sampling_grid(xmin, xmax)); |
---|
405 | float_type intg=weighted.partial_integrals(grid); |
---|
406 | |
---|
407 | return *new c2_scaled_function<float_type>(*this, std::sqrt(norm/intg)); |
---|
408 | } |
---|
409 | |
---|
410 | template <typename float_type> float_type c2_function<float_type>::partial_integrals( |
---|
411 | std::vector<float_type> xgrid, std::vector<float_type> *partials, |
---|
412 | float_type abs_tol, float_type rel_tol, int derivs, bool adapt, bool extrapolate) const |
---|
413 | { |
---|
414 | int np=xgrid.size(); |
---|
415 | |
---|
416 | struct c2_integrate_fblock f0, f2; |
---|
417 | struct c2_integrate_recur rb; |
---|
418 | rb.rel_tol=rel_tol; |
---|
419 | rb.extrapolate=extrapolate; |
---|
420 | rb.adapt=adapt; |
---|
421 | rb.derivs=derivs; |
---|
422 | |
---|
423 | reset_evaluations(); // counter returns with total evaluations needed for this integral |
---|
424 | |
---|
425 | if(partials) partials->resize(np-1); |
---|
426 | |
---|
427 | float_type sum=0.0; |
---|
428 | |
---|
429 | f2.x=xgrid[0]; |
---|
430 | f2.y=value_with_derivatives(f2.x, &f2.yp, &f2.ypp); |
---|
431 | increment_evaluations(); |
---|
432 | |
---|
433 | for(int i=0; i<np-1; i++) { |
---|
434 | f0=f2; // copy upper bound to lower before computing new upper bound |
---|
435 | |
---|
436 | f2.x=xgrid[i+1]; |
---|
437 | f2.y=value_with_derivatives(f2.x, &f2.yp, &f2.ypp); |
---|
438 | increment_evaluations(); |
---|
439 | |
---|
440 | rb.depth=0; |
---|
441 | rb.abs_tol=abs_tol; |
---|
442 | rb.f0=&f0; rb.f1=&f2; rb.f2=&f2; // we are really only using the left half for the top level |
---|
443 | rb.lr=0; // pointer is meaningless; will be filled in in recursion |
---|
444 | float_type ps=integrate_step(rb); |
---|
445 | sum+=ps; |
---|
446 | if(partials) (*partials)[i]=ps; |
---|
447 | } |
---|
448 | return sum; |
---|
449 | } |
---|
450 | |
---|
451 | // declare singleton functions for most common c2_function instances |
---|
452 | #define c2_singleton(X) template <typename float_type> const c2_##X<float_type> c2_##X<float_type>::X=c2_##X(); |
---|
453 | c2_singleton(sin) |
---|
454 | c2_singleton(cos) |
---|
455 | c2_singleton(tan) |
---|
456 | c2_singleton(log) |
---|
457 | c2_singleton(exp) |
---|
458 | c2_singleton(sqrt) |
---|
459 | c2_singleton(identity) |
---|
460 | |
---|
461 | // reciprocal is actually parametric (a/x), but make singleton 1/x |
---|
462 | template <typename float_type> const c2_recip<float_type> c2_recip<float_type>::recip=c2_recip(1.0); |
---|
463 | |
---|
464 | #undef c2_singleton |
---|
465 | |
---|
466 | // generate a sampling grid at points separated by dx=5, which is intentionally |
---|
467 | // incommensurate with pi and 2*pi so grid errors are somewhat randomized |
---|
468 | template <typename float_type> std::vector<float_type> &c2_sin<float_type>::get_sampling_grid(float_type xmin, float_type xmax) |
---|
469 | { |
---|
470 | std::vector<float_type> *result=new std::vector<float_type>; |
---|
471 | |
---|
472 | for(; xmin < xmax; xmin+=5.0) result->push_back(xmin); |
---|
473 | result->push_back(xmax); |
---|
474 | this->preen_sampling_grid(result); |
---|
475 | return *result; |
---|
476 | } |
---|
477 | |
---|
478 | template <typename float_type> float_type Identity(float_type x) { return x; } // a useful function |
---|
479 | template <typename float_type> float_type f_one(float_type) { return 1.0; } // the first derivative of identity |
---|
480 | template <typename float_type> float_type f_zero(float_type) { return 0.0; } // the second derivative of identity |
---|
481 | |
---|
482 | // The constructor |
---|
483 | template <typename float_type> void interpolating_function<float_type>::init( |
---|
484 | const std::vector<float_type> &x, const std::vector<float_type> &f, |
---|
485 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
486 | bool upperSlopeNatural, float_type upperSlope, |
---|
487 | float_type (*inputXConversion)(float_type), |
---|
488 | float_type (*inputXConversionPrime)(float_type), |
---|
489 | float_type (*inputXConversionDPrime)(float_type), |
---|
490 | float_type (*inputYConversion)(float_type), |
---|
491 | float_type (*inputYConversionPrime)(float_type), |
---|
492 | float_type (*inputYConversionDPrime)(float_type), |
---|
493 | float_type (*outputYConversion)(float_type) |
---|
494 | ) throw(c2_exception) |
---|
495 | { |
---|
496 | X= x; |
---|
497 | F= f; |
---|
498 | |
---|
499 | // Xraw is useful in some of the arithmetic operations between interpolating functions |
---|
500 | Xraw=x; |
---|
501 | |
---|
502 | set_domain(std::min(Xraw.front(), Xraw.back()),std::max(Xraw.front(), Xraw.back())); |
---|
503 | |
---|
504 | fXin=inputXConversion; |
---|
505 | fXinPrime=inputXConversionPrime; |
---|
506 | fXinDPrime=inputXConversionDPrime; |
---|
507 | fYin=inputYConversion; |
---|
508 | fYinPrime=inputYConversionPrime; |
---|
509 | fYinDPrime=inputYConversionDPrime; |
---|
510 | fYout=outputYConversion; |
---|
511 | |
---|
512 | if(x.size() != f.size()) { |
---|
513 | throw c2_exception("interpolating_function::init() -- x & y inputs are of different size"); |
---|
514 | } |
---|
515 | |
---|
516 | size_t np=X.size(); // they are the same now, so lets take a short cut |
---|
517 | |
---|
518 | if(np < 2) { |
---|
519 | throw c2_exception("interpolating_function::init() -- input < 2 elements "); |
---|
520 | } |
---|
521 | |
---|
522 | bool xraw_rev=check_monotonicity(Xraw, |
---|
523 | "interpolating_function::init() non-monotonic raw x input"); // which way does raw X point? sampling grid MUST be increasing |
---|
524 | |
---|
525 | if(!xraw_rev) { // we can use pointer to raw X values if they are in the right order |
---|
526 | set_sampling_grid_pointer(Xraw); // our intial grid of x values is certainly a good guess for 'interesting' points |
---|
527 | } else { |
---|
528 | set_sampling_grid(Xraw); // make a copy of it, and assure it is in right order |
---|
529 | } |
---|
530 | |
---|
531 | if(fXin) { // check if X scale is nonlinear, and if so, do transform |
---|
532 | if(!lowerSlopeNatural) lowerSlope /= fXinPrime(X[0]); |
---|
533 | if(!upperSlopeNatural) upperSlope /= fXinPrime(X[np-1]); |
---|
534 | for(size_t i=0; i<np; i++) X[i]=fXin(X[i]); |
---|
535 | } else { |
---|
536 | fXin=Identity<float_type>; |
---|
537 | fXinPrime=f_one<float_type>; |
---|
538 | fXinDPrime=f_zero<float_type>; |
---|
539 | } |
---|
540 | |
---|
541 | if(inputYConversion) { // check if Y scale is nonlinear, and if so, do transform |
---|
542 | if(!lowerSlopeNatural) lowerSlope *= fYinPrime(F[0]); |
---|
543 | if(!upperSlopeNatural) upperSlope *= fYinPrime(F[np-1]); |
---|
544 | for(size_t i=0; i<np; i++) F[i]=inputYConversion(F[i]); |
---|
545 | } else { |
---|
546 | fYin=Identity<float_type>; |
---|
547 | fYinPrime=f_one<float_type>; |
---|
548 | fYinDPrime=f_zero<float_type>; |
---|
549 | fYout=Identity<float_type>; |
---|
550 | } |
---|
551 | |
---|
552 | xInverted=check_monotonicity(X, |
---|
553 | "interpolating_function::init() non-monotonic transformed x input"); |
---|
554 | |
---|
555 | // construct spline tables here. |
---|
556 | // this code is a re-translation of the pythonlabtools spline algorithm from pythonlabtools.sourceforge.net |
---|
557 | |
---|
558 | std::vector<float_type> u(np), dy(np-1), dx(np-1), dxi(np-1), dx2i(np-2), siga(np-2), dydx(np-1); |
---|
559 | |
---|
560 | std::transform(X.begin()+1, X.end(), X.begin(), dx.begin(), std::minus<float_type>() ); // dx=X[1:] - X [:-1] |
---|
561 | for(size_t i=0; i<dxi.size(); i++) dxi[i]=1.0/dx[i]; // dxi = 1/dx |
---|
562 | for(size_t i=0; i<dx2i.size(); i++) dx2i[i]=1.0/(X[i+2]-X[i]); |
---|
563 | |
---|
564 | std::transform(F.begin()+1, F.end(), F.begin(), dy.begin(), std::minus<float_type>() ); // dy = F[i+1]-F[i] |
---|
565 | std::transform(dx2i.begin(), dx2i.end(), dx.begin(), siga.begin(), std::multiplies<float_type>()); // siga = dx[:-1]*dx2i |
---|
566 | std::transform(dxi.begin(), dxi.end(), dy.begin(), dydx.begin(), std::multiplies<float_type>()); // dydx=dy/dx |
---|
567 | |
---|
568 | // u[i]=(y[i+1]-y[i])/float(x[i+1]-x[i]) - (y[i]-y[i-1])/float(x[i]-x[i-1]) |
---|
569 | std::transform(dydx.begin()+1, dydx.end(), dydx.begin(), u.begin()+1, std::minus<float_type>() ); // incomplete rendering of u = dydx[1:]-dydx[:-1] |
---|
570 | |
---|
571 | y2.resize(np,0.0); |
---|
572 | |
---|
573 | if(lowerSlopeNatural) { |
---|
574 | y2[0]=u[0]=0.0; |
---|
575 | } else { |
---|
576 | y2[0]= -0.5; |
---|
577 | u[0]=(3.0*dxi[0])*(dy[0]*dxi[0] -lowerSlope); |
---|
578 | } |
---|
579 | |
---|
580 | for(size_t i=1; i < np -1; i++) { // the inner loop |
---|
581 | float_type sig=siga[i-1]; |
---|
582 | float_type p=sig*y2[i-1]+2.0; |
---|
583 | y2[i]=(sig-1.0)/p; |
---|
584 | u[i]=(6.0*u[i]*dx2i[i-1] - sig*u[i-1])/p; |
---|
585 | } |
---|
586 | |
---|
587 | float_type qn, un; |
---|
588 | |
---|
589 | if(upperSlopeNatural) { |
---|
590 | qn=un=0.0; |
---|
591 | } else { |
---|
592 | qn= 0.5; |
---|
593 | un=(3.0*dxi[dxi.size()-1])*(upperSlope- dy[dy.size()-1]*dxi[dxi.size()-1] ); |
---|
594 | } |
---|
595 | |
---|
596 | y2[np-1]=(un-qn*u[np-2])/(qn*y2[np-2]+1.0); |
---|
597 | for (size_t k=np-1; k != 0; k--) y2[k-1]=y2[k-1]*y2[k]+u[k-1]; |
---|
598 | |
---|
599 | lastKLow=-1; // flag new X search required for next evaluation |
---|
600 | } |
---|
601 | |
---|
602 | // This function is the reason for this class to exist |
---|
603 | // it computes the interpolated function, and (if requested) its proper first and second derivatives including all coordinate transforms |
---|
604 | template <typename float_type> float_type interpolating_function<float_type>::value_with_derivatives( |
---|
605 | float_type x, float_type *yprime, float_type *yprime2) const throw(c2_exception) |
---|
606 | { |
---|
607 | if(x < this->xmin() || x > this->xmax()) { |
---|
608 | std::ostringstream outstr; |
---|
609 | outstr << "Interpolating function argument " << x << " out of range " << this->xmin() << " -- " << this ->xmax() << ": bailing"; |
---|
610 | throw c2_exception(outstr.str().c_str()); |
---|
611 | } |
---|
612 | |
---|
613 | float_type xraw=x; |
---|
614 | |
---|
615 | // template here is impossible! if(fXin && fXin != (Identity<float_type>) ) |
---|
616 | x=fXin(x); // save time by explicitly testing for identity function here |
---|
617 | |
---|
618 | int klo=0, khi=X.size()-1; |
---|
619 | |
---|
620 | if(!xInverted) { // select search depending on whether transformed X is increasing or decreasing |
---|
621 | if(lastKLow >=0 && (X[lastKLow] <= x) && (X[lastKLow+1] >= x) ) { // already bracketed |
---|
622 | klo=lastKLow; |
---|
623 | } else if(lastKLow >=0 && (X[lastKLow+1] <= x) && (X[lastKLow+2] > x)) { // in next bracket to the right |
---|
624 | klo=lastKLow+1; |
---|
625 | } else if(lastKLow > 0 && (X[lastKLow-1] <= x) && (X[lastKLow] > x)) { // in next bracket to the left |
---|
626 | klo=lastKLow-1; |
---|
627 | } else { // not bracketed, not close, start over |
---|
628 | // search for new KLow |
---|
629 | while(khi-klo > 1) { |
---|
630 | int km=(khi+klo)/2; |
---|
631 | if(X[km] > x) khi=km; |
---|
632 | else klo=km; |
---|
633 | } |
---|
634 | } |
---|
635 | } else { |
---|
636 | if(lastKLow >=0 && (X[lastKLow] >= x) && (X[lastKLow+1] <= x) ) { // already bracketed |
---|
637 | klo=lastKLow; |
---|
638 | } else if(lastKLow >=0 && (X[lastKLow+1] >= x) && (X[lastKLow+2] < x)) { // in next bracket to the right |
---|
639 | klo=lastKLow+1; |
---|
640 | } else if(lastKLow > 0 && (X[lastKLow-1] >= x) && (X[lastKLow] < x)) { // in next bracket to the left |
---|
641 | klo=lastKLow-1; |
---|
642 | } else { // not bracketed, not close, start over |
---|
643 | // search for new KLow |
---|
644 | while(khi-klo > 1) { |
---|
645 | int km=(khi+klo)/2; |
---|
646 | if(X[km] < x) khi=km; |
---|
647 | else klo=km; |
---|
648 | } |
---|
649 | } |
---|
650 | } |
---|
651 | |
---|
652 | khi=klo+1; |
---|
653 | lastKLow=klo; |
---|
654 | |
---|
655 | float_type h=X[khi]-X[klo]; |
---|
656 | |
---|
657 | float_type a=(X[khi]-x)/h; |
---|
658 | float_type b=1.0-a; |
---|
659 | float_type ylo=F[klo], yhi=F[khi], y2lo=y2[klo], y2hi=y2[khi]; |
---|
660 | float_type y=a*ylo+b*yhi+((a*a*a-a)*y2lo+(b*b*b-b)*y2hi)*(h*h)/6.0; |
---|
661 | |
---|
662 | // template here is impossible! if(fYin && fYin != Identity) |
---|
663 | y=fYout(y); // save time by explicitly testing for identity function here |
---|
664 | |
---|
665 | if(yprime || yprime2) { |
---|
666 | float_type fpi=1.0/fYinPrime(y); |
---|
667 | float_type gp=fXinPrime(xraw); |
---|
668 | float_type yp0=(yhi-ylo)/h+((3*b*b-1)*y2hi-(3*a*a-1)*y2lo)*h/6.0; // the derivative in interpolating table coordinates |
---|
669 | |
---|
670 | // from Mathematica Dt[InverseFunction[f][y[g[x]]], x] |
---|
671 | if(yprime) *yprime=gp*yp0*fpi; // the real derivative of the inverse transformed output |
---|
672 | if(yprime2) { |
---|
673 | float_type ypp0=b*y2hi+a*y2lo; |
---|
674 | float_type fpp=fYinDPrime(y); |
---|
675 | float_type gpp=fXinDPrime(xraw); |
---|
676 | // also from Mathematica Dt[InverseFunction[f][y[g[x]]], {x,2}] |
---|
677 | if(yprime2) *yprime2=(gp*gp*ypp0 + yp0*gpp - gp*gp*yp0*yp0*fpp*fpi*fpi)*fpi; |
---|
678 | } |
---|
679 | } |
---|
680 | |
---|
681 | return y; |
---|
682 | } |
---|
683 | |
---|
684 | template <typename float_type> void interpolating_function<float_type>::set_lower_extrapolation(float_type bound) |
---|
685 | { |
---|
686 | int kl = 0 ; |
---|
687 | int kh=kl+1; |
---|
688 | float_type xx=fXin(bound); |
---|
689 | float_type h0=X[kh]-X[kl]; |
---|
690 | float_type h1=xx-X[kl]; |
---|
691 | float_type yextrap=F[kl]+((F[kh]-F[kl])/h0 - h0*(y2[kl]+2.0*y2[kh])/6.0)*h1+y2[kl]*h1*h1/2.0; |
---|
692 | |
---|
693 | X.insert(X.begin(), xx); |
---|
694 | F.insert(F.begin(), yextrap); |
---|
695 | y2.insert(y2.begin(), y2.front()); // duplicate first or last element |
---|
696 | Xraw.insert(Xraw.begin(), bound); |
---|
697 | if (bound < this->fXMin) this->fXMin=bound; // check for reversed data |
---|
698 | else this->fXMax=bound; |
---|
699 | |
---|
700 | //printf("%10.4f %10.4f %10.4f %10.4f %10.4f\n", bound, xx, h0, h1, yextrap); |
---|
701 | //for(int i=0; i<X.size(); i++) printf("%4d %10.4f %10.4f %10.4f %10.4f \n", i, Xraw[i], X[i], F[i], y2[i]); |
---|
702 | } |
---|
703 | |
---|
704 | template <typename float_type> void interpolating_function<float_type>::set_upper_extrapolation(float_type bound) |
---|
705 | { |
---|
706 | int kl = X.size()-2 ; |
---|
707 | int kh=kl+1; |
---|
708 | float_type xx=fXin(bound); |
---|
709 | float_type h0=X[kh]-X[kl]; |
---|
710 | float_type h1=xx-X[kl]; |
---|
711 | float_type yextrap=F[kl]+((F[kh]-F[kl])/h0 - h0*(y2[kl]+2.0*y2[kh])/6.0)*h1+y2[kl]*h1*h1/2.0; |
---|
712 | |
---|
713 | X.insert(X.end(), xx); |
---|
714 | F.insert(F.end(), yextrap); |
---|
715 | y2.insert(y2.end(), y2.back()); // duplicate first or last element |
---|
716 | Xraw.insert(Xraw.end(), bound); |
---|
717 | if (bound < this->fXMin) this->fXMin=bound; // check for reversed data |
---|
718 | else this->fXMax=bound; |
---|
719 | //printf("%10.4f %10.4f %10.4f %10.4f %10.4f\n", bound, xx, h0, h1, yextrap); |
---|
720 | //for(int i=0; i<X.size(); i++) printf("%4d %10.4f %10.4f %10.4f %10.4f \n", i, Xraw[i], X[i], F[i], y2[i]); |
---|
721 | } |
---|
722 | |
---|
723 | // move derivatives into our internal coordinates (use splint to go the other way!) |
---|
724 | template <typename float_type> void interpolating_function<float_type>::localize_derivatives( |
---|
725 | float_type xraw, float_type y, float_type yp, float_type ypp, float_type *y0, float_type *yprime, float_type *yprime2) const |
---|
726 | { |
---|
727 | float_type fp=fYinPrime(y); |
---|
728 | float_type gp=fXinPrime(xraw); |
---|
729 | float_type fpp=fYinDPrime(y); |
---|
730 | float_type gpp=fXinDPrime(xraw); |
---|
731 | |
---|
732 | if(y0) *y0=fYin(y); |
---|
733 | if(yprime) *yprime=yp*fp/gp; // Mathematica Dt[f[y[InverseFunction[g][x]]], x] |
---|
734 | if(yprime2) *yprime2=( yp*yp*fpp - fp*yp*gpp/gp + fp*ypp )/(gp*gp) ; // Mathematica Dt[f[y[InverseFunction[g][x]]], {x,2}] |
---|
735 | } |
---|
736 | |
---|
737 | // return a new interpolating_function which is the unary function of an existing interpolating_function |
---|
738 | // can also be used to generate a resampling of another c2_function on a different grid |
---|
739 | // by creating a=interpolating_function(x,x) |
---|
740 | // and doing b=a.unary_operator(c) where c is a c2_function (probably another interpolating_function) |
---|
741 | |
---|
742 | template <typename float_type> interpolating_function<float_type>& |
---|
743 | interpolating_function<float_type>::unary_operator(const c2_function<float_type> &source) const |
---|
744 | { |
---|
745 | size_t np=X.size(); |
---|
746 | std::vector<float_type>yv(np); |
---|
747 | c2_composed_function<float_type> comp(source, *this); |
---|
748 | float_type yp0, yp1, ypp; |
---|
749 | |
---|
750 | for(size_t i=0; i<np; i++) { |
---|
751 | yv[i]=source(fYout(F[i])); // copy pointwise the function of our data values |
---|
752 | } |
---|
753 | |
---|
754 | comp(Xraw.front(), &yp0, &ypp); // get derivative at front |
---|
755 | comp(Xraw.back(), &yp1, &ypp); // get derivative at back |
---|
756 | |
---|
757 | return *new interpolating_function(Xraw, yv, false, yp0, false, yp1, |
---|
758 | fXin, fXinPrime, fXinDPrime, |
---|
759 | fYin, fYinPrime, fYinDPrime, fYout); |
---|
760 | } |
---|
761 | |
---|
762 | template <typename float_type> void |
---|
763 | interpolating_function<float_type>::get_data(std::vector<float_type> &xvals, std::vector<float_type> &yvals) const throw() |
---|
764 | { |
---|
765 | |
---|
766 | xvals=Xraw; |
---|
767 | yvals.resize(F.size()); |
---|
768 | |
---|
769 | for(size_t i=0; i<F.size(); i++) yvals[i]=fYout(F[i]); |
---|
770 | } |
---|
771 | |
---|
772 | template <typename float_type> interpolating_function<float_type> & |
---|
773 | interpolating_function<float_type>::binary_operator(const c2_function<float_type> &rhs, |
---|
774 | c2_binary_function<float_type> *combining_stub) const |
---|
775 | { |
---|
776 | size_t np=X.size(); |
---|
777 | std::vector<float_type> yv(np); |
---|
778 | c2_constant<float_type> fval; |
---|
779 | c2_constant<float_type> yval; |
---|
780 | float_type yp0, yp1, ypp; |
---|
781 | |
---|
782 | for(size_t i=0; i<np; i++) { |
---|
783 | fval.reset(fYout(F[i])); // update the constant function pointwise |
---|
784 | yval.reset(rhs(Xraw[i])); |
---|
785 | yv[i]=(*combining_stub).combine(fval, yval, Xraw[i], (float_type *)0, (float_type *)0); // compute rhs & combine without derivatives |
---|
786 | } |
---|
787 | |
---|
788 | (*combining_stub).combine(*this, rhs, Xraw.front(), &yp0, &ypp); // get derivative at front |
---|
789 | (*combining_stub).combine(*this, rhs, Xraw.back(), &yp1, &ypp); // get derivative at back |
---|
790 | |
---|
791 | delete combining_stub; |
---|
792 | |
---|
793 | return *new interpolating_function(Xraw, yv, false, yp0, false, yp1, |
---|
794 | fXin, fXinPrime, fXinDPrime, |
---|
795 | fYin, fYinPrime, fYinDPrime, fYout); |
---|
796 | } |
---|
797 | |
---|
798 | template <typename float_type> float_type c2_f_logprime(float_type x) { return 1.0/x; } // the derivative of log(x) |
---|
799 | template <typename float_type> float_type c2_f_logprime2(float_type x) { return -1.0/(x*x); } // the second derivative of log(x) |
---|
800 | |
---|
801 | template <typename float_type> log_lin_interpolating_function<float_type>::log_lin_interpolating_function( |
---|
802 | const std::vector<float_type> &x, const std::vector<float_type> &f, |
---|
803 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
804 | bool upperSlopeNatural, float_type upperSlope) |
---|
805 | : interpolating_function<float_type>() |
---|
806 | { |
---|
807 | init(x, f, lowerSlopeNatural, lowerSlope, upperSlopeNatural, upperSlope, |
---|
808 | (float_type (*)(float_type)) (std::log), c2_f_logprime, c2_f_logprime2, 0, 0, 0, 0); |
---|
809 | } |
---|
810 | |
---|
811 | template <typename float_type> lin_log_interpolating_function<float_type>::lin_log_interpolating_function( |
---|
812 | const std::vector<float_type> &x, const std::vector<float_type> &f, |
---|
813 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
814 | bool upperSlopeNatural, float_type upperSlope) |
---|
815 | : interpolating_function<float_type>() |
---|
816 | { |
---|
817 | init(x, f, lowerSlopeNatural, lowerSlope, upperSlopeNatural, upperSlope, |
---|
818 | 0, 0, 0, |
---|
819 | (float_type (*)(float_type)) (std::log), c2_f_logprime, c2_f_logprime2, |
---|
820 | (float_type (*)(float_type)) (std::exp) ); |
---|
821 | } |
---|
822 | |
---|
823 | template <typename float_type> log_log_interpolating_function<float_type>::log_log_interpolating_function( |
---|
824 | const std::vector<float_type> &x, const std::vector<float_type> &f, |
---|
825 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
826 | bool upperSlopeNatural, float_type upperSlope) |
---|
827 | : interpolating_function<float_type>() |
---|
828 | { |
---|
829 | init(x, f, lowerSlopeNatural, lowerSlope, upperSlopeNatural, upperSlope, |
---|
830 | (float_type (*)(float_type)) (std::log), c2_f_logprime, c2_f_logprime2, |
---|
831 | (float_type (*)(float_type)) (std::log), c2_f_logprime, c2_f_logprime2, |
---|
832 | (float_type (*)(float_type)) (std::exp) ); |
---|
833 | } |
---|
834 | |
---|
835 | template <typename float_type> float_type c2_f_recip(float_type x) { return 1.0/x; } |
---|
836 | template <typename float_type> float_type c2_f_recipprime(float_type x) { return -1.0/(x*x); } // the derivative of 1/x |
---|
837 | template <typename float_type> float_type c2_f_recipprime2(float_type x) { return 2.0/(x*x*x); } // the second derivative of 1/x |
---|
838 | |
---|
839 | template <typename float_type> arrhenius_interpolating_function<float_type>::arrhenius_interpolating_function( |
---|
840 | const std::vector<float_type> &x, const std::vector<float_type> &f, |
---|
841 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
842 | bool upperSlopeNatural, float_type upperSlope) |
---|
843 | : interpolating_function<float_type>() |
---|
844 | { |
---|
845 | init(x, f, lowerSlopeNatural, lowerSlope, upperSlopeNatural, upperSlope, |
---|
846 | c2_f_recip, c2_f_recipprime, c2_f_recipprime2, |
---|
847 | (float_type (*)(float_type)) (std::log), c2_f_logprime, c2_f_logprime2, |
---|
848 | (float_type (*)(float_type)) (std::exp) ); |
---|
849 | } |
---|
850 | |
---|
851 | template <typename float_type> c2_inverse_function<float_type>::c2_inverse_function(const c2_function<float_type> &source) |
---|
852 | : c2_plugin_function<float_type>(source) |
---|
853 | { |
---|
854 | float_type l=source.xmin(); |
---|
855 | float_type r=source.xmax(); |
---|
856 | start_hint=(l+r)*0.5; // guess that we start in the middle |
---|
857 | // compute our domain assuming the function is monotonic so its values on its domain boundaries are our domain |
---|
858 | float_type ly=source(l); |
---|
859 | float_type ry=source(r); |
---|
860 | if (ly > ry) { |
---|
861 | float_type t=ly; ly=ry; ry=t; |
---|
862 | } |
---|
863 | set_domain(ly, ry); |
---|
864 | } |
---|
865 | |
---|
866 | template <typename float_type> float_type c2_inverse_function<float_type>::value_with_derivatives( |
---|
867 | float_type x, float_type *yprime, float_type *yprime2 |
---|
868 | ) const throw(c2_exception) |
---|
869 | { |
---|
870 | float_type l=this->func->xmin(); |
---|
871 | float_type r=this->func->xmax(); |
---|
872 | float_type yp, ypp; |
---|
873 | float_type y=this->func->find_root(l, r, get_start_hint(x), x, 0, &yp, &ypp); |
---|
874 | start_hint=y; |
---|
875 | if(yprime) *yprime=1.0/yp; |
---|
876 | if(yprime2) *yprime2=-ypp/(yp*yp*yp); |
---|
877 | return y; |
---|
878 | } |
---|
879 | |
---|
880 | //accumulated_histogram starts with binned data, generates the integral, and generates a piecewise linear interpolating_function |
---|
881 | //If drop_zeros is true, it merges empty bins together before integration |
---|
882 | //Note that the resulting interpolating_function is guaranteed to be increasing (if drop_zeros is false) |
---|
883 | // or stricly increasing (if drop_zeros is true) |
---|
884 | //If inverse_function is true, it drop zeros, integrates, and returns the inverse function which is useful |
---|
885 | // for random number generation based on the input distribution. |
---|
886 | //If normalize is true, the big end of the integral is scaled to 1. |
---|
887 | //If the data are passed in reverse order (large X first), the integral is carried out from the big end, |
---|
888 | // and then the data are reversed to the result in in increasing X order. |
---|
889 | template <typename float_type> accumulated_histogram<float_type>::accumulated_histogram( |
---|
890 | const std::vector<float_type>binedges, const std::vector<float_type> binheights, |
---|
891 | bool normalize, bool inverse_function, bool drop_zeros) |
---|
892 | { |
---|
893 | |
---|
894 | int np=binheights.size(); |
---|
895 | |
---|
896 | std::vector<float_type> be, bh; |
---|
897 | if(drop_zeros || inverse_function) { //inverse functions cannot have any zero bins or they have vertical sections |
---|
898 | if(binheights[0] || !inverse_function) { // conserve lower x bound if not an inverse function |
---|
899 | be.push_back(binedges[0]); |
---|
900 | bh.push_back(binheights[0]); |
---|
901 | } |
---|
902 | for(int i=1; i<np-1; i++) { |
---|
903 | if(binheights[i]) { |
---|
904 | be.push_back(binedges[i]); |
---|
905 | bh.push_back(binheights[i]); |
---|
906 | } |
---|
907 | } |
---|
908 | if(binheights[np-1] || !inverse_function) { |
---|
909 | bh.push_back(binheights[np-1]); |
---|
910 | be.push_back(binedges[np-1]); |
---|
911 | be.push_back(binedges[np]); // push both sides of the last bin if needed |
---|
912 | } |
---|
913 | np=bh.size(); // set np to compressed size of bin array |
---|
914 | } else { |
---|
915 | be=binedges; |
---|
916 | bh=binheights; |
---|
917 | } |
---|
918 | std::vector<float_type> cum(np+1, 0.0); |
---|
919 | for(int i=1; i<=np; i++) cum[i]=bh[i]*(be[i]-be[i-1])+cum[i-1]; // accumulate bins, leaving bin 0 as 0 |
---|
920 | if(be[1] < be[0]) { // if bins passed in backwards, reverse them |
---|
921 | std::reverse(be.begin(), be.end()); |
---|
922 | std::reverse(cum.begin(), cum.end()); |
---|
923 | for(unsigned int i=0; i<cum.size(); i++) cum[i]*=-1; // flip sign on reversed data |
---|
924 | } |
---|
925 | if(normalize) { |
---|
926 | float_type m=1.0/std::max(cum[0], cum[np]); |
---|
927 | for(int i=0; i<=np; i++) cum[i]*=m; |
---|
928 | } |
---|
929 | if(inverse_function) interpolating_function<float_type>(cum, be); // use cum as x axis in inverse function |
---|
930 | else interpolating_function<float_type>(be, cum); // else use lower bin edge as x axis |
---|
931 | std::fill(this->y2.begin(), this->y2.end(), 0.0); // clear second derivatives, to we are piecewise linear |
---|
932 | } |
---|
933 | |
---|
934 | template <typename float_type> c2_piecewise_function<float_type>::c2_piecewise_function() |
---|
935 | : c2_function<float_type>(), lastKLow(-1) |
---|
936 | { |
---|
937 | this->sampling_grid=new std::vector<float_type>; // this always has a smapling grid |
---|
938 | } |
---|
939 | |
---|
940 | template <typename float_type> c2_piecewise_function<float_type>::~c2_piecewise_function() |
---|
941 | { |
---|
942 | size_t np=functions.size(); |
---|
943 | for(size_t i=0; i<np; i++) if(owns[i]) delete functions[i]; |
---|
944 | } |
---|
945 | |
---|
946 | template <typename float_type> float_type c2_piecewise_function<float_type>::value_with_derivatives( |
---|
947 | float_type x, float_type *yprime, float_type *yprime2 |
---|
948 | ) const throw(c2_exception) |
---|
949 | { |
---|
950 | |
---|
951 | size_t np=functions.size(); |
---|
952 | if(!np) throw c2_exception("attempting to evaluate an empty piecewise function"); |
---|
953 | |
---|
954 | if(x < this->xmin() || x > this->xmax()) { |
---|
955 | std::ostringstream outstr; |
---|
956 | outstr << "piecewise function argument " << x << " out of range " << this->xmin() << " -- " << this->xmax(); |
---|
957 | throw c2_exception(outstr.str().c_str()); |
---|
958 | } |
---|
959 | |
---|
960 | int klo=0; |
---|
961 | |
---|
962 | if(lastKLow >= 0 && functions[lastKLow]->xmin() <= x && functions[lastKLow]->xmax() > x) { |
---|
963 | klo=lastKLow; |
---|
964 | } else { |
---|
965 | int khi=np; |
---|
966 | while(khi-klo > 1) { |
---|
967 | int km=(khi+klo)/2; |
---|
968 | if(functions[km]->xmin() > x) khi=km; |
---|
969 | else klo=km; |
---|
970 | } |
---|
971 | } |
---|
972 | lastKLow=klo; |
---|
973 | return functions[klo]->value_with_derivatives(x, yprime, yprime2); |
---|
974 | } |
---|
975 | |
---|
976 | template <typename float_type> void c2_piecewise_function<float_type>::append_function( |
---|
977 | c2_function<float_type> &func, bool pass_ownership) throw(c2_exception) |
---|
978 | { |
---|
979 | if(functions.size()) { // check whether there are any gaps to fill, etc. |
---|
980 | c2_function<float_type> &tail=*(functions.back()); |
---|
981 | float_type x0=tail.xmax(); |
---|
982 | float_type x1=func.xmin(); |
---|
983 | if(x0 < x1) { |
---|
984 | // must insert a connector if x0 < x1 |
---|
985 | float_type y0=tail(x0); |
---|
986 | float_type y1=func(x1); |
---|
987 | c2_function<float_type> *connector=new c2_linear<float_type>(x0, y0, (y1-y0)/(x1-x0)); |
---|
988 | connector->set_domain(x0,x1); |
---|
989 | functions.push_back(connector); |
---|
990 | owns.push_back(true); |
---|
991 | this->sampling_grid->push_back(x1); |
---|
992 | } else if(x0>x1) throw c2_exception("function domains not increasing in c2_piecewise_function"); |
---|
993 | } |
---|
994 | functions.push_back(&func); |
---|
995 | owns.push_back(pass_ownership); |
---|
996 | // extend our domain to include all known functions |
---|
997 | this->set_domain(functions.front()->xmin(), functions.back()->xmax()); |
---|
998 | // extend our sampling grid with the new function's grid, with the first point dropped to avoid duplicates |
---|
999 | std::vector<float_type> &newgrid=func.get_sampling_grid(func.xmin(), func.xmax()); |
---|
1000 | this->sampling_grid->insert(this->sampling_grid->end(), newgrid.begin()+1, newgrid.end()); |
---|
1001 | delete &newgrid; |
---|
1002 | } |
---|
1003 | |
---|
1004 | template <typename float_type> c2_connector_function<float_type>::c2_connector_function( |
---|
1005 | const c2_function<float_type> &f1, const c2_function<float_type> &f2, float_type x0, float_type x2, |
---|
1006 | bool auto_center, float_type y1) |
---|
1007 | |
---|
1008 | : c2_function<float_type>() |
---|
1009 | { |
---|
1010 | float_type y0, yp0, ypp0, y2, yp2, ypp2; |
---|
1011 | fdx=(x2-x0)/2.0; |
---|
1012 | fhinv=1.0/fdx; |
---|
1013 | fx1=(x0+x2)/2.0; |
---|
1014 | |
---|
1015 | y0=f1.value_with_derivatives(x0, &yp0, &ypp0); // get left wall values from conventional computation |
---|
1016 | y2=f2.value_with_derivatives(x2, &yp2, &ypp2); // get right wall values from conventional computation |
---|
1017 | |
---|
1018 | // scale derivs to put function on [-1,1] since mma solution is done this way |
---|
1019 | yp0*=fdx; |
---|
1020 | yp2*=fdx; |
---|
1021 | ypp0*=fdx*fdx; |
---|
1022 | ypp2*=fdx*fdx; |
---|
1023 | |
---|
1024 | float_type ff0=(8*(y0 + y2) + 5*(yp0 - yp2) + ypp0 + ypp2)/16.0; |
---|
1025 | if(auto_center) y1=ff0; // forces ff to be 0 if we are auto-centering |
---|
1026 | |
---|
1027 | // y[x_] = y1 + x (a + b x) + (x-1) x (x+1) (c + d x + e x^2 + f x^3) |
---|
1028 | fy1=y1; |
---|
1029 | fa=-(y0 - y2)/2.; |
---|
1030 | fb=(y0 - 2*y1 + y2)/2.; |
---|
1031 | fc=(7*(y0 - y2 + yp0 + yp2) + ypp0 - ypp2)/16.; |
---|
1032 | fd=(32*y1 - 16*(y2 + y0) + 9*(yp2 - yp0) - ypp0 - ypp2)/16.; |
---|
1033 | fe=(3*(y2 - y0 - yp0 - yp2) - ypp0 + ypp2)/16.; |
---|
1034 | ff=(ff0 - y1); |
---|
1035 | // y'[x] = a + 2 b x + (3x^2 - 1) (c + d x + e x^2 + f x^3) + (x-1) x (x+1) (d + 2 e x + 3 f x^2 ) |
---|
1036 | // y''[x] = 2b + (x-1) x (x+1) (2 e + 6 f x) + 2 (3 x^2 -1) (d + 2 e x + 3 f x^2 ) + 6 x (c + d x + e x^2 + f x^3) |
---|
1037 | this->set_domain(x0,x2); // this is where the function is valid |
---|
1038 | } |
---|
1039 | |
---|
1040 | template <typename float_type> c2_connector_function<float_type>::~c2_connector_function() |
---|
1041 | { |
---|
1042 | } |
---|
1043 | |
---|
1044 | template <typename float_type> float_type c2_connector_function<float_type>::value_with_derivatives( |
---|
1045 | float_type x, float_type *yprime, float_type *yprime2 |
---|
1046 | ) const throw(c2_exception) |
---|
1047 | { |
---|
1048 | |
---|
1049 | float_type dx=(x-fx1)*fhinv; |
---|
1050 | float_type q1=fc + dx*(fd + dx*(fe + dx*ff)); |
---|
1051 | float_type xp1=(dx-1)*(dx+1)*dx; |
---|
1052 | |
---|
1053 | float_type y= fy1 + dx*(fa+fb*dx) + xp1*q1; |
---|
1054 | if(yprime || yprime2) { |
---|
1055 | float_type q2 =fd + dx*(2*fe + dx*3*ff); |
---|
1056 | float_type q3=2*fe+6*ff*dx; |
---|
1057 | float_type xp2=(3*dx*dx-1); |
---|
1058 | if(yprime) *yprime=(fa + 2*fb*dx + xp2*q1 + xp1*q2)*fhinv; |
---|
1059 | if(yprime2) *yprime2=(2*fb+xp1*q3+2*xp2*q2+6*dx*q1)*fhinv*fhinv; |
---|
1060 | } |
---|
1061 | return y; |
---|
1062 | } |
---|