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24 | // ******************************************************************** |
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25 | // |
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26 | // |
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27 | /** |
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28 | * \file |
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29 | * \brief Provides code for the general c2_function algebra which supports |
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30 | * fast, flexible operations on piecewise-twice-differentiable functions |
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31 | * |
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32 | * \author Created by R. A. Weller and Marcus H. Mendenhall on 7/9/05. |
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33 | * \author Copyright 2005 __Vanderbilt University__. All rights reserved. |
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34 | * |
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35 | * \version c2_function.cc,v 1.169 2008/05/22 12:45:19 marcus Exp |
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36 | */ |
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37 | |
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38 | #include <iostream> |
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39 | #include <vector> |
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40 | #include <algorithm> |
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41 | #include <cstdlib> |
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42 | #include <numeric> |
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43 | #include <functional> |
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44 | #include <iterator> |
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45 | #include <cmath> |
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46 | #include <limits> |
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47 | #include <sstream> |
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48 | |
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49 | template <typename float_type> const std::string c2_function<float_type>::cvs_file_vers() const |
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50 | { return "c2_function.cc,v 1.169 2008/05/22 12:45:19 marcus Exp"; } |
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51 | |
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52 | // find a pre-bracketed root of a c2_function, which is a MUCH easier job than general root finding |
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53 | // since the derivatives are known exactly, and smoothness is guaranteed. |
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54 | // this searches for f(x)=value, to make life a little easier than always searching for f(x)=0 |
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55 | template <typename float_type> float_type c2_function<float_type>::find_root(float_type lower_bracket, float_type upper_bracket, |
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56 | float_type start, float_type value, int *error, |
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57 | float_type *final_yprime, float_type *final_yprime2) const throw(c2_exception) |
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58 | { |
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59 | // find f(x)=value within the brackets, using the guarantees of smoothness associated with a c2_function |
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60 | // can use local f(x)=a*x**2 + b*x + c and solve quadratic to find root, then iterate |
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61 | |
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62 | float_type yp, yp2; // we will make unused pointers point here, to save null checks later |
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63 | if (!final_yprime) final_yprime=&yp; |
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64 | if (!final_yprime2) final_yprime2=&yp2; |
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65 | |
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66 | float_type ftol=5*(std::numeric_limits<float_type>::epsilon()*std::abs(value)+std::numeric_limits<float_type>::min()); |
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67 | 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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68 | |
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69 | float_type root=start; // start looking in the middle |
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70 | if(error) *error=0; // start out with error flag set to OK, if it is expected |
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71 | |
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72 | float_type c, b; |
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73 | |
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74 | if(!root_info) { |
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75 | root_info=new struct c2_root_info; |
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76 | root_info->inited=false; |
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77 | } |
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78 | // this new logic is to keep track of where we were before, and lower the number of |
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79 | // function evaluations if we are searching inside the same bracket as before. |
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80 | // Since this root finder has, very often, the bracket of the entire domain of the function, |
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81 | // this makes a big difference, especially to c2_inverse_function |
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82 | if(!root_info->inited || upper_bracket != root_info->upper.x || lower_bracket != root_info->lower.x) { |
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83 | root_info->upper.x=upper_bracket; |
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84 | fill_fblock(root_info->upper); |
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85 | root_info->lower.x=lower_bracket; |
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86 | fill_fblock(root_info->lower); |
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87 | root_info->inited=true; |
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88 | } |
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89 | |
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90 | float_type clower=root_info->lower.y-value; |
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91 | if(!clower) { |
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92 | *final_yprime=root_info->lower.yp; |
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93 | *final_yprime2=root_info->lower.ypp; |
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94 | return lower_bracket; |
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95 | } |
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96 | |
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97 | float_type cupper=root_info->upper.y-value; |
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98 | if(!cupper) { |
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99 | *final_yprime=root_info->upper.yp; |
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100 | *final_yprime2=root_info->upper.ypp; |
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101 | return upper_bracket; |
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102 | } |
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103 | const float_type lower_sign = (clower < 0) ? -1 : 1; |
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104 | |
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105 | if(lower_sign*cupper >0) { |
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106 | // argh, no sign change in here! |
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107 | if(error) { *error=1; return 0.0; } |
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108 | else { |
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109 | std::ostringstream outstr; |
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110 | outstr << "unbracketed root in find_root at xlower= " << lower_bracket << ", xupper= " << upper_bracket; |
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111 | outstr << ", value= " << value << ": bailing"; |
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112 | throw c2_exception(outstr.str().c_str()); |
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113 | } |
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114 | } |
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115 | |
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116 | float_type delta=upper_bracket-lower_bracket; // first error step |
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117 | c=value_with_derivatives(root, final_yprime, final_yprime2)-value; // compute initial values |
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118 | b=*final_yprime; // make a local copy for readability |
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119 | increment_evaluations(); |
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120 | |
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121 | while( |
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122 | std::abs(delta) > xtol && // absolute x step check |
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123 | std::abs(c) > ftol && // absolute y tolerance |
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124 | std::abs(c) > xtol*std::abs(b) // comparison to smallest possible Y step from derivative |
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125 | ) |
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126 | { |
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127 | float_type a=(*final_yprime2)/2; // second derivative is 2*a |
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128 | float_type disc=b*b-4*a*c; |
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129 | // std::cout << std::endl << "find_root_debug a,b,c,d " << a << " " << b << " " << c << " " << disc << std::endl; |
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130 | |
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131 | if(disc >= 0) { |
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132 | float_type q=-0.5*((b>=0)?(b+std::sqrt(disc)):(b-std::sqrt(disc))); |
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133 | 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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134 | else delta=q/a; |
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135 | root+=delta; |
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136 | } |
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137 | |
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138 | if(disc < 0 || root<lower_bracket || root>upper_bracket || |
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139 | std::abs(delta) >= 0.5*(upper_bracket-lower_bracket)) { |
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140 | // if we jump out of the bracket, or aren't converging well, bisect |
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141 | root=0.5*(lower_bracket+upper_bracket); |
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142 | delta=upper_bracket-lower_bracket; |
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143 | } |
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144 | c=value_with_derivatives(root, final_yprime, final_yprime2)-value; // compute initial values |
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145 | if(c2_isnan(c)) { |
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146 | bad_x_point=root; |
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147 | return c; // return the nan if a computation failed |
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148 | } |
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149 | b=*final_yprime; // make a local copy for readability |
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150 | increment_evaluations(); |
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151 | |
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152 | // now, close in bracket on whichever side this still brackets |
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153 | if(c*lower_sign < 0.0) { |
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154 | cupper=c; |
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155 | upper_bracket=root; |
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156 | } else { |
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157 | clower=c; |
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158 | lower_bracket=root; |
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159 | } |
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160 | // std::cout << "find_root_debug x, y, dx " << root << " " << c << " " << delta << std::endl; |
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161 | } |
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162 | return root; |
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163 | } |
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164 | |
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165 | /* def partial_integrals(self, xgrid): |
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166 | Return the integrals of a function between the sampling points xgrid. The sum is the definite integral. |
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167 | This method uses an exact integration of the polynomial which matches the values and derivatives at the |
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168 | endpoints of a segment. Its error scales as h**6, if the input functions really are smooth. |
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169 | This could very well be used as a stepper for adaptive Romberg integration. |
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170 | For InterpolatingFunctions, it is likely that the Simpson's rule integrator is sufficient. |
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171 | #the weights come from an exact mathematica solution to the 5th order polynomial with the given values & derivatives |
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172 | #yint = (y0+y1)*dx/2 + dx^2*(yp0-yp1)/10 + dx^3 * (ypp0+ypp1)/120 ) |
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173 | */ |
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174 | |
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175 | // the recursive part of the integrator is agressively designed to minimize copying of data... lots of pointers |
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176 | template <typename float_type> float_type c2_function<float_type>::integrate_step(c2_integrate_recur &rb) const throw(c2_exception) |
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177 | { |
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178 | std::vector< recur_item > &rb_stack=*rb.rb_stack; // heap-based stack of data for recursion |
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179 | rb_stack.clear(); |
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180 | |
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181 | recur_item top; |
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182 | top.depth=0; top.done=false; top.f0index=0; top.f2index=0; top.step_sum=0; |
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183 | |
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184 | // push storage for our initial elements |
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185 | rb_stack.push_back(top); |
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186 | rb_stack.back().f1=*rb.f0; |
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187 | rb_stack.back().done=true; // this element will never be evaluated further |
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188 | |
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189 | rb_stack.push_back(top); |
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190 | rb_stack.back().f1=*rb.f1; |
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191 | rb_stack.back().done=true; // this element will never be evaluated further |
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192 | |
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193 | if(!rb.inited) { |
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194 | switch(rb.derivs) { |
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195 | case 0: |
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196 | rb.eps_scale=0.1; rb.extrap_coef=16; break; |
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197 | case 1: |
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198 | rb.eps_scale=0.1; rb.extrap_coef=64; break; |
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199 | case 2: |
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200 | rb.eps_scale=0.02; rb.extrap_coef=1024; break; |
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201 | default: |
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202 | throw c2_exception("derivs must be 0, 1 or 2 in partial_integrals"); |
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203 | } |
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204 | |
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205 | rb.extrap2=1.0/(rb.extrap_coef-1.0); |
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206 | rb.dx_tolerance=10.0*std::numeric_limits<float_type>::epsilon(); |
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207 | rb.abs_tol_min=10.0*std::numeric_limits<float_type>::min(); |
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208 | rb.inited=true; |
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209 | } |
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210 | |
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211 | // now, push our first real element |
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212 | top.f0index=0; // left element is stack[0] |
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213 | top.f2index=1; // right element is stack[1] |
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214 | top.abs_tol=rb.abs_tol; |
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215 | rb_stack.push_back(top); |
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216 | |
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217 | while(rb_stack.size() > 2) { |
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218 | recur_item &back=rb_stack.back(); |
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219 | if(back.done) { |
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220 | float_type sum=back.step_sum; |
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221 | rb_stack.pop_back(); |
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222 | rb_stack.back().step_sum+=sum; // bump our sum up to the parent |
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223 | continue; |
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224 | } |
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225 | back.done=true; |
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226 | |
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227 | c2_fblock<float_type> &f0=rb_stack[back.f0index].f1, &f2=rb_stack[back.f2index].f1; |
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228 | c2_fblock<float_type> &f1=back.f1; // will hold new middle values |
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229 | size_t f1index=rb_stack.size()-1; // our current offset |
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230 | float_type abs_tol=back.abs_tol; |
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231 | |
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232 | f1.x=0.5*(f0.x + f2.x); // center of interval |
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233 | float_type dx2=0.5*(f2.x - f0.x); |
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234 | |
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235 | // check for underflow on step size, which prevents us from achieving specified accuracy. |
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236 | if(std::abs(dx2) < std::abs(f1.x)*rb.dx_tolerance || std::abs(dx2) < rb.abs_tol_min) { |
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237 | std::ostringstream outstr; |
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238 | outstr << "Step size underflow in adaptive_partial_integrals at depth=" << back.depth << ", x= " << f1.x; |
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239 | throw c2_exception(outstr.str().c_str()); |
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240 | } |
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241 | |
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242 | fill_fblock(f1); |
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243 | if(c2_isnan(f1.y)) { |
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244 | bad_x_point=f1.x; |
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245 | return f1.y; // can't go any further if a nan has appeared |
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246 | } |
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247 | |
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248 | bool yptrouble=f0.ypbad || f2.ypbad || f1.ypbad; |
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249 | bool ypptrouble=f0.yppbad || f2.yppbad || f1.yppbad; |
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250 | |
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251 | // select the real derivative count based on whether we are at a point where derivatives exist |
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252 | int derivs = std::min(rb.derivs, (yptrouble||ypptrouble)?(yptrouble?0:1):2); |
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253 | |
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254 | if(!back.depth) { // top level, total has not been initialized yet |
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255 | switch(derivs) { // create estimate of next lower order for first try |
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256 | case 0: |
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257 | back.previous_estimate=(f0.y+f2.y)*dx2; break; |
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258 | case 1: |
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259 | back.previous_estimate=(f0.y+4.0*f1.y+f2.y)*dx2/3.0; break; |
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260 | case 2: |
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261 | back.previous_estimate=( (14*f0.y + 32*f1.y + 14*f2.y) + 2*dx2 * (f0.yp - f2.yp) ) * dx2 /30.; break; |
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262 | default: |
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263 | back.previous_estimate=0.0; // just to suppress missing default warnings |
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264 | } |
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265 | } |
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266 | |
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267 | float_type left, right; |
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268 | |
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269 | // pre-compute constants so all multiplies use a small dynamic range |
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270 | // constants for 0 derivative integrator |
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271 | static const float_type c0c1=5./12., c0c2=8./12., c0c3=-1./12.; |
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272 | // constants for 1 derivative integrator |
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273 | static const float_type c1c1=101./240., c1c2=128./240., c1c3=11./240., |
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274 | c1c4=13./240., c1c5=-40./240., c1c6=-3./240.; |
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275 | // constants for 2 derivative integrator |
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276 | static const float_type c2c1=169./40320., c2c2=1024./ 40320., c2c3=-41./40320., |
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277 | c2c4=2727./40320., c2c5=-5040./40320., c2c6=423./40320., |
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278 | c2c7=17007./40320., c2c8=24576./40320., c2c9=-1263./40320.; |
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279 | |
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280 | switch(derivs) { |
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281 | case 2: |
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282 | // use ninth-order estimates for each side, from full set of all values (!) (Thanks, Mathematica!) |
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283 | left= ( ( (c2c1*f0.ypp + c2c2*f1.ypp + c2c3*f2.ypp)*dx2 + |
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284 | (c2c4*f0.yp + c2c5*f1.yp + c2c6*f2.yp) )*dx2 + |
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285 | (c2c7*f0.y + c2c8*f1.y + c2c9*f2.y) )* dx2; |
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286 | right= ( ( (c2c1*f2.ypp + c2c2*f1.ypp + c2c3*f0.ypp)*dx2 - |
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287 | (c2c4*f2.yp + c2c5*f1.yp + c2c6*f0.yp) )*dx2 + |
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288 | (c2c7*f2.y + c2c8*f1.y + c2c9*f0.y) )* dx2; |
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289 | // std::cout << f0.x << " " << f1.x << " " << f2.x << std::endl ; |
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290 | // std::cout << f0.y << " " << f1.y << " " << f2.y << " " << left << " " << right << " " << total << std::endl ; |
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291 | break; |
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292 | case 1: |
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293 | left= ( (c1c1*f0.y + c1c2*f1.y + c1c3*f2.y) + dx2*(c1c4*f0.yp + c1c5*f1.yp + c1c6*f2.yp) ) * dx2 ; |
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294 | right= ( (c1c1*f2.y + c1c2*f1.y + c1c3*f0.y) - dx2*(c1c4*f2.yp + c1c5*f1.yp + c1c6*f0.yp) ) * dx2 ; |
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295 | break; |
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296 | case 0: |
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297 | left= (c0c1*f0.y + c0c2*f1.y + c0c3*f2.y)*dx2; |
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298 | right= (c0c1*f2.y + c0c2*f1.y + c0c3*f0.y)*dx2; |
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299 | break; |
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300 | default: |
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301 | left=right=0.0; // suppress warnings about missing default |
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302 | break; |
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303 | } |
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304 | |
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305 | float_type lrsum=left+right; |
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306 | |
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307 | bool extrapolate=back.depth && rb.extrapolate && (derivs==rb.derivs); // only extrapolate if no trouble with derivs |
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308 | float_type eps=std::abs(back.previous_estimate-lrsum)*rb.eps_scale; |
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309 | if(extrapolate) eps*=rb.eps_scale; |
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310 | |
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311 | if(rb.adapt && eps > abs_tol && eps > std::abs(lrsum)*rb.rel_tol) { |
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312 | // tolerance not met, subdivide & recur |
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313 | if(abs_tol > rb.abs_tol_min) abs_tol=abs_tol*0.5; // each half has half the error budget |
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314 | top.abs_tol=abs_tol; |
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315 | top.depth=back.depth+1; |
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316 | |
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317 | // save the last things we need from back before a push happens, in case |
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318 | // the push causes a reallocation and moves the whole stack. |
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319 | size_t f0index=back.f0index, f2index=back.f2index; |
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320 | |
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321 | top.f0index=f1index; top.f2index=f2index; // insert pointers to right side data into our recursion block |
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322 | top.previous_estimate=right; |
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323 | rb_stack.push_back(top); |
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324 | |
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325 | top.f0index=f0index; top.f2index=f1index; // insert pointers to left side data into our recursion block |
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326 | top.previous_estimate=left; |
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327 | rb_stack.push_back(top); |
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328 | |
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329 | } else if(extrapolate) { |
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330 | // extrapolation only happens on leaf nodes, where the tolerance was met. |
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331 | back.step_sum+=(rb.extrap_coef*lrsum - back.previous_estimate)*rb.extrap2; |
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332 | } else { |
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333 | back.step_sum+=lrsum; |
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334 | } |
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335 | } |
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336 | return rb_stack.back().step_sum; // last element on the stack holds the sum |
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337 | } |
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338 | |
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339 | template <typename float_type> bool c2_function<float_type>::check_monotonicity( |
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340 | const std::vector<float_type> &data, const char message[]) const throw(c2_exception) |
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341 | { |
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342 | size_t np=data.size(); |
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343 | if(np < 2) return false; // one point has no direction! |
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344 | |
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345 | bool rev=(data[1] < data[0]); // which way do data point? |
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346 | size_t i; |
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347 | |
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348 | if(!rev) for(i = 2; i < np && (data[i-1] < data[i]) ; i++); |
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349 | else for(i = 2; i < np &&(data[i-1] > data[i]) ; i++); |
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350 | |
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351 | if(i != np) throw c2_exception(message); |
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352 | |
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353 | return rev; |
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354 | } |
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355 | |
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356 | 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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357 | { |
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358 | bool rev=check_monotonicity(grid, "set_sampling_grid: sampling grid not monotonic"); |
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359 | |
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360 | if(!sampling_grid || no_overwrite_grid) sampling_grid=new std::vector<float_type>; |
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361 | (*sampling_grid)=grid; no_overwrite_grid=0; |
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362 | |
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363 | if(rev) std::reverse(sampling_grid->begin(), sampling_grid->end()); // make it increasing |
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364 | } |
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365 | |
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366 | template <typename float_type> void c2_function<float_type>:: |
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367 | get_sampling_grid(float_type xmin, float_type xmax, std::vector<float_type> &grid) const |
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368 | { |
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369 | std::vector<float_type> *result=&grid; |
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370 | result->clear(); |
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371 | |
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372 | if( !(sampling_grid) || !(sampling_grid->size()) || (xmax <= sampling_grid->front()) || (xmin >= sampling_grid->back()) ) { |
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373 | // nothing is known about the function in this region, return xmin and xmax |
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374 | result->push_back(xmin); |
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375 | result->push_back(xmax); |
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376 | } else { |
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377 | std::vector<float_type> &sg=*sampling_grid; // just a shortcut |
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378 | int np=sg.size(); |
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379 | int klo=0, khi=np-1, firstindex=0, lastindex=np-1; |
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380 | |
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381 | result->push_back(xmin); |
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382 | |
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383 | if(xmin > sg.front() ) { |
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384 | // hunt through table for position bracketing starting point |
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385 | while(khi-klo > 1) { |
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386 | int km=(khi+klo)/2; |
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387 | if(sg[km] > xmin) khi=km; |
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388 | else klo=km; |
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389 | } |
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390 | khi=klo+1; |
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391 | // khi now points to first point definitively beyond our first point, or last point of array |
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392 | firstindex=khi; |
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393 | khi=np-1; // restart upper end of search |
---|
394 | } |
---|
395 | |
---|
396 | if(xmax < sg.back()) { |
---|
397 | // hunt through table for position bracketing starting point |
---|
398 | while(khi-klo > 1) { |
---|
399 | int km=(khi+klo)/2; |
---|
400 | if(sg[km] > xmax) khi=km; |
---|
401 | else klo=km; |
---|
402 | } |
---|
403 | khi=klo+1; |
---|
404 | // khi now points to first point definitively beyond our last point, or last point of array |
---|
405 | lastindex=klo; |
---|
406 | } |
---|
407 | |
---|
408 | int initsize=result->size(); |
---|
409 | result->resize(initsize+(lastindex-firstindex+2)); |
---|
410 | std::copy(sg.begin()+firstindex, sg.begin()+lastindex+1, result->begin()+initsize); |
---|
411 | result->back()=xmax; |
---|
412 | |
---|
413 | // this is the unrefined sampling grid... now check for very close points on front & back and fix if needed. |
---|
414 | preen_sampling_grid(result); |
---|
415 | } |
---|
416 | } |
---|
417 | |
---|
418 | template <typename float_type> void c2_function<float_type>::preen_sampling_grid(std::vector<float_type> *result) const |
---|
419 | { |
---|
420 | // this is the unrefined sampling grid... now check for very close points on front & back and fix if needed. |
---|
421 | if(result->size() > 2) { // may be able to prune dangerously close points near the ends if there are at least 3 points |
---|
422 | bool deleteit=false; |
---|
423 | float_type x0=(*result)[0], x1=(*result)[1]; |
---|
424 | float_type dx1=x1-x0; |
---|
425 | |
---|
426 | float_type ftol=10.0*(std::numeric_limits<float_type>::epsilon()*(std::abs(x0)+std::abs(x1))+std::numeric_limits<float_type>::min()); |
---|
427 | if(dx1 < ftol) deleteit=true; |
---|
428 | float_type dx2=(*result)[2]-x0; |
---|
429 | if(dx1/dx2 < 0.1) deleteit=true; // endpoint is very close to internal interesting point |
---|
430 | |
---|
431 | if(deleteit) result->erase(result->begin()+1); // delete redundant interesting point |
---|
432 | } |
---|
433 | |
---|
434 | if(result->size() > 2) { // may be able to prune dangerously close points near the ends if there are at least 3 points |
---|
435 | bool deleteit=false; |
---|
436 | int pos=result->size()-3; |
---|
437 | float_type x0=(*result)[pos+1], x1=(*result)[pos+2]; |
---|
438 | float_type dx1=x1-x0; |
---|
439 | |
---|
440 | float_type ftol=10.0*(std::numeric_limits<float_type>::epsilon()*(std::abs(x0)+std::abs(x1))+std::numeric_limits<float_type>::min()); |
---|
441 | if(dx1 < ftol) deleteit=true; |
---|
442 | float_type dx2=x1-(*result)[pos]; |
---|
443 | if(dx1/dx2 < 0.1) deleteit=true; // endpoint is very close to internal interesting point |
---|
444 | |
---|
445 | if(deleteit) result->erase(result->end()-2); // delete redundant interesting point |
---|
446 | } |
---|
447 | } |
---|
448 | |
---|
449 | template <typename float_type> void c2_function<float_type>:: |
---|
450 | refine_sampling_grid(std::vector<float_type> &grid, size_t refinement) const |
---|
451 | { |
---|
452 | size_t np=grid.size(); |
---|
453 | size_t count=(np-1)*refinement + 1; |
---|
454 | float_type dxscale=1.0/refinement; |
---|
455 | |
---|
456 | std::vector<float_type> result(count); |
---|
457 | |
---|
458 | for(size_t i=0; i<(np-1); i++) { |
---|
459 | float_type x=grid[i]; |
---|
460 | float_type dx=(grid[i+1]-x)*dxscale; |
---|
461 | for(size_t j=0; j<refinement; j++, x+=dx) result[i*refinement+j]=x; |
---|
462 | } |
---|
463 | result.back()=grid.back(); |
---|
464 | grid=result; // copy the expanded grid back to the input |
---|
465 | } |
---|
466 | |
---|
467 | template <typename float_type> float_type c2_function<float_type>::integral(float_type xmin, float_type xmax, std::vector<float_type> *partials, |
---|
468 | float_type abs_tol, float_type rel_tol, int derivs, bool adapt, bool extrapolate) const throw(c2_exception) |
---|
469 | { |
---|
470 | if(xmin==xmax) { |
---|
471 | if(partials) partials->clear(); |
---|
472 | return 0.0; |
---|
473 | } |
---|
474 | std::vector<float_type> grid; |
---|
475 | get_sampling_grid(xmin, xmax, grid); |
---|
476 | float_type intg=partial_integrals(grid, partials, abs_tol, rel_tol, derivs, adapt, extrapolate); |
---|
477 | return intg; |
---|
478 | } |
---|
479 | |
---|
480 | template <typename float_type> c2_function<float_type> &c2_function<float_type>::normalized_function(float_type xmin, float_type xmax, float_type norm) |
---|
481 | const throw(c2_exception) |
---|
482 | { |
---|
483 | float_type intg=integral(xmin, xmax); |
---|
484 | return *new c2_scaled_function_p<float_type>(*this, norm/intg); |
---|
485 | } |
---|
486 | |
---|
487 | template <typename float_type> c2_function<float_type> &c2_function<float_type>::square_normalized_function(float_type xmin, float_type xmax, float_type norm) |
---|
488 | const throw(c2_exception) |
---|
489 | { |
---|
490 | c2_ptr<float_type> mesquared((*new c2_quadratic_p<float_type>(0., 0., 0., 1.))(*this)); |
---|
491 | |
---|
492 | std::vector<float_type> grid; |
---|
493 | get_sampling_grid(xmin, xmax, grid); |
---|
494 | float_type intg=mesquared->partial_integrals(grid); |
---|
495 | |
---|
496 | return *new c2_scaled_function_p<float_type>(*this, std::sqrt(norm/intg)); |
---|
497 | } |
---|
498 | |
---|
499 | template <typename float_type> c2_function<float_type> &c2_function<float_type>::square_normalized_function( |
---|
500 | float_type xmin, float_type xmax, const c2_function<float_type> &weight, float_type norm) |
---|
501 | const throw(c2_exception) |
---|
502 | { |
---|
503 | c2_ptr<float_type> weighted((*new c2_quadratic_p<float_type>(0., 0., 0., 1.))(*this) * weight); |
---|
504 | |
---|
505 | std::vector<float_type> grid; |
---|
506 | get_sampling_grid(xmin, xmax, grid); |
---|
507 | float_type intg=weighted->partial_integrals(grid); |
---|
508 | |
---|
509 | return *new c2_scaled_function_p<float_type>(*this, std::sqrt(norm/intg)); |
---|
510 | } |
---|
511 | |
---|
512 | template <typename float_type> float_type c2_function<float_type>::partial_integrals( |
---|
513 | std::vector<float_type> xgrid, std::vector<float_type> *partials, |
---|
514 | float_type abs_tol, float_type rel_tol, int derivs, bool adapt, bool extrapolate) |
---|
515 | const throw(c2_exception) |
---|
516 | { |
---|
517 | int np=xgrid.size(); |
---|
518 | |
---|
519 | c2_fblock<float_type> f0, f2; |
---|
520 | struct c2_integrate_recur rb; |
---|
521 | rb.rel_tol=rel_tol; |
---|
522 | rb.extrapolate=extrapolate; |
---|
523 | rb.adapt=adapt; |
---|
524 | rb.derivs=derivs; |
---|
525 | std::vector< recur_item > rb_stack; |
---|
526 | rb_stack.reserve(20); // enough for most operations |
---|
527 | rb.rb_stack=&rb_stack; |
---|
528 | rb.inited=false; |
---|
529 | float_type dx_inv=1.0/std::abs(xgrid.back()-xgrid.front()); |
---|
530 | |
---|
531 | if(partials) partials->resize(np-1); |
---|
532 | |
---|
533 | float_type sum=0.0; |
---|
534 | |
---|
535 | f2.x=xgrid[0]; |
---|
536 | fill_fblock(f2); |
---|
537 | if(c2_isnan(f2.y)) { |
---|
538 | bad_x_point=f2.x; |
---|
539 | return f2.y; // can't go any further if a nan has appeared |
---|
540 | } |
---|
541 | |
---|
542 | for(int i=0; i<np-1; i++) { |
---|
543 | f0=f2; // copy upper bound to lower before computing new upper bound |
---|
544 | |
---|
545 | f2.x=xgrid[i+1]; |
---|
546 | fill_fblock(f2); |
---|
547 | if(c2_isnan(f2.y)) { |
---|
548 | bad_x_point=f2.x; |
---|
549 | return f2.y; // can't go any further if a nan has appeared |
---|
550 | } |
---|
551 | |
---|
552 | rb.abs_tol=abs_tol*std::abs(f2.x-f0.x)*dx_inv; // distribute error tolerance over whole domain |
---|
553 | rb.f0=&f0; rb.f1=&f2; |
---|
554 | float_type ps=integrate_step(rb); |
---|
555 | sum+=ps; |
---|
556 | if(partials) (*partials)[i]=ps; |
---|
557 | if(c2_isnan(ps)) break; // NaN stops integration |
---|
558 | } |
---|
559 | return sum; |
---|
560 | } |
---|
561 | |
---|
562 | // generate a sampling grid at points separated by dx=5, which is intentionally |
---|
563 | // incommensurate with pi and 2*pi so grid errors are somewhat randomized |
---|
564 | template <typename float_type> void c2_sin_p<float_type>:: |
---|
565 | get_sampling_grid(float_type xmin, float_type xmax, std::vector<float_type> &grid) const |
---|
566 | { |
---|
567 | grid.clear(); |
---|
568 | for(; xmin < xmax; xmin+=5.0) grid.push_back(xmin); |
---|
569 | grid.push_back(xmax); |
---|
570 | this->preen_sampling_grid(&grid); |
---|
571 | } |
---|
572 | |
---|
573 | template <typename float_type> float_type c2_function_transformation<float_type>::evaluate( |
---|
574 | float_type xraw, |
---|
575 | float_type y, float_type yp0, float_type ypp0, |
---|
576 | float_type *yprime, float_type *yprime2) const |
---|
577 | { |
---|
578 | y=Y.fHasStaticTransforms ? Y.pOut(y) : Y.fOut(y); |
---|
579 | |
---|
580 | if(yprime || yprime2) { |
---|
581 | |
---|
582 | float_type yp, yp2; |
---|
583 | if(X.fHasStaticTransforms && Y.fHasStaticTransforms) { |
---|
584 | float_type fpi=1.0/Y.pInPrime(y); |
---|
585 | float_type gp=X.pInPrime(xraw); |
---|
586 | // from Mathematica Dt[InverseFunction[f][y[g[x]]], x] |
---|
587 | yp=gp*yp0*fpi; // transformed derivative |
---|
588 | yp2=(gp*gp*ypp0 + X.pInDPrime(xraw)*yp0 - Y.pInDPrime(y)*yp*yp )*fpi; |
---|
589 | } else { |
---|
590 | float_type fpi=1.0/Y.fInPrime(y); |
---|
591 | float_type gp=X.fInPrime(xraw); |
---|
592 | // from Mathematica Dt[InverseFunction[f][y[g[x]]], x] |
---|
593 | yp=gp*yp0*fpi; // transformed derivative |
---|
594 | yp2=(gp*gp*ypp0 + X.fInDPrime(xraw)*yp0 - Y.fInDPrime(y)*yp*yp )*fpi; |
---|
595 | } |
---|
596 | if(yprime) *yprime=yp; |
---|
597 | if(yprime2) *yprime2=yp2; |
---|
598 | } |
---|
599 | return y; |
---|
600 | } |
---|
601 | |
---|
602 | // The constructor |
---|
603 | template <typename float_type> interpolating_function_p<float_type> & interpolating_function_p<float_type>::load( |
---|
604 | const std::vector<float_type> &x, const std::vector<float_type> &f, |
---|
605 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
606 | bool upperSlopeNatural, float_type upperSlope, |
---|
607 | bool splined |
---|
608 | ) throw(c2_exception) |
---|
609 | { |
---|
610 | c2_ptr<float_type> keepme(*this); |
---|
611 | X= x; |
---|
612 | F= f; |
---|
613 | |
---|
614 | // Xraw is useful in some of the arithmetic operations between interpolating functions |
---|
615 | Xraw=x; |
---|
616 | |
---|
617 | set_domain(std::min(Xraw.front(), Xraw.back()),std::max(Xraw.front(), Xraw.back())); |
---|
618 | |
---|
619 | if(x.size() != f.size()) { |
---|
620 | throw c2_exception("interpolating_function::init() -- x & y inputs are of different size"); |
---|
621 | } |
---|
622 | |
---|
623 | size_t np=X.size(); // they are the same now, so lets take a short cut |
---|
624 | |
---|
625 | if(np < 2) { |
---|
626 | throw c2_exception("interpolating_function::init() -- input < 2 elements "); |
---|
627 | } |
---|
628 | |
---|
629 | bool xraw_rev=check_monotonicity(Xraw, |
---|
630 | "interpolating_function::init() non-monotonic raw x input"); // which way does raw X point? sampling grid MUST be increasing |
---|
631 | |
---|
632 | if(!xraw_rev) { // we can use pointer to raw X values if they are in the right order |
---|
633 | set_sampling_grid_pointer(Xraw); // our intial grid of x values is certainly a good guess for 'interesting' points |
---|
634 | } else { |
---|
635 | set_sampling_grid(Xraw); // make a copy of it, and assure it is in right order |
---|
636 | } |
---|
637 | |
---|
638 | if(fTransform.X.fTransformed) { // check if X scale is nonlinear, and if so, do transform |
---|
639 | if(!lowerSlopeNatural) lowerSlope /= fTransform.X.fInPrime(X[0]); |
---|
640 | if(!upperSlopeNatural) upperSlope /= fTransform.X.fInPrime(X[np-1]); |
---|
641 | for(size_t i=0; i<np; i++) X[i]=fTransform.X.fIn(X[i]); |
---|
642 | } |
---|
643 | if(fTransform.Y.fTransformed) { // check if Y scale is nonlinear, and if so, do transform |
---|
644 | if(!lowerSlopeNatural) lowerSlope *= fTransform.Y.fInPrime(F[0]); |
---|
645 | if(!upperSlopeNatural) upperSlope *= fTransform.Y.fInPrime(F[np-1]); |
---|
646 | for(size_t i=0; i<np; i++) F[i]=fTransform.Y.fIn(F[i]); |
---|
647 | } |
---|
648 | |
---|
649 | xInverted=check_monotonicity(X, |
---|
650 | "interpolating_function::init() non-monotonic transformed x input"); |
---|
651 | |
---|
652 | if(splined) spline(lowerSlopeNatural, lowerSlope, upperSlopeNatural, upperSlope); |
---|
653 | else y2.assign(np,0.0); |
---|
654 | |
---|
655 | lastKLow=0; |
---|
656 | keepme.release_for_return(); |
---|
657 | return *this; |
---|
658 | } |
---|
659 | |
---|
660 | /* |
---|
661 | // The constructor |
---|
662 | template <typename float_type> interpolating_function_p<float_type> & interpolating_function_p<float_type>::load_pairs( |
---|
663 | std::vector<std::pair<float_type, float_type> > &data, |
---|
664 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
665 | bool upperSlopeNatural, float_type upperSlope, |
---|
666 | bool splined |
---|
667 | ) throw(c2_exception) |
---|
668 | { |
---|
669 | c2_ptr<float_type> keepme(*this); |
---|
670 | |
---|
671 | size_t np=data.size(); |
---|
672 | if(np < 2) { |
---|
673 | throw c2_exception("interpolating_function::init() -- input < 2 elements "); |
---|
674 | } |
---|
675 | |
---|
676 | // sort into ascending order |
---|
677 | std::sort(data.begin(), data.end(), comp_pair); |
---|
678 | |
---|
679 | std::vector<float_type> xtmp, ytmp; |
---|
680 | xtmp.reserve(np); |
---|
681 | ytmp.reserve(np); |
---|
682 | for (size_t i=0; i<np; i++) { |
---|
683 | xtmp.push_back(data[i].first); |
---|
684 | ytmp.push_back(data[i].second); |
---|
685 | } |
---|
686 | this->load(xtmp, ytmp, lowerSlopeNatural, lowerSlope, upperSlopeNatural, upperSlope, splined); |
---|
687 | |
---|
688 | keepme.release_for_return(); |
---|
689 | return *this; |
---|
690 | } |
---|
691 | |
---|
692 | template <typename float_type> interpolating_function_p<float_type> & |
---|
693 | interpolating_function_p<float_type>::load_random_generator_function( |
---|
694 | const std::vector<float_type> &bincenters, const c2_function<float_type> &binheights) |
---|
695 | throw(c2_exception) |
---|
696 | { |
---|
697 | c2_ptr<float_type> keepme(*this); |
---|
698 | |
---|
699 | std::vector<float_type> integral; |
---|
700 | c2_const_ptr<float_type> keepit(binheights); // manage function... not really needed here, but always safe. |
---|
701 | // integrate from first to last bin in original order, leaving results in integral |
---|
702 | // ask for relative error of 1e-6 on each bin, with absolute error set to 0 (since we don't know the data scale). |
---|
703 | float_type sum=binheights.partial_integrals(bincenters, &integral, 0.0, 1e-6); |
---|
704 | // the integral vector now has partial integrals... it must be accumulated by summing |
---|
705 | integral.insert(integral.begin(), 0.0); // integral from start to start is 0 |
---|
706 | float_type scale=1.0/sum; |
---|
707 | for(size_t i=1; i<integral.size(); i++) integral[i]=integral[i]*scale + integral[i-1]; |
---|
708 | integral.back()=1.0; // force exact value on boundary |
---|
709 | |
---|
710 | this->load(integral, bincenters, |
---|
711 | false, 1.0/(scale*binheights(bincenters.front() )), |
---|
712 | false, 1.0/(scale*binheights(bincenters.back() )) |
---|
713 | ); // use integral as x axis in inverse function |
---|
714 | keepme.release_for_return(); |
---|
715 | return *this; |
---|
716 | } |
---|
717 | |
---|
718 | template <typename float_type> interpolating_function_p<float_type> & |
---|
719 | interpolating_function_p<float_type>::load_random_generator_bins( |
---|
720 | const std::vector<float_type> &bins, const std::vector<float_type> &binheights) |
---|
721 | throw(c2_exception) |
---|
722 | { |
---|
723 | c2_ptr<float_type> keepme(*this); |
---|
724 | |
---|
725 | size_t np=binheights.size(); |
---|
726 | std::vector<float_type> integral(np+1), bin_edges(np+1); |
---|
727 | |
---|
728 | // compute the integral based on estimates of the bin edges from the given bin centers... |
---|
729 | // except for bin 0 & final bin, the edge of a bin is halfway between then center of the |
---|
730 | // bin and the center of the previous/next bin. |
---|
731 | // This gives width[n] = (center[n+1]+center[n])/2 - (center[n]+center[n-1])/2 = (center[n+1]-center[n-1])/2 |
---|
732 | // for the edges, assume a bin of width (center[1]-center[0]) or (center[np-1]-center[np-2]) |
---|
733 | // be careful that absolute values are used in case data are reversed. |
---|
734 | |
---|
735 | if(bins.size() == binheights.size()+1) { |
---|
736 | bin_edges=bins; // edges array was passed in |
---|
737 | } else if (bins.size() == binheights.size()) { |
---|
738 | bin_edges.front()=bins[0] - (bins[1]-bins[0])*0.5; // edge bin |
---|
739 | for(size_t i=1; i<np; i++) { |
---|
740 | bin_edges[i]=(bins[i]+bins[i-1])*0.5; |
---|
741 | } |
---|
742 | bin_edges.back()=bins[np-1] + (bins[np-1]-bins[np-2])*0.5; // edge bin |
---|
743 | } else { |
---|
744 | throw c2_exception("inconsistent bin vectors passed to load_random_generator_bins"); |
---|
745 | } |
---|
746 | |
---|
747 | float_type running_sum=0.0; |
---|
748 | for(size_t i=0; i<np; i++) { |
---|
749 | integral[i]=running_sum; |
---|
750 | if(!binheights[i]) throw c2_exception("empty bin passed to load_random_generator_bins"); |
---|
751 | running_sum+=binheights[i]*std::abs(bin_edges[i+1]-bin_edges[i]); |
---|
752 | } |
---|
753 | float_type scale=1.0/running_sum; |
---|
754 | for(size_t i=0; i<np; i++) integral[i]*=scale; |
---|
755 | integral.back()=1.0; // force exactly correct value on boundary |
---|
756 | this->load(integral, bin_edges, |
---|
757 | false, 1.0/(scale*binheights.front()), |
---|
758 | false, 1.0/(scale*binheights.back()) |
---|
759 | ); // use integral as x axis in inverse function |
---|
760 | keepme.release_for_return(); |
---|
761 | return *this; |
---|
762 | } |
---|
763 | */ |
---|
764 | |
---|
765 | // The spline table generator |
---|
766 | template <typename float_type> void interpolating_function_p<float_type>::spline( |
---|
767 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
768 | bool upperSlopeNatural, float_type upperSlope |
---|
769 | ) throw(c2_exception) |
---|
770 | { |
---|
771 | // construct spline tables here. |
---|
772 | // this code is a re-translation of the pythonlabtools spline algorithm from pythonlabtools.sourceforge.net |
---|
773 | size_t np=X.size(); |
---|
774 | std::vector<float_type> u(np), dy(np-1), dx(np-1), dxi(np-1), dx2i(np-2), siga(np-2), dydx(np-1); |
---|
775 | |
---|
776 | std::transform(X.begin()+1, X.end(), X.begin(), dx.begin(), std::minus<float_type>() ); // dx=X[1:] - X [:-1] |
---|
777 | for(size_t i=0; i<dxi.size(); i++) dxi[i]=1.0/dx[i]; // dxi = 1/dx |
---|
778 | for(size_t i=0; i<dx2i.size(); i++) dx2i[i]=1.0/(X[i+2]-X[i]); |
---|
779 | |
---|
780 | std::transform(F.begin()+1, F.end(), F.begin(), dy.begin(), std::minus<float_type>() ); // dy = F[i+1]-F[i] |
---|
781 | std::transform(dx2i.begin(), dx2i.end(), dx.begin(), siga.begin(), std::multiplies<float_type>()); // siga = dx[:-1]*dx2i |
---|
782 | std::transform(dxi.begin(), dxi.end(), dy.begin(), dydx.begin(), std::multiplies<float_type>()); // dydx=dy/dx |
---|
783 | |
---|
784 | // u[i]=(y[i+1]-y[i])/float(x[i+1]-x[i]) - (y[i]-y[i-1])/float(x[i]-x[i-1]) |
---|
785 | std::transform(dydx.begin()+1, dydx.end(), dydx.begin(), u.begin()+1, std::minus<float_type>() ); // incomplete rendering of u = dydx[1:]-dydx[:-1] |
---|
786 | |
---|
787 | y2.resize(np,0.0); |
---|
788 | |
---|
789 | if(lowerSlopeNatural) { |
---|
790 | y2[0]=u[0]=0.0; |
---|
791 | } else { |
---|
792 | y2[0]= -0.5; |
---|
793 | u[0]=(3.0*dxi[0])*(dy[0]*dxi[0] -lowerSlope); |
---|
794 | } |
---|
795 | |
---|
796 | for(size_t i=1; i < np -1; i++) { // the inner loop |
---|
797 | float_type sig=siga[i-1]; |
---|
798 | float_type p=sig*y2[i-1]+2.0; |
---|
799 | y2[i]=(sig-1.0)/p; |
---|
800 | u[i]=(6.0*u[i]*dx2i[i-1] - sig*u[i-1])/p; |
---|
801 | } |
---|
802 | |
---|
803 | float_type qn, un; |
---|
804 | |
---|
805 | if(upperSlopeNatural) { |
---|
806 | qn=un=0.0; |
---|
807 | } else { |
---|
808 | qn= 0.5; |
---|
809 | un=(3.0*dxi[dxi.size()-1])*(upperSlope- dy[dy.size()-1]*dxi[dxi.size()-1] ); |
---|
810 | } |
---|
811 | |
---|
812 | y2[np-1]=(un-qn*u[np-2])/(qn*y2[np-2]+1.0); |
---|
813 | for (size_t k=np-1; k != 0; k--) y2[k-1]=y2[k-1]*y2[k]+u[k-1]; |
---|
814 | } |
---|
815 | |
---|
816 | template <typename float_type> interpolating_function_p<float_type> &interpolating_function_p<float_type>::sample_function( |
---|
817 | const c2_function<float_type> &func, |
---|
818 | float_type xmin, float_type xmax, float_type abs_tol, float_type rel_tol, |
---|
819 | bool lowerSlopeNatural, float_type lowerSlope, |
---|
820 | bool upperSlopeNatural, float_type upperSlope |
---|
821 | ) throw(c2_exception) |
---|
822 | { |
---|
823 | c2_ptr<float_type> keepme(*this); |
---|
824 | |
---|
825 | const c2_transformation<float_type> &XX=fTransform.X, &YY=fTransform.Y; // shortcuts |
---|
826 | |
---|
827 | // set up our params to look like the samplng function for now |
---|
828 | sampler_function=func; |
---|
829 | std::vector<float_type> grid; |
---|
830 | func.get_sampling_grid(xmin, xmax, grid); |
---|
831 | size_t gsize=grid.size(); |
---|
832 | if(XX.fTransformed) for(size_t i=0; i<gsize; i++) grid[i]=XX.fIn(grid[i]); |
---|
833 | set_sampling_grid_pointer(grid); |
---|
834 | |
---|
835 | // float_type xmin1=fXin(xmin), xmax1=fXin(xmax); // bounds in transformed space |
---|
836 | // get a list of points needed in transformed space, directly into our tables |
---|
837 | this->adaptively_sample(grid.front(), grid.back(), 8*abs_tol, 8*rel_tol, 0, &X, &F); |
---|
838 | // clear the sampler function now, since otherwise our value_with_derivatives is broken |
---|
839 | sampler_function.unset_function(); |
---|
840 | |
---|
841 | xInverted=check_monotonicity(X, |
---|
842 | "interpolating_function::init() non-monotonic transformed x input"); |
---|
843 | |
---|
844 | size_t np=X.size(); |
---|
845 | |
---|
846 | // Xraw is useful in some of the arithmetic operations between interpolating functions |
---|
847 | if(!XX.fTransformed) Xraw=X; |
---|
848 | else { |
---|
849 | Xraw.resize(np); |
---|
850 | for (size_t i=1; i<np-1; i++) Xraw[i]=XX.fOut(X[i]); |
---|
851 | Xraw.front()=xmin; |
---|
852 | Xraw.back()=xmax; |
---|
853 | } |
---|
854 | |
---|
855 | bool xraw_rev=check_monotonicity(Xraw, |
---|
856 | "interpolating_function::init() non-monotonic raw x input"); |
---|
857 | // which way does raw X point? sampling grid MUST be increasing |
---|
858 | |
---|
859 | if(!xraw_rev) { // we can use pointer to raw X values if they are in the right order |
---|
860 | set_sampling_grid_pointer(Xraw); |
---|
861 | // our intial grid of x values is certainly a good guess for 'interesting' points |
---|
862 | } else { |
---|
863 | set_sampling_grid(Xraw); // make a copy of it, and assure it is in right order |
---|
864 | } |
---|
865 | |
---|
866 | if(XX.fTransformed) { // check if X scale is nonlinear, and if so, do transform |
---|
867 | if(!lowerSlopeNatural) lowerSlope /= XX.fInPrime(xmin); |
---|
868 | if(!upperSlopeNatural) upperSlope /= XX.fInPrime(xmax); |
---|
869 | } |
---|
870 | if(YY.fTransformed) { // check if Y scale is nonlinear, and if so, do transform |
---|
871 | if(!lowerSlopeNatural) lowerSlope *= YY.fInPrime(func(xmin)); |
---|
872 | if(!upperSlopeNatural) upperSlope *= YY.fInPrime(func(xmax)); |
---|
873 | } |
---|
874 | // note that each of ends has 3 points with two equal gaps, since they were obtained by bisection |
---|
875 | // so the step sizes are easy to get |
---|
876 | // the 'natural slope' option for sampled functions has a different meaning than |
---|
877 | // for normal splines. In this case, the derivative is adjusted to make the |
---|
878 | // second derivative constant on the last two points at each end |
---|
879 | // which is consistent with the error sampling technique we used to get here |
---|
880 | if(lowerSlopeNatural) { |
---|
881 | float_type hlower=X[1]-X[0]; |
---|
882 | lowerSlope=0.5*(-F[2]-3*F[0]+4*F[1])/hlower; |
---|
883 | lowerSlopeNatural=false; // it's not the usual meaning of natural any more |
---|
884 | } |
---|
885 | if(upperSlopeNatural) { |
---|
886 | float_type hupper=X[np-1]-X[np-2]; |
---|
887 | upperSlope=0.5*(F[np-3]+3*F[np-1]-4*F[np-2])/hupper; |
---|
888 | upperSlopeNatural=false; // it's not the usual meaning of natural any more |
---|
889 | } |
---|
890 | this->set_domain(xmin, xmax); |
---|
891 | |
---|
892 | spline(lowerSlopeNatural, lowerSlope, upperSlopeNatural, upperSlope); |
---|
893 | lastKLow=0; |
---|
894 | keepme.release_for_return(); |
---|
895 | return *this; |
---|
896 | } |
---|
897 | |
---|
898 | // This function is the reason for this class to exist |
---|
899 | // it computes the interpolated function, and (if requested) its proper first and second derivatives including all coordinate transforms |
---|
900 | template <typename float_type> float_type interpolating_function_p<float_type>::value_with_derivatives( |
---|
901 | float_type x, float_type *yprime, float_type *yprime2) const throw(c2_exception) |
---|
902 | { |
---|
903 | if(sampler_function.valid()) { |
---|
904 | // if this is non-null, we are sampling data for later, so just return raw function |
---|
905 | // however, transform it into our sampling space, first. |
---|
906 | if(yprime) *yprime=0; |
---|
907 | if(yprime2) *yprime2=0; |
---|
908 | sampler_function->increment_evaluations(); |
---|
909 | return fTransform.Y.fIn(sampler_function(fTransform.X.fOut(x))); // derivatives are completely undefined |
---|
910 | } |
---|
911 | |
---|
912 | if(x < this->xmin() || x > this->xmax()) { |
---|
913 | std::ostringstream outstr; |
---|
914 | outstr << "Interpolating function argument " << x << " out of range " << this->xmin() << " -- " << this ->xmax() << ": bailing"; |
---|
915 | throw c2_exception(outstr.str().c_str()); |
---|
916 | } |
---|
917 | |
---|
918 | float_type xraw=x; |
---|
919 | |
---|
920 | if(fTransform.X.fTransformed) x=fTransform.X.fHasStaticTransforms? |
---|
921 | fTransform.X.pIn(x) : fTransform.X.fIn(x); // save time by explicitly testing for identity function here |
---|
922 | |
---|
923 | int klo=0, khi=X.size()-1; |
---|
924 | |
---|
925 | if(khi < 0) throw c2_exception("Uninitialized interpolating function being evaluated"); |
---|
926 | |
---|
927 | const float_type *XX=&X[lastKLow]; // make all fast checks short offsets from here |
---|
928 | |
---|
929 | if(!xInverted) { // select search depending on whether transformed X is increasing or decreasing |
---|
930 | if((XX[0] <= x) && (XX[1] >= x) ) { // already bracketed |
---|
931 | klo=lastKLow; |
---|
932 | } else if((XX[1] <= x) && (XX[2] >= x)) { // in next bracket to the right |
---|
933 | klo=lastKLow+1; |
---|
934 | } else if(lastKLow > 0 && (XX[-1] <= x) && (XX[0] >= x)) { // in next bracket to the left |
---|
935 | klo=lastKLow-1; |
---|
936 | } else { // not bracketed, not close, start over |
---|
937 | // search for new KLow |
---|
938 | while(khi-klo > 1) { |
---|
939 | int km=(khi+klo)/2; |
---|
940 | if(X[km] > x) khi=km; |
---|
941 | else klo=km; |
---|
942 | } |
---|
943 | } |
---|
944 | } else { |
---|
945 | if((XX[0] >= x) && (XX[1] <= x) ) { // already bracketed |
---|
946 | klo=lastKLow; |
---|
947 | } else if((XX[1] >= x) && (XX[2] <= x)) { // in next bracket to the right |
---|
948 | klo=lastKLow+1; |
---|
949 | } else if(lastKLow > 0 && (XX[-1] >= x) && (XX[0] <= x)) { // in next bracket to the left |
---|
950 | klo=lastKLow-1; |
---|
951 | } else { // not bracketed, not close, start over |
---|
952 | // search for new KLow |
---|
953 | while(khi-klo > 1) { |
---|
954 | int km=(khi+klo)/2; |
---|
955 | if(X[km] < x) khi=km; |
---|
956 | else klo=km; |
---|
957 | } |
---|
958 | } |
---|
959 | } |
---|
960 | |
---|
961 | khi=klo+1; |
---|
962 | lastKLow=klo; |
---|
963 | |
---|
964 | float_type h=X[khi]-X[klo]; |
---|
965 | |
---|
966 | float_type a=(X[khi]-x)/h; |
---|
967 | float_type b=1.0-a; |
---|
968 | float_type ylo=F[klo], yhi=F[khi], y2lo=y2[klo], y2hi=y2[khi]; |
---|
969 | float_type y=a*ylo+b*yhi+((a*a*a-a)*y2lo+(b*b*b-b)*y2hi)*(h*h)/6.0; |
---|
970 | |
---|
971 | float_type yp0=0; // the derivative in interpolating table coordinates |
---|
972 | float_type ypp0=0; // second derivative |
---|
973 | |
---|
974 | if(yprime || yprime2) { |
---|
975 | yp0=(yhi-ylo)/h+((3*b*b-1)*y2hi-(3*a*a-1)*y2lo)*h/6.0; // the derivative in interpolating table coordinates |
---|
976 | ypp0=b*y2hi+a*y2lo; // second derivative |
---|
977 | } |
---|
978 | |
---|
979 | if(fTransform.isIdentity) { |
---|
980 | if(yprime) *yprime=yp0; |
---|
981 | if(yprime2) *yprime2=ypp0; |
---|
982 | return y; |
---|
983 | } else return fTransform.evaluate(xraw, y, yp0, ypp0, yprime, yprime2); |
---|
984 | } |
---|
985 | |
---|
986 | template <typename float_type> void interpolating_function_p<float_type>::set_lower_extrapolation(float_type bound) |
---|
987 | { |
---|
988 | int kl = 0 ; |
---|
989 | int kh=kl+1; |
---|
990 | float_type xx=fTransform.X.fIn(bound); |
---|
991 | float_type h0=X[kh]-X[kl]; |
---|
992 | float_type h1=xx-X[kl]; |
---|
993 | 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; |
---|
994 | |
---|
995 | X.insert(X.begin(), xx); |
---|
996 | F.insert(F.begin(), yextrap); |
---|
997 | y2.insert(y2.begin(), y2.front()); // duplicate first or last element |
---|
998 | Xraw.insert(Xraw.begin(), bound); |
---|
999 | if (bound < this->fXMin) this->fXMin=bound; // check for reversed data |
---|
1000 | else this->fXMax=bound; |
---|
1001 | |
---|
1002 | //printf("%10.4f %10.4f %10.4f %10.4f %10.4f\n", bound, xx, h0, h1, yextrap); |
---|
1003 | //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]); |
---|
1004 | } |
---|
1005 | |
---|
1006 | template <typename float_type> void interpolating_function_p<float_type>::set_upper_extrapolation(float_type bound) |
---|
1007 | { |
---|
1008 | int kl = X.size()-2 ; |
---|
1009 | int kh=kl+1; |
---|
1010 | float_type xx=fTransform.X.fIn(bound); |
---|
1011 | float_type h0=X[kh]-X[kl]; |
---|
1012 | float_type h1=xx-X[kl]; |
---|
1013 | 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; |
---|
1014 | |
---|
1015 | X.insert(X.end(), xx); |
---|
1016 | F.insert(F.end(), yextrap); |
---|
1017 | y2.insert(y2.end(), y2.back()); // duplicate first or last element |
---|
1018 | Xraw.insert(Xraw.end(), bound); |
---|
1019 | if (bound < this->fXMin) this->fXMin=bound; // check for reversed data |
---|
1020 | else this->fXMax=bound; |
---|
1021 | //printf("%10.4f %10.4f %10.4f %10.4f %10.4f\n", bound, xx, h0, h1, yextrap); |
---|
1022 | //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]); |
---|
1023 | } |
---|
1024 | |
---|
1025 | // return a new interpolating_function which is the unary function of an existing interpolating_function |
---|
1026 | // can also be used to generate a resampling of another c2_function on a different grid |
---|
1027 | // by creating a=interpolating_function(x,x) |
---|
1028 | // and doing b=a.unary_operator(c) where c is a c2_function (probably another interpolating_function) |
---|
1029 | |
---|
1030 | template <typename float_type> interpolating_function_p<float_type>& |
---|
1031 | interpolating_function_p<float_type>::unary_operator(const c2_function<float_type> &source) const |
---|
1032 | { |
---|
1033 | size_t np=X.size(); |
---|
1034 | std::vector<float_type>yv(np); |
---|
1035 | c2_ptr<float_type> comp(source(*this)); |
---|
1036 | float_type yp0, yp1, ypp; |
---|
1037 | |
---|
1038 | for(size_t i=1; i<np-1; i++) { |
---|
1039 | yv[i]=source(fTransform.Y.fOut(F[i])); // copy pointwise the function of our data values |
---|
1040 | } |
---|
1041 | |
---|
1042 | yv.front()=comp(Xraw.front(), &yp0, &ypp); // get derivative at front |
---|
1043 | yv.back()= comp(Xraw.back(), &yp1, &ypp); // get derivative at back |
---|
1044 | |
---|
1045 | interpolating_function_p ©=clone(); |
---|
1046 | copy.load(this->Xraw, yv, false, yp0, false, yp1); |
---|
1047 | |
---|
1048 | return copy; |
---|
1049 | } |
---|
1050 | |
---|
1051 | template <typename float_type> void |
---|
1052 | interpolating_function_p<float_type>::get_data(std::vector<float_type> &xvals, std::vector<float_type> &yvals) const throw() |
---|
1053 | { |
---|
1054 | |
---|
1055 | xvals=Xraw; |
---|
1056 | yvals.resize(F.size()); |
---|
1057 | |
---|
1058 | for(size_t i=0; i<F.size(); i++) yvals[i]=fTransform.Y.fOut(F[i]); |
---|
1059 | } |
---|
1060 | |
---|
1061 | template <typename float_type> interpolating_function_p<float_type> & |
---|
1062 | interpolating_function_p<float_type>::binary_operator(const c2_function<float_type> &rhs, |
---|
1063 | const c2_binary_function<float_type> *combining_stub) const |
---|
1064 | { |
---|
1065 | size_t np=X.size(); |
---|
1066 | std::vector<float_type> yv(np); |
---|
1067 | c2_constant_p<float_type> fval(0); |
---|
1068 | float_type yp0, yp1, ypp; |
---|
1069 | |
---|
1070 | c2_const_ptr<float_type> stub(*combining_stub); // manage ownership |
---|
1071 | |
---|
1072 | for(size_t i=1; i<np-1; i++) { |
---|
1073 | fval.reset(fTransform.Y.fOut(F[i])); // update the constant function pointwise |
---|
1074 | yv[i]=combining_stub->combine(fval, rhs, Xraw[i], (float_type *)0, (float_type *)0); // compute rhs & combine without derivatives |
---|
1075 | } |
---|
1076 | |
---|
1077 | yv.front()=combining_stub->combine(*this, rhs, Xraw.front(), &yp0, &ypp); // get derivative at front |
---|
1078 | yv.back()= combining_stub->combine(*this, rhs, Xraw.back(), &yp1, &ypp); // get derivative at back |
---|
1079 | |
---|
1080 | interpolating_function_p ©=clone(); |
---|
1081 | copy.load(this->Xraw, yv, false, yp0, false, yp1); |
---|
1082 | |
---|
1083 | return copy; |
---|
1084 | } |
---|
1085 | |
---|
1086 | template <typename float_type> c2_inverse_function_p<float_type>::c2_inverse_function_p(const c2_function<float_type> &source) |
---|
1087 | : c2_function<float_type>(), func(source) |
---|
1088 | { |
---|
1089 | float_type l=source.xmin(); |
---|
1090 | float_type r=source.xmax(); |
---|
1091 | start_hint=(l+r)*0.5; // guess that we start in the middle |
---|
1092 | // compute our domain assuming the function is monotonic so its values on its domain boundaries are our domain |
---|
1093 | float_type ly=source(l); |
---|
1094 | float_type ry=source(r); |
---|
1095 | if (ly > ry) { |
---|
1096 | float_type t=ly; ly=ry; ry=t; |
---|
1097 | } |
---|
1098 | set_domain(ly, ry); |
---|
1099 | } |
---|
1100 | |
---|
1101 | template <typename float_type> float_type c2_inverse_function_p<float_type>::value_with_derivatives( |
---|
1102 | float_type x, float_type *yprime, float_type *yprime2 |
---|
1103 | ) const throw(c2_exception) |
---|
1104 | { |
---|
1105 | float_type l=this->func->xmin(); |
---|
1106 | float_type r=this->func->xmax(); |
---|
1107 | float_type yp, ypp; |
---|
1108 | float_type y=this->func->find_root(l, r, get_start_hint(x), x, 0, &yp, &ypp); |
---|
1109 | start_hint=y; |
---|
1110 | if(yprime) *yprime=1.0/yp; |
---|
1111 | if(yprime2) *yprime2=-ypp/(yp*yp*yp); |
---|
1112 | return y; |
---|
1113 | } |
---|
1114 | |
---|
1115 | //accumulated_histogram starts with binned data, generates the integral, and generates a piecewise linear interpolating_function |
---|
1116 | //If drop_zeros is true, it merges empty bins together before integration |
---|
1117 | //Note that the resulting interpolating_function is guaranteed to be increasing (if drop_zeros is false) |
---|
1118 | // or stricly increasing (if drop_zeros is true) |
---|
1119 | //If inverse_function is true, it drop zeros, integrates, and returns the inverse function which is useful |
---|
1120 | // for random number generation based on the input distribution. |
---|
1121 | //If normalize is true, the big end of the integral is scaled to 1. |
---|
1122 | //If the data are passed in reverse order (large X first), the integral is carried out from the big end, |
---|
1123 | // and then the data are reversed to the result in in increasing X order. |
---|
1124 | template <typename float_type> accumulated_histogram<float_type>::accumulated_histogram( |
---|
1125 | const std::vector<float_type>binedges, const std::vector<float_type> binheights, |
---|
1126 | bool normalize, bool inverse_function, bool drop_zeros) |
---|
1127 | { |
---|
1128 | |
---|
1129 | int np=binheights.size(); |
---|
1130 | |
---|
1131 | std::vector<float_type> be, bh; |
---|
1132 | if(drop_zeros || inverse_function) { //inverse functions cannot have any zero bins or they have vertical sections |
---|
1133 | if(binheights[0] || !inverse_function) { // conserve lower x bound if not an inverse function |
---|
1134 | be.push_back(binedges[0]); |
---|
1135 | bh.push_back(binheights[0]); |
---|
1136 | } |
---|
1137 | for(int i=1; i<np-1; i++) { |
---|
1138 | if(binheights[i]) { |
---|
1139 | be.push_back(binedges[i]); |
---|
1140 | bh.push_back(binheights[i]); |
---|
1141 | } |
---|
1142 | } |
---|
1143 | if(binheights[np-1] || !inverse_function) { |
---|
1144 | bh.push_back(binheights[np-1]); |
---|
1145 | be.push_back(binedges[np-1]); |
---|
1146 | be.push_back(binedges[np]); // push both sides of the last bin if needed |
---|
1147 | } |
---|
1148 | np=bh.size(); // set np to compressed size of bin array |
---|
1149 | } else { |
---|
1150 | be=binedges; |
---|
1151 | bh=binheights; |
---|
1152 | } |
---|
1153 | std::vector<float_type> cum(np+1, 0.0); |
---|
1154 | 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 |
---|
1155 | if(be[1] < be[0]) { // if bins passed in backwards, reverse them |
---|
1156 | std::reverse(be.begin(), be.end()); |
---|
1157 | std::reverse(cum.begin(), cum.end()); |
---|
1158 | for(unsigned int i=0; i<cum.size(); i++) cum[i]*=-1; // flip sign on reversed data |
---|
1159 | } |
---|
1160 | if(normalize) { |
---|
1161 | float_type m=1.0/std::max(cum[0], cum[np]); |
---|
1162 | for(int i=0; i<=np; i++) cum[i]*=m; |
---|
1163 | } |
---|
1164 | if(inverse_function) interpolating_function_p<float_type>(cum, be); // use cum as x axis in inverse function |
---|
1165 | else interpolating_function_p<float_type>(be, cum); // else use lower bin edge as x axis |
---|
1166 | std::fill(this->y2.begin(), this->y2.end(), 0.0); // clear second derivatives, to we are piecewise linear |
---|
1167 | } |
---|
1168 | |
---|
1169 | template <typename float_type> c2_piecewise_function_p<float_type>::c2_piecewise_function_p() |
---|
1170 | : c2_function<float_type>(), lastKLow(-1) |
---|
1171 | { |
---|
1172 | this->sampling_grid=new std::vector<float_type>; // this always has a smapling grid |
---|
1173 | } |
---|
1174 | |
---|
1175 | template <typename float_type> c2_piecewise_function_p<float_type>::~c2_piecewise_function_p() |
---|
1176 | { |
---|
1177 | } |
---|
1178 | |
---|
1179 | template <typename float_type> float_type c2_piecewise_function_p<float_type>::value_with_derivatives( |
---|
1180 | float_type x, float_type *yprime, float_type *yprime2 |
---|
1181 | ) const throw(c2_exception) |
---|
1182 | { |
---|
1183 | |
---|
1184 | size_t np=functions.size(); |
---|
1185 | if(!np) throw c2_exception("attempting to evaluate an empty piecewise function"); |
---|
1186 | |
---|
1187 | if(x < this->xmin() || x > this->xmax()) { |
---|
1188 | std::ostringstream outstr; |
---|
1189 | outstr << "piecewise function argument " << x << " out of range " << this->xmin() << " -- " << this->xmax(); |
---|
1190 | throw c2_exception(outstr.str().c_str()); |
---|
1191 | } |
---|
1192 | |
---|
1193 | int klo=0; |
---|
1194 | |
---|
1195 | if(lastKLow >= 0 && functions[lastKLow]->xmin() <= x && functions[lastKLow]->xmax() > x) { |
---|
1196 | klo=lastKLow; |
---|
1197 | } else { |
---|
1198 | int khi=np; |
---|
1199 | while(khi-klo > 1) { |
---|
1200 | int km=(khi+klo)/2; |
---|
1201 | if(functions[km]->xmin() > x) khi=km; |
---|
1202 | else klo=km; |
---|
1203 | } |
---|
1204 | } |
---|
1205 | lastKLow=klo; |
---|
1206 | return functions[klo]->value_with_derivatives(x, yprime, yprime2); |
---|
1207 | } |
---|
1208 | |
---|
1209 | template <typename float_type> void c2_piecewise_function_p<float_type>::append_function( |
---|
1210 | const c2_function<float_type> &func) throw(c2_exception) |
---|
1211 | { |
---|
1212 | c2_const_ptr<float_type> keepfunc(func); // manage function before we can throw any exceptions |
---|
1213 | if(functions.size()) { // check whether there are any gaps to fill, etc. |
---|
1214 | const c2_function<float_type> &tail=functions.back(); |
---|
1215 | float_type x0=tail.xmax(); |
---|
1216 | float_type x1=func.xmin(); |
---|
1217 | if(x0 < x1) { |
---|
1218 | // must insert a connector if x0 < x1 |
---|
1219 | float_type y0=tail(x0); |
---|
1220 | float_type y1=func(x1); |
---|
1221 | c2_function<float_type> &connector=*new c2_linear_p<float_type>(x0, y0, (y1-y0)/(x1-x0)); |
---|
1222 | connector.set_domain(x0,x1); |
---|
1223 | functions.push_back(c2_const_ptr<float_type>(connector)); |
---|
1224 | this->sampling_grid->push_back(x1); |
---|
1225 | } else if(x0>x1) throw c2_exception("function domains not increasing in c2_piecewise_function"); |
---|
1226 | } |
---|
1227 | functions.push_back(keepfunc); |
---|
1228 | // extend our domain to include all known functions |
---|
1229 | this->set_domain(functions.front()->xmin(), functions.back()->xmax()); |
---|
1230 | // extend our sampling grid with the new function's grid, with the first point dropped to avoid duplicates |
---|
1231 | std::vector<float_type> newgrid; |
---|
1232 | func.get_sampling_grid(func.xmin(), func.xmax(), newgrid); |
---|
1233 | this->sampling_grid->insert(this->sampling_grid->end(), newgrid.begin()+1, newgrid.end()); |
---|
1234 | } |
---|
1235 | |
---|
1236 | template <typename float_type> c2_connector_function_p<float_type>::c2_connector_function_p( |
---|
1237 | float_type x0, const c2_function<float_type> &f0, float_type x2, const c2_function<float_type> &f2, |
---|
1238 | bool auto_center, float_type y1) |
---|
1239 | : c2_function<float_type>() |
---|
1240 | { |
---|
1241 | c2_const_ptr<float_type> left(f0), right(f2); // make sure if these are unowned, they get deleted |
---|
1242 | c2_fblock<float_type> fb0, fb2; |
---|
1243 | fb0.x=x0; |
---|
1244 | f0.fill_fblock(fb0); |
---|
1245 | fb2.x=x2; |
---|
1246 | f2.fill_fblock(fb2); |
---|
1247 | init(fb0, fb2, auto_center, y1); |
---|
1248 | } |
---|
1249 | |
---|
1250 | template <typename float_type> c2_connector_function_p<float_type>::c2_connector_function_p( |
---|
1251 | float_type x0, float_type y0, float_type yp0, float_type ypp0, |
---|
1252 | float_type x2, float_type y2, float_type yp2, float_type ypp2, |
---|
1253 | bool auto_center, float_type y1) |
---|
1254 | : c2_function<float_type>() |
---|
1255 | { |
---|
1256 | c2_fblock<float_type> fb0, fb2; |
---|
1257 | fb0.x=x0; fb0.y=y0; fb0.yp=yp0; fb0.ypp=ypp0; |
---|
1258 | fb2.x=x2; fb2.y=y2; fb2.yp=yp2; fb2.ypp=ypp2; |
---|
1259 | init(fb0, fb2, auto_center, y1); |
---|
1260 | } |
---|
1261 | |
---|
1262 | template <typename float_type> c2_connector_function_p<float_type>::c2_connector_function_p( |
---|
1263 | const c2_fblock<float_type> &fb0, |
---|
1264 | const c2_fblock<float_type> &fb2, |
---|
1265 | bool auto_center, float_type y1) |
---|
1266 | : c2_function<float_type>() |
---|
1267 | { |
---|
1268 | init(fb0, fb2, auto_center, y1); |
---|
1269 | } |
---|
1270 | |
---|
1271 | template <typename float_type> void c2_connector_function_p<float_type>::init( |
---|
1272 | const c2_fblock<float_type> &fb0, |
---|
1273 | const c2_fblock<float_type> &fb2, |
---|
1274 | bool auto_center, float_type y1) |
---|
1275 | { |
---|
1276 | float_type dx=(fb2.x-fb0.x)/2.0; |
---|
1277 | fhinv=1.0/dx; |
---|
1278 | |
---|
1279 | // scale derivs to put function on [-1,1] since mma solution is done this way |
---|
1280 | float_type yp0=fb0.yp*dx; |
---|
1281 | float_type yp2=fb2.yp*dx; |
---|
1282 | float_type ypp0=fb0.ypp*dx*dx; |
---|
1283 | float_type ypp2=fb2.ypp*dx*dx; |
---|
1284 | |
---|
1285 | float_type ff0=(8*(fb0.y + fb2.y) + 5*(yp0 - yp2) + ypp0 + ypp2)*0.0625; |
---|
1286 | if(auto_center) y1=ff0; // forces ff to be 0 if we are auto-centering |
---|
1287 | |
---|
1288 | // y[x_] = y1 + x (a + b x) + x [(x-1) (x+1)] (c + d x) + x (x-1)^2 (x+1)^2 (e + f x) |
---|
1289 | // y' = a + 2 b x + d x [(x+1)(x-1)] + (c + d x)(3x^2-1) + f x [(x+1)(x-1)]^2 + (e + f x)[(x+1)(x-1)](5x^2-1) |
---|
1290 | // y'' = 2 b + 6x(c + d x) + 2d(3x^2-1) + 4x(e + f x)(5x^2-3) + 2f(x^2-1)(5x^2-1) |
---|
1291 | fy1=y1; |
---|
1292 | fa=(fb2.y - fb0.y)*0.5; |
---|
1293 | fb=(fb0.y + fb2.y)*0.5 - y1; |
---|
1294 | fc=(yp2+yp0-2.*fa)*0.25; |
---|
1295 | fd=(yp2-yp0-4.*fb)*0.25; |
---|
1296 | fe=(ypp2-ypp0-12.*fc)*0.0625; |
---|
1297 | ff=(ff0 - y1); |
---|
1298 | this->set_domain(fb0.x, fb2.x); // this is where the function is valid |
---|
1299 | } |
---|
1300 | |
---|
1301 | template <typename float_type> c2_connector_function_p<float_type>::~c2_connector_function_p() |
---|
1302 | { |
---|
1303 | } |
---|
1304 | |
---|
1305 | template <typename float_type> float_type c2_connector_function_p<float_type>::value_with_derivatives( |
---|
1306 | float_type x, float_type *yprime, float_type *yprime2 |
---|
1307 | ) const throw(c2_exception) |
---|
1308 | { |
---|
1309 | float_type x0=this->xmin(), x2=this->xmax(); |
---|
1310 | float_type dx=(x-(x0+x2)*0.5)*fhinv; |
---|
1311 | float_type q1=(x-x0)*(x-x2)*fhinv*fhinv; // exactly vanish all bits at both ends |
---|
1312 | float_type q2=dx*q1; |
---|
1313 | |
---|
1314 | float_type r1=fa+fb*dx; |
---|
1315 | float_type r2=fc+fd*dx; |
---|
1316 | float_type r3=fe+ff*dx; |
---|
1317 | |
---|
1318 | float_type y=fy1+dx*r1+q2*r2+q1*q2*r3; |
---|
1319 | |
---|
1320 | if(yprime || yprime2) { |
---|
1321 | float_type q3=3*q1+2; |
---|
1322 | float_type q4=5*q1+4; |
---|
1323 | if(yprime) *yprime=(fa+2*fb*dx+fd*q2+r2*q3+ff*q1*q2+q1*q4*r3)*fhinv; |
---|
1324 | if(yprime2) *yprime2=2*(fb+fd*q3+3*dx*r2+ff*q1*q4+r3*(2*dx*(5*q1+2)))*fhinv*fhinv; |
---|
1325 | } |
---|
1326 | return y; |
---|
1327 | } |
---|
1328 | |
---|
1329 | // the recursive part of the sampler is agressively designed to minimize copying of data... lots of pointers |
---|
1330 | template <typename float_type> void c2_function<float_type>::sample_step(c2_sample_recur &rb) const throw(c2_exception) |
---|
1331 | { |
---|
1332 | std::vector< recur_item > &rb_stack=*rb.rb_stack; // heap-based stack of data for recursion |
---|
1333 | rb_stack.clear(); |
---|
1334 | |
---|
1335 | recur_item top; |
---|
1336 | top.depth=0; top.done=false; top.f0index=0; top.f2index=0; |
---|
1337 | |
---|
1338 | // push storage for our initial elements |
---|
1339 | rb_stack.push_back(top); |
---|
1340 | rb_stack.back().f1=*rb.f0; |
---|
1341 | rb_stack.back().done=true; |
---|
1342 | |
---|
1343 | rb_stack.push_back(top); |
---|
1344 | rb_stack.back().f1=*rb.f1; |
---|
1345 | rb_stack.back().done=true; |
---|
1346 | |
---|
1347 | if(!rb.inited) { |
---|
1348 | rb.dx_tolerance=10.0*std::numeric_limits<float_type>::epsilon(); |
---|
1349 | rb.abs_tol_min=10.0*std::numeric_limits<float_type>::min(); |
---|
1350 | rb.inited=true; |
---|
1351 | } |
---|
1352 | |
---|
1353 | // now, push our first real element |
---|
1354 | top.f0index=0; // left element is stack[0] |
---|
1355 | top.f2index=1; // right element is stack[1] |
---|
1356 | rb_stack.push_back(top); |
---|
1357 | |
---|
1358 | while(rb_stack.size() > 2) { |
---|
1359 | recur_item &back=rb_stack.back(); |
---|
1360 | if(back.done) { |
---|
1361 | rb_stack.pop_back(); |
---|
1362 | continue; |
---|
1363 | } |
---|
1364 | back.done=true; |
---|
1365 | |
---|
1366 | c2_fblock<float_type> &f0=rb_stack[back.f0index].f1, &f2=rb_stack[back.f2index].f1; |
---|
1367 | c2_fblock<float_type> &f1=back.f1; // will hold new middle values |
---|
1368 | size_t f1index=rb_stack.size()-1; // our current offset |
---|
1369 | |
---|
1370 | // std::cout << "processing: " << rb_stack.size() << " " << |
---|
1371 | // (&back-&rb_stack.front()) << " " << back.depth << " " << f0.x << " " << f2.x << std::endl; |
---|
1372 | |
---|
1373 | f1.x=0.5*(f0.x + f2.x); // center of interval |
---|
1374 | float_type dx2=0.5*(f2.x - f0.x); |
---|
1375 | |
---|
1376 | // check for underflow on step size, which prevents us from achieving specified accuracy. |
---|
1377 | if(std::abs(dx2) < std::abs(f1.x)*rb.dx_tolerance || std::abs(dx2) < rb.abs_tol_min) { |
---|
1378 | std::ostringstream outstr; |
---|
1379 | outstr << "Step size underflow in adaptive_sampling at depth=" << back.depth << ", x= " << f1.x; |
---|
1380 | throw c2_exception(outstr.str().c_str()); |
---|
1381 | } |
---|
1382 | |
---|
1383 | fill_fblock(f1); |
---|
1384 | |
---|
1385 | if(c2_isnan(f1.y) || f1.ypbad || f1.yppbad) { |
---|
1386 | // can't go any further if a nan has appeared |
---|
1387 | bad_x_point=f1.x; |
---|
1388 | throw c2_exception("NaN encountered while sampling function"); |
---|
1389 | } |
---|
1390 | |
---|
1391 | float_type eps; |
---|
1392 | if(rb.derivs==2) { |
---|
1393 | // this is code from connector_function to compute the value at the midpoint |
---|
1394 | // it is re-included here to avoid constructing a complete c2connector |
---|
1395 | // just to find out if we are close enough |
---|
1396 | float_type ff0=(8*(f0.y + f2.y) + 5*(f0.yp - f2.yp)*dx2 + (f0.ypp+f2.ypp)*dx2*dx2)*0.0625; |
---|
1397 | // we are converging as at least x**5 and bisecting, so real error on final step is smaller |
---|
1398 | eps=std::abs(ff0-f1.y)/32.0; |
---|
1399 | } else { |
---|
1400 | // there are two tolerances to meet... the shift in the estimate of the actual point, |
---|
1401 | // and the difference between the current points and the extremum |
---|
1402 | // build all the coefficients needed to construct the local parabola |
---|
1403 | float_type ypcenter, ypp; |
---|
1404 | if (rb.derivs==1) { |
---|
1405 | // linear extrapolation error using exact derivs |
---|
1406 | eps = (std::abs(f0.y+f0.yp*dx2-f1.y)+std::abs(f2.y-f2.yp*dx2-f1.y))*0.125; |
---|
1407 | ypcenter=2*f1.yp*dx2; // first deriv scaled so this interval is on [-1,1] |
---|
1408 | ypp=2*(f2.yp-f0.yp)*dx2*dx2; // second deriv estimate scaled so this interval is on [-1,1] |
---|
1409 | } else { |
---|
1410 | // linear interpolation error without derivs if we are at top level |
---|
1411 | // or 3-point parabolic interpolation estimates from previous level, if available |
---|
1412 | ypcenter=(f2.y-f0.y)*0.5; // derivative estimate at center |
---|
1413 | ypp=(f2.y+f0.y-2*f1.y); // second deriv estimate |
---|
1414 | if(back.depth==0) eps=std::abs((f0.y+f2.y)*0.5 - f1.y)*2; // penalize first step |
---|
1415 | else eps=std::abs(f1.y-back.previous_estimate)*0.25; |
---|
1416 | } |
---|
1417 | float_type ypleft=ypcenter-ypp; // derivative at left edge |
---|
1418 | float_type ypright=ypcenter+ypp; // derivative at right edge |
---|
1419 | float_type extremum_eps=0; |
---|
1420 | if((ypleft*ypright) <=0) // y' changes sign if we have an extremum |
---|
1421 | { |
---|
1422 | // compute position and value of the extremum this way |
---|
1423 | float_type xext=-ypcenter/ypp; |
---|
1424 | float_type yext=f1.y + xext*ypcenter + 0.5*xext*xext*ypp; |
---|
1425 | // and then find the the smallest offset of it from a point, looking in the left or right side |
---|
1426 | if(xext <=0) extremum_eps=std::min(std::abs(f0.y-yext), std::abs(f1.y-yext)); |
---|
1427 | else extremum_eps=std::min(std::abs(f2.y-yext), std::abs(f1.y-yext)); |
---|
1428 | } |
---|
1429 | eps=std::max(eps, extremum_eps); // if previous shot was really bad, keep trying |
---|
1430 | } |
---|
1431 | |
---|
1432 | if(eps < rb.abs_tol || eps < std::abs(f1.y)*rb.rel_tol) { |
---|
1433 | if(rb.out) { |
---|
1434 | // we've met the tolerance, and are building a function, append two connectors |
---|
1435 | rb.out->append_function( |
---|
1436 | *new c2_connector_function_p<float_type>(f0, f1, true, 0.0) |
---|
1437 | ); |
---|
1438 | rb.out->append_function( |
---|
1439 | *new c2_connector_function_p<float_type>(f1, f2, true, 0.0) |
---|
1440 | ); |
---|
1441 | } |
---|
1442 | if(rb.xvals && rb.yvals) { |
---|
1443 | rb.xvals->push_back(f0.x); |
---|
1444 | rb.xvals->push_back(f1.x); |
---|
1445 | rb.yvals->push_back(f0.y); |
---|
1446 | rb.yvals->push_back(f1.y); |
---|
1447 | // the value at f2 will get pushed in the next segment... it is not forgotten |
---|
1448 | } |
---|
1449 | } else { |
---|
1450 | top.depth=back.depth+1; // increment depth counter |
---|
1451 | |
---|
1452 | // save the last things we need from back before a push happens, in case |
---|
1453 | // the push causes a reallocation and moves the whole stack. |
---|
1454 | size_t f0index=back.f0index, f2index=back.f2index; |
---|
1455 | float_type left=0, right=0; |
---|
1456 | if(rb.derivs==0) { |
---|
1457 | // compute three-point parabolic interpolation estimate of right-hand and left-hand midpoint |
---|
1458 | left=(6*f1.y + 3*f0.y - f2.y) * 0.125; |
---|
1459 | right=(6*f1.y + 3*f2.y - f0.y) * 0.125; |
---|
1460 | } |
---|
1461 | |
---|
1462 | top.f0index=f1index; top.f2index=f2index; // insert pointers to right side data into our recursion block |
---|
1463 | top.previous_estimate=right; |
---|
1464 | rb_stack.push_back(top); |
---|
1465 | |
---|
1466 | top.f0index=f0index; top.f2index=f1index; // insert pointers to left side data into our recursion block |
---|
1467 | top.previous_estimate=left; |
---|
1468 | rb_stack.push_back(top); |
---|
1469 | } |
---|
1470 | } |
---|
1471 | } |
---|
1472 | |
---|
1473 | template <typename float_type> c2_piecewise_function_p<float_type> * |
---|
1474 | c2_function<float_type>::adaptively_sample( |
---|
1475 | float_type xmin, float_type xmax, |
---|
1476 | float_type abs_tol, float_type rel_tol, |
---|
1477 | int derivs, std::vector<float_type> *xvals, std::vector<float_type> *yvals) const throw(c2_exception) |
---|
1478 | { |
---|
1479 | c2_fblock<float_type> f0, f2; |
---|
1480 | c2_sample_recur rb; |
---|
1481 | std::vector< recur_item > rb_stack; |
---|
1482 | rb_stack.reserve(20); // enough for most operations |
---|
1483 | rb.rb_stack=&rb_stack; |
---|
1484 | rb.out=0; |
---|
1485 | if(derivs==2) rb.out=new c2_piecewise_function_p<float_type>(); |
---|
1486 | c2_ptr<float_type> pieces(*rb.out); // manage this function, if any, so it deletes on an exception |
---|
1487 | rb.rel_tol=rel_tol; |
---|
1488 | rb.abs_tol=abs_tol; |
---|
1489 | rb.xvals=xvals; |
---|
1490 | rb.yvals=yvals; |
---|
1491 | rb.derivs=derivs; |
---|
1492 | rb.inited=false; |
---|
1493 | |
---|
1494 | if(xvals && yvals) { |
---|
1495 | xvals->clear(); |
---|
1496 | yvals->clear(); |
---|
1497 | } |
---|
1498 | |
---|
1499 | // create xgrid as a automatic-variable copy of the sampling grid so the exception handler correctly |
---|
1500 | // disposes of it. |
---|
1501 | std::vector<float_type> xgrid; |
---|
1502 | get_sampling_grid(xmin, xmax, xgrid); |
---|
1503 | int np=xgrid.size(); |
---|
1504 | |
---|
1505 | f2.x=xgrid[0]; |
---|
1506 | fill_fblock(f2); |
---|
1507 | if(c2_isnan(f2.y) || f2.ypbad || f2.yppbad) { |
---|
1508 | // can't go any further if a nan has appeared |
---|
1509 | bad_x_point=f2.x; |
---|
1510 | throw c2_exception("NaN encountered while sampling function"); |
---|
1511 | } |
---|
1512 | |
---|
1513 | for(int i=0; i<np-1; i++) { |
---|
1514 | f0=f2; // copy upper bound to lower before computing new upper bound |
---|
1515 | |
---|
1516 | f2.x=xgrid[i+1]; |
---|
1517 | fill_fblock(f2); |
---|
1518 | if(c2_isnan(f2.y) || f2.ypbad || f2.yppbad) { |
---|
1519 | // can't go any further if a nan has appeared |
---|
1520 | bad_x_point=f2.x; |
---|
1521 | throw c2_exception("NaN encountered while sampling function"); |
---|
1522 | } |
---|
1523 | |
---|
1524 | rb.f0=&f0; rb.f1=&f2; |
---|
1525 | sample_step(rb); |
---|
1526 | } |
---|
1527 | if(xvals && yvals) { // push final point in vector |
---|
1528 | xvals->push_back(f2.x); |
---|
1529 | yvals->push_back(f2.y); |
---|
1530 | } |
---|
1531 | |
---|
1532 | if(rb.out) rb.out->set_sampling_grid(xgrid); // reflect old sampling grid, which still should be right |
---|
1533 | pieces.release_for_return(); // unmanage the piecewise_function so we can return it |
---|
1534 | return rb.out; |
---|
1535 | } |
---|
1536 | |
---|
1537 | template <typename float_type, typename Final> |
---|
1538 | interpolating_function_p<float_type> & inverse_integrated_density_function( |
---|
1539 | const std::vector<float_type> &bincenters, const c2_function<float_type> &binheights) |
---|
1540 | throw(c2_exception) |
---|
1541 | { |
---|
1542 | return (new Final())->load_random_generator_function(bincenters, binheights); |
---|
1543 | } |
---|
1544 | |
---|
1545 | template <typename float_type, typename Final> |
---|
1546 | interpolating_function_p<float_type> & inverse_integrated_density_bins( |
---|
1547 | const std::vector<float_type> &bins, const std::vector<float_type> &binheights) |
---|
1548 | throw(c2_exception) |
---|
1549 | { |
---|
1550 | return (new Final())->load_random_generator_bins(bins, binheights); |
---|
1551 | } |
---|