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