1 | // |
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2 | // Lol Engine - Sample math program: Chebyshev polynomials |
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3 | // |
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4 | // Copyright: (c) 2005-2011 Sam Hocevar <sam@hocevar.net> |
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5 | // This program is free software; you can redistribute it and/or |
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6 | // modify it under the terms of the Do What The Fuck You Want To |
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7 | // Public License, Version 2, as published by Sam Hocevar. See |
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8 | // http://sam.zoy.org/projects/COPYING.WTFPL for more details. |
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9 | // |
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10 | |
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11 | #if defined HAVE_CONFIG_H |
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12 | # include "config.h" |
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13 | #endif |
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14 | |
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15 | #include <cstring> |
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16 | #include <cstdio> |
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17 | |
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18 | #include "core.h" |
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19 | |
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20 | using namespace lol; |
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21 | using namespace std; |
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22 | |
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23 | /* The order of the approximation we're looking for */ |
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24 | static int const ORDER = 4; |
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25 | |
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26 | /* The function we want to approximate */ |
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27 | static real myfun(real const &x) |
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28 | { |
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29 | return exp(x); |
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30 | //if (!x) |
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31 | // return real::R_PI_2; |
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32 | //return sin(x * real::R_PI_2) / x; |
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33 | } |
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34 | |
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35 | static real myerror(real const &x) |
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36 | { |
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37 | return myfun(x); |
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38 | //return real::R_1; |
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39 | } |
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40 | |
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41 | /* Naive matrix inversion */ |
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42 | template<int N> struct Matrix |
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43 | { |
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44 | inline Matrix() {} |
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45 | |
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46 | Matrix(real x) |
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47 | { |
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48 | for (int j = 0; j < N; j++) |
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49 | for (int i = 0; i < N; i++) |
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50 | if (i == j) |
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51 | m[i][j] = x; |
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52 | else |
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53 | m[i][j] = 0; |
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54 | } |
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55 | |
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56 | Matrix<N> inv() const |
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57 | { |
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58 | Matrix a = *this, b((real)1.0); |
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59 | |
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60 | /* Inversion method: iterate through all columns and make sure |
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61 | * all the terms are 1 on the diagonal and 0 everywhere else */ |
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62 | for (int i = 0; i < N; i++) |
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63 | { |
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64 | /* If the expected coefficient is zero, add one of |
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65 | * the other lines. The first we meet will do. */ |
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66 | if ((double)a.m[i][i] == 0.0) |
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67 | { |
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68 | for (int j = i + 1; j < N; j++) |
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69 | { |
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70 | if ((double)a.m[i][j] == 0.0) |
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71 | continue; |
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72 | /* Add row j to row i */ |
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73 | for (int n = 0; n < N; n++) |
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74 | { |
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75 | a.m[n][i] += a.m[n][j]; |
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76 | b.m[n][i] += b.m[n][j]; |
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77 | } |
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78 | break; |
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79 | } |
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80 | } |
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81 | |
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82 | /* Now we know the diagonal term is non-zero. Get its inverse |
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83 | * and use that to nullify all other terms in the column */ |
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84 | real x = (real)1.0 / a.m[i][i]; |
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85 | for (int j = 0; j < N; j++) |
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86 | { |
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87 | if (j == i) |
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88 | continue; |
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89 | real mul = x * a.m[i][j]; |
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90 | for (int n = 0; n < N; n++) |
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91 | { |
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92 | a.m[n][j] -= mul * a.m[n][i]; |
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93 | b.m[n][j] -= mul * b.m[n][i]; |
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94 | } |
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95 | } |
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96 | |
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97 | /* Finally, ensure the diagonal term is 1 */ |
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98 | for (int n = 0; n < N; n++) |
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99 | { |
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100 | a.m[n][i] *= x; |
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101 | b.m[n][i] *= x; |
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102 | } |
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103 | } |
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104 | |
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105 | return b; |
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106 | } |
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107 | |
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108 | void print() const |
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109 | { |
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110 | for (int j = 0; j < N; j++) |
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111 | { |
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112 | for (int i = 0; i < N; i++) |
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113 | printf("%9.5f ", (double)m[j][i]); |
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114 | printf("\n"); |
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115 | } |
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116 | } |
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117 | |
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118 | real m[N][N]; |
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119 | }; |
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120 | |
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121 | |
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122 | static int cheby[ORDER + 1][ORDER + 1]; |
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123 | |
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124 | /* Fill the Chebyshev tables */ |
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125 | static void cheby_init() |
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126 | { |
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127 | memset(cheby, 0, sizeof(cheby)); |
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128 | |
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129 | cheby[0][0] = 1; |
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130 | cheby[1][1] = 1; |
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131 | |
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132 | for (int i = 2; i < ORDER + 1; i++) |
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133 | { |
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134 | cheby[i][0] = -cheby[i - 2][0]; |
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135 | for (int j = 1; j < ORDER + 1; j++) |
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136 | cheby[i][j] = 2 * cheby[i - 1][j - 1] - cheby[i - 2][j]; |
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137 | } |
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138 | } |
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139 | |
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140 | static void cheby_coeff(real *coeff, real *bn) |
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141 | { |
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142 | for (int i = 0; i < ORDER + 1; i++) |
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143 | { |
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144 | bn[i] = 0; |
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145 | for (int j = 0; j < ORDER + 1; j++) |
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146 | if (cheby[j][i]) |
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147 | bn[i] += coeff[j] * (real)cheby[j][i]; |
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148 | } |
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149 | } |
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150 | |
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151 | static real cheby_eval(real *coeff, real const &x) |
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152 | { |
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153 | real ret = 0.0, xn = 1.0; |
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154 | |
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155 | for (int i = 0; i < ORDER + 1; i++) |
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156 | { |
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157 | real mul = 0; |
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158 | for (int j = 0; j < ORDER + 1; j++) |
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159 | if (cheby[j][i]) |
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160 | mul += coeff[j] * (real)cheby[j][i]; |
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161 | ret += mul * xn; |
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162 | xn *= x; |
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163 | } |
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164 | |
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165 | return ret; |
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166 | } |
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167 | |
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168 | static void remez_init(real *coeff, real *zeroes) |
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169 | { |
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170 | /* Pick up x_i where error will be 0 and compute f(x_i) */ |
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171 | real fxn[ORDER + 1]; |
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172 | for (int i = 0; i < ORDER + 1; i++) |
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173 | { |
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174 | zeroes[i] = (real)(2 * i - ORDER) / (real)(ORDER + 1); |
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175 | fxn[i] = myfun(zeroes[i]); |
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176 | } |
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177 | |
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178 | /* We build a matrix of Chebishev evaluations: row i contains the |
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179 | * evaluations of x_i for polynomial order n = 0, 1, ... */ |
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180 | Matrix<ORDER + 1> mat; |
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181 | for (int i = 0; i < ORDER + 1; i++) |
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182 | { |
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183 | /* Compute the powers of x_i */ |
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184 | real powers[ORDER + 1]; |
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185 | powers[0] = 1.0; |
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186 | for (int n = 1; n < ORDER + 1; n++) |
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187 | powers[n] = powers[n - 1] * zeroes[i]; |
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188 | |
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189 | /* Compute the Chebishev evaluations at x_i */ |
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190 | for (int n = 0; n < ORDER + 1; n++) |
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191 | { |
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192 | real sum = 0.0; |
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193 | for (int k = 0; k < ORDER + 1; k++) |
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194 | if (cheby[n][k]) |
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195 | sum += (real)cheby[n][k] * powers[k]; |
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196 | mat.m[i][n] = sum; |
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197 | } |
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198 | } |
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199 | |
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200 | /* Solve the system */ |
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201 | mat = mat.inv(); |
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202 | |
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203 | /* Compute interpolation coefficients */ |
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204 | for (int j = 0; j < ORDER + 1; j++) |
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205 | { |
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206 | coeff[j] = 0; |
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207 | for (int i = 0; i < ORDER + 1; i++) |
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208 | coeff[j] += mat.m[j][i] * fxn[i]; |
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209 | } |
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210 | } |
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211 | |
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212 | static void remez_findzeroes(real *coeff, real *zeroes, real *control) |
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213 | { |
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214 | for (int i = 0; i < ORDER + 1; i++) |
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215 | { |
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216 | real a = control[i]; |
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217 | real ea = cheby_eval(coeff, a) - myfun(a); |
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218 | real b = control[i + 1]; |
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219 | real eb = cheby_eval(coeff, b) - myfun(b); |
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220 | |
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221 | while (fabs(a - b) > (real)1e-140) |
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222 | { |
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223 | real c = (a + b) * (real)0.5; |
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224 | real ec = cheby_eval(coeff, c) - myfun(c); |
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225 | |
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226 | if ((ea < (real)0 && ec < (real)0) |
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227 | || (ea > (real)0 && ec > (real)0)) |
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228 | { |
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229 | a = c; |
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230 | ea = ec; |
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231 | } |
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232 | else |
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233 | { |
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234 | b = c; |
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235 | eb = ec; |
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236 | } |
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237 | } |
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238 | |
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239 | zeroes[i] = a; |
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240 | } |
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241 | } |
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242 | |
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243 | static void remez_finderror(real *coeff, real *zeroes, real *control) |
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244 | { |
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245 | real final = 0; |
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246 | |
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247 | for (int i = 0; i < ORDER + 2; i++) |
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248 | { |
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249 | real a = -1, b = 1; |
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250 | if (i > 0) |
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251 | a = zeroes[i - 1]; |
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252 | if (i < ORDER + 1) |
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253 | b = zeroes[i]; |
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254 | |
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255 | printf("Error for [%g..%g]: ", (double)a, (double)b); |
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256 | for (;;) |
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257 | { |
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258 | real c = a, delta = (b - a) / (real)10.0; |
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259 | real maxerror = 0; |
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260 | int best = -1; |
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261 | for (int k = 0; k <= 10; k++) |
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262 | { |
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263 | real e = fabs(cheby_eval(coeff, c) - myfun(c)); |
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264 | if (e > maxerror) |
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265 | { |
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266 | maxerror = e; |
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267 | best = k; |
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268 | } |
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269 | c += delta; |
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270 | } |
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271 | |
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272 | if (best == 0) |
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273 | best = 1; |
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274 | if (best == 10) |
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275 | best = 9; |
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276 | |
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277 | b = a + (real)(best + 1) * delta; |
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278 | a = a + (real)(best - 1) * delta; |
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279 | |
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280 | if (b - a < (real)1e-15) |
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281 | { |
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282 | if (maxerror > final) |
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283 | final = maxerror; |
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284 | control[i] = (a + b) * (real)0.5; |
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285 | printf("%g (at %g)\n", (double)maxerror, (double)control[i]); |
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286 | break; |
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287 | } |
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288 | } |
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289 | } |
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290 | |
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291 | printf("Final error: %g\n", (double)final); |
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292 | } |
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293 | |
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294 | static void remez_step(real *coeff, real *control) |
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295 | { |
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296 | /* Pick up x_i where error will be 0 and compute f(x_i) */ |
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297 | real fxn[ORDER + 2]; |
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298 | for (int i = 0; i < ORDER + 2; i++) |
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299 | fxn[i] = myfun(control[i]); |
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300 | |
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301 | /* We build a matrix of Chebishev evaluations: row i contains the |
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302 | * evaluations of x_i for polynomial order n = 0, 1, ... */ |
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303 | Matrix<ORDER + 2> mat; |
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304 | for (int i = 0; i < ORDER + 2; i++) |
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305 | { |
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306 | /* Compute the powers of x_i */ |
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307 | real powers[ORDER + 1]; |
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308 | powers[0] = 1.0; |
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309 | for (int n = 1; n < ORDER + 1; n++) |
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310 | powers[n] = powers[n - 1] * control[i]; |
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311 | |
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312 | /* Compute the Chebishev evaluations at x_i */ |
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313 | for (int n = 0; n < ORDER + 1; n++) |
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314 | { |
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315 | real sum = 0.0; |
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316 | for (int k = 0; k < ORDER + 1; k++) |
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317 | if (cheby[n][k]) |
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318 | sum += (real)cheby[n][k] * powers[k]; |
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319 | mat.m[i][n] = sum; |
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320 | } |
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321 | if (i & 1) |
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322 | mat.m[i][ORDER + 1] = fabs(myerror(control[i])); |
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323 | else |
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324 | mat.m[i][ORDER + 1] = -fabs(myerror(control[i])); |
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325 | } |
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326 | |
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327 | /* Solve the system */ |
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328 | mat = mat.inv(); |
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329 | |
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330 | /* Compute interpolation coefficients */ |
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331 | for (int j = 0; j < ORDER + 1; j++) |
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332 | { |
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333 | coeff[j] = 0; |
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334 | for (int i = 0; i < ORDER + 2; i++) |
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335 | coeff[j] += mat.m[j][i] * fxn[i]; |
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336 | } |
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337 | |
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338 | /* Compute the error */ |
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339 | real error = 0; |
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340 | for (int i = 0; i < ORDER + 2; i++) |
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341 | error += mat.m[ORDER + 1][i] * fxn[i]; |
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342 | } |
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343 | |
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344 | int main(int argc, char **argv) |
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345 | { |
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346 | cheby_init(); |
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347 | |
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348 | /* ORDER + 1 chebyshev coefficients and 1 error value */ |
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349 | real coeff[ORDER + 2]; |
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350 | /* ORDER + 1 zeroes of the error function */ |
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351 | real zeroes[ORDER + 1]; |
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352 | /* ORDER + 2 control points */ |
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353 | real control[ORDER + 2]; |
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354 | |
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355 | real bn[ORDER + 1]; |
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356 | |
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357 | remez_init(coeff, zeroes); |
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358 | |
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359 | cheby_coeff(coeff, bn); |
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360 | for (int j = 0; j < ORDER + 1; j++) |
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361 | printf("%s%12.10gx^%i", j ? "+" : "", (double)bn[j], j); |
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362 | printf("\n"); |
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363 | |
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364 | for (int n = 0; n < 200; n++) |
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365 | { |
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366 | remez_finderror(coeff, zeroes, control); |
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367 | remez_step(coeff, control); |
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368 | |
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369 | cheby_coeff(coeff, bn); |
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370 | for (int j = 0; j < ORDER + 1; j++) |
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371 | printf("%s%12.10gx^%i", j ? "+" : "", (double)bn[j], j); |
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372 | printf("\n"); |
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373 | |
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374 | remez_findzeroes(coeff, zeroes, control); |
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375 | } |
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376 | |
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377 | remez_finderror(coeff, zeroes, control); |
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378 | remez_step(coeff, control); |
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379 | |
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380 | cheby_coeff(coeff, bn); |
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381 | for (int j = 0; j < ORDER + 1; j++) |
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382 | printf("%s%12.10gx^%i", j ? "+" : "", (double)bn[j], j); |
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383 | printf("\n"); |
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384 | |
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385 | return EXIT_SUCCESS; |
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386 | } |
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387 | |
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