1 #include <math.h> 2 /************************************************* 3 Function: solve_quadratic_equation 4 Description: 求一元二次方程(a*x^2 + b*x + c = 0)的所有实数根 5 Input: 方程的系数 p = {c, b, a} 6 Output: 方程的所有实数根x 7 Return: 实数根的个数 8 Author: 枫箫 9 Version: 1.0 10 Date: 2017.7.8 11 Others: 如有疑问、意见或建议, 请多多交流, 多多指教!邮箱:gxb31415926@163.com 12 *************************************************/ 13 int solve_quadratic_equation(float p[], float x[]) 14 { 15 #define EPS 1.0e-30 16 #define ZERO 1.0e-30 17 float a, b, c, delta, sqrtDelta; 18 19 a = p[2]; 20 b = p[1]; 21 c = p[0]; 22 23 if (fabs(a - 0.0) < EPS) 24 { 25 if (fabs(b - 0.0) < EPS) 26 { 27 return 0; 28 } 29 else 30 { 31 x[0] = -c / b; 32 return 1; 33 } 34 } 35 else 36 { 37 delta = b * b - 4.0 * a * c; 38 if (delta > ZERO) 39 { 40 if (fabs(c - 0.0) < EPS) //若c = 0,由于计算误差,sqrt(b*b - 4*a*c）不等于b 41 { 42 x[0] = 0.0; 43 x[1] = -b / a; 44 } 45 else 46 { 47 sqrtDelta = sqrt(delta); 48 if (b > 0.0) 49 { 50 x[0] = (-2.0 * c) / (b + sqrtDelta); //避免两个很接近的数相减,导致精度丢失 51 x[1] = (-b - sqrtDelta) / (2.0 * a); 52 } 53 else 54 { 55 x[0] = (-b + sqrtDelta) / (2.0 * a); 56 x[1] = (-2.0 * c) / (b - sqrtDelta); //避免两个很接近的数相减,导致精度丢失 57 } 58 } 59 return 2; 60 } 61 else if (fabs(delta - 0.0) < EPS) 62 { 63 x[0] = x[1] = -b / (2.0 * a); 64 } 65 else 66 { 67 return 0; 68 } 69 } 70 #undef EPS 71 #undef ZERO 72 } 73 74 75 /************************************************* 76 Function: solve_cubic_equation 77 Description: 盛金公式求一元三次方程(a*x^3 + b*x^2 + c*x + d = 0)的所有实数根 78 A = b * b - 3.0 * a * c; 79 B = b * c - 9.0 * a * d; 80 C = c * c - 3.0 * b * d; 81 (1)当A = B = 0时，方程有一个三重实根 82 (2)当Δ = B^2－4AC>0时，方程有一个实根和一对共轭虚根 83 (3)当Δ = B^2－4AC = 0时，方程有三个实根，其中有一个两重根 84 (4)当Δ = B^2－4AC<0时，方程有三个不相等的实根 85 Input: 方程的系数 p = {d, c, b, a} 86 Output: 方程的所有实数根x 87 Return: 实数根的个数 88 Author: 枫箫 89 Version: 1.0 90 Date: 2017.7.8 91 Others: 如有疑问、意见或建议, 请多多交流, 多多指教!邮箱:gxb31415926@163.com 92 *************************************************/ 93 int solve_cubic_equation(float p[], float x[]) 94 { 95 #define EPS 1.0e-30 96 #define ZERO 1.0e-30 97 float a, b, c, d, A, B, C, delta; 98 float Y1, Y2, Z1, Z2, K, parm[3], roots[2], theta, T; 99 100 a = p[3] ; 101 b = p[2] ; 102 c = p[1] ; 103 d = p[0] ; 104 105 if (fabs(a - 0.0) < EPS) 106 { 107 parm[2] = b; 108 parm[1] = c; 109 parm[0] = d; 110 111 return solve_quadratic_equation(parm, x); 112 } 113 else 114 { 115 A = b * b - 3.0 * a * c; 116 B = b * c - 9.0 * a * d; 117 C = c * c - 3.0 * b * d; 118 119 delta = B * B - 4.0 * A * C; 120 121 if (fabs(A - 0.0) < EPS && fabs(B - 0.0) < EPS) 122 { 123 x[0] = x[1] = x[2] = -b / (3.0 * a); 124 return 3; 125 } 126 127 if (delta > ZERO) 128 { 129 parm[2] = 1.0; 130 parm[1] = B; 131 parm[0] = A * C; 132 133 solve_quadratic_equation(parm, roots); 134 Z1 = roots[0]; 135 Z2 = roots[1]; 136 137 Y1 = A * b + 3.0 * a * Z1; 138 Y2 = A * b + 3.0 * a * Z2; 139 140 if (Y1 < 0.0 && Y2 < 0.0) //pow函数的底数必须为非负数,必须分类讨论 141 { 142 x[0] = (-b + pow(-Y1, 1.0 / 3.0) + pow(-Y2, 1.0 / 3.0)) / (3.0*a); 143 } 144 else if (Y1 < 0.0 && Y2 > 0.0) 145 { 146 x[0] = (-b + pow(-Y1, 1.0 / 3.0) - pow(Y2, 1.0 / 3.0)) / (3.0*a); 147 } 148 else if (Y1 > 0.0 && Y2 < 0.0) 149 { 150 x[0] = (-b - pow(Y1, 1.0 / 3.0) + pow(-Y2, 1.0 / 3.0)) / (3.0*a); 151 } 152 else 153 { 154 x[0] = (-b - pow(Y1, 1.0 / 3.0) - pow(Y2, 1.0 / 3.0)) / (3.0*a); 155 } 156 return 1; 157 } 158 else if (fabs(delta - 0.0) < EPS) 159 { 160 if (fabs(A - 0.0) > EPS) 161 { 162 K = B / A; 163 x[0] = -b / a + K; 164 x[1] = x[2] = -0.5 * K; 165 return 3; 166 } 167 else 168 { 169 return 0; 170 } 171 } 172 else 173 { 174 if (A > 0.0) 175 { 176 T = (2.0 * A * b - 3.0 * a * B) / (2.0 * pow(A, 3.0 / 2.0)); 177 if (T > 1.0) //由于计算误差,T的值可能略大于1(如1.0000001) 178 { 179 T = 1.0; 180 } 181 if (T < -1.0) 182 { 183 T = -1.0; 184 } 185 theta = acos(T); 186 x[0] = (-b - 2.0 * sqrt(A) * cos(theta / 3.0)) / (3.0 * a); 187 x[1] = (-b + sqrt(A) * (cos(theta / 3.0) + sqrt(3.0) * sin(theta / 3.0))) / (3.0 * a); 188 x[2] = (-b + sqrt(A) * (cos(theta / 3.0) - sqrt(3.0) * sin(theta / 3.0))) / (3.0 * a); 189 return 3; 190 } 191 else 192 { 193 return 0; 194 } 195 } 196 } 197 #undef EPS 198 #undef ZERO 199 } 200 201 202 /************************************************* 203 Function: solve_quartic_equation 204 Description: 费拉里法求一元四次方程(a*x^4 + b*x^3 + c*x^2 + d*x + e = 0)的所有实数根 205 Input: 方程的系数 p = {e, d, c, b, a} 206 Output: 方程的所有实数根x 207 Return: 实数根的个数 208 Author: 枫箫 209 Version: 1.0 210 Date: 2017.7.8 211 Others: 如有疑问、意见或建议, 请多多交流, 多多指教!邮箱:gxb31415926@163.com 212 *************************************************/ 213 int solve_quartic_equation(float p[], float x[]) 214 { 215 #define EPS 1.0e-30 216 217 float a, b, c, d, e; 218 float parm[4], roots[3]; 219 float y, M, N; 220 float x1[2], x2[2]; 221 int rootCount1, rootCount2, rootCount, i; 222 float MSquareTemp, MSquare, yTemp; 223 224 a = p[4]; 225 b = p[3]; 226 c = p[2]; 227 d = p[1]; 228 e = p[0]; 229 230 if (fabs(a - 0.0) < EPS) 231 { 232 if (fabs(b - 0.0) < EPS) 233 { 234 parm[2] = c; 235 parm[1] = d; 236 parm[0] = e; 237 return solve_quadratic_equation(parm, x); 238 } 239 else 240 { 241 parm[3] = b; 242 parm[2] = c; 243 parm[1] = d; 244 parm[0] = e; 245 return solve_cubic_equation(parm, x); 246 } 247 } 248 else 249 { 250 b = b / a; 251 c = c / a; 252 d = d / a; 253 e = e / a; 254 255 parm[3] = 8.0; 256 parm[2] = -4.0 * c; 257 parm[1] = 2.0 * (b * d - 4.0 * e); 258 parm[0] = -e * (b * b - 4.0 * c) - d * d; 259 260 if (rootCount = solve_cubic_equation(parm, roots)) 261 { 262 y = roots[0]; 263 MSquare = 8.0 * y + b * b - 4.0 * c; 264 for (i = 1; i < rootCount; i++) 265 { 266 yTemp = roots[i]; 267 MSquareTemp = 8.0 * yTemp + b * b - 4.0 * c; 268 if (MSquareTemp > MSquare) 269 { 270 MSquare = MSquareTemp; 271 y = yTemp; 272 } 273 } 274 275 if (MSquare > 0.0) 276 { 277 M = sqrt(MSquare); 278 N = b * y - d; 279 parm[2] = 2.0; 280 parm[1] = b + M; 281 parm[0] = 2.0 * (y + N / M); 282 rootCount1 = solve_quadratic_equation(parm, x1); 283 284 parm[2] = 2.0; 285 parm[1] = b - M; 286 parm[0] = 2.0 * (y - N / M); 287 rootCount2 = solve_quadratic_equation(parm, x2); 288 289 if (rootCount1 == 2) 290 { 291 x[0] = x1[0]; 292 x[1] = x1[1]; 293 x[2] = x2[0]; 294 x[3] = x2[1]; 295 } 296 else 297 { 298 x[0] = x2[0]; 299 x[1] = x2[1]; 300 x[2] = x1[0]; 301 x[3] = x1[1]; 302 } 303 return rootCount1 + rootCount2; 304 } 305 else 306 { 307 return 0; 308 } 309 } 310 else 311 { 312 return 0; 313 } 314 } 315 #undef EPS 316 } 317 318 319 void main() 320 { 321 float x[2], xx[3], xxx[4], p[5]; 322 int rootCount; 323 float breakPointHere, x1, x2, a, b, c, d, e; 324 325 //(1)一元二次方程测试 326 //x^2 - 1000000.000001*x + 1 = 0 327 //0*x^2 - 10*x + 1 = 0 328 //0 * x ^ 2 - 10 * x + 1 = 0 329 //x^2 - 10000000*x + 0.01 = 0 330 //1.0e-20*x^2 - 2.0e-20*x + 1.0e-20 = 0 331 332 a = 1; 333 b = -1000000.000001; 334 c = 1; 335 p[0] = c; 336 p[1] = b; 337 p[2] = a; 338 rootCount = solve_quadratic_equation(p, x); 339 x1 = (-b + sqrt(b * b - 4.0 * a * c)) / (2.0 * a); 340 x2 = (-b - sqrt(b * b - 4.0 * a * c)) / (2.0 * a); 341 breakPointHere = 1.0; 342 343 //(2)一元三次方程测试 344 //(x-1)*(x^2+1)=0 (x^3 - x^2 + x - 1 = 0) 345 //(x-1)^3 = 0 (x^3 - 3*x^2 + 3*x - 1 = 0) 346 //(x-1)^2*(x-2)=0 (x^3 - 4*x^2 + 5*x - 2 = 0) 347 //(x-1)*(x-2)*(x-3) = 0 (x^3 - 6*x^2 + 11*x - 6 = 0) 348 //0*x^3 + x^2 - 2*x + 1 = 0 349 //0*x^3 + 0*x^2 - 2*x + 1 = 0 350 //0*x^3 + 0*x^2 + 0*x + 1 = 0 351 352 a = 1; 353 b = -6; 354 c = 11; 355 d = -6; 356 p[0] = d; 357 p[1] = c; 358 p[2] = b; 359 p[3] = a; 360 rootCount = solve_cubic_equation(p, xx); 361 breakPointHere = 1.0; 362 363 //(3)一元四次方程测试 364 //(x-1)*(x-2)*(x^2 + 1)=0 (x^4 - 3*x^3 + 3*x^2 - 3*x + 2 = 0) 365 //(x-1)^2*(x^2 + 1)=0 (x^4 - 2*x^3 + 2*x^2 - 2*x + 1 = 0) 366 //(x-1)*(x-2)*(x-3)*(x-4)=0 (x^4 - 10*x^3 + 35*x^2 - 50*x + 24 = 0) 367 //(x-1)^2*(x-2)^2 = 0 (x^4 - 6*x^3 + 13*x^2 - 12*x + 4 = 0) 368 //0*x^4 + x^3 - 3*x^2 + 3*x - 1 = 0 369 //0*x^4 + 0*x^3 + x^2 - 2*x + 1 = 0 370 //0*x^4 + 0*x^3 + 0*x^2 - 2*x + 1 = 0 371 a = 1; 372 b = -10; 373 c = 35; 374 d = -50; 375 e = 24; 376 p[0] = e; 377 p[1] = d; 378 p[2] = c; 379 p[3] = b; 380 p[4] = a; 381 rootCount = solve_quartic_equation(p, xxx); 382 breakPointHere = 1.0; 383 }
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