1 #include2 /************************************************* 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 else114 {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 else153 {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 else168 {169 return 0;170 }171 }172 else173 {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 else192 {193 return 0;194 }195 }196 }197 #undef EPS198 #undef ZERO199 }200 201 202 /*************************************************203 Function: solve_quartic_equation204 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x207 Return: 实数根的个数208 Author: 枫箫209 Version: 1.0210 Date: 2017.7.8211 Others: 如有疑问、意见或建议, 请多多交流, 多多指教!邮箱:gxb31415926@163.com212 *************************************************/213 int solve_quartic_equation(float p[], float x[])214 {215 #define EPS 1.0e-30216 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 else240 {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 else249 {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 else297 {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 else306 {307 return 0;308 }309 }310 else311 {312 return 0;313 }314 }315 #undef EPS316 }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 = 0327 //0*x^2 - 10*x + 1 = 0328 //0 * x ^ 2 - 10 * x + 1 = 0329 //x^2 - 10000000*x + 0.01 = 0330 //1.0e-20*x^2 - 2.0e-20*x + 1.0e-20 = 0331 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 = 0349 //0*x^3 + 0*x^2 - 2*x + 1 = 0350 //0*x^3 + 0*x^2 + 0*x + 1 = 0351 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 = 0369 //0*x^4 + 0*x^3 + x^2 - 2*x + 1 = 0370 //0*x^4 + 0*x^3 + 0*x^2 - 2*x + 1 = 0371 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 }