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高斯算法（Gaussian elimination）是一种常用的数值方法，用于求解线性方程组。以下是使用Objective-C实现高斯算法的详细步骤和示例代码。
高斯算法通过消元的方法将一个大型线性方程组分解为更小的子问题，逐步求解。其核心步骤包括：
导入必要的头文件
首先，确保在代码中导入了必要的头文件：
#import#import
创建矩阵和向量
创建一个系数矩阵和常数向量：
double A[3][3] = { {2.0, 3.0, 1.0}, {4.0, 2.0, 3.0}, {6.0, 1.0, 2.0}};double b[3] = {10.0, 16.0, 13.0};实现高斯消元
实现高斯消元的主要步骤：
- (void)gaussianElimination { // 选择主元列 int n = 3; int maxRow = 0; double maxElem = 0; for (int i = 0; i < n; i++) { for (int row = 0; row < n; row++) { if (A[row][i] > maxElem) { maxElem = A[row][i]; maxRow = row; } } if (maxRow != 0) { // 行交换 id temp = A[0]; A[0] = A[maxRow]; A[maxRow] = temp; // 重新计算maxRow for (int col = 0; col < n; col++) { temp = A[0][col]; A[0][col] = A[maxRow][col]; A[maxRow][col] = temp; } } } // 消去主元以外的列 for (int col = 0; col < n; col++) { if (A[0][col] == 0) { // 无解 return; } for (int row = 1; row < n; row++) { if (A[row][col] != 0) { double factor = A[row][col] / A[0][col]; for (int c = 0; c < n; c++) { A[row][c] -= factor * A[0][c]; } } } } // 回代 for (int row = n-1; row >= 0; row--) { double x = b[row]; for (int col = 0; col < n; col++) { if (col != row && A[row][col] != 0) { x -= A[row][col] * b[col]; } } b[row] = x; }}求解未知数
通过高斯消元后的矩阵，可以直接读取未知数的值：
double x1 = b[0];double x2 = b[1];double x3 = b[2];NSLog(@"解为：x1 = %.6f, x2 = %.6f, x3 = %.6f", x1, x2, x3);
#import#import @interface GaussianSolver : NSObject { double *A; double *b; int n;}- (void)initializeWithMatrix:(double **)matrix andConstants:(double **)constants;- (void)gaussianElimination;- (void)backSubstitution;- (void)printSolution;@end@implementation GaussianSolver- (void)initializeWithMatrix:(double **)matrix andConstants:(double **)constants { self->A = matrix; self->b = constants; self->n = 3;}- (void)gaussianElimination { int maxRow; double maxElement; // 选择主元列 for (int col = 0; col < self->n; col++) { maxElement = 0.0; maxRow = -1; for (int row = 0; row < self->n; row++) { if (self->A[row][col] != 0) { if (self->A[row][col] > maxElement) { maxElement = self->A[row][col]; maxRow = row; } } } if (maxRow == -1) { // 无解 return; } // 行交换 if (maxRow != 0) { id temp = self->A[0]; self->A[0] = self->A[maxRow]; self->A[maxRow] = temp; // 重新计算 for (int c = 0; c < self->n; c++) { temp = self->A[0][c]; self->A[0][c] = self->A[maxRow][c]; self->A[maxRow][c] = temp; } } // 消去其他行 for (int row = 0; row < self->n; row++) { if (row != 0 && self->A[row][col] != 0) { double factor = self->A[row][col] / self->A[0][col]; for (int c = 0; c < self->n; c++) { self->A[row][c] -= factor * self->A[0][c]; } } } } // 回代 for (int row = self->n - 1; row >= 0; row--) { double x = self->b[row]; for (int col = 0; col < self->n; col++) { if (col != row && self->A[row][col] != 0) { x -= self->A[row][col] * self->b[col]; } } self->b[row] = x; }}- (void)printSolution { NSLog(@"解为：x1 = %.6f, x2 = %.6f, x3 = %.6f", self->b[0], self->b[1], self->b[2]);}// 创建单例static GaussianSolver *solver = nil;+ (GaussianSolver *)solver { if (solver == nil) { solver = [[GaussianSolver alloc] init]; } return solver;}// 使用示例int main() { double A[3][3] = { {2.0, 3.0, 1.0}, {4.0, 2.0, 3.0}, {6.0, 1.0, 2.0} }; double b[3] = {10.0, 16.0, 13.0}; GaussianSolver *solver = [GaussianSolver solver]; [solver initializeWithMatrix:A andConstants:b]; [solver gaussianElimination]; [solver printSolution]; return 0;}
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