Feat: hw2 done

HW2_111550013/HW2_111550013_code/problem1.cpp

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+// Problem 1: Gaussian elimination with partial pivoting
+#include <iostream>
+#include <iomanip>
+#include <cmath>
+using namespace std;
+
+const int N = 4;
+
+int main() {
+    double A[N][N+1] = {
+        {4, 2, -2, -1, 7},
+        {0, 4, 1, 2, 10},
+        {3, -2, 1, 2, 2},
+        {2, 0, 3, -5, 3}
+    };
+
+    // Forward elimination with partial pivoting
+    for (int k = 0; k < N - 1; k++) {
+        // Find pivot
+        int maxRow = k;
+        for (int i = k + 1; i < N; i++)
+            if (fabs(A[i][k]) > fabs(A[maxRow][k])) maxRow = i;
+
+        if (maxRow != k) {
+            cout << "Column " << (k+1) << ": swap row " << (k+1) << " with row " << (maxRow+1) << "\n";
+            for (int j = 0; j <= N; j++) swap(A[k][j], A[maxRow][j]);
+        }
+
+        // Eliminate
+        for (int i = k + 1; i < N; i++) {
+            double m = A[i][k] / A[k][k];
+            for (int j = k; j <= N; j++) A[i][j] -= m * A[k][j];
+        }
+    }
+
+    // Back substitution
+    double x[N];
+    for (int i = N - 1; i >= 0; i--) {
+        x[i] = A[i][N];
+        for (int j = i + 1; j < N; j++) x[i] -= A[i][j] * x[j];
+        x[i] /= A[i][i];
+    }
+
+    cout << "\nSolution:\n";
+    for (int i = 0; i < N; i++)
+        cout << "x" << (i+1) << " = " << fixed << setprecision(6) << x[i] << "\n";
+
+    return 0;
+}

HW2_111550013/HW2_111550013_code/problem2.cpp

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+// Problem 2: Scaled partial pivoting
+#include <iostream>
+#include <iomanip>
+#include <cmath>
+#include <vector>
+using namespace std;
+
+double roundSig(double val, int sig) {
+    if (val == 0) return 0;
+    double d = ceil(log10(fabs(val)));
+    double p = pow(10, sig - d);
+    return round(val * p) / p;
+}
+
+void solve(vector<vector<double>> A, int n, int sig) {
+    cout << "\n" << sig << " significant digits:\n";
+
+    vector<double> s(n);
+    for (int i = 0; i < n; i++)
+        for (int j = 0; j < n; j++) s[i] = max(s[i], fabs(A[i][j]));
+
+    vector<int> idx(n);
+    for (int i = 0; i < n; i++) idx[i] = i;
+
+    for (int k = 0; k < n - 1; k++) {
+        int maxRow = k;
+        double maxRatio = fabs(A[idx[k]][k]) / s[idx[k]];
+        for (int i = k + 1; i < n; i++) {
+            double ratio = fabs(A[idx[i]][k]) / s[idx[i]];
+            if (ratio > maxRatio) { maxRatio = ratio; maxRow = i; }
+        }
+        swap(idx[k], idx[maxRow]);
+
+        for (int i = k + 1; i < n; i++) {
+            double m = roundSig(A[idx[i]][k] / A[idx[k]][k], sig);
+            for (int j = k; j <= n; j++)
+                A[idx[i]][j] = roundSig(A[idx[i]][j] - m * A[idx[k]][j], sig);
+        }
+    }
+
+    vector<double> x(n);
+    for (int i = n - 1; i >= 0; i--) {
+        x[i] = A[idx[i]][n];
+        for (int j = i + 1; j < n; j++)
+            x[i] = roundSig(x[i] - A[idx[i]][j] * x[j], sig);
+        x[i] = roundSig(x[i] / A[idx[i]][i], sig);
+    }
+
+    for (int i = 0; i < n; i++)
+        cout << "x" << (i+1) << " = " << fixed << setprecision(6) << x[i] << "\n";
+}
+
+int main() {
+    vector<vector<double>> A = {
+        {4.13, -2.20, 0.95, 3.02},
+        {6.14, 4.45, -1.45, -4.02},
+        {1.03, 1.86, 0.44, 5.22}
+    };
+    solve(A, 3, 6);
+    solve(A, 3, 3);
+    return 0;
+}

HW2_111550013/HW2_111550013_code/problem3.cpp

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+// Problem 3: Symmetric tridiagonal system
+#include <iostream>
+#include <iomanip>
+#include <vector>
+using namespace std;
+
+int main() {
+    int n = 6;
+    vector<double> d = {4, 4, 4, 4, 4, 4};      // diagonal
+    vector<double> a = {-1, -1, -1, -1, -1};    // off-diagonal
+    vector<double> b = {100, 200, 200, 200, 200, 100};
+
+    // Forward elimination
+    for (int i = 1; i < n; i++) {
+        double m = a[i-1] / d[i-1];
+        d[i] -= m * a[i-1];
+        b[i] -= m * b[i-1];
+    }
+
+    // Back substitution
+    vector<double> x(n);
+    x[n-1] = b[n-1] / d[n-1];
+    for (int i = n - 2; i >= 0; i--)
+        x[i] = (b[i] - a[i] * x[i+1]) / d[i];
+
+    cout << "Solution:\n";
+    for (int i = 0; i < n; i++)
+        cout << "x" << (i+1) << " = " << fixed << setprecision(6) << x[i] << "\n";
+
+    cout << "\nOperations: 7N - 6 = " << (7*n - 6) << " for N = " << n << "\n";
+    return 0;
+}

HW2_111550013/HW2_111550013_code/problem4.cpp

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+// Problem 4: Jacobi and Gauss-Seidel methods
+#include <iostream>
+#include <iomanip>
+#include <cmath>
+#include <vector>
+#include <algorithm>
+using namespace std;
+
+const double TOL = 1e-6;
+const int MAX_ITER = 100;
+
+int main() {
+    // Rearranged system (weakly diagonally dominant):
+    // 7x1 - 3x2 + 4x3 = 6
+    // 2x1 + 5x2 + 3x3 = -5
+    // -3x1 + 2x2 + 6x3 = 2
+
+    double b[3] = {6, -5, 2};
+
+    // Jacobi
+    cout << "Jacobi Method:\n";
+    vector<double> x = {0, 0, 0};
+    for (int iter = 1; iter <= MAX_ITER; iter++) {
+        vector<double> x_old = x;
+        x[0] = (b[0] + 3*x_old[1] - 4*x_old[2]) / 7;
+        x[1] = (b[1] - 2*x_old[0] - 3*x_old[2]) / 5;
+        x[2] = (b[2] + 3*x_old[0] - 2*x_old[1]) / 6;
+
+        double maxDiff = max({fabs(x[0]-x_old[0]), fabs(x[1]-x_old[1]), fabs(x[2]-x_old[2])});
+        if (maxDiff < TOL) {
+            cout << "Converged in " << iter << " iterations\n";
+            break;
+        }
+    }
+    for (int i = 0; i < 3; i++)
+        cout << "x" << (i+1) << " = " << fixed << setprecision(6) << x[i] << "\n";
+
+    // Gauss-Seidel
+    cout << "\nGauss-Seidel Method:\n";
+    x = {0, 0, 0};
+    for (int iter = 1; iter <= MAX_ITER; iter++) {
+        vector<double> x_old = x;
+        x[0] = (b[0] + 3*x[1] - 4*x[2]) / 7;
+        x[1] = (b[1] - 2*x[0] - 3*x[2]) / 5;
+        x[2] = (b[2] + 3*x[0] - 2*x[1]) / 6;
+
+        double maxDiff = max({fabs(x[0]-x_old[0]), fabs(x[1]-x_old[1]), fabs(x[2]-x_old[2])});
+        if (maxDiff < TOL) {
+            cout << "Converged in " << iter << " iterations\n";
+            break;
+        }
+    }
+    for (int i = 0; i < 3; i++)
+        cout << "x" << (i+1) << " = " << fixed << setprecision(6) << x[i] << "\n";
+
+    return 0;
+}