UF_MTX_to_tridiag(A, n) 函数和UF_MTX_tqli(T, w, v, n)函数和UF_MTX_from_tri_eig(A, T, w, v, n)分别实现这三个函数

1. UF_MTX_to_tridiag(A, n)函数：将一个$n\times n$的对称矩阵$A$转化为对称三对角矩阵$T$。代码如下：

void UF_MTX_to_tridiag(double **A, int n, double **T) {
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            T[i][j] = A[i][j];
        }
    }
    for (int k = 0; k < n - 2; k++) {
        double *x = new double[n - k];
        double *v = new double[n - k];
        for (int i = k + 1; i < n; i++) {
            x[i - k - 1] = T[i][k];
        }
        double alpha = 0.0;
        for (int i = 0; i < n - k - 1; i++) {
            alpha += x[i] * x[i];
        }
        alpha = sqrt(alpha);
        if (x[0] > 0) {
            alpha = -alpha;
        }
        double beta = 1.0 / (alpha * (alpha - x[0]));
        T[k + 1][k] = -alpha;
        for (int i = k + 2; i < n; i++) {
            T[i][k] = 0.0;
        }
        for (int j = k + 1; j < n; j++) {
            double sum = 0.0;
            for (int i = k + 1; i < n; i++) {
                sum += T[i][j] * x[i - k - 1];
            }
            double tau = beta * sum;
            for (int i = k + 1; i < n; i++) {
                T[i][j] -= tau * x[i - k - 1];
            }
        }
        delete[] x;
        delete[] v;
    }
}

2. UF_MTX_tqli(T, w, v, n)函数：对对称三对角矩阵$T$进行求解，得到其特征值和特征向量。代码如下：

void UF_MTX_tqli(double **T, double *w, double **v, int n) {
    for (int i = 0; i < n; i++) {
        v[i][i] = 1.0;
    }
    for (int i = 1; i < n; i++) {
        w[i - 1] = T[i - 1][i];
        T[i - 1][i - 1] = T[i][i - 1] = 0.0;
        double alpha = T[i][i];
        double beta = sqrt(alpha * alpha + w[i - 1] * w[i - 1]);
        if (beta == 0.0) {
            continue;
        }
        if (alpha < 0.0) {
            beta = -beta;
        }
        w[i - 1] = (w[i - 1] / alpha) / beta;
        T[i][i - 1] = -beta;
        T[i][i] = alpha / beta;
        for (int j = i + 1; j < n; j++) {
            double sum = 0.0;
            for (int k = i; k < n; k++) {
                sum += T[k][i - 1] * T[k][j];
            }
            double tau = sum / T[i][i - 1];
            for (int k = i; k < n; k++) {
                T[k][j] -= tau * T[k][i - 1];
            }
        }
    }
    w[n - 1] = T[n - 1][n - 1];
    T[n - 1][n - 1] = 0.0;
    for (int i = 0; i < n; i++) {
        double sum = 0.0;
        for (int j = i; j < n; j++) {
            sum += T[i][j] * v[j][n - 1];
        }
        double tau = sum / w[n - 1];
        for (int j = i; j < n; j++) {
            v[j][i] = v[j][n - 1] - tau * T[i][j];
        }
    }
}

3. UF_MTX_from_tri_eig(A, T, w, v, n)函数：利用特征值和特征向量构造出对称矩阵$A$。代码如下：

void UF_MTX_from_tri_eig(double **A, double **T, double *w, double **v, int n) {
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            A[i][j] = 0.0;
            for (int k = 0; k < n; k++) {
                A[i][j] += v[i][k] * w[k] * v[j][k];
            }
        }
    }
}