7.6 课后习题

练习7.1 证明引理7.1，即

(提示: 用归纳法证明 $v_{k} \in \mathcal{K}_{k}, k = 1,2,\ldots,m$ 即可)

练习7.2 证明定理7.3, 即由Lanczos过程生成的向量是相互正交的.

练习7.3 证明定理7.5中的必要性

练习7.4 设 $H_{m+1,m}$ 不可约, 试证明 (7.8) 中的 $R_m$ 非奇异

练习7.5 设 $A \in \mathbb{R}^{n \times n}$ 对称正定, 非零向量 $p_1, p_2, \ldots, p_m$ 是 $A$ -共轭的, 即当 $i \neq j$ 时有 $p_i^\top A p_j = 0$ . 试证明: $p_1, p_2, \ldots, p_m$ 线性无关.

练习7.6 用右预处理方式推导PCG算法

练习7.7 设 $A \in \mathbb{R}^{n \times n}$ 可对角化且特征值都是实数, 试证明: 存在某内积 $(\cdot, \cdot)$ , 使得在此内积下 $A$ 是自伴随的.

以下为实践题

练习7.8 编程实现GMRES方法

练习7.9 编程实现CG算法

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