# Conjugate gradient method - direct method

For solving $Ax=b$, where $A$ is a square symmetric positive definite matrix.

## Assumptions:

$A\in M_{n\times n}$ is a symmetric positive definite matrix.

## Definition:

• A-orthogonal (A-conjugate) 假設有兩個向量 $u_1$ 跟 $u_2$ 皆非 $0$ 且 $u_1 \neq u_2$，若這兩個向量滿足 $$\langle{u_1},A{u_2}\rangle = {u_1}^TA{u_2} = 0,$$ 則稱之為 A-orthogonal (或 A-conjugate).
• A-orthonormal 假設有兩個向量 $u_1$ 跟 $u_2$ 為 A-orthogonal, 並且 $\langle{u_i}, {u_i}\rangle_A = 1$ for $1\le i\le 2$, 則稱之為 A-orthonormal.

Recall : ${u_1}$ and ${u_2}$ are orthogonal if ${u_1}^T{u_2} = 0$.

Note : We can define $$\langle{u_1}, {u_2}\rangle_A = \langle{u_1},A{u_2}\rangle= \langle A{u_1},{u_2}\rangle,$$ then $\langle \cdot\rangle_A$ is an inner product.

### Lemma

#### pf:

Note : 如果 ${u_0, \cdots, u_{n-1}}\subset\mathbb{R}^n$ 是個 A-orthogonal set, 則他們也是 $\mathbb{R}^n$ 的一組 basis.

## CG as a direct method

### Theorem:

#### pf:

$$Ax = \sum^{n-1}_{i=0} c_i Au_i.$$

##### Te-Sheng Lin (林得勝)
###### Professor

The focus of my research is concerned with the development of analytical and computational tools, and further to communicate with scientists from other disciplines to solve engineering problems in practice.