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Definition: Conjugate Direction Methods

The conjugate direction methods can be viewed as being intermediate between the method of steepest descent and Newton’s method. The conjugate direction methods have the following properties:

  1. Solve quadratics of $n$ variables in $n$ steps.

  2. The usual implementation, the conjugate gradient algorithm, requires no Hessian matrix evaluations.

  3. No matrix inversion and no storage of an $n \times n $ matrix required.

In this chapter, we will talk about two conjugate direction methods: the conjugate direction algorithm and the conjugate gradient algorithm.

Definition: Q-Conjugate

Let $\b{Q} $ be a real symmetric $n \times n$ matrix. The directions $\b{d}^{(0)}, \b{d}^{(1)}, \b{d}^{(2)}, …, \b{d}^{(m)} $ are $Q$-conjugate if, for all $i != j $, we have $\transpose{ d^{(i)}} \b{Q} \b{d}^{(j)} = 0 $.

Lemma

Let $\b{Q} $ be a symmetric positive definite $n \times n $ matrix. If the directions $\b{d}^{0}, \b{d}^{(1)}, …, \b{d}^{(k)} \in \R^n, k \leq n - 1$, are nonzero and $Q$-conjugate, then they are linearly independent.

Proof 
  </span>
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<span class="proof__expand"><a>[expand]</a></span>

Let $\alpha_0, … \alpha_k $ be scalars such that

$$\alpha_0 \b{d}^{(0)} + \alpha_1 \b{d}^{(1)} + … + \alpha_k \b{d}^{k} = \b{0}$$

Premultiplying the above equality by $\transpose{ \b{d}^{(j)}} \b{Q}, 0 \leq j \leq k $, yields

$$\alpha_j \transpose{ \b{d}^{(j)}} \b{Q} \b{d}^{(j)} = 0$$

Because all other terms $\transpose{ \b{d}^{(j)}} \b{Q} \b{d}^{(i)} = 0, i \neq j$, by $Q$-conjugacy. But $\b{Q} = \transpose{ \b{Q}} > 0 $ and $\b{d}^{(j)} \neq \b{0} $; hence $\alpha_j = 0, j = 0, 1, …, k$. Therefore, $\b{d}^{(0)}, \b{d}^{(1)}, …, \b{d}^{(k)}, k \leq n -1$, are linearly independent.