前処理

ここでは共役勾配法に用いる前処理について説明する。共役勾配法では係数行列(式(B.5)のA) に対して事前に処理をすることで収束速度を向上させることができることがしられている。この操 作を前処理といい、前処理を用いた共役勾配法を一般的に前処理付き共役勾配法と言う。

B.3.1 対角スケーリング

これは本研究でも用いている手法であり、もっとも簡単な前処理である。次のような前処理行列 P を作用させ、係数行列Aの対角成分をすべて1にすることで収束速度が向上する。

P =







 1/√

a11

1/√a22

. .. 1/√

ann







 (B.13)

Ax=b P AP P⁻¹x=P b

A˜˜x=˜b (B.14)

ここでA˜=P AP、x˜=P⁻¹x、˜b=P bである。また、P⁻¹はP の対角成分の逆数をとった行 列である。これより、式(B.14)から共役勾配法によってx˜を計算し、x=Px˜から最終的な解を 求めることで、対角スケーリングによる前処理付き共役勾配法を計算できる。

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