biconjugate gradient method について

Words near each other

Dictionary Lists

mini英和辞書

翻訳と辞書　辞書検索 [ 開発暫定版 ]

スポンサードリンク

biconjugate gradient method ：ウィキペディア英語版

biconjugate gradient method

In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations
:

A x= b.\,

Unlike the conjugate gradient method, this algorithm does not require the matrix

A

to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose .
==The algorithm==

# Choose initial guess

x_0\,

, two other vectors

x_0^*

and

b^*\,

M\,

r_0 \leftarrow b-A\, x_0\,

r_0^* \leftarrow b^*-x_0^*\, A^T

p_0 \leftarrow M^ r_0\,

p_0^* \leftarrow r_0^*M^\,

# for

k=0, 1, \ldots

do
##

\alpha_k \leftarrow \,

x_ \leftarrow x_k + \alpha_k \cdot p_k\,

x_^* \leftarrow x_k^* + \overline\cdot p_k^*\,

r_ \leftarrow r_k - \alpha_k \cdot A p_k\,

r_^* \leftarrow r_k^*- \overline \cdot p_k^*\, A

\beta_k \leftarrow  r_ \over r_k^* M^ r_k}\,

p_ \leftarrow M^ r_ + \beta_k \cdot p_k\,

p_^* \leftarrow r_^*M^  + \overline\cdot p_k^*\,

In the above formulation, the computed

r_k\,

and

r_k^*

satisfy
:

r_k = b - A x_k,\,

r_k^* = b^* - x_k^*\, A

and thus are the respective residuals corresponding to

x_k\,

and

x_k^*

, as approximate solutions to the systems
:

A x = b,\,

x^*\, A = b^*\,;

x^*

is the adjoint, and

\overline

翻訳と辞書 : 翻訳のためのインターネットリソース