BICG¶

约 160 个字预计阅读时间 1 分钟

Initialization:

Choose an initial guess \(x_0\).
Compute the initial residual \(r_0=b−Ax_0\).
Set the initial shadow residual \(r'_0=r_0\)(or another suitable initial residual vector).
Initialize the search directions \(p_0=r_0\)and \(p'_0=r'_0\).

Iteration: For \(k = 0, 1, 2, \ldots\) until convergence:

Compute Matrix-Vector Products:
- \(v = A*p_k\)
- \(A^T*p'_k\)
Compute the Alpha Coefficient:
- \(\alpha_k = \frac{r_k^T r_k'}{(A p_k)^T p_k'}\)
Update the Solution:
- \(x_{k+1} = x_k + \alpha_k p_k\)
Update the Residuals:
- \(r_{k+1} = r_k - \alpha_k A p_k\)
- \(r_{k+1}' = r_k' - \alpha_k (A^T p_k')\)
Check for Convergence (e.g., if \(\|r_{k+1}\|\) is sufficiently small, stop the iteration).
Compute the Beta Coefficient:
- \(\beta_k = \frac{r_{k+1}^T r_{k+1}'}{r_k^T r_k'}\)
Update the Search Directions:
- \(p_{k+1} = r_{k+1} + \beta_kp_k\)
- \(p_{k+1}' = r_{k+1}' + \beta_kp'_k \)

Output: