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Quantitative Finance · Begrippenlijst

Wat is Conjugate gradient method, preconditioner?

Ook bekend als: conjugate gradient method · preconditioner

Definition 25.9 Quantitative Methods · Hoofdstuk 25 — Floating Point and Numerical Linear Algebra

The conjugate gradient method (Hestenes and Stiefel, 1952) solves Ax=bAx = b for symmetric positive-definite AA by minimising 12x⊤Ax−b⊤x\frac12x^\top Ax - b^\top x along mutually AA-conjugate directions built from the residuals. A preconditioner M≈AM \approx A replaces the system by one with M−1AM^{-1}A, of smaller condition number.

Conjugate gradient on a badly scaled symmetric positive-definite system of dimension 400 (condition number 1.1 × 105), with and without a diagonal preconditioner: relative residual against the iteration (the preconditioned run stops at 102). Data: the chapter’s tutorial, seeded.
Figure 25.5. Conjugate gradient on a badly scaled symmetric positive-definite system of dimension 400 (condition number 1.1×1051.1 \times 10^5), with and without a diagonal preconditioner: relative residual against the iteration (the preconditioned run stops at 102). Data: the chapter’s tutorial, seeded.
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