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Conjugate gradient method

For a positive-definite matrix

A

, the conjugate gradient method chooses mutually

A

-conjugate search directions. Starting with

r_{0} = b - A x_{0}

and

p_{0} = r_{0}

, it uses

α_{k} = \frac{r _{k}^{T} r _{k}}{p _{k}^{T} A p _{k}}, x_{k + 1} = x_{k} + α_{k} p_{k}, r_{k + 1} = r_{k} - α_{k} A p_{k},

and

p_{k + 1} = r_{k + 1} + β_{k} p_{k}

with

β_{k} = r_{k + 1}^{T} r_{k + 1} / (r_{k}^{T} r_{k})

.

Table of contents 75 2
- Krylov subspace Conjugate gradient method 13
- Finite termination of the conjugate gradient method Conjugate gradient method 32

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