# Conjugate Gradient

### Motivation

Solve large systems of linear equations

where

### Why symmetric and Why positive

In short, if

##### This linear system problem can be transfer to an optimization problem

- We have a quadratic function of vector

- Gradient of
is - If
is symmetric Therefore, is critical point of . ( ) - If
is positive-define, that is for every nonzero vector , , is an increasing function and can be solved by finding an that minimizes .

Note

Different Situations w.r.t. positive or not

![Pasted image 20221225214120.png](/img/user/attachment/Pasted image 20221225214120.png)

### CG -- The Method of Steepest Descent

### CG -- The Method of Conjugate Directions

### The Method of Conjugate Gradients

#### High level idea

In CG -- The Method of Conjugate Directions we can set

#### Reason

#### Compute \beta

We don't need to memory all previous vectors like CG -- Gram-Schmidt Conjugation#Difficulties.

### CG procedure

HPCG (Understand the preconditioned conjugate gradient method )