# 2.3 共轭斜量法 （ Conjugate Gradient Methods) 属于一种迭代法，但如果不考虑计算过程的舍入误 差， CG 算法只用有限步就收敛于方程组的精确解.

## Presentation on theme: "2.3 共轭斜量法 （ Conjugate Gradient Methods) 属于一种迭代法，但如果不考虑计算过程的舍入误 差， CG 算法只用有限步就收敛于方程组的精确解."— Presentation transcript:

2.3 共轭斜量法 （ Conjugate Gradient Methods) 属于一种迭代法，但如果不考虑计算过程的舍入误 差， CG 算法只用有限步就收敛于方程组的精确解

Outline  Background  Steepest Descent  Conjugate Gradient

1 Background The min(max) problem: But we learned in calculus how to solve that kind of question!

“real world” problem Connectivity shapes (isenburg,gumhold,gotsman) What do we get only from C without geometry?

Motivation- “real world” problem First we introduce error functionals and then try to minimize them:

Motivation- “real world” problem  Then we minimize:  High dimension non-linear problem.  Conjugate gradient method is maybe the most popular optimization technique based on what we’ll see here.

Directional Derivatives: first, the one dimension derivative:

Directional Derivatives : Along the Axes…

Directional Derivatives : In general direction…

Directional Derivatives

In the plane The Gradient: Definition in

2 最速下降法 （ Steepest Descent ） （ 1 ）概念：将 点的修正方向取为该点的负 梯度方向 ，即为最速下降 方向，该方法进而称之为最速下降法. （ 2 ）计算公式：任意取定初始向量，

Steepest Descent

3 共轭斜量法 （ Conjugate Gradient ）  Modern optimization methods : “conjugate direction” methods.  A method to solve quadratic function minimization: (A is symmetric and positive definite)

Conjugate Gradient Originally aimed to solve linear problems: Later extended to general functions under rational of quadratic approximation to a function is quite accurate.

Conjugate Gradient  The basic idea: decompose the n-dimensional quadratic problem into n problems of 1-dimension  This is done by exploring the function in “conjugate directions”.  Definition: A-conjugate vectors:

Conjugate Gradient If there is an A-conjugate basis then: N problems in 1-dimension (simple smiling quadratic) The global minimizer is calculated sequentially starting from x 0 :