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# Cojugate Gradient Method - PowerPoint PPT Presentation

2010 年教学实践周 7.12-7.16. Cojugate Gradient Method. Zhengru Zhang ( 张争茹 ) [email protected] Office: Math. Building 413(West). Outline. Aim Method of Gauss Elimination Basic Iterative Methods Conjugate Gradient Method Derivation Theory Algorithm References Homework & Project. Aim.

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2010年教学实践周7.12-7.16

Zhengru Zhang (张争茹)

Office: Math. Building 413(West)

• Aim

• Method of Gauss Elimination

• Basic Iterative Methods

• Derivation

• Theory

• Algorithm

• References

• Homework & Project

Solve linear algebraic system like

a11 x1 + a12 x2 + ... + a1n xn = b1

a21 x1 + a22 x2 + ... + a2n xn = b2

...

an1 x1 + an2 x2 + ... + ann xn = bn

Using matrix, the above system can be written as

Ax=b

A is a N x N matrix, b is a N x 1 vector

Consider the case: A is large and sparse

without Pivoting

U = A, L = I

for k = 1 to N-1

for j = k +1 to N

ljk= ujk/ukk

uj,k:m = uj,k:m – ljkuk,k:m

• LU Factorization, let A=LU

• Solve Ly=b

• Solve Ux=y

• Gauss Elimination and Back Substitution

• There are 3 loops

• There are 2 flops per entry

• For eachk, the inner loop is repeated for rows k +1, …, N

without Pivoting

Examples

A2=

A1=

• Pivoting

• Partial Pivoting

• Complete Pivoting

Remedy

with Partial Pivoting

U = A, L = I

For k = 1 to N-1

for j = k +1 to N

ljk= ujk/ukk

uj,k:m = uj,k:m – ljkuk,k:m

• How to construct iterative sequence?

• Convergence? Conditions?

• Convergence rate?

Gauss Seidel iteration

X[k+1] = D-1(L+U) X[k] + D-1 b

B = D-1(L+U)

X[k+1] = (D-L)-1 U X[k] + (D-L)-1 b

B = (D-L)-1 U

• Iterative method X[k+1] = BX[k]+g converges if and only if

• (B) < 1

• Convergence rate

• ||X[k]-X*||  ||X[1]-X[0]||, whereq =||B||<1

• Consider the case: A is symmetric positive definite

• (x)= xTAx - 2bTx

• The solution of Ax=b is equivalent to find the minimizer

• of the functional(x)

• Method of optimization: find a direction pk and a step k

Determine pkand k

• Suppose that pk is determined. Let’s start from xk

• Let f() =(xk +  pk)

• = (xk +  pk)TA(xk + pk)-2bT(xk + pk)

• = 2pkTApk - 2 rkTpk +(xk)

• whererk = b - Axk(Residual)

• By calculas f’() = 2pkTApk- 2rkTpk =0

• Then let xk+1 = xk + k pk

• Verify(xk+1) - (xk) =(xk +k pk) - (xk)

• = k2pkTApk - 2k rkTpk

• How to determine the directionpk ?

• take as the negative gradientpk = rk

Algorithm

Suppose the eigenvalues of A

then there holds

where

Derivation

• Negative gradient direction rk is the locally steepest

• decent direction, but it may not be the global one

• Consider a new direction: combination of rk and pk-1

• Initially, take p0 = r0 , x1 = x0 + 0p0

• For step k +1, choose  and  to minimize

• By calculas

0

Where  and  are obtained

in a simple form

• Operations involved:

• Transpose,

• Scalar Multiply,

• Matrix Multiply

Orthogonal

properties

• Theoretically, CG method is an exact method. Actually,

• works as an iterative method.

• Convergence rate:

• where

• 徐树方，高立，张平文，

• 数值线性代数，北京大学出版社，北京，2007

• 袁亚湘，孙文瑜，

• 最优化理论与方法，科学出版社，北京，2000

• Iterative Methods for Sparse Linear Systems, 2000

Due at the end of this week

Problem: Minimize the functional E(u)=∫(|u|2+u2-2fu )dx

The corresponding Euler-Lagrange equation is

E/u=-2u+2u-2f=0 or -u+u=f

• Solve the following linear systems using CG method

-u xx + u =f 0<x<1

f=(1+42)sin2x

u(0)=u(1)=0

where

• Set n = 100, 200, 300, 400, 500

• Use Matlab to graph the solution (j, uj)

Due at the end of this week

• Solve the following linear systems using CG method

• The unknowns can be ordered as below

Where

S Tridiagonal matrix with

diagonal entry:

other entry:

• Set n=20,40,80,100. Find the solution

• Use Matlab to graph the solution (i, j, uij)

-u+u=f (x,y)(0,1)(0,1)

u(x,y)=100(x2-x)(y2-y)

f=200(y-y2) + 200(x-x2) + 100(x2-x)(y2-y)