School

University of WaterlooDepartment

Mechanical EngineeringCourse Code

ME303Professor

Serhiy Yarusevych Convergence of the direct iteration method

The )(xgx = rearrangement that we choose can have a strong effect on convergence of the

iteration. We can show graphically that the iteration will converge only if the function )(xg on

the right hand side satisfies 1<

dx

dg near the root.

(i) 1<

dx

dg

Calculations:

1) guess x1

2) compute g(x1)

3) x2= g(x1), i.e., the length corresponding to x2 equals to the length given by g(x1), as shown on

the sketch above. Thus, the point P(x=xnew, y=g(xold) ) falls on the line y=x. In this case,

repeating the process moves P along y=x towards the root and the iterations eventually converge,

as shown on the sketch.

(ii) 1≥

dx

dg

Calculations here are the same as above. In this case, however, repeating the process moves P

along y=x away from the root and the iterations do not converge on the root, as shown in the

sketch.

The Rule: always choose an )(xgx

=

form corresponding to 1<

dx

dg near the root.

x2

length=x2

x

y

y=g(x)

P

root

0

g(x1)

y=x (slope=1)

x1 x

y

root

0

x1

y=g(x)

y=x

length=x2

x

y

x1

y=g(x)

x2

P

root

0

g(x1)

y=x (slope=1) y y=x

root

x1

y=g(x)

**Unlock Document**

###### Document Summary

Convergence of the direct iteration method x = We can show graphically that the iteration will converge only if the function rearrangement that we choose can have a strong effect on convergence of the on. )(xg the right hand side satisfies dg dx. P y=x (slope=1) y=g(x) y y=x y=g(x) root. Calculations: guess x1, compute g(x1, x2= g(x1), i. e. , the length corresponding to x2 equals to the length given by g(x1), as shown on the sketch above. Thus, the point p(x=xnew, y=g(xold) ) falls on the line y=x. In this case, repeating the process moves p along y=x towards the root and the iterations eventually converge, as shown on the sketch. (ii) 1 dg dx y=g(x) y=g(x) y y=x (slope=1) y length=x2 root. In this case, however, repeating the process moves p along y=x away from the root and the iterations do not converge on the root, as shown in the sketch.

## Classmates also unlocked

###### ME 559 - Finite Element Methods pp. 1~77.pdf

Lecture Note

###### ME 564 Aerodynamics Notes - Prof. Lien Winter 2013.pdf

Lecture Note

###### ME 360 - Intro to Control Systems - FULL COURSE LECTURE NOTES - Prof Jeon - Winter 2013

Lecture Note

###### ME219 Final: Me 219 Notes.pdf

Exam Note

###### ME 203 Lecture 24: 24-Method of undetermined coefficients - Polynomial.pdf

Lecture Note

###### ME 203 Lecture 20: 20-Method of Undetermined Coefficients.pdf

Lecture Note

###### ME201 Final: Me 201 Notes.pdf

Exam Note

###### ENGG 2230 Midterm W13 - equation and properties sheet.pdf

Exam Note

###### ME 353 - Heat Transfer 1 - Prof. X. Li - Winter 2013.pdf

Lecture Note

###### ME 203 Lecture 9: 9-first Order Linear ODE, Examples.pdf

Lecture Note

###### ME 203 Lecture 8: 8-Integrating factor example, 1st order Linear ODE.pdf

Lecture Note

###### ME250 Study Guide - Gas Constant, Joule, Italian General Confederation Of Labour

Exam Note

###### ME 203 Lecture 21: 21-midterm solutions.pdf

Lecture Note

###### Getting a Time Response using Transfer Function

Lecture Note

###### ME 203 Lecture 19: 19-DAlemberts examples.pdf

Lecture Note