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Lecture Notes 5
Solving nonlinear systems of
equations
The core of modern macroeconomics lies in the concept of equilibrium, which
is usually expressed as a system of plausibly nonlinear equations which can
be either viewed as finding the zero of a a give
The Simplex Method in Tabular Form
In its original algebraic form, our problem is:
Maximize
Subject to:
z
z 4x1 3x2
2x1 +3x2 +s1
3x1 +2x2
+s2
2x2
+s3
2x1 +x2
+s4
=
=
=
=
=
0
6
3
5
4
(0)
(1)
(2)
(3)
(4)
x1 , x2 , s1 , s2 , s3 , s4 0 .
Since the objective f
301
Documenta Math.
Broyden Updating, the Good and the Bad!
Andreas Griewank
Abstract.
2010 Mathematics Subject Classification: 65H10, 49M99, 65F30
Keywords and Phrases: Quasi-Newton, secant condition, least change,
bounded deterioration, superlinear conv
A Stationary Newton Method for Nonlinear Functional Equations
Author(s): J. E. Dennis, Jr.
Source: SIAM Journal on Numerical Analysis, Vol. 4, No. 2 (Jun., 1967), pp. 222-232
Published by: Society for Industrial and Applied Mathematics
Stable URL: http:/w
Practical Quasi-Newton algorithms for singular
nonlinear systems
Sandra Buhmiler
Natasa Krejic
Zorana Luzanin
January 25, 2010
Abstract
Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots c
Mathematical Programming 35 (1986) 71-82
North-Holland
BROYDEN'S
METHOD
IN HILBERT SPACE
Ekkehard W. SACHS
Department of Mathematics, Box 8205, North Carolina State University, Raleigh,
NC 27695-8205, USA
Received 6 July 1984
Revised manuscript received 1
Broyden's Method
Outline
What is
Broyden's
method?
How is it
used?
Newton's
method
algorithm
Broyden's
method
algorithm
Comparison
of both
methods
Application of
Broyden's
method
Broyden's Method
Broyden's Method is a method for solving
F(x)=0.
F(x) could
Computing58, 69-89 (1997)
~ 1 " 1 ~
Springer-Verlag 1997
Printed in Austria
Numerical Experience with Newton-like Methods
for Nonlinear Algebraic Systems*
E. Spedicato and Z. Huang*, Bergamo
Received June 6, 1994; revised March 9, 1996
Abstract - - Zusam
A QUASI-NEWTON METHOD FOR SOLVING SMALL NONLINEAR
SYSTEMS OF ALGEBRAIC EQUATIONS
JEFFREY L. MARTIN
Abstract. Finding roots of nonlinear algebraic systems is paramount to many applications in
applied mathematics because the physical laws of interest are us
Notes: Rate of Convergence
Suppose we have a sequence of real numbers that converge to some point x:
x 0 , x1 , x 2 , . . . x
Is there a way to talk about how fast the numbers are converging to x?
Definition: If
|xn+1 x|
=<
|xn x|
then the sequence conver
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Quiz 4 Solutions
1. a. Let us rst nd the present value of the award. This is
30
0
20000 e0.06x dx
Evaluation the integral, we get
1
20000 e0.06x |30 = 278, 233
0
0.06
So the prize is worth $278,233. Now, how much money ve years from now is
worth that? We
QUIZ #5 SOLUTIONS
MATH 42
Winter 2001
1. a) Let B be Sallys balance. Then dB is the rate at which she
dt
recieves interest minus the rate at which she withdraws money:
dB
= (.05)B 1500.
dt
The only equilibrium is found by setting dB = 0 to get B = 1500/(.
Quiz 6 Solutions
1a. (i) First check whether the nth term goes to 0:
1
1
sin( n )
sin( x )
1
sin y
lim n sin( ) = lim
= lim
= lim
=1
1
1
n
n
x
y 0 y
n
n
x
by lHospitals Rule. Since the nth term does not have limit 0, the series diverges.
(ii) The series i
QUIZ 4
Math 42, Winter 2001. You have 25 minutes. No notes, no books. YOU MUST SHOW ALL WORK TO RECEIVE CREDIT Good luck! Name
1.
(/20 points)
2.
(/20 points)
Total
(/40 points)
1
1. a. After a lawsuit involving people injured by falling ball bearings, Ma
QUIZ 1
Math 42, Winter 2001. You have 25 minutes. No notes, no books. YOU MUST SHOW ALL WORK TO RECEIVE CREDIT Good luck! Name
1.
(/20 points)
2.
(/20 points)
Total
(/40 points)
1. a) Sally Spendthrift starts college with a modest endowment bestowed upon
QUIZ 6
Math 42, Winter 2001. You have 25 minutes. No notes, no books. YOU MUST SHOW ALL WORK TO RECEIVE CREDIT Good luck! Name
1.
(/20 points)
2.
(/20 points)
Total
(/40 points)
1
1a. Determine whether each of the following series converges or diverges. C
k
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5 11id 3 3
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$7 ' S
k
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6 11ie 4 4 13
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)
!
Convergence of Newtons method
Newtons method uses a linear Taylor approximation to the function f to
approximate its root. Let p be such that f (p) = 0, and let pk be an approximation
to p. Let us use the abbreviation fk f (pk ) throughout. If we take as