Show me the steps of deremine red and inf is here i need evey I need all the details step by step and inf is her Motivated by difference equations and their systems, we consider the followingsystem of difference equationsXnУп, Yn+1 = A + B-Уп-1Xn+1 = A + B-(1)Xn-1where A and B are positive numbers and the initial values are positive numbers. InFrom this, system (1) transform into following system:Zn, Zn+1 = 1 +p,n-1tn+1 = 1+P7(2)%3Dn-1where p = > 0. From now on, we study the system (2).Now we study the rate of convergence of system (2). Hence, we consider the followingsystem:En+1 = (A + B (n)) En,(14)where En is a k-dimensional vector, A e Ckxk is a constant matrix, and B : Z+ →Ckxk is a matrix function satisfying||B(n)|| → 0,(15) Theorem 9 Suppose that 0

Given, tn+1-t¯=∑i=01αi(tn-i-t¯)+∑i=01βi(zn-i-z¯)…

Answered: where En is a k-dimensional vector, A e…

where En is a k-dimensional vector, A e Ckxk is a constant matrix, and B : Z+ Ckxk is a matrix function satisfying ||B(n)|| → 0, (15)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.2: Matrix Algebra

Problem 21EQ: Prove the half of Theorem 3.3 (e) that was not proved in the text.

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Question

Show me the steps of deremine red and inf is here i need evey I need all the details step by step and inf is her

Motivated by difference equations and their systems, we consider the following
system of difference equations
Xn
Уп
, Yn+1 = A + B-
Уп-1
Xn+1 = A + B-
(1)
Xn-1
where A and B are positive numbers and the initial values are positive numbers. In
From this, system (1) transform into following system:
Zn
, Zn+1 = 1 +p,
n-1
tn+1 = 1+P7
(2)
%3D
n-1
where p = > 0. From now on, we study the system (2).
Now we study the rate of convergence of system (2). Hence, we consider the following
system:
En+1 = (A + B (n)) En,
(14)
where En is a k-dimensional vector, A e Ckxk is a constant matrix, and B : Z+ →
Ckxk is a matrix function satisfying
||B(n)|| → 0,
(15)

Theorem 9 Suppose that 0 <p <
such that lim tn =i and lim z, = 7. Then the error vector
5 and {(tn, Zn)} be a solution of the system (2)
n-00
n-00
tn - i
tn-1 - i
Zn - Z
en
En =
n-1
en
n-1
Zn-1 – 2
of every solution of system (2) satisfies both of the following asymptotic relations:
lim VE,| = |à1,2,3,4F1 (7, 7)|,
n-00
||En+1||
lim
= |à1,2,3,4 F1(T, 2)|,
n→00 || En||
where A1,2,3,4FJ (7, 7) are the characteristic roots of the Jacobian matrix Fj(i, 7).
Proof To find the error terms, we set
1
1
In+l -i =a; (tn-i – i) +Bi (Zn-i – 7) ,
i=0
i=0
1
1
Zn+1 – ē =
Eri (tn-i - i) + ôi (zn-i – 2),
i=0
i=0
and e = t, – i, e? = zn – 7. Thus we have
1
1
%3D
i=0
i=0
1
n-i
i=0
i=0
where
a0 = a1 = 0,
-p (7 + Zn-1)
Bo =
B1
%3D
Zn-1
-p(i + fn-1).
Yo =
, Yi =
12
'n-1
8o = 81 = 0.
Now we take the limits
lim ao = lim aj = 0,
n-00
n-00
lim Bo =
n-00
-2p
lim B1 =
n00
-2p
lim yo =
n-00
lim
Yi =
12'n-00
lim 8o = lim 81 = 0.
n-00
n-00
Hence
-2p
Bo = + an, Pi =
+ bn,
-2p
+ Cn, Y1 =
+ dn,
YO =
where a, → 0, b, → 0, cn → 0, d, → 0 as n → 00. Therefore, we obtain the
system of the form (14)
En+1
= (A + B (n)) En
where
1
A =
1
0 0 an bn
100 0
B(n) =
Cn dn 0 0
0 0 1 0
io o o

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