Solve the tridiagonal system: J C - 2x1 + x2 = -x1 + 2x2 + x3 = 2 -x2 + 2x3 + x4 = 2 -x3 +2x4+x5 = 2 -x4 + 2x5=1 = 3
Solve the tridiagonal system: J C - 2x1 + x2 = -x1 + 2x2 + x3 = 2 -x2 + 2x3 + x4 = 2 -x3 +2x4+x5 = 2 -x4 + 2x5=1 = 3
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.8: Determinants
Problem 34E
Related questions
Question
100%
Please solve the two questions. 4 and 5
![Problem 4. Solve the tridiagonal system:
Problem 5. Define the matrix An € Rn×n by
An
2x1 + x₂ = 3
-x1 + 2x₂ + x3 = 2
-x₂ + 2x3 + x4 = 2
−x3 +2x4+x5 = 2
-x4 + 2x5=1
=
1 −1 -1
0
-1
00
⠀
-1
1
0
1. Show that the entry of the inverse of An at row i and column j has value:
0,
if i > j
1,
if i = J
2j-i-1,
if i < j
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c0b55c5-f8cf-47d3-9eba-1726b3768700%2F07464729-5797-4e65-a441-96a9f1ca529d%2Fbuywv6_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 4. Solve the tridiagonal system:
Problem 5. Define the matrix An € Rn×n by
An
2x1 + x₂ = 3
-x1 + 2x₂ + x3 = 2
-x₂ + 2x3 + x4 = 2
−x3 +2x4+x5 = 2
-x4 + 2x5=1
=
1 −1 -1
0
-1
00
⠀
-1
1
0
1. Show that the entry of the inverse of An at row i and column j has value:
0,
if i > j
1,
if i = J
2j-i-1,
if i < j
1
![2. What is the condition number of An? (The condition number of a matrix A is defined to be
|| A||∞ || A¯¹||·) Is matrix An ill-conditioned?
3. Let
The solution of Anx = bis
Perturb
Ď
7 = (−n+2 −n+3
to
J
= (1 1
b = b + (0
What is â? (Hint: î
Let â be the solution to Anî = 6.
4. Show that the values of 7, 6, 7, î satisfy
1)T.
0 €) T.
−1 0 1)¹.
=
À + Añ¹(ô − ¯').)
||☎ − â||∞ ≤ || An||∞ || Añ¹² ||00
|| 7² ||00
n
∞
- bll00
FL
81100](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c0b55c5-f8cf-47d3-9eba-1726b3768700%2F07464729-5797-4e65-a441-96a9f1ca529d%2Fbosf116_processed.png&w=3840&q=75)
Transcribed Image Text:2. What is the condition number of An? (The condition number of a matrix A is defined to be
|| A||∞ || A¯¹||·) Is matrix An ill-conditioned?
3. Let
The solution of Anx = bis
Perturb
Ď
7 = (−n+2 −n+3
to
J
= (1 1
b = b + (0
What is â? (Hint: î
Let â be the solution to Anî = 6.
4. Show that the values of 7, 6, 7, î satisfy
1)T.
0 €) T.
−1 0 1)¹.
=
À + Añ¹(ô − ¯').)
||☎ − â||∞ ≤ || An||∞ || Añ¹² ||00
|| 7² ||00
n
∞
- bll00
FL
81100
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