Answer each question: Section 7.3 The Jacobi and Gauss-Siedel Iterative Techniques 1) Find the first two iterations of the Jacobi method for the following linear systems, using x (0) = 0.: d. 4x 1 –x...

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Answer each question:
Section 7.3 The Jacobi and Gauss-Siedel Iterative Techniques
1)
Find the first two iterations of the Jacobi method for the following linear systems, using x(0)
= 0.:
d. 4x1
–x2- x4
= 0,
-x1
+4x2
–x3 _x5
=5,


-x2
– 4x3
– x6
= 0,
-x1
+4x4
–x5
= 6,
-x2
– x4
+4x5
–x6
= -2,
-x3
– x5
+4x6
=6
9)
The linear system 2x1
- x2
+ x3
= -1,
2x1
+2x2
+ 2x3
= 4,
-x1
–x2
+ 2x3
= -5
Has the solution (1, 2, -1)t
.

  1. Show that p(Tj) = /2 > 1.

  2. Show that Jacobi method with x(0)
    = 0 fails to give a good approximation after 25 iterations.

  3. Show that p(Tg) = 1/2

  4. Use the Gauss-Seidel method with x(0)
    = 0 to approximate the solution to the linear system to within 10-5
    in the ?norm.



10)
The linear system x1
+2x2
– 2x3
= 7,
x1
+ x2
+x3
=2,
2x1
+2x2
+x3
= 5
Has the solution (1, 2, -1)t
.

  1. Show that p(Tj) =0

  2. Use the Jacobi method with x(0)
    = 0 to approximate the solution to the linear system to within 10-5
    in the ?norm.

  3. Show that p(Tg) = 2

  4. Show that the Gauss-Seidel method applied as in part (b) fails to give a good approximation in 25 iterations.



Section 8.1 Discrete Least Square Approximation
3)
Find the least squares polynomials of degree 1, 2, and 3 for the data in the following table. Compute the error E in each case. Graph the data and the polynomials.
xi
0 0.15 0.31 0.5 0.6 0.75
yi
1.0 1.004 1.031 1.117 1.223 1.422
5)
Given the data:
xi
1.0 1.1 1.3 1.5 1.9 2.1
yi
1.84 1.96 2.21 2.45 2.94 3.18


  1. Construct the least squares polynomial of degree 1, and compute the error.

  2. Construct the least squares polynomial of degree 2, and compute the error.

  3. Construct the least squares polynomial of degree 3, and compute the error.

  4. Construct the least squares approximation of the form beax
    , and compute the error.

  5. Construct the least squares approximation of the form bxa
    , and compute the error.



Section 8.2 Chebyshev Polynomials and Economization of Power Series
11)
Use the Gram-Schmidt procedure to calculate L1
, L2
and L3
where { L0(x), L1(x), L2(x), L3(x) } is an orthogonal set of polynomials on (0, ) with respect to the weight functions (x) = e-x
and L0(x) 1. The polynomials obtained from this procedure are called the Laguerre polynomials.
Answered Same DayDec 20, 2021

Answer To: Answer each question: Section 7.3 The Jacobi and Gauss-Siedel Iterative Techniques 1) Find the first...

David answered on Dec 20 2021
121 Votes
Section 7.3 The Jacobi and Gauss-Siedel Iterative Techniques
1)
Find the first two iterations of the Jacobi method for the following linear systems, using x(0)
= 0.:
d. 4x1 –x2- x4 = 0,
-x1 +4x2 –x3 _x5 =5,
-x2 – 4x3 – x6 = 0,
-x1 +4x4 –x5 = 6,
-x2 – x4 +4x5 –x6 = -2,
-x3 – x5 +4x6 =6
Solution:
Jacob method the sequence is
xk = Tx(k-1) + c




























































Given for initial approximation x(0) = 0, Then x(1) is given by

( )


( )


( )

( )


( )


( )


( )



( )


( )


( )

( )


( )


( )



( )


( )


( )


( )



( )


( )


( )
Similarly second iterate is calculated

( )

( )


( )

( )

( )

( )
The first two iteration values of the linear system are as follows:
k 0 1 2

( ) 0 0 0.6875

( ) 0 1.25 1.375

( ) 0 0 0.3125

( ) 0 1.5 1.625

( ) 0 0.5 0.1875

( ) 0 0
9)
The linear system 2x1 - x2 + x3 = -1,
2x1 +2x2 + 2x3 = 4,
-x1 –x2 + 2x3 = -5
Has the solution (1, 2, -1)t .
a. Show that p(Tj) = /2 > 1.
b. Show that Jacobi method with x(0) = 0 fails to give a good approximation after 25
iterations.
c. Show that p(Tg) = 1/2
d. Use the Gauss-Seidel method with x(0) = 0 to approximate the solution to the linear
system to within 10-5 in the ɭ norm.
Solution:
a. 2x1 - x2 + x3 = -1,
2x1 +2x2 + 2x3 = 4,
-x1 –x2 + 2x3 = -5
A [



]
We split A = D+L+U
Where D is diagonal matrix
-L is lower triangular part of A
-U is upper triangular part of A
A = [



] [



] [



]
The equation Ax = b or (D-L-U)x = b is then transformed into
Dx = (L+U)x + b
x = D-1 (L+U)x + D-1b
Tj = D-1 (L+U)
L + U = [



]
D-1 = [



]
Tj = D-1 (L+U)
Tj =[



] [



]
Tj =[



]
det(Tj – λI) = [



] [



]
= [



]
= -λ(λ2 + 0.5) – 0.5(λ+0.5) – 0.5(-0.5+0.5λ)
= -λ(λ2 + 0.5) – 0.5 λ – 0.25 + 0.25 – 0.25 λ
= -λ(λ2 + 0.5 + 0.75)
= -λ(λ2 + 1.25)
[ρ(Tj)]2 = 1.25 = 5/4
ρ(Tj) = √5/2 > 1
b. Jacob method the sequence is
xk = Tx(k-1) + c











Given for initial approximation x(0) = 0, Then x(1) is given by


















Similarly following iterations are done:
k
( )
( )
( )
0 0 0 0
1 -0.5 2 -2.5
2 1.75 5 -1.75
3 2.875 2 0.875
4 0.0625 -1.75 -0.0625
5 -1.3438 2 -3.3438
6 2.1719 6.6875 -2.1719
7 3.9297 2 1.9297
8 -0.4648 -3.8594 0.4648
9 2.6621 2 -4.6621
10 2.8311 9.3242 -2.8311
11 5.5776 2 3.5776
12 -1.2888 -7.1553 1.2888
13 -4.7220 2 -6.7220
14 3.8610 13.4441 -3.8610
15 8.1526 2 6.1526
16 -2.5763 -12.3051 2.5763
17 -7.9407 2 -9.9407
18 5.4703 19.8814 -5.4703
19 12.1759 2 10.1759
20 -4.5879 -20.3517 4.5879
21 -12.9698 2 -14.9698
22 7.9849 29.9397 -7.9849
23 18.4623 2 16.4623
24 -7.7311 -32.9246 7.7311
25 -20.8279 -22.8279
c. 2x1 - x2 + x3 = -1,
2x1 +2x2 + 2x3 = 4,
-x1 –x2 + 2x3 = -5
A [



]
We split A = D+L+U
Where D is diagonal matrix
-L is lower triangular part of A
-U is upper triangular part of A
A = [



] [



] [



]
The equation Ax = b or (D-L-U)x = b is then transformed into
(D-L)x = Ux + b
x = (D-L)-1 Ux + (D-L)-1b
Tg = (D-L)-1 U
D-L = [



] [



]
D-L = [



]
(D-L)-1=[



]
Tg =[



] [



]
Tg =[



]
det(Tg – λI) = [



] [



]
det(Tg – λI) =[



]
=-λ(0.5+ λ)2-0.5(0-0)-0.5(0-0)
=-λ(0.5+ λ)2
λ = 0.5
d. Gauss-Seidel method the sequence is


















Given for initial approximation x(0) = 0, we generate the Gauss-Seidel iterates in the
following Table:
k
( )

( )

( )

0 0 0 0
1 -0.5 2.5 -1.5
2 1.5 2 -0.75
3 0.875 1.875 -1.125
4 1 2.125 -0.9375
5 1.03125 1.90625 -1.03125
6 0.96875 2.0625 -0.98438
7...
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