Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b.

(a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Instruction: Apply three decimal place rounding to calculations where applicable. (a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b. (b) The 3 x 3 system below has solution ₁=1, x₂ = -2 and x3 = 0. 4x1 + 2x₂ - 2x3 0 2x16x22x3 = 14 3x₁x₂ + 4x3 = 5 - (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (ii) Using the initial approximation z(0) = 0.75, x= -1.75 and r = -0.25, apply three iterations of the Jacobi Method to the above system. (iii) Taking the initial approximation (0) = 0.75, x) = -1.75 and 20) = -0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
Transcribed Image Text:Instruction: Apply three decimal place rounding to calculations where applicable. (a) Explain why the Jacobi and Gauss-Seidel methods may sometimes be preferred over direct methods (such as elimination and factorization) when solving a 3 x 3 linear system Ax = b. (b) The 3 x 3 system below has solution ₁=1, x₂ = -2 and x3 = 0. 4x1 + 2x₂ - 2x3 0 2x16x22x3 = 14 3x₁x₂ + 4x3 = 5 - (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (ii) Using the initial approximation z(0) = 0.75, x= -1.75 and r = -0.25, apply three iterations of the Jacobi Method to the above system. (iii) Taking the initial approximation (0) = 0.75, x) = -1.75 and 20) = -0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
(b) The 3 x 3 system below has solution ₁ = 1, 2 = -2 and x3 = 0. 0 4x1 + 2x2 - 2x3 2x16x22x3 = 14 3x1x₂ + 4x3 = 5 (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (0) (ii) Using the initial approximation z = 0.75, x=-1.75 and three iterations of the Jacobi Method to the above system. = = -0.25, apply (iii) Taking the initial approximation (0) = 0.75, x = -1.75 and r=-0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
Transcribed Image Text:(b) The 3 x 3 system below has solution ₁ = 1, 2 = -2 and x3 = 0. 0 4x1 + 2x2 - 2x3 2x16x22x3 = 14 3x1x₂ + 4x3 = 5 (i) Explain why this system is not strictly diagonally dominant (hereafter SDD). (0) (0) (0) (ii) Using the initial approximation z = 0.75, x=-1.75 and three iterations of the Jacobi Method to the above system. = = -0.25, apply (iii) Taking the initial approximation (0) = 0.75, x = -1.75 and r=-0.25, apply three iterations of the Gauss-Seidel Method to the system. (iv) Will the Jacobi and Gauss-Seidel iterates for all 3 x 3 linear systems that are not SDD always diverge? If not, give an example of a system that is not SDD, but for which the Jacobi and Gauss-Seidel iterates converge to the solution of the system.
Solution
Bartleby Expert
SEE SOLUTION
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS