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e a tridiagonal matrix (possibly by matrix are all real. If we (0,2), both converge or both diverge simul they converge, the function w p -1 for

e a tridiagonal matrix (possibly by matrix are all real. If we (0,2), both converge or both diverge simul they converge, the function w p -1 for

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
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Let A be a tridiagonal matrix (possibly by blocks), and assume that the eigenvalues of the Jacobi matrix are all real. If we (0,2), then the method of Jacobi and the method of relaxation both converge or both diverge simultaneously (even when A is tridiagonal by blocks). When they converge, the function wp(Lw) (for w€ (0,2)) has a unique minimum equal to wo 1 for 2 Wo= 1+√1 - (p(J))²¹ where 1 < wo < 2 if p(J) > 0. We also have p(L₁) = (p(J))², as before.
Transcribed Image Text:Let A be a tridiagonal matrix (possibly by blocks), and assume that the eigenvalues of the Jacobi matrix are all real. If we (0,2), then the method of Jacobi and the method of relaxation both converge or both diverge simultaneously (even when A is tridiagonal by blocks). When they converge, the function wp(Lw) (for w€ (0,2)) has a unique minimum equal to wo 1 for 2 Wo= 1+√1 - (p(J))²¹ where 1 < wo < 2 if p(J) > 0. We also have p(L₁) = (p(J))², as before.
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