aConsider the following linear system,[a1011x = 31a1a) Construct the iteration formula using the Jacobi method.b) Let a = 4, perform one iteration with initial guess x, = [1,1,1]", anda.compute the error. (The exact solution is xc) Without doing the actual computation:i) Using an initial guess x; = [10,-10,10]", will it take moreiterations to converge, when compared to (b)?ii) If a is very large, say 1000, how many iterations does one need toachieve an error of less than 0.01? And why? (Using the initialguess XA)%3DL8'2'8

Given the following system, α101α101αxyz=233 , x is assumed to be xyz.

Answered: Consider the following linear system,…

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

Similar questions

Derive the equation for βi that is used by the Newton algorithm.
7.5 SOR with Jacobi iterations (a). Design an SOR method based on Jacobi iteration. Write out the iteration step, i.e., the formula to compute +¹ for (i = 1,...,n) when given x for (i = 1,...,n). (b). Write the matrix formula for this SOR method.
VII. Run two iterations using normalized power iteration method for the following matrix and initial vector. Roughly predict the dominate eigenvalue and corresponding eigenvector. A = [ ²₁ 2] ₁ x₁ = 1 []
Consider the following system. x+5y=9 4x-y=15 Start with P0=0 and use Gauss-Seidel Iteration to find Pk (k=1,2). Options for the value of x1, y1, x2, y2 include: 3.75 1.05 9 21 1.0002 4.0005 0.9975 -96 -399 Will Gauss-Seidel Iteration converge to the solution? A) Yes B) No
four decimal places
1. (a) Let A E Mmxn (F) and let Are be the reduced echelon form of A. Prove that Ax = 0 if and only if Arex = 0. (b) Suppose A € M3×4(R) is such that Ax = 0 has the two solutions X1 = and x2 = 0 and any other solution to Ax = 0 is a linear combination of x₁ and x2. i. Find the reduced echelon form Are of A. Make sure to explain/justify your answer! ii. Find the dimensions of the four fundamental subspaces of A.
The same disease is spreading through two populations, say Pi and P2, with the same size. You may assume that the spread of the disease is well described by the SIR model. dSa - BaSaIa dt dla - dt dRa YaIa dt with Sa(0) + I.(0) + R.(0) = N where N denotes the fixed population size. The subscript a identifies the population P or P2. For example, if a = 1, the variables are related to P1. Assume that S1 (0) = S2 (0) and I1(0) = I½(0) and that no interventions such as quarantine or vaccination have been implemented. If the difference in the spread of the disease is due only to the poor over-all health of a population, which population has the best over-all health of the two populations? - Susceptible Population 1 - Susceptible Population 2 700 600 500 400 300 200 100 10 15 20 t P1. P2. Susceptible
Find the third iteration value of z using the Gauss-Seidel method with an initial guess of (1, 1, 2). Round off your final answer to nine decimal places. Do not round off in preliminary calculations.
In this question, we want to develop a version of iterative refinement (as seen in the Week 5 computer lab) for iterative methods for solving Ax = b. The main steps in iterative refinement for Gaussian elimination were as follows: 1. Solve Ax = b with Gaussian elimination, to obtain computed solution 2. Compute the residual r = b - AX 3. Solve Ae = r with Gaussian elimination, to obtain computed solution ê 4. Update = x + ê. In step 4, we are using the fact that the error e = x - in the computed solution satisfies Ae = Ax - Ax = b - Ax = r. The solve in step 3 can be done cheaply, in O(n²) cost, since we can re-use the LU factorisation that was already computed in step 1. So the additional cost of adding steps 2-4 is negligible, but they significantly increase the accuracy of the computed solution. For iterative methods, such as the Jacobi and Gauss-Seidel methods, both solves in steps 1 and 3 would have (²) cost, and so the "refinement" steps 2-4 would substantially increase the cost.…

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

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…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

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…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS