MTH603 Quiz 1 Solution and Discussion
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Which of the following rearrangement make strictly diagonal dominant, the system of linear
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if the relaxation method is applied on the system
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Please Madam Answer This question.
Total Marks 5
Starting Date Friday, May 03, 2024
Closing Date Wednesday, May 08, 2024
Status Open
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while using jacobi method for the matrix a=[2 0 0 02 1 0 1 2] o=pi/4
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I’m slightly confused about how to use the power method and the steps to calculate an eigenvalue. - I understand that the power method is defined as U(x+1) = AU(x)/a(x) where “a” is the first component of U(x). I do not understand at all what “U” is. Are we picking any vector we want that minimizes the error? What would I do given the practice problem below?
Apply the power method to
132−4
1 2 3 −4to obtain three approximations of the largest eigenvalue of A. What is the limiting vector u∞?
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Here is a table of iterations. The actual vector U is horizontal written as entry 1 and 2, and the current approximation to the first eigenvalue is in the third column. The first(largest in absolute value) eigenvalue is negative, so the system needs a certain time to reduce oscillations:
U[1] U[2] eigenvalue 1 0 0 1 3 1 1 -9/7 7 1 -57/11 -11/7 1 -261/103 -103/11 1 -1353/419 -419/103 1 -6669/2287 -2287/419 -
@cyberian said in MTH603 Quiz 1 Solution and Discussion:
Here is a table of iterations. The actual vector U is horizontal written as entry 1 and 2, and the current approximation to the first eigenvalue is in the third column. The first(largest in absolute value) eigenvalue is negative, so the system needs a certain time to reduce oscillations:
U[1] U[2] eigenvalue 1 0 0 1 3 1 1 -9/7 7 1 -57/11 -11/7 1 -261/103 -103/11 1 -1353/419 -419/103 1 -6669/2287 -2287/419 in float reprecentation
U[1] U[2] eigenvalue 1.0000000 0 0 1.0000000 3.0000000 1.0000000 1.0000000 -1.2857143 7.0000000 1.0000000 -5.1818182 -1.5714286 1.0000000 -2.5339806 -9.3636364 1.0000000 -3.2291169 -4.0679612 1.0000000 -2.9160472 -5.4582339 1.0000000 -3.0347480 -4.8320944 1.0000000 -2.9862913 -5.0694960 1.0000000 -3.0055137 -4.9725827 1.0000000 -2.9977994 -5.0110274 1.0000000 -3.0008810 -4.9955987 -
Using the Jacobi method find all the eigenvalues and the corresponding 2 eigenvectors of the matrix A = 2 1 2 Iterate till the oR- ~1 2 diagonal elements, in magnitude; are less than 0.0005
Using the Jacobi method find all the eigenvalues and the corresponding 2 eigenvectors of the matrix A = 2 1 2 Iterate till the oR- ~1 2 diagonal elements, in magnitude; are less than 0.0005 -
while using jacobi method fot the matrix A=[1 1/4 1/4 1/4 1/3 1/2 1/3 1/2 1/5]
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Let A be an n × n matrix. Then λ = 0 is an eigenvalue of A if and only if there exists a non-zero vector v ∈ Rn such that Av = λv = 0. In other words, 0 is an eigenvalue of A if and only if the vector equation Ax = 0 has a non-zero solution x ∈ Rn.
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an eigenvector v is said to be normalized if the coordinate of largest magnitude is equal to aero?
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Two similar matrices have the same eigenvalues, even though they will usually have different eigenvectors. Said more precisely, if B = Ai’AJ. I and x is an eigenvector of A, then M’x is an eigenvector of B = M’AM. So, A1’x is an eigenvector for B, with eigenvalue ).
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b) The eigenvalues of a real symmetric matrix need not be positive. They can be positive, negative, or even zero, depending on the elements and the specific structure of the matrix.
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Jacobian Method in Matrix Form
Let the n system of linear equations be Ax = b. Let us decompose matrix A into a diagonal component D and remainder R such that A = D + R. Iteratively the solution will be obtained using the below equation. -
for a function; y=f(x), if y0, y1 are 2,3