Answer: The Jacobi iteration method is an iterative method used to solve a system of linear equations Ax = b. This method is based on the idea that an approximate solution can be obtained by iteratively solving each equation for one of its unknowns while the other unknowns are kept fixed. In order for the Jacobi iteration method to converge, we must prove that it is convergent for any initial x (0) provided that A is strictly diagonally dominant.
Let A be an n-by-n matrix and b be a vector in Rn. We assume that A is strictly diagonally dominant. This means that the absolute value of each diagonal element of A is greater than the sum of the absolute values of the non-diagonal elements in the same row. This can be expressed mathematically as:
|a_jj| > ∑ |a_ij| , where i ≠ j and i, j = 1,2, ..., n.
Now, let x(0) be the initial vector in Rn. The Jacobi iteration method for solving Ax = b is given by:
x_j^{k+1} = (b_j - ∑_{i=1,i≠j}^{n}a_ijx_i^k) / a_jj , where j = 1,2, ..., n.
We can prove that the Jacobi iteration method is convergent for any initial x (0), provided that A is strictly diagonally dominant, by using the following theorem.
Theorem: Let A be an n-by-n matrix and b be a vector in Rn. Assume that A is strictly diagonally dominant and let x(0) be the initial vector in Rn. Then, the Jacobi iteration method is convergent for any initial x (0).
Proof: We will prove the theorem by using the Banach fixed point theorem. Let X be the set of all vectors x in Rn and define a mapping T : X → X as follows:
T(x) = (b_1 - ∑{i=1,i≠1}^{n}a_1ix_i) / a_11 , (b_2 - ∑{i=1,i≠2}^{n}a_2ix_i) / a_22 , ... , (b_n - ∑_{i=1,i≠n}^{n}a_nix_i) / a_nn .
We will prove that T is a contraction mapping. To do this, we need to show that there exists a constant c >= 0 such that for all x, y in X, we have
||T(x) - T(y)|| ≤ c||x - y|| , where ||.|| is the Euclidean norm.
From the definition of T, we have
T(x) - T(y) = (b_1 - ∑{i=1,i≠1}^{n}a_1ix_i) / a_11 - (b_1 - ∑{i=1,i≠1}^{n}a_1iy_i) / a_11 , (b_2 - ∑{i=1,i≠2}^{n}a_2ix_i) / a_22 - (b_2 - ∑{i=1,i≠2}^{n}a_2iy_i) / a_22 , ... , (b_n - ∑{i=1,i≠n}^{n}a_nix_i) / a_nn - (b_n - ∑{i=1,i≠n}^{n}a_niy_i) / a_nn .
Now, we can use the triangle inequality to get
||T(x) - T(y)|| ≤ ∑{j=1}^{n} |(b_j - ∑{i=1,i≠j}^{n}a_jix_i) / a_jj - (b_j - ∑_{i=1,i≠j}^{n}a_jiy_i) / a_jj| .
Using the definition of T and the fact that A is strictly diagonally dominant, we can further simplify this to
||T(x) - T(y)|| ≤ ∑{j=1}^{n} |a_jj(x_j - y_j)| / |a_jj| ≤ ∑{j=1}^{n} |a_jj||x_j - y_j| / |a_jj| ≤ ∑_{j=1}^{n} |x_j - y_j| = ||x - y|| .
Thus, we have shown that ||T(x) - T(y)|| ≤ ||x - y||, which implies that T is a contraction mapping. Therefore, by the Banach fixed point theorem, the Jacobi iteration method is convergent for any initial x (0).
This completes the proof.
Some one might be more dominant/stronger on one side making them swim diagonally.
Jesse Barlow has written: 'Computing accurate eigensystems of scaled diagonally dominant matrices' -- subject(s): Accessible book
I have had the best results by starting with a quick sketch to compose the piece then starting in the upper left corner (I am right hand dominant in my art work) applying only a small amount of detail in the background and as I move diagonally across the surface I add much more detail to the main focus. This greatly reduces the amount of unwanted smudging of the finished piece.
You can add interesting effect to the picture with photo filter or remove color cast by applying photo filter of opposite color from dominant color which causes color cast.
dominant genes for freckles
It is dominant.
In my opinion it is dominant.
autosomal dominant
A non-dominant group is the group with less power.. For example women are non-dominant, men are dominant, heterosexuals are dominant, gays are non-dominant. The group that sets the polices, laws and "standards" OS dominant. The group with power is the dominant group.
A non-dominant group is the group with less power.. For example women are non-dominant, men are dominant, heterosexuals are dominant, gays are non-dominant. The group that sets the polices, laws and "standards" OS dominant. The group with power is the dominant group.
Huntingdons chorea is caused by a dominant allele.
it is dominant