Quantitative Methods Answers

for the linear system of equations [1 2 - 2,1 1 1, 2 2 1][x y z] =[1 3 5] set up the gauss - jacobi and gauss - seidel iteration schemes in matrix form. also check the convergence of the two schemes.

find the dominant eigenvalue and the corresponding eigenvector for the matrix A =[-4 14 0, - 5 13 0, - 1 0 2]using five itrations of the power method and taking y(0)=[1 1 1] as the initial vector.

find the inverse of the matrix A =[1 - 1 1, 1 - 2 4, 1 2 2] by gauss Jordan method.

estimate the eigenvalues of the matrix [1 - 2 3] [6 - 13 18] [4 - 10 14] using the Gershgorin bounds.draw a rough sketch of the region where the eigenvalues lie

what is the identity element in multiplication?

Determine the spacing h in a table of equally spaced values for the function f(x)= (2+x)^4 , 1≤x≤2 so that the quadratic interpolation in this table satisfies | error|≤ 10^-6

determine a unique polynomial f(x) of degree<=3 such that
f(x_0)=1, f'(x_0)=2, f(x_1)=2, f'(x_1)=3 where x_1 - x_0 = h

Using Xo = 0 find an approximation to one of the zeros of x^3 − 4x +1 = 0 by using Birge Vieta Method. Perform two iterations

Are the following TRUE or FALSE. explain
n2 = O(n2)
n3 = O(n2)
n log n = O(n2)
n2 = O(n log2 n)

Consider the graph G0 with 3 components which are triangles. G0 has 9 vertices labeled A to I and 9 edges (A, B), (B, C) … as shown below.
If each vertex of G0 is assigned a red or a green color, then we say that an edge is colored if its ends have different colors.
Ajai and Rekha color the vertices of G0 in the following manner. Ajai proposes a color (red or green) and Rekha chooses the vertex to apply this color. After 9 turns, all the vertices of G0 are colored and the number of colored edges is counted.
Suppose Ajai would like to maximize the number of colored edges while Rekha would like to minimize the number of colored edges. Assuming optimal play from both players, how many edges will be colored? Explain your reasoning.