Quantitative Methods Answers

sum <-- 2 * n + 404
for (int i==1; i < n + 1; i++ ) do
sum <-- sum + i + MyA[i] + 404
for (int j = 0; j <= i; j++) do
sum <-- sum * MyA[j] + i + 404 * n
end-for (j)
for (int k = 0; k <= n; k++) do
sum <-- sum * k + MyA[k]
end-for (k)
sum <-- sum + 303
end-for(i)
sum <-- sum + 303 * n

Find an explicit expression for T(n), the count function for the number of times that your operation is executed (is a function of n). It likely starts with one more summations, and ends with a polynomial in n in standardized form (i.e. simplest terms). Show work.

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.

Give the most compact theta notation for the number of times the statement
x = x + 1 is executed in the following pseudo-code:
for i = 1 to i = 3n − 1{
for j = 1 to j = n{
x = x + 1
}
}

Let f(n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following:
If f(n) = Ω(g(n)), then g(n) = O(f(n)).

b) Set up the Gaussi-Jacobi iteration scheme in matrix form for the linear system of equations

4 3
4 2
4 3
2 3
1 2 3
1 2
− + =
− + − =
− =
x x
x x x
x x

Show that the iteration scheme is convergent. Hence find the rate of convergence of this
method.

2. a) Find by Newton’s method the roots of the following equations correct to three places of
decimals.
i) log .4 772393 x 10 x = near x = 6 .
ii) f (x) = x − 2sin x near x = 2 .