peace951

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Use the Symmetric Power method to approximate the most dominant eigenvalue of the matrix A=Matrix([[4, 2, -1], [2, 0, 2], [-1, 2, 0]]) , Use x_0 = (-1,0, 1)^t.  Iterate until a tolerance of 10^-4 is achieved or until the number of iterations exceeds 25. 

I tried but failed to get full evaluation (numbers only).

with(LinearAlgebra);

A := Matrix([[4, 2, -1], [2, 0, 2], [-1, 2, 0]]);
x_0 := <-1, 0, 1>;
Tol := 0.0001;
k := 15;
y_k := evalf(A*x[k - 1]);
mu_k := evalf(Transpose(x_k - 1)*y_k);
x_k = y_k/Norm(y_k, 2);
 

Q5_Ch_9.3.pdf

 

Solve the following problems using computer. Use Algorithm from  Algorithm 4.4 in the textbook with 1) n = 4,m = 8, 2) n = 8,m = 4, and 3) n = m = 6 to approximate the following double integral and compare the results to the exact solution
∫∫cos ydydx (interior integral from 0 to x and exterior integral is from 0 to pi)

Using maple, approximate integral of 2/(x^2-4) from 0 to .30 by using

(a) the Trapezoidal rule: n=1, h=(b-a)/n, inegral of f(x) from a to b=(h/2)(f(x0)+f(x1))

(b) Simpson’s rule:

(c) the Midpoint rule: 

how to compute the integral of (1+lnx)/x in the interval from 1 to e by using substitution teqnique. Substitute u=1+lnx

 1. Consider this code line: > a := b; b := c; c := matrix([[1, 2, 3], [3, 4, 5]]);
    • if you ask Maple to return the values assigned to a, b and c, which values will be printed?
    • Are these values the ones you would expect? Why? Which kind of evaluation rule is utilized by Maple to return these values?
   • Show the single steps of the evaluation process and briefly explain the outcome of each evaluation step (hint: use eval (a, 1/2/3), to return each evaluation step).
   • How can you “force” Maple to return the “expected” values for a, b and c? • Extract the first row of the matrix in such a way to have [1, 2, 3] as output. Hint: use the command row/Row and its right syntax (it needs two arguments: the name of the matrix and the number of the row to extract). Be careful to load the linalg/LinearAlgebra package before using the command. Instruction: apply the row/Row command to a, b and c.
  • Can you explain what Maple does in the previous step? Can you explain, in other words, why you get the expected outcome? Hint: remember which are the “drivers” of the evaluation process in Maple (position of the assignment/level of evaluation; type of object / data-structure / datacontainer / procedures / local versus global variables / use of built-in commands, etc.).

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