Question 1- Use Newton's method to find three roots of x3-0.1x2-4x+1 = 0 accurate to within 10-14. Indicate clearly the initial approximations and the number of iterations needed to achieve the required accuracy.
Question 2- Determine the cubic polynomial p(x) = a3x3 + a2x2 + a1x + a0 satisfying the following conditions and then plot the graph of y = p(x) on [-3, 3].
p(-2) = 2.9, p(-0.5) = 0.8, p(0.5) = -0.8, p(1.5) = 3.
Question 3- For the following matrix A use Matlab to 1) calculate its eigenvalues and the corresponding eigenvectors; 2) calculate A2, AT, A-1, rref(A); and 3) calculate det(A),det(A2), det(2A).
Question 4- For the matrix A = magic(8), find its reduced row echelon form. Then 1) find a basis for its column space, and 2) find a basis for its null space.
Question 5- Let a1, a2 and b be as follows. 1) Show that the system x1a1 + x2a2 = b has no solution. 2) Find the least squares solution x1^, x2^for the system. 3) Calculate p = x1^a1 + x2^a2, the projection of b onto the plane spanned by a1, a2. 4) Verify that b - p is perpendicular to both a1, a2.
