1. Consider Shamir's Lagrange interpolating polynomial key threshold scheme. Let t p-11, K-Tand Compute shadows for 1, 2, 3, 4, 5, 6 and 7. Reconstruct h(x) from the shadows for x-1, 3, 5 and 7. 1. Consider Shamir's Lagrange interpolating polynomial key threshold scheme. Let t p-11, K-Tand Compute shadows for 1, 2, 3, 4, 5, 6 and 7. Reconstruct h(x) from the shadows for x-1, 3, 5 and 7.
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the abscissae to find the corresponding ordinates (a) (8 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2(x) of f(x) at the given points. State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2...
Compute, using divided differences, the value of the piecewise cubic Her- mite interpolating polynomial at x = 11=10 given nodes at xi = i, for i = 1; : : : ; 10, and values and derivatives at the nodes from the function f(x) = 1=x. Remember iterative formula for divided differences: 2. (25 pts) Compute, using divided differences, the value of the piecewise cubic Her mite interpolating polynomial at x-11/10 given nodes at ai-i, for i-1, , 10. and...
Find the quotient Q(x) and remainder R(x) when the polynomial P(x) is divided by the polynomial D(x). P(x) = 4x5 + 9x4 − 5x3 + x2 + x − 25; D(x) = x4 + x3 − 4x − 5 Q(x) = R(x) = Use the Factor Theorem to show that x − c is a factor of P(x) for the given values of c. P(x) = 2x4 − 13x3 − 3x2 + 117x − 135; c = −3, c = 3...
er Lagrange ,Divided difference and Hermitewatnejed, Jnp 1.5, and x2-2, andf (x)ssin(x) * Given the point sx.-1, a) Find its Lagrange interpolation P on these points b) Write its newton's divided difference P, polynomial c)Write Hermite Hs by Using part a outcomes d) Write Hermite Hi by Using part b outcomes Rules: Lagrange form of Hermite polynomial of degre at most 2n-+1 Here, L., (r) denotes the Lagrange coefficient polynomial of degree n. If ec la.bl, then the error formula...
(5) (a) Let p(2) be a polynomial of the form r3 + ax2 + bx + c. What can you say about p() if you plug in a very, very large value for x? What about plugging in a very large negative number? (b) Give a justification for why p(x) must have a root. Hint: Try to draw it without drawing a root. (C) Show that every 3 x 3 matrix has an eigenvector. (d) Can you generalize your argument...
0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue 0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue
The polynomial of degree 4 The polynomial of degree 4, P(x) has a root of multiplicity 2 at x = 4 and roots of multiplicity 1 at x = 0 and x = – 2. It goes through the point (5, 7). Find a formula for P(x). P(x) =
Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) = +-12+} Find the trace and determinant of A. 2 e: tr(4) and det(A) = 0 12: tr(A) = 0 and det(A) 2 3 2 T: tr(A) = 0 and det(A) 3 : None of the other answers 01 OW
I need help with these questions. 1. Answer the questions below. (a) For the polynomial P(x) = 3.r* - 2x2 + 2x – 4, identify the following. i. (2 points) The degree of P ii. (2 points) The constant term of P iii. (2 points) The leading coefficient of P iv. (4 points) The possible rational zeros of P (b) (5 points) Find the quotient and remainder of when Q(x) = * - 223 +22 - 3 is divided by...