1. Apply the the model function Gauss-Newton method to the least squares problem using xit x2 + t for the data set y = t₁ 2 4 6 8 Yi 5 6 7 8 starting with x = (1,1)T. Don't compute the solution at the first set, write only the equations for the Gauss-Newton iteration.
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- A function, z = ax + by, is to be optimized subject to the constraint, x2 + y2=1 where a and b are positive constants. Use Lagrange multipliers to show that this problem has only one solution in the positive quadrant (i.e. in the region x > 0, y > 0) and that the optimal value of z is √a2 +b2.Can u proof the Newton's method would converge in this nonlinear system??Find a formula for the least-squares solution of Ax =b when the columns of A are orthonormal.
- 1. How can you measure the quality of the linear approximation offered by the above model?What is meant by R 2 = 1 and R 2 = 0?Suppose measurements of y versus x give three points in the x-y plane withcoordinates (0,6), (1,0), (2,0). (a) Show that there is no straight line that intersects all three points. (b) Find the linear function y = c0 + c1x that best fits this data in the least squares sense, by finding the coefficients c0, c1 using the method of least squares. (c) Show that the solution is the best possible in the sense that it minimizes the length of the residual.Show that the equation x2 + y2 = 2003 has no solutions in theintegers.
- The total cost function of a firm that produces its product on two assembly lines is given as: subject to the constraint: TC= 3X2+6y2 - XY subject to the constraint: X+Y = 20 The problem facing the firm is to determine the least-cost combination of output on assembly lines X and Y subject to the side condition that total output equal 20 units. (A)Use the Lagrangian multiplier method to determine X and Y that lead to the minimization of TC. (B) Verify that the values of X and Y minimize the TC, then Find the TC given the values of X and Y. (C)Interpret the value of the Lagrangian multiplier.The coefficient matrix is not strictly diagonally dominant, nor can the equations be rearranged to make it so. However, both the Jacobi and the Gauss-Seidel method converge anyway. Demonstrate that this is true of the Gauss-Seidel method, starting with the zero vector as the initial approximation and obtaining a solution that is accurate to within 0.01.4. Find the orthogonal trajectories of y^2=cx^3.
- Show that this model has an unbounded solution by Big M Method. Max Z= 3x1 + 6x2 s.to 3x1 + 4x2 ≥ 12 -2x1+ x2 ≤ 4 x1, x2 ≥ 0Can I get some assistance with this coordinatization problem?Given the system of equationsx − 5y − z = −84x + y − z = 132x − y − 6z = −2Start with P0 = (0, 0, 0) and use Gauss-Seidel iteration to fi nd Pk for k = 1, 2, 3.Will Gauss-Seidel iteration converge to the solution