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(b) Consider the nonlinear system (10) IP +y = 9 r? + y2 = 25, *20....
QUESTION 4 (10) (a) A natural cubic spline that fits the data given by f(3.0) =...
QUESTION 4 (10) (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, f(3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system (10) 2+y=9, 22 + y2 = 25, 2,y> 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as the initial solution. [20]
its a numerical analysis question QUESTION 4 (a) A natural cubic spline that fits the data...
its a numerical analysis question
QUESTION 4 (a) A natural cubic spline that fits the data given by (10) f(3.0) = -5.6790, f(3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system (10) z+ y = 9, 1² + y2 = 25, 2, y > 0. Perform one iteration of Newton's method to approximate the solution,...
QUESTION 4 (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790,...
QUESTION 4 (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, F (3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system x2 + y = 9, 22 + y2 = 25, x, y > 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as...
4. [Newton's Method, 10 pts] Consider the system of two nonlinear equations 2y- sinz 0 (o)...
4. [Newton's Method, 10 pts] Consider the system of two nonlinear equations 2y- sinz 0 (o) Write dowrn the vector-valued function whome zero f (x,y) g(x, )Whose zero we seek. (b) Calculate the Jacobian of F. (c) Apply one step of Newton's Method to initial iterate Po 0o to obtain the next iterate pi
Problem Two: (Based on Chapra, Problems 12.9 Consider the simultaneous nonlinear equations: 2-5-y y+i- 1. Plot...
Problem Two: (Based on Chapra, Problems 12.9 Consider the simultaneous nonlinear equations: 2-5-y y+i- 1. Plot the equations and identify the solution graphically. Page 1 of 2 2. Solve the system of equations using successive substitution, starting with the initial guess xo-y-1.5. Show two complete iterations. Evaluate &s for the second iteration. 3. Redo Part 2 using Newton-Raphson method . Automate the solutions in Parts 2 and 3 using MATLAB scripts 5. Solve the system of nonlinear equations by calling...
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change...
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to polar coordinates and find the solution r(t) and (t) of the system (b) As t goes to infinity, (x(t). y(t)) will follow the circle trajectory. Find the radius and period of the circle trajectory. (limit behavior of the solution (a))
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to...
Using the "Newton's Method" Write a MATLAB script to solve for the following nonlinear system of...
Using the "Newton's Method" Write a MATLAB script to solve for the following nonlinear system of equations: x2 + y2 + z2 = 3 x2 + y2 - z = 1 x + y + z =3 using the initial guess (x,y,z) = (1,0,1), tolerance tol = 1e-7, and maximum number of iterations maxiter = 20.
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system...
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system t' =-y+px(x² + y) (4) y = 1+ y(x² + y2), where is a parameter. Obviously, the origin x* = (0,0) is a fixed point of (4). (e) The solution of the ODE for o(t) is obvious - the angle o increases at a constant rate. Without solving the ODE for r(t), explain how r(t) behaves when t o in the cases H<0,1 =...
hele please 2. Will the 14 -1 01 (e) Let A = -1 4 -1, b...
hele please
2. Will the 14 -1 01 (e) Let A = -1 4 -1, b E R3. The eigenvalues of A are 4, 4 0-14 Jacobi iteration converge to a solution of Ax=b? Explain. (f) Consider the 2 x 2 nonlinear system of equations for x = 0, y): cy=1 =y What are the two solutions of this system of equations? What is the set of all starting points for which Newton's method will immediately fail?
4. (20 pts) In this problem, we combine the Steepest Descent method with Newton's method for solv...
4. (20 pts) In this problem, we combine the Steepest Descent method with Newton's method for solving the following nonlinear system. en +en-13 = 0, 12-2113 = 4. Use the Steepest Descent method with initial approximation x0,0,0 three iterations x(1), x(2), and x(3) to find the first ·Use x(3) fron the above the result as the initial approximation for Newton's iteration. Use the stopping criteria X(k)-s(k 1) < tol = 10 9 Display the results for each iteration in the...
QUESTION 4 (10) (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, f(3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system (10) 2+y=9, 22 + y2 = 25, 2,y> 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as the initial solution. [20]
its a numerical analysis question
QUESTION 4 (a) A natural cubic spline that fits the data given by (10) f(3.0) = -5.6790, f(3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system (10) z+ y = 9, 1² + y2 = 25, 2, y > 0. Perform one iteration of Newton's method to approximate the solution,...
QUESTION 4 (a) A natural cubic spline that fits the data given by f(3.0) = -5.6790, F (3.1) = -3.6674, f(3.2) = -2.2178 is to be constructed. Write down explicitly the system of equations that need to be used to construct the required natural cubic spline. (b) Consider the nonlinear system x2 + y = 9, 22 + y2 = 25, x, y > 0. Perform one iteration of Newton's method to approximate the solution, starting with (2, 4) as...
4. [Newton's Method, 10 pts] Consider the system of two nonlinear equations 2y- sinz 0 (o) Write dowrn the vector-valued function whome zero f (x,y) g(x, )Whose zero we seek. (b) Calculate the Jacobian of F. (c) Apply one step of Newton's Method to initial iterate Po 0o to obtain the next iterate pi
Problem Two: (Based on Chapra, Problems 12.9 Consider the simultaneous nonlinear equations: 2-5-y y+i- 1. Plot the equations and identify the solution graphically. Page 1 of 2 2. Solve the system of equations using successive substitution, starting with the initial guess xo-y-1.5. Show two complete iterations. Evaluate &s for the second iteration. 3. Redo Part 2 using Newton-Raphson method . Automate the solutions in Parts 2 and 3 using MATLAB scripts 5. Solve the system of nonlinear equations by calling...
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to polar coordinates and find the solution r(t) and (t) of the system (b) As t goes to infinity, (x(t). y(t)) will follow the circle trajectory. Find the radius and period of the circle trajectory. (limit behavior of the solution (a))
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to...
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system t' =-y+px(x² + y) (4) y = 1+ y(x² + y2), where is a parameter. Obviously, the origin x* = (0,0) is a fixed point of (4). (e) The solution of the ODE for o(t) is obvious - the angle o increases at a constant rate. Without solving the ODE for r(t), explain how r(t) behaves when t o in the cases H<0,1 =...
hele please
2. Will the 14 -1 01 (e) Let A = -1 4 -1, b E R3. The eigenvalues of A are 4, 4 0-14 Jacobi iteration converge to a solution of Ax=b? Explain. (f) Consider the 2 x 2 nonlinear system of equations for x = 0, y): cy=1 =y What are the two solutions of this system of equations? What is the set of all starting points for which Newton's method will immediately fail?
4. (20 pts) In this problem, we combine the Steepest Descent method with Newton's method for solving the following nonlinear system. en +en-13 = 0, 12-2113 = 4. Use the Steepest Descent method with initial approximation x0,0,0 three iterations x(1), x(2), and x(3) to find the first ·Use x(3) fron the above the result as the initial approximation for Newton's iteration. Use the stopping criteria X(k)-s(k 1) < tol = 10 9 Display the results for each iteration in the...
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