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b) Suppose we wish to find the solution of a nonlinear equation of the form sin(x-0.36)--sin(0.36)e.xo 38 e-x/0.38 Describe briefly how you can use a numerical optimization method to find a solution...
numerical analysis ANSWER ALL QUESTIONS 1) Suppose we are looking for a solution of the equation:...
numerical analysis
ANSWER ALL QUESTIONS 1) Suppose we are looking for a solution of the equation: e a) Show that there is a solution in the interval [0, 1. 25x2 b) How many iterations of the bisection method would be required to approximate the solution with an error less than .001? c) Suppose we wrote the equation in the form : x= g(x) = In(25x2) and tried to find the solution by iterating x.l g(xm). Would the sequence converge with...
»lem 2(*): Suppose that we want to find the best equation of the form y -c c2t + 2 C3 sin(nt to d...
»lem 2(*): Suppose that we want to find the best equation of the form y -c c2t + 2 C3 sin(nt to describe some observed data we are given the data points IA , , за , 0 where each entry is of the form Our goal is to find the best solutions in the least squares sense. » Set up the system of equations in variables c1, c2, c3 determined by the data points Write the system in matrix...
Consider Newton's method for solving the scalar nonlinear equation f(x) = 0. Suppose we replace the...
Consider Newton's method for solving the scalar nonlinear equation f(x) = 0. Suppose we replace the derivative f'(xx) with a constant value d and use the iteration (a) Under what condition for d will this iteration be locally convergent? (b) What is the convergence rate in general? (c) Is there a value for d that would lead to quadratic convergence?
(1 point) Suppose that we use Euler's method to approximate the solution to the differential equation...
(1 point) Suppose that we use Euler's method to approximate the solution to the differential equation dyr. dzvi y(0.4) = 9. Let f(x, y) = 25/y. We let Xo = 0.4 and yo = 9 and pick a step size h=0.2. Euler's method is the the following algorithm. From In and Yn, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing In+1 = xin + h Y n+1 =...
In this exercise we consider the second order linear equation y" therefore has a power series solution in the form 4y = 0. This equation has an ordinary point at x = 0 and We learned how to easil...
In this exercise we consider the second order linear equation y" therefore has a power series solution in the form 4y = 0. This equation has an ordinary point at x = 0 and We learned how to easily solve problems like this in several different ways but here we want to consider the power series method (1) Insert the formal power series into the differential equation and derive the recurrence relation Cn-2 for n - 2, 3, NOTE co...
In the lectures, we introduced Gradient Descent, an optimization method to find the minimum value of...
In the lectures, we introduced Gradient Descent, an optimization method to find the minimum value of a function. In this problem we try to solve a fairly simple optimization problem: min f(x) = x2 TER That is, finding the minimum value of x2 over the real line. Of course you know it is when x = 0, but this time we do it with gradient descent. Recall that to perform gradient descent, you start at an arbitrary initial point xo,...
1. Numerical Integration The integral of a function f(x) for a s x S b can be interpreted as the ...
1. Numerical Integration The integral of a function f(x) for a s x S b can be interpreted as the area between the f(x) curve and the x axis, bounded by the limits x- a and x b. If we denote this area by A, then we can write A as A-f(x)dx A sophisticated method to find the area under a curve is to split the area into trapezoidal elements. Each trapezoid is called a panel. 1.2 0.2 1.2 13...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solutio...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
You are given that there exists a solution to the equation et = 4 sin (2)...
You are given that there exists a solution to the equation et = 4 sin (2) in the interval [0, 0.5]. a) Use the iterative method explained during the course to find this solution up to 3 decimal places. Make sure you include enough details of the calculation and that your cal- culator is set in radians for this problem. [3 marks] b) Now estimate the same solution using the Maclaurin expansion of the function f(x) = e" – 4...
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each of the variables is defined implicitly as a function of the others. 2 a) If F and z(x, y) are both assumed to b...
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each of the variables is defined implicitly as a function of the others. 2 a) If F and z(x, y) are both assumed to be differentiable, fnd in terms of partial derivatives of F. b) Under similar assumptions on the other variables, find
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each...
numerical analysis
ANSWER ALL QUESTIONS 1) Suppose we are looking for a solution of the equation: e a) Show that there is a solution in the interval [0, 1. 25x2 b) How many iterations of the bisection method would be required to approximate the solution with an error less than .001? c) Suppose we wrote the equation in the form : x= g(x) = In(25x2) and tried to find the solution by iterating x.l g(xm). Would the sequence converge with...
»lem 2(*): Suppose that we want to find the best equation of the form y -c c2t + 2 C3 sin(nt to describe some observed data we are given the data points IA , , за , 0 where each entry is of the form Our goal is to find the best solutions in the least squares sense. » Set up the system of equations in variables c1, c2, c3 determined by the data points Write the system in matrix...
Consider Newton's method for solving the scalar nonlinear equation f(x) = 0. Suppose we replace the derivative f'(xx) with a constant value d and use the iteration (a) Under what condition for d will this iteration be locally convergent? (b) What is the convergence rate in general? (c) Is there a value for d that would lead to quadratic convergence?
(1 point) Suppose that we use Euler's method to approximate the solution to the differential equation dyr. dzvi y(0.4) = 9. Let f(x, y) = 25/y. We let Xo = 0.4 and yo = 9 and pick a step size h=0.2. Euler's method is the the following algorithm. From In and Yn, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing In+1 = xin + h Y n+1 =...
In this exercise we consider the second order linear equation y" therefore has a power series solution in the form 4y = 0. This equation has an ordinary point at x = 0 and We learned how to easily solve problems like this in several different ways but here we want to consider the power series method (1) Insert the formal power series into the differential equation and derive the recurrence relation Cn-2 for n - 2, 3, NOTE co...
In the lectures, we introduced Gradient Descent, an optimization method to find the minimum value of a function. In this problem we try to solve a fairly simple optimization problem: min f(x) = x2 TER That is, finding the minimum value of x2 over the real line. Of course you know it is when x = 0, but this time we do it with gradient descent. Recall that to perform gradient descent, you start at an arbitrary initial point xo,...
1. Numerical Integration The integral of a function f(x) for a s x S b can be interpreted as the area between the f(x) curve and the x axis, bounded by the limits x- a and x b. If we denote this area by A, then we can write A as A-f(x)dx A sophisticated method to find the area under a curve is to split the area into trapezoidal elements. Each trapezoid is called a panel. 1.2 0.2 1.2 13...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
You are given that there exists a solution to the equation et = 4 sin (2) in the interval [0, 0.5]. a) Use the iterative method explained during the course to find this solution up to 3 decimal places. Make sure you include enough details of the calculation and that your cal- culator is set in radians for this problem. [3 marks] b) Now estimate the same solution using the Maclaurin expansion of the function f(x) = e" – 4...
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each of the variables is defined implicitly as a function of the others. 2 a) If F and z(x, y) are both assumed to be differentiable, fnd in terms of partial derivatives of F. b) Under similar assumptions on the other variables, find
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each...
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