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Consider the following equation: el – 2 – 7 = 0 and the numerical approximation: X,...
Consider the following equation: et -x-7= 0 and the numerical approximation: x, = -6.99904573 of a...
Consider the following equation: et -x-7= 0 and the numerical approximation: x, = -6.99904573 of a solution of this equation Compute the backward error to one significant digit Answer: Consider the following equation: ex -x- 7 = 0 and the numerical approximation: x, = -6.99904573 of a solution of this equation Estimate the error |-x, to one significant digit ifr is the true root of the equation. Answer:
u(x,t) is solution to heat equation, ,with following parameters for numerical approximation: 0 < x <...
u(x,t) is solution to heat equation, ,with following parameters
for numerical approximation:
0 < x < 2, 0 < t < 0.1, n = 20, m =
100, c =1. Boundary conditions: u(0,t) =0, and u(2,0) = 0.
Initial conditions: u (x,0) =30o for 0<x<=1
0o
for 1<x<2
Set the approximate difference equation for this equation.
Do you think this equation converges to a numerical
solution.
Continuing with problem 1, calculate u(0.1,0.001) by
iteration
Continuing with problem 1, calculate u(0.2,0.001)...
this is numerical analysis QUESTION 1 (a) Apart from 1 = 0 the equation f(1) =...
this is numerical analysis
QUESTION 1 (a) Apart from 1 = 0 the equation f(1) = x2 - 4 sin r = 0 has another root in (1, 2.5). Perform three (10) iterations of the bisection method to approximate the root. State the accuracy of the root after the three iterations. (b) Perform three iterations of Newton's method for the function in (a) above, using x) = 1.5 as the initial (10) solution. Compare the error from the Newton's approximation...
2. (25 pts) Numerical differentiation. Numerical implementation. a. Compute the forward, central, and backward numerical first...
2. (25 pts) Numerical differentiation. Numerical implementation. a. Compute the forward, central, and backward numerical first derivative using, 2, 3, and 4 points for the function y = cos x at x = 7/4 using step size h = /12. Provide the results in the hard copy. Note that the central differences can only be apply for odd number of points ). b. Provide the analytic form of the derivatives, as well as table of the computed relative error for...
Consider the function /(x) = -3x - 2x2, the true value of its root is x*...
Consider the function /(x) = -3x - 2x2, the true value of its root is x* -0.392474. (1) Use the Newton-Raphson Method starting with x = 0, the new approximation to the root after one iteration is .x2 = __ __blank The true absolute error is EP =____blank 2__ (2) The new approximation to the root after one more iteration is x3 = ___ _blank 3 Use six significant digits for x; to calculate the true absolute error E -_...
ME/AE 342 - Numerical Methods in Engineering with Applications Homework 2 1. Use finite difference approximation...
ME/AE 342 - Numerical Methods in Engineering with Applications Homework 2 1. Use finite difference approximation to compute f'(2.36), /'(2.37) and /"(2.37) from the data 2.36 2.37 2.38 2 .39 0.85866 0.86289 0.86710 0.87129 1() 2. Estimate f'(1) from the following data using either forward or backward dif- ference 1 f(0) 0 .97 0.85040 1.00 1.05 0.84147 0.82612 3. Use polynomial interpolation to compute f' and l" at r = 0, using the data be- low. Hint: Find the Lagrange...
write the equation of the line that represents the linear approximation to the following function at...
write the equation of the line that represents the
linear approximation to the following function at the given point
a. answer a, b and c.
a. Write the equation of the line that represents the linear approximation to the following function at the given point a b. Use the linear approximation to estimate the given quantity approximation - exact C. Compute the percent error in the approximation, 100. where the exact value is given by a calculator exact f(x) =...
MATLAB HELP 3. Consider the equation y′ = y2 − 3x, where y(0) = 1. USE THE EULER AND RUNGE-KUTTA APPROXIMATION SCRIPTS...
MATLAB HELP 3. Consider the equation y′ = y2 − 3x, where y(0) =
1. USE THE EULER AND RUNGE-KUTTA APPROXIMATION SCRIPTS
PROVIDED IN THE PICTURES
a. Use a Euler approximation with a step size of 0.25 to
approximate y(2).
b. Use a Runge-Kutta approximation with a step size of 0.25 to
approximate y(2).
c. Graph both approximation functions in the same window as a
slope field for the differential equation.
d. Find a formula for the actual solution (not...
(3) Consider the expressions (a) Write down the Runge-Kutta method for the numerical solution to a differential equation Oy (b) Show that if f is independent of y, i.e. f(x, y) g(x) for some g, t...
(3) Consider the expressions (a) Write down the Runge-Kutta method for the numerical solution to a differential equation Oy (b) Show that if f is independent of y, i.e. f(x, y) g(x) for some g, then the Runge-Kutta method on the interval n n + h] becomes Simpson's Rule for the numerical approximation of the integral g(x) dr. In this case, what is the global error, in terms of O(hk) for some k>0?
(3) Consider the expressions (a) Write down...
Multiple choice - numerical methods Multiple-Choice Test Measuring Errors I. True error is defined as a)...
Multiple choice
- numerical methods
Multiple-Choice Test Measuring Errors I. True error is defined as a) Present Approximation Previous ) True Value- Approximate Value oabs (True Value- Approximate Value) D) abs (Present Approximation-Previous 2 The expression for true error in calculating the derivative of-er) at … 4 by using the approximate expression EA の 间 The relative approximate error at the end of an iteration to find the root of an equation is ome· The least number of significant digits...
Consider the following equation: et -x-7= 0 and the numerical approximation: x, = -6.99904573 of a solution of this equation Compute the backward error to one significant digit Answer: Consider the following equation: ex -x- 7 = 0 and the numerical approximation: x, = -6.99904573 of a solution of this equation Estimate the error |-x, to one significant digit ifr is the true root of the equation. Answer:
u(x,t) is solution to heat equation, ,with following parameters
for numerical approximation:
0 < x < 2, 0 < t < 0.1, n = 20, m =
100, c =1. Boundary conditions: u(0,t) =0, and u(2,0) = 0.
Initial conditions: u (x,0) =30o for 0<x<=1
0o
for 1<x<2
Set the approximate difference equation for this equation.
Do you think this equation converges to a numerical
solution.
Continuing with problem 1, calculate u(0.1,0.001) by
iteration
Continuing with problem 1, calculate u(0.2,0.001)...
this is numerical analysis
QUESTION 1 (a) Apart from 1 = 0 the equation f(1) = x2 - 4 sin r = 0 has another root in (1, 2.5). Perform three (10) iterations of the bisection method to approximate the root. State the accuracy of the root after the three iterations. (b) Perform three iterations of Newton's method for the function in (a) above, using x) = 1.5 as the initial (10) solution. Compare the error from the Newton's approximation...
2. (25 pts) Numerical differentiation. Numerical implementation. a. Compute the forward, central, and backward numerical first derivative using, 2, 3, and 4 points for the function y = cos x at x = 7/4 using step size h = /12. Provide the results in the hard copy. Note that the central differences can only be apply for odd number of points ). b. Provide the analytic form of the derivatives, as well as table of the computed relative error for...
Consider the function /(x) = -3x - 2x2, the true value of its root is x* -0.392474. (1) Use the Newton-Raphson Method starting with x = 0, the new approximation to the root after one iteration is .x2 = __ __blank The true absolute error is EP =____blank 2__ (2) The new approximation to the root after one more iteration is x3 = ___ _blank 3 Use six significant digits for x; to calculate the true absolute error E -_...
ME/AE 342 - Numerical Methods in Engineering with Applications Homework 2 1. Use finite difference approximation to compute f'(2.36), /'(2.37) and /"(2.37) from the data 2.36 2.37 2.38 2 .39 0.85866 0.86289 0.86710 0.87129 1() 2. Estimate f'(1) from the following data using either forward or backward dif- ference 1 f(0) 0 .97 0.85040 1.00 1.05 0.84147 0.82612 3. Use polynomial interpolation to compute f' and l" at r = 0, using the data be- low. Hint: Find the Lagrange...
write the equation of the line that represents the
linear approximation to the following function at the given point
a. answer a, b and c.
a. Write the equation of the line that represents the linear approximation to the following function at the given point a b. Use the linear approximation to estimate the given quantity approximation - exact C. Compute the percent error in the approximation, 100. where the exact value is given by a calculator exact f(x) =...
MATLAB HELP 3. Consider the equation y′ = y2 − 3x, where y(0) =
1. USE THE EULER AND RUNGE-KUTTA APPROXIMATION SCRIPTS
PROVIDED IN THE PICTURES
a. Use a Euler approximation with a step size of 0.25 to
approximate y(2).
b. Use a Runge-Kutta approximation with a step size of 0.25 to
approximate y(2).
c. Graph both approximation functions in the same window as a
slope field for the differential equation.
d. Find a formula for the actual solution (not...
(3) Consider the expressions (a) Write down the Runge-Kutta method for the numerical solution to a differential equation Oy (b) Show that if f is independent of y, i.e. f(x, y) g(x) for some g, then the Runge-Kutta method on the interval n n + h] becomes Simpson's Rule for the numerical approximation of the integral g(x) dr. In this case, what is the global error, in terms of O(hk) for some k>0?
(3) Consider the expressions (a) Write down...
Multiple choice
- numerical methods
Multiple-Choice Test Measuring Errors I. True error is defined as a) Present Approximation Previous ) True Value- Approximate Value oabs (True Value- Approximate Value) D) abs (Present Approximation-Previous 2 The expression for true error in calculating the derivative of-er) at … 4 by using the approximate expression EA の 间 The relative approximate error at the end of an iteration to find the root of an equation is ome· The least number of significant digits...
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