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4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree...
Given the following quadrature formula: (x)dx = c0) + f(a) Find a 4 and 2 so...
Given the following quadrature formula: (x)dx = c0) + f(a) Find a 4 and 2 so that the quadrature formula has the highest degree of precision. Clearly, state its degree. ſrcade -
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where...
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where P2(x) is the polynomial of degree at most two which interpolates f at 0, 1/2, and 1. (a) Write P2(x) in Lagrange form and prove that Q[F] o [s0 f(0) + 4f 45 (2) +scn)] (2) (b) Consider now a general interval [a, b] and the integral só f(x)dx. Do the change of variables x = a + (b − a)t to transform the...
this is numerical analysis. Please do all the questions 3. (a) Consider the quadrature rule path...
this is numerical analysis. Please do all the questions
3. (a) Consider the quadrature rule path ( * s(a)dx = Af (a – 1) + Bf(a) + Cf(a+h). Find A, B, C which maximize the degree of precision. Hint: First derive the rule for a = 0 and then use a change of variable. (b) State this degree of precision and verify it is not any higher. (c) Suppase g is a function whose 3rd divided differences are all the...
Problem 1. Consider an integral of the form 1 (f) = f0 /if (x) dx. Show that the one-point Gaussian quadrature rule for integrals of the form Jo VRf (x) dx has the node x1and weight w3 Probl...
Problem 1. Consider an integral of the form 1 (f) = f0 /if (x) dx. Show that the one-point Gaussian quadrature rule for integrals of the form Jo VRf (x) dx has the node x1and weight w3
Problem 1. Consider an integral of the form 1 (f) = f0 /if (x) dx. Show that the one-point Gaussian quadrature rule for integrals of the form Jo VRf (x) dx has the node x1and weight w3
Problem 1 (1) Derive a basic quadrature rule RM(f) to approximate I= f(r)dr by integrating an...
Problem 1 (1) Derive a basic quadrature rule RM(f) to approximate I= f(r)dr by integrating an interpolating polynomial po(r) of degree 0 that interpolates one data point generated by f (x) at the node (a+b)/2. (2) Give a geomet- ric interpretation of the rule and then derive the rule using the geometric interpretation.
6. Compute four Legendre polynomials degree 0, 1, 2 and 3, respectively. You can assume that...
6. Compute four Legendre polynomials degree 0, 1, 2 and 3, respectively. You can assume that these polynomials endre polynomial to construct a Gaussian quadrature. Approximate the value of the integral are monic. Use the roots of the cubic Leg- sin(2x) dx using your quadrature rule.
6. Compute four Legendre polynomials degree 0, 1, 2 and 3, respectively. You can assume that these polynomials endre polynomial to construct a Gaussian quadrature. Approximate the value of the integral are monic. Use...
how to do question 3? "normal equations" for the line's coefficients from the Error Function E. 3. Le (x) = VX + 1 . Use Adaptive Quadrature Simpson's Rule with n = 4 to 2 and n estim...
how to do question 3?
"normal equations" for the line's coefficients from the Error Function E. 3. Le (x) = VX + 1 . Use Adaptive Quadrature Simpson's Rule with n = 4 to 2 and n estimate J f Cr)dx and find the Absolute and Estimated Errors. 2 20p 0 in initial value probler
"normal equations" for the line's coefficients from the Error Function E. 3. Le (x) = VX + 1 . Use Adaptive Quadrature Simpson's Rule with...
3. (1 point) This is 2-point Gaussian Quadrature for any f(x) dx. The weights and the...
3. (1 point) This is 2-point Gaussian Quadrature for any f(x) dx. The weights and the nodes do not change when we integrate a new function. So...does it work? Does this actually lead to a method that is good for intergating functions in general? Use 2-point Gaussian quadrature to approximate the following integral: I e-ra da. -1 The exact value of this integral rounded to 2 decimal places is 1.49. Show your work when computing the approximation. Report the answer...
Consider the differential equation: (7y sin(xy) + 2 sec x) dx = (2 lny – 4x...
Consider the differential equation: (7y sin(xy) + 2 sec x) dx = (2 lny – 4x sin(xy))dy Note: Do not use square brackets in your response, use normal parantheses if you have to, i.e "0" Then aM ду and ƏN ax Is this equation exact? Yes No Consider the differential equation: sin(x)dx + 5y cos(x)dy = 0 Which of the following can be an integrating factor to make the equation exact? Select all that apply. On=e-54 On=tan(x) Ju=e-542/2 On =...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-x...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n.
Problem 4 Let V be the vector space of functions of...
Given the following quadrature formula: (x)dx = c0) + f(a) Find a 4 and 2 so that the quadrature formula has the highest degree of precision. Clearly, state its degree. ſrcade -
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where P2(x) is the polynomial of degree at most two which interpolates f at 0, 1/2, and 1. (a) Write P2(x) in Lagrange form and prove that Q[F] o [s0 f(0) + 4f 45 (2) +scn)] (2) (b) Consider now a general interval [a, b] and the integral só f(x)dx. Do the change of variables x = a + (b − a)t to transform the...
this is numerical analysis. Please do all the questions
3. (a) Consider the quadrature rule path ( * s(a)dx = Af (a – 1) + Bf(a) + Cf(a+h). Find A, B, C which maximize the degree of precision. Hint: First derive the rule for a = 0 and then use a change of variable. (b) State this degree of precision and verify it is not any higher. (c) Suppase g is a function whose 3rd divided differences are all the...
Problem 1. Consider an integral of the form 1 (f) = f0 /if (x) dx. Show that the one-point Gaussian quadrature rule for integrals of the form Jo VRf (x) dx has the node x1and weight w3
Problem 1. Consider an integral of the form 1 (f) = f0 /if (x) dx. Show that the one-point Gaussian quadrature rule for integrals of the form Jo VRf (x) dx has the node x1and weight w3
Problem 1 (1) Derive a basic quadrature rule RM(f) to approximate I= f(r)dr by integrating an interpolating polynomial po(r) of degree 0 that interpolates one data point generated by f (x) at the node (a+b)/2. (2) Give a geomet- ric interpretation of the rule and then derive the rule using the geometric interpretation.
6. Compute four Legendre polynomials degree 0, 1, 2 and 3, respectively. You can assume that these polynomials endre polynomial to construct a Gaussian quadrature. Approximate the value of the integral are monic. Use the roots of the cubic Leg- sin(2x) dx using your quadrature rule.
6. Compute four Legendre polynomials degree 0, 1, 2 and 3, respectively. You can assume that these polynomials endre polynomial to construct a Gaussian quadrature. Approximate the value of the integral are monic. Use...
how to do question 3?
"normal equations" for the line's coefficients from the Error Function E. 3. Le (x) = VX + 1 . Use Adaptive Quadrature Simpson's Rule with n = 4 to 2 and n estimate J f Cr)dx and find the Absolute and Estimated Errors. 2 20p 0 in initial value probler
"normal equations" for the line's coefficients from the Error Function E. 3. Le (x) = VX + 1 . Use Adaptive Quadrature Simpson's Rule with...
3. (1 point) This is 2-point Gaussian Quadrature for any f(x) dx. The weights and the nodes do not change when we integrate a new function. So...does it work? Does this actually lead to a method that is good for intergating functions in general? Use 2-point Gaussian quadrature to approximate the following integral: I e-ra da. -1 The exact value of this integral rounded to 2 decimal places is 1.49. Show your work when computing the approximation. Report the answer...
Consider the differential equation: (7y sin(xy) + 2 sec x) dx = (2 lny – 4x sin(xy))dy Note: Do not use square brackets in your response, use normal parantheses if you have to, i.e "0" Then aM ду and ƏN ax Is this equation exact? Yes No Consider the differential equation: sin(x)dx + 5y cos(x)dy = 0 Which of the following can be an integrating factor to make the equation exact? Select all that apply. On=e-54 On=tan(x) Ju=e-542/2 On =...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n.
Problem 4 Let V be the vector space of functions of...
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