MA5509/18讲解、辅导Java/Python编程语言、讲解c/c++、data留学生辅导
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UNIVERSITY OF KENT
FACULTY OF SCIENCES
LEVEL 5 EXAMINATION
NUMERICAL METHODS
Saturday, 19 May 2018: 2:00pm – 4:00pm
This paper is divided into TWO sections as follows:
Section A: Six short questions each marked out of 10.
Candidates should attempt ALL SIX questions.
Section B: Two longer questions each marked out of 20.
Candidates should attempt BOTH questions.
Candidates are advised to show their working on their
scripts. Marks might then be allocated for use of a correct
method, even if the numerical or algebraic result is incorrect.
Calculators: Approved calculators are permitted.
Stationery: Yellow Answer Booklet.
Turn overMA5509/18 2
SECTION A
These questions will each be marked out of 10. Candidates
should attempt ALL SIX questions.
1. Given the data
(x0, y0) = (0, 0), (x1, y1) = (1, ?1), (x2, y2) = (2, 0), (x3, y3) = (3, 2),
construct the Lagrange interpolating polynomial P3(x) that interpolates these points in two
different ways: using Newton’s divided differences and using the barycentric formula.
[5+5 marks]
2. (a) Write the formula for the composite trapezoidal rule with n subintervals to approximate
the value of the integral
[2 marks]
(b) Is the composite trapezoidal rule an open or closed quadrature rule? Justify your answer.
[2 marks]
(c) We apply this rule to approximate the value of the following integral:
If the error of the composite trapezoidal rule is
with h = (b a)/n, then determine how many subintervals we need to take (at least) to
approximate the value of I with an absolute error smaller than 10?4
.[6 marks]
3. We want to approximate the solution of the initial value problem
y(x) = f(x, y(x)), y(x0) = y0
using the trapezoidal rule:
yn+1 = yn +
h
2
(f(xn, yn) + f(xn+1, yn+1)), n = 0, 1, . . . ,
where h = xn+1 xn.
(a) Is this method explicit or implicit? Justify your answer.
[2 marks]
(b) Show that this numerical method is of order 2, by studying the local truncation error.3 MA5509/18
[8 marks]
4. We consider the following function:f(x) = 1 + 2x + sin(x).
(a) Show that it has one root in the interval [0, π].
[3 marks]
(b) Using the derivative f0
(x), show that this root is unique.
[3 marks]
(c) We apply two different numerical methods to generate sequences {xn}n≥1 and approximate
the root, which is α ≈ 0.335418. The absolute errors ek = |α ? xn| are the following:
Iteration Error method 1 Error method 2
Based on these observations, would you say that the sequences generated with these methods
converge or diverge to the root? If they converge, what is the order in each case?
[4 marks]
5. We consider the initial value problem
y0(x) = f(x, y(x)), y(x0) = y0.
(a) Write down the explicit Euler method to solve such problem.
[4 marks]
(b) In the case f(x, y(x)) = 1 + y
2
(x), take y(0) = 0, h = 0.1 and calculate an approximate
value for y(0.5) using the explicit Euler method. What is the absolute error with respect to
the value of the exact solution at that point? (Hint: the exact solution is y(x) = tan x).
[6 marks]
6. (a) Find the coefficients A, B and C that maximise the order in the following centred
difference formula for the second derivative f
00(x):
D2[f, h] = Af(x + h) + Bf(x) + Cf(x ? h).
[6 marks]
(b) Determine the order of the resulting formula.
[4 marks]
Turn overMA5509/18 4
SECTION B
These questions will each be marked out of 20. Candidates
should attempt BOTH questions.
7. (a) Explain how the Newton–Raphson method is constructed to solve a general root-finding
problem f(x) = 0.
[5 marks]
(b) Let k ≥ 2, we apply the method of Newton-Raphson to the following function:
to compute the value of the root p = 21/k. Show that in this example, the Newton–Raphson
method is equivalent to the fixed point iteration x = g(x), where
[3 marks]
(c) If k = 3, use the general theorem about convergence of fixed point iteration to prove that
there is convergence to the root p = 21/3
starting with any initial value in the interval [1, 2].
[4 marks]
(d) What is the order of convergence of the fixed point iteration to p = 21/3 with this function
g(x)?[4 marks]
(e) Compute the approximations x1, . . . , x4 to p = 21/3
, using the fixed point iteration
xn+1 = g(xn) and starting from x0 = 2.
[4 marks]
8. (a) Define the Lagrange interpolating polynomial and the associated interpolation error for
a given function f(x) that is known at a set of points x0, . . . , xn in an interval [a, b].
[5 marks]
(b) Construct the Lagrange polynomial L2(x) with nodes x0 = 0, x1 = 1/2 and x2 = 1 for
the function
[3 marks]
(c) Integrate this Lagrange polynomial between 0 and 1 to get an approximate value Q of
the integral
(d) Calculate the exact value of the integral I and give the absolute and relative errors when
approximating I by Q.5 MA5509/18
[4 marks]
(e) If the Gauss–Legendre nodes and weights in [?1, 1] are given by
give the Gauss-Legendre nodes and weights on [0, 1] and calculate an approximation QGL for
I using them. What are the absolute and relative errors with respect to I this time?
[5 marks]
UNIVERSITY OF KENT
FACULTY OF SCIENCES
LEVEL 5 EXAMINATION
NUMERICAL METHODS
Saturday, 19 May 2018: 2:00pm – 4:00pm
This paper is divided into TWO sections as follows:
Section A: Six short questions each marked out of 10.
Candidates should attempt ALL SIX questions.
Section B: Two longer questions each marked out of 20.
Candidates should attempt BOTH questions.
Candidates are advised to show their working on their
scripts. Marks might then be allocated for use of a correct
method, even if the numerical or algebraic result is incorrect.
Calculators: Approved calculators are permitted.
Stationery: Yellow Answer Booklet.
Turn overMA5509/18 2
SECTION A
These questions will each be marked out of 10. Candidates
should attempt ALL SIX questions.
1. Given the data
(x0, y0) = (0, 0), (x1, y1) = (1, ?1), (x2, y2) = (2, 0), (x3, y3) = (3, 2),
construct the Lagrange interpolating polynomial P3(x) that interpolates these points in two
different ways: using Newton’s divided differences and using the barycentric formula.
[5+5 marks]
2. (a) Write the formula for the composite trapezoidal rule with n subintervals to approximate
the value of the integral
[2 marks]
(b) Is the composite trapezoidal rule an open or closed quadrature rule? Justify your answer.
[2 marks]
(c) We apply this rule to approximate the value of the following integral:
If the error of the composite trapezoidal rule is
with h = (b a)/n, then determine how many subintervals we need to take (at least) to
approximate the value of I with an absolute error smaller than 10?4
.[6 marks]
3. We want to approximate the solution of the initial value problem
y(x) = f(x, y(x)), y(x0) = y0
using the trapezoidal rule:
yn+1 = yn +
h
2
(f(xn, yn) + f(xn+1, yn+1)), n = 0, 1, . . . ,
where h = xn+1 xn.
(a) Is this method explicit or implicit? Justify your answer.
[2 marks]
(b) Show that this numerical method is of order 2, by studying the local truncation error.3 MA5509/18
[8 marks]
4. We consider the following function:f(x) = 1 + 2x + sin(x).
(a) Show that it has one root in the interval [0, π].
[3 marks]
(b) Using the derivative f0
(x), show that this root is unique.
[3 marks]
(c) We apply two different numerical methods to generate sequences {xn}n≥1 and approximate
the root, which is α ≈ 0.335418. The absolute errors ek = |α ? xn| are the following:
Iteration Error method 1 Error method 2
Based on these observations, would you say that the sequences generated with these methods
converge or diverge to the root? If they converge, what is the order in each case?
[4 marks]
5. We consider the initial value problem
y0(x) = f(x, y(x)), y(x0) = y0.
(a) Write down the explicit Euler method to solve such problem.
[4 marks]
(b) In the case f(x, y(x)) = 1 + y
2
(x), take y(0) = 0, h = 0.1 and calculate an approximate
value for y(0.5) using the explicit Euler method. What is the absolute error with respect to
the value of the exact solution at that point? (Hint: the exact solution is y(x) = tan x).
[6 marks]
6. (a) Find the coefficients A, B and C that maximise the order in the following centred
difference formula for the second derivative f
00(x):
D2[f, h] = Af(x + h) + Bf(x) + Cf(x ? h).
[6 marks]
(b) Determine the order of the resulting formula.
[4 marks]
Turn overMA5509/18 4
SECTION B
These questions will each be marked out of 20. Candidates
should attempt BOTH questions.
7. (a) Explain how the Newton–Raphson method is constructed to solve a general root-finding
problem f(x) = 0.
[5 marks]
(b) Let k ≥ 2, we apply the method of Newton-Raphson to the following function:
to compute the value of the root p = 21/k. Show that in this example, the Newton–Raphson
method is equivalent to the fixed point iteration x = g(x), where
[3 marks]
(c) If k = 3, use the general theorem about convergence of fixed point iteration to prove that
there is convergence to the root p = 21/3
starting with any initial value in the interval [1, 2].
[4 marks]
(d) What is the order of convergence of the fixed point iteration to p = 21/3 with this function
g(x)?[4 marks]
(e) Compute the approximations x1, . . . , x4 to p = 21/3
, using the fixed point iteration
xn+1 = g(xn) and starting from x0 = 2.
[4 marks]
8. (a) Define the Lagrange interpolating polynomial and the associated interpolation error for
a given function f(x) that is known at a set of points x0, . . . , xn in an interval [a, b].
[5 marks]
(b) Construct the Lagrange polynomial L2(x) with nodes x0 = 0, x1 = 1/2 and x2 = 1 for
the function
[3 marks]
(c) Integrate this Lagrange polynomial between 0 and 1 to get an approximate value Q of
the integral
(d) Calculate the exact value of the integral I and give the absolute and relative errors when
approximating I by Q.5 MA5509/18
[4 marks]
(e) If the Gauss–Legendre nodes and weights in [?1, 1] are given by
give the Gauss-Legendre nodes and weights on [0, 1] and calculate an approximation QGL for
I using them. What are the absolute and relative errors with respect to I this time?
[5 marks]