(1) Consider the Fixed Point iteration algorithm defined by the formula In+1 = 9(xn), where g(x) = x - a+ 2ae-. Here a E R is a parameter. (a) Find the fixed point, p. (b) Does there exist a value of a for which the iterations could converge quadratically? If yes, find it and explain your answer.
(1) Consider the Fixed Point iteration algorithm defined by the formula In+1 = 9(xn), where g(x) = x - a+ 2ae-. Here a E R is a parameter. (a) Find the fixed point, p. (b) Does there exist a value of a for which the iterations could converge quadratically? If yes, find it and explain your answer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(1) Consider the Fixed Point iteration algorithm defined by the formula xn+1 = g(xn), where g(x)
x – a + 2ae*. Here a E R is a parameter.
(a) Find the fixed point, p.
(b) Does there exist a value of a for which the iterations could converge quadratically? If yes, find
it and explain your answer.
(2) Consider the following table of values of f(x) = e".
| 0.0
f(x)| 1.0 1.22140
0.2
0.6
1.82212
(a) Approximate f(0.5) using the given data and the Newton forward divided difference formula.
(b) Find the best upper bound for the error of the approximation in (a).
(3) Consider approximating
e2r sin(3x) dx.
(a) Evaluate the integral.
(b) Approximate the integral by
(i) Composite Trapezoidal rule with n = 2
(ii) Composite Simpson's rule with n = 2
(iii) Composite Midpoint rule with n = 2
(y² + y)
dy
(4) Use the following methods to approximate y(1.2) for the initial value problem
dt
%3D
t
1<t<1.2, y(1) = -2 and h = 0.1.
(a) Euler method
(b) Taylor method of order 2
(c) Runge-Kutta method of order 4
(5) Consider the Adams-Bashforth method
wo = a; wi = a1;
Wi+1 = Wi + (3f(t;, wi) – f(ii-1, Wi–1)]
to solve the equation y' = f(t, y), a <t<b, y(a) = a.
(a) Give the local truncation error at the (i+ 1)st step. Eliminate any reference to f in your answer.
(b) The local truncation error in part (a) is of order O(hP) for a certain p. By expanding the terms in
the numerator in suitable Taylor polynomials (or some other way) simplify the local truntion error
and determine p.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff12449fc-9597-482d-8ea2-3c43f22083d9%2Feeb056d8-b73f-4ad9-976d-d4b712617502%2Frb2eszs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(1) Consider the Fixed Point iteration algorithm defined by the formula xn+1 = g(xn), where g(x)
x – a + 2ae*. Here a E R is a parameter.
(a) Find the fixed point, p.
(b) Does there exist a value of a for which the iterations could converge quadratically? If yes, find
it and explain your answer.
(2) Consider the following table of values of f(x) = e".
| 0.0
f(x)| 1.0 1.22140
0.2
0.6
1.82212
(a) Approximate f(0.5) using the given data and the Newton forward divided difference formula.
(b) Find the best upper bound for the error of the approximation in (a).
(3) Consider approximating
e2r sin(3x) dx.
(a) Evaluate the integral.
(b) Approximate the integral by
(i) Composite Trapezoidal rule with n = 2
(ii) Composite Simpson's rule with n = 2
(iii) Composite Midpoint rule with n = 2
(y² + y)
dy
(4) Use the following methods to approximate y(1.2) for the initial value problem
dt
%3D
t
1<t<1.2, y(1) = -2 and h = 0.1.
(a) Euler method
(b) Taylor method of order 2
(c) Runge-Kutta method of order 4
(5) Consider the Adams-Bashforth method
wo = a; wi = a1;
Wi+1 = Wi + (3f(t;, wi) – f(ii-1, Wi–1)]
to solve the equation y' = f(t, y), a <t<b, y(a) = a.
(a) Give the local truncation error at the (i+ 1)st step. Eliminate any reference to f in your answer.
(b) The local truncation error in part (a) is of order O(hP) for a certain p. By expanding the terms in
the numerator in suitable Taylor polynomials (or some other way) simplify the local truntion error
and determine p.
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