= 1. (a) Show that the function f(x) exp(x) has only one zero, x*. Write down the general step of Newton's method for solving f(x) = 0. Starting with an initial guess of xo = 0, calculate the next three approximations to the solution of f(x) = exp(x) — 3 = 0 using Newton's method. For each iteration calculate the error from the exact solution. = Write down the map xn+1 g(xn) corresponding to the Newton's method solution of f(x) = exp(x) − 3 = 0 and show that g'(x) 0 at the fixed point. = State the fixed point theorem giving the conditions that guarantee the iteration scheme xn+1 = = g(xn) is stable and has a unique fixed point in the interval [a, b]. Show that g(x) satisfies the conditions of this theorem in a neighbourhood of x*. (b) Write down the Lagrange form of the interpolating polynomial P(x) that satisfies P(x) = f(x) for x = xi, i = 1,...,n and x1 < X2 < · · · < Xn• (1) = 1 through the points x Find the form of the polynomial that interpolates f(x) x1 = 0.5, x2 1 and x3 = 3, then simplify it. Use this polynomial to estimate the value of f(2) and find the error from the actual value. Using the error formula for Lagrange interpolation, n f(x) = P(x) + X n! dr f xi)- (§), dxn for some ε = (x1,xn), (2) i=1 find upper and lower bounds for the error in your estimate above. How does this compare with the actual error?
= 1. (a) Show that the function f(x) exp(x) has only one zero, x*. Write down the general step of Newton's method for solving f(x) = 0. Starting with an initial guess of xo = 0, calculate the next three approximations to the solution of f(x) = exp(x) — 3 = 0 using Newton's method. For each iteration calculate the error from the exact solution. = Write down the map xn+1 g(xn) corresponding to the Newton's method solution of f(x) = exp(x) − 3 = 0 and show that g'(x) 0 at the fixed point. = State the fixed point theorem giving the conditions that guarantee the iteration scheme xn+1 = = g(xn) is stable and has a unique fixed point in the interval [a, b]. Show that g(x) satisfies the conditions of this theorem in a neighbourhood of x*. (b) Write down the Lagrange form of the interpolating polynomial P(x) that satisfies P(x) = f(x) for x = xi, i = 1,...,n and x1 < X2 < · · · < Xn• (1) = 1 through the points x Find the form of the polynomial that interpolates f(x) x1 = 0.5, x2 1 and x3 = 3, then simplify it. Use this polynomial to estimate the value of f(2) and find the error from the actual value. Using the error formula for Lagrange interpolation, n f(x) = P(x) + X n! dr f xi)- (§), dxn for some ε = (x1,xn), (2) i=1 find upper and lower bounds for the error in your estimate above. How does this compare with the actual error?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![=
1. (a) Show that the function f(x) exp(x) has only one zero, x*. Write down
the general step of Newton's method for solving f(x)
=
0. Starting with an
initial guess of xo
=
0, calculate the next three approximations to the solution of
f(x) = exp(x) — 3 = 0 using Newton's method. For each iteration calculate the
error from the exact solution.
=
Write down the map xn+1
g(xn) corresponding to the Newton's method solution
of f(x) = exp(x) − 3 = 0 and show that g'(x) 0 at the fixed point.
=
State the fixed point theorem giving the conditions that guarantee the iteration
scheme xn+1 = = g(xn) is stable and has a unique fixed point in the interval [a, b].
Show that g(x) satisfies the conditions of this theorem in a neighbourhood of x*.
(b) Write down the Lagrange form of the interpolating polynomial P(x) that satisfies
P(x) = f(x) for x = xi,
i
=
1,...,n and x1 < X2 < · · · < Xn• (1)
=
1 through the points
x
Find the form of the polynomial that interpolates f(x)
x1 = 0.5, x2 1 and x3 = 3, then simplify it. Use this polynomial to estimate the
value of f(2) and find the error from the actual value.
Using the error formula for Lagrange interpolation,
n
f(x) = P(x) +
X
n!
dr f
xi)- (§),
dxn
for some ε = (x1,xn),
(2)
i=1
find upper and lower bounds for the error in your estimate above. How does this
compare with the actual error?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F06f4b026-5638-4cdd-a9f8-3f908edb6873%2Fjplmfd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:=
1. (a) Show that the function f(x) exp(x) has only one zero, x*. Write down
the general step of Newton's method for solving f(x)
=
0. Starting with an
initial guess of xo
=
0, calculate the next three approximations to the solution of
f(x) = exp(x) — 3 = 0 using Newton's method. For each iteration calculate the
error from the exact solution.
=
Write down the map xn+1
g(xn) corresponding to the Newton's method solution
of f(x) = exp(x) − 3 = 0 and show that g'(x) 0 at the fixed point.
=
State the fixed point theorem giving the conditions that guarantee the iteration
scheme xn+1 = = g(xn) is stable and has a unique fixed point in the interval [a, b].
Show that g(x) satisfies the conditions of this theorem in a neighbourhood of x*.
(b) Write down the Lagrange form of the interpolating polynomial P(x) that satisfies
P(x) = f(x) for x = xi,
i
=
1,...,n and x1 < X2 < · · · < Xn• (1)
=
1 through the points
x
Find the form of the polynomial that interpolates f(x)
x1 = 0.5, x2 1 and x3 = 3, then simplify it. Use this polynomial to estimate the
value of f(2) and find the error from the actual value.
Using the error formula for Lagrange interpolation,
n
f(x) = P(x) +
X
n!
dr f
xi)- (§),
dxn
for some ε = (x1,xn),
(2)
i=1
find upper and lower bounds for the error in your estimate above. How does this
compare with the actual error?
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