In this exercise will derive Simpson's rule using Taylor's theorem. Suppose ƒ € C¹([a, b]) and consider three equispaced points {x0, x₁, x2} with x。 = a, x₁ = a + h and x₂ = b, where h = (b − a)/2. - (a) Use Taylor's theorem to write f(x) as a third order polynomial plus a remainder term using the point of expansion x₁. (b) Insert the expression for f(x) from (a) into the expression 5² ƒ(x)dx and explicitly compute the integral of each term in the Taylor polynomial to derive x2 h³ Cx2 [** f (x) dx = 2hf (3₁) + h_ƒ"(x₁) + 2/14 [** ƒ^(¹) ({(x)) (x − x₁) ªdx - 3 XO (hint: two of the integrals will vanish identically because of parity, i.e. be- cause they are antisymmetric functions) (c) Justify that the Weighted Mean Value Theorem from given in lecture can be used for the remainder term. Then, apply the theorem to obtain: x2 1 ½¼/4 *^* = ƒ(4) (§(x)) (x − x₁) ªdx 24 ΤΟ f" (x₁) for some c = (a, b). (d) Recall the centered difference formula for the second derivative shown in HW5, Q4: = f(4) (c) 60 f(x₂) - 2f(x1) + f(xo) h² h² -h5 -ƒ(4) (5) 12
In this exercise will derive Simpson's rule using Taylor's theorem. Suppose ƒ € C¹([a, b]) and consider three equispaced points {x0, x₁, x2} with x。 = a, x₁ = a + h and x₂ = b, where h = (b − a)/2. - (a) Use Taylor's theorem to write f(x) as a third order polynomial plus a remainder term using the point of expansion x₁. (b) Insert the expression for f(x) from (a) into the expression 5² ƒ(x)dx and explicitly compute the integral of each term in the Taylor polynomial to derive x2 h³ Cx2 [** f (x) dx = 2hf (3₁) + h_ƒ"(x₁) + 2/14 [** ƒ^(¹) ({(x)) (x − x₁) ªdx - 3 XO (hint: two of the integrals will vanish identically because of parity, i.e. be- cause they are antisymmetric functions) (c) Justify that the Weighted Mean Value Theorem from given in lecture can be used for the remainder term. Then, apply the theorem to obtain: x2 1 ½¼/4 *^* = ƒ(4) (§(x)) (x − x₁) ªdx 24 ΤΟ f" (x₁) for some c = (a, b). (d) Recall the centered difference formula for the second derivative shown in HW5, Q4: = f(4) (c) 60 f(x₂) - 2f(x1) + f(xo) h² h² -h5 -ƒ(4) (5) 12
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
Simpson's rule
![[2] [Derivation of Simpon's Rule]
In this exercise will derive Simpson's rule using Taylor's theorem. Suppose
f = C¹([a, b]) and consider three equispaced points {x0, x₁, x2} with xo = a,
x₁ = a + h and x2 = b, where h
=
(b-a)/2.
(a) Use Taylor's theorem to write f(x) as a third order polynomial plus
a remainder term using the point of expansion ₁.
(b) Insert the expression for f(x) from (a) into the expression ² f(x) dx
and explicitly compute the integral of each term in the Taylor polynomial to
derive
h³
X2
1
[** f(x)dx = 2hf(x1) +
: 2h f(x₁) + ƒ" (x₁) +
kª ƒ" (#1) + 2/4 *²* ƒ^(¹) (§ (x)) (x − x1)ªda
3
хо
XO
(hint: two of the integrals will vanish identically because of parity, i.e. be-
cause they are antisymmetric functions)
(c) Justify that the Weighted Mean Value Theorem from given in lecture
can be used for the remainder term. Then, apply the theorem to obtain:
Cx2
1
=
24 *²* ƒ^(¹) (E(x)) (x − x₁)³¹dx =
xo
f(4) (c).
60
f" (x₁) =
for some c = (a, b).
(d) Recall the centered difference formula for the second derivative shown
in HW5, Q4:
f(x₂) 2ƒ(x₁) + f(xo) h²
h²
-h5
-
12 √(4) (6)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2Ff8b36a57-c94d-4cb7-b89f-4554e6461ba7%2Fvuumuaq_processed.png&w=3840&q=75)
Transcribed Image Text:[2] [Derivation of Simpon's Rule]
In this exercise will derive Simpson's rule using Taylor's theorem. Suppose
f = C¹([a, b]) and consider three equispaced points {x0, x₁, x2} with xo = a,
x₁ = a + h and x2 = b, where h
=
(b-a)/2.
(a) Use Taylor's theorem to write f(x) as a third order polynomial plus
a remainder term using the point of expansion ₁.
(b) Insert the expression for f(x) from (a) into the expression ² f(x) dx
and explicitly compute the integral of each term in the Taylor polynomial to
derive
h³
X2
1
[** f(x)dx = 2hf(x1) +
: 2h f(x₁) + ƒ" (x₁) +
kª ƒ" (#1) + 2/4 *²* ƒ^(¹) (§ (x)) (x − x1)ªda
3
хо
XO
(hint: two of the integrals will vanish identically because of parity, i.e. be-
cause they are antisymmetric functions)
(c) Justify that the Weighted Mean Value Theorem from given in lecture
can be used for the remainder term. Then, apply the theorem to obtain:
Cx2
1
=
24 *²* ƒ^(¹) (E(x)) (x − x₁)³¹dx =
xo
f(4) (c).
60
f" (x₁) =
for some c = (a, b).
(d) Recall the centered difference formula for the second derivative shown
in HW5, Q4:
f(x₂) 2ƒ(x₁) + f(xo) h²
h²
-h5
-
12 √(4) (6)
Transcribed Image Text:x2 = x₁ + h and xo = x₁ − h). Insert both this expression and your answer
from part (c) into the result from part (b) to conclude:
h
[*^* f(x) dx = '/3 (ƒ(x0) + 4ƒ (x1) + f(x2)) =
xo
h5
17/12 ( 1 ƒ (*) (4) - — ƒ (*) (c) )
The right hand side consists of Simpson's rule and a O(h5) error term.
(e) Based on the error term, what is the degree of exactness of Simpson's
rule? That is, what is the largest value d such that Simpson's rule is exact
(no error) for all polynomials of degree d or less?
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