Consider the data points (0, 1), (1, 1), (2, 5). (a) Find the piecewise function P(x) = that interpolates the given data points, where Spo(x), z = [0, 1]. P₁(x), x € [1,2], Po(x)= a + be, P₁(x) = c+dx², for some constants a, b, c, d to be determined. (b) Find the natural cubic spline Si(2) = S(x) = [so(x), x = [0, 1]. ε $1(x), x € [1,2], that interpolates the given data points, where so, 81 are cubic functions within their respective intervals. Express the resulting polynomials in monomial form, that is, so (x)= = ao + box + cox² + dox³, $₁(x) = a₁ + b₁x + ₁x³ +d₁x³, for some ao, bo, co, do, a1, b₁, C₁, d₁. Hint: Recall the formula we derived in class for the cubic splines, 1 ; [(Xi+1 − x)³M; + (x − x;)³Mi+1] − hi [(xi+1 − x)M; + (x − x;)Mi+1] 6h₂ + + 1 / [(²²+1 - − z)f(zi) + (z – xi)f(i+z)] for x = [xi,i+1], and i = 0,..., n-1. Solve for the values Mo, M₁,... by setting up the appropriate system of equations, and use the formula for s; to obtain the desired cubic spline.
Consider the data points (0, 1), (1, 1), (2, 5). (a) Find the piecewise function P(x) = that interpolates the given data points, where Spo(x), z = [0, 1]. P₁(x), x € [1,2], Po(x)= a + be, P₁(x) = c+dx², for some constants a, b, c, d to be determined. (b) Find the natural cubic spline Si(2) = S(x) = [so(x), x = [0, 1]. ε $1(x), x € [1,2], that interpolates the given data points, where so, 81 are cubic functions within their respective intervals. Express the resulting polynomials in monomial form, that is, so (x)= = ao + box + cox² + dox³, $₁(x) = a₁ + b₁x + ₁x³ +d₁x³, for some ao, bo, co, do, a1, b₁, C₁, d₁. Hint: Recall the formula we derived in class for the cubic splines, 1 ; [(Xi+1 − x)³M; + (x − x;)³Mi+1] − hi [(xi+1 − x)M; + (x − x;)Mi+1] 6h₂ + + 1 / [(²²+1 - − z)f(zi) + (z – xi)f(i+z)] for x = [xi,i+1], and i = 0,..., n-1. Solve for the values Mo, M₁,... by setting up the appropriate system of equations, and use the formula for s; to obtain the desired cubic spline.
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.
Consider the data points (0, 1), (1, 1), (2, 5).
(a) Find the piecewise function
P(x) =
that interpolates the given data points, where
Spo(x), x € [0,1].
(p₁(x), x= [1,2],
Po(x) = a + be,
P₁(x) = c+dx²,
for some constants a, b, c, d to be determined.
(b) Find the natural cubic spline
Si(2)
S(x) =
Jso(x), x= [0,1].
$1(x), x= [1,2],
that interpolates the given data points, where so, 81 are cubic functions within their respective
intervals. Express the resulting polynomials in monomial form, that is,
so (x) = ao + box + cox² +dox³,
$₁(x) = a₁ + b₁x + ₁x³ + ₁x³,
for some ao, bo, co, do, a1, b1, C₁, d₁.
Hint: Recall the formula we derived in class for the cubic splines,
1
hi
=
oh [(+– z)*M + (z – zi)®M+] - * [+- z)M + (z − )Min]
Xi
+
* [(2 - 2)f(z) + (z – zi)f(+)]
for x = [xi,i+1], and i = 0,..., n-1. Solve for the values Mo, M₁, ... by setting up the appropriate
system of equations, and use the formula for si to obtain the desired cubic spline.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1dbcb96-b484-476e-adb0-126482b446f4%2F662e3514-c282-4ced-90e9-4a2f940ceed9%2Fl64f5o9_processed.png&w=3840&q=75)
Transcribed Image Text:1.
Consider the data points (0, 1), (1, 1), (2, 5).
(a) Find the piecewise function
P(x) =
that interpolates the given data points, where
Spo(x), x € [0,1].
(p₁(x), x= [1,2],
Po(x) = a + be,
P₁(x) = c+dx²,
for some constants a, b, c, d to be determined.
(b) Find the natural cubic spline
Si(2)
S(x) =
Jso(x), x= [0,1].
$1(x), x= [1,2],
that interpolates the given data points, where so, 81 are cubic functions within their respective
intervals. Express the resulting polynomials in monomial form, that is,
so (x) = ao + box + cox² +dox³,
$₁(x) = a₁ + b₁x + ₁x³ + ₁x³,
for some ao, bo, co, do, a1, b1, C₁, d₁.
Hint: Recall the formula we derived in class for the cubic splines,
1
hi
=
oh [(+– z)*M + (z – zi)®M+] - * [+- z)M + (z − )Min]
Xi
+
* [(2 - 2)f(z) + (z – zi)f(+)]
for x = [xi,i+1], and i = 0,..., n-1. Solve for the values Mo, M₁, ... by setting up the appropriate
system of equations, and use the formula for si to obtain the desired cubic spline.
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