Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

6.3. B SPLINES 85
Exercises
(6.1) Is the following function a linear spline on [0, 4]? Why or why not?
{
3x + 2 : 0 ≤ x < 1
S(x) =
−2x + 4 : 1 ≤ x ≤ 4
(6.2) Is the following function a linear spline on [0, 2]? Why or why not?
{
x + 3 : 0 ≤ x < 1
S(x) =
3 : 1 ≤ x ≤ 2
(6.3) Is the following function a linear spline on [0, 4]? Why or why not?
{
x 2 + 3 : 0 ≤ x < 3
S(x) =
5x − 6 : 3 ≤ x ≤ 4
(6.4) Find constants, α, β such that the following is a linear spline on [0, 5].
⎧
⎪⎨ 4x − 2 : 0 ≤ x < 1
S(x) = αx + β : 1 ≤ x < 3
⎪⎩
−2x + 10 : 3 ≤ x ≤ 5
(6.5) Is the following function a quadratic spline on [0, 4]? Why or why not?
{
x 2 + 3 : 0 ≤ x < 3
Q(x) =
5x − 6 : 3 ≤ x ≤ 4
(6.6) Is the following function a quadratic spline on [0, 2]? Why or why not?
{
x 2 + 3x + 2 : 0 ≤ x < 1
Q(x) =
2x 2 + x + 3 : 1 ≤ x ≤ 2
(6.7) Find constants, α, β, γ such that the following is a quadratic spline on [0, 5].
⎧
1
⎪⎨ 2 x2 + 2x + 3 2
: 0 ≤ x < 1
Q(x) = αx
⎪⎩
2 + βx + γ : 1 ≤ x < 3
3x 2 − 7x + 12 : 3 ≤ x ≤ 5
(6.8) Find the quadratic spline that interpolates the following data:
t 0 1 4
y 1 −2 1
To resolve the single degree of freedom, assume that Q ′ (0) = −Q ′ (4). Assume your solution
takes the form
{
α 1 (x − 1) 2 + β 1 (x − 1) − 2 : 0 ≤ x < 1
Q(x) =
α 2 (x − 1) 2 + β 2 (x − 1) − 2 : 1 ≤ x ≤ 4
Find the constants α 1 , β 1 , α 2 , β 2 .