Exercise · ZW-CUBIC-SPLINES-007 · ★★☆ Intermediate
For the natural spline through \((0,0),(1,1),(2,0)\), what is \(S''(1)\) using the \(c_1\) form of the system?
Choose one option.
Show answer
- B. \(-3\)
Explanation
With \(c_0=c_2=0\) the system gives \(4c_1=3\,(f[x_1,x_2]-f[x_0,x_1])=-6\), so \(c_1=-\tfrac{3}{2}\) and \(S''(x_1)=2c_1=-3\).
Review the concept: Cubic splines
A cubic spline \(S\) is a piecewise function, each piece a polynomial of degree three, that interpolates the data \((x_i,y_i)\) and is of class \(C^2\) at the nodes.
\[S_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i)^2+d_i(x-x_i)^3\]
More practice problems: cubic splines
- ZW-CUBIC-SPLINES-001 What degree is each piece of a cubic spline?
- ZW-CUBIC-SPLINES-002 What class of continuity does a cubic spline have at interior nodes?
- ZW-CUBIC-SPLINES-003 Which boundary condition defines the natural spline?
- ZW-CUBIC-SPLINES-004 How many cubic pieces does a spline have over n intervals?
- ZW-CUBIC-SPLINES-005 What type of linear system appears when computing the coefficients c_i?
- ZW-CUBIC-SPLINES-006 A natural spline with n=3 intervals has 4·3=12 unknowns. How many interpolation and continuity equa…
- ZW-CUBIC-SPLINES-008 A clamped spline imposes S'(x_0)=f'(x_0). Which of these statements about it is correct?
- ZW-CUBIC-SPLINES-009 If S is a natural cubic spline with S''(x_0)=0, what happens to c_0 in the piecewise representation?