Cubic spline solving equation

Question

$$S(x)=\begin{cases} x^3 +4x^2 -2x +7 & \text{ if } -1\leq x\leq 0, \\ x^3 - 2x^2 +4x +5& \text{ if } 1\leq x\leq 2, \end{cases}$$

is a cubic spline with knots $\{-1, 0, 1, 2\}$

Find $s2(x)$ on the interval $0\leq x\leq 1$

So $S'(x) = 3x^2 +8x -2$ if $-1\leq x\leq 0$

$S'(x)=3x^2 - 4x +4$ if $1\leq x\leq 2$

I know We need $s2(0) = 7$, $s2(1) = 8$ and $s2'(0) = -2$ , $s2'(1) = 3$

But I don't know what to actually put into the matrix to solve this.. The answer given in class was

$$s2(x) = -x^3 + 4x^2 -2x +7$$

but I'm not sure how he got that.

Answer 1

$s2(x)=ax^3+bx²+cx+d$

$s2'(x)=3ax²+2bx+c$

You just have to replace $x$ with $\{0,1\}$ and you will get 4 equations with 4 unknowns...

Answer 2

The goal in matching a polynomial to points and slopes is to write a system of equations that, when solved, gives the coefficients. There are two basic equations, shown here for cubics: $$y=ax^3+bx^2+cx+d$$ $$y'=3ax^2+2bx+c$$ These then can be converted to matrix rows, with $a$, $b$, $c$, $d$ as the targets: $$\left[\begin{array}{cccc|c} x^3&x^2&x&1&y\\ 3x^2&2x&1&0&y'\\ \end{array}\right]$$

So to find the whole polynomial we set up a row for each constraint.

$$\left[\begin{array}{cccc|c} 0&0&0&1&7\\ 1&1&1&1&8\\ 0&0&1&0&-2\\ 3&2&1&0&3\\ \end{array}\right]$$

Turn this to RREF we get the last column as $[-1,4,-2,7]$, which are the coefficients of the interpolating cubic:

$$y=-x^3+4x^2-2x+7$$

Two things of note:

1. if you have more constraints, you can include them; you need the order to go up by one for each additional constraint you have: five constraints means you're finding a quartic, six means you're finding a quintic, etc. At least one of the constraints must be on the actual value and not the derivative.

2. if you're going to be doing a lot of this sort of solving and your $x$ coordinates can be easily shifted to $0$ and $1$, you can simply take the inverse of the above matrix and apply it to your $y$ values. Once done this is a lot faster than trying to run rref every time.