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  "body": "\n## Definition\n\nLet $k \\ge 1$ and let $a = t_0 \u003c t_1 \u003c \\cdots \u003c t_k = b$ be knots.\nA *cubic spline* with these knots is a function $s \\colon [a, b] \\to \\R$ such that:\n\n1. for each $i = 1, \\dots, k$, the restriction of $s$ to $[t_{i-1}, t_i]$ is a [[polynomial function|1G0B73WPX8]] of degree at most $3$;\n2. $s$ is twice continuously differentiable on $[a, b]$.\n\nA cubic spline is *natural* if moreover $s''(a) = s''(b) = 0$.\n\n## Remarks\n\nThis is the [[spline|84TJ8KQ926]] of degree $3$.\nWriting $s_i$ for the piece on $[t_{i-1}, t_i]$, clause 2 says that $s_i(t_i) = s_{i+1}(t_i)$, $s_i'(t_i) = s_{i+1}'(t_i)$ and $s_i''(t_i) = s_{i+1}''(t_i)$ for $1 \\le i \\le k - 1$, where $'$ is the [[derivative|0V1EJCT8G6]].\nThe cubic splines with fixed knots form a vector space of dimension $k + 3$.\nGiven values at all knots, there is exactly one natural cubic spline taking them.\n\nExample: knots $-1 \u003c 0 \u003c 1$ and $s(x) = 0$ for $x \\le 0$, $s(x) = x^3$ for $x \\ge 0$.\nAt $0$ both pieces have value, first derivative and second derivative $0$, so $s$ is a cubic spline.\nIt is not natural, since $s''(1) = 6 \\ne 0$.\n\nNon-example: $s(x) = \\abs{x}$ on $[-1, 1]$ with the knots $-1 \u003c 0 \u003c 1$ satisfies clause 1, since its pieces $-x$ and $x$ are polynomials.\nIt fails clause 2: its first derivative jumps from $-1$ to $1$ at $0$.\n",
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