43.08.03 · numerical-analysis / 08-interpolation-approximation

Hermite interpolation and piecewise / cubic spline interpolation

shipped3 tiersLean: none

Anchor (Master): de Boor 2001 *A Practical Guide to Splines* revised ed. (Springer) Ch. V (the minimum-curvature variational characterisation, the $O(h^4)$ convergence, and the spline projector) and Ch. IX-X; Hall-Meyer 1976 *J. Approx. Theory* 16(2) (the sharp $5/384$ constant in the clamped cubic-spline error bound); Holladay 1957 *Quart. Appl. Math.* 15 (the original minimum-bending-energy theorem for the natural cubic spline)

Intuition Beginner

The last unit ended on a warning. If you sample a smooth curve at many evenly spaced points and force a single high-degree polynomial through all of them, the fit can swing wildly near the ends — the Runge phenomenon. One polynomial stretched across a wide interval is simply too stiff in the middle and too floppy at the edges. This unit gives two repairs, and both come from the same instinct: stop asking one polynomial to do everything.

The first repair sharpens what "passing through a point" means. So far you matched the height of the curve at each node. But you often know more — you know the slope there too. Matching both the height and the slope at each node is called Hermite interpolation. It is like a handshake that agrees not only on where to meet but on which direction both curves are heading, so the fit hugs the true curve more snugly around each point.

The second repair is the one that defeats Runge for good. Instead of one tall polynomial, lay down many short low-degree pieces, one per gap between neighbouring points, and glue them together. Each piece is mild and cannot misbehave. The art is in the gluing. If you join cubic pieces so that the heights, the slopes, and even the bending all match where two pieces meet, the seams disappear and the whole curve looks perfectly smooth. That glued-up curve is a cubic spline.

The name is borrowed from the drafting tool: a thin flexible strip of wood that draughtsmen bent through pegs to draw a fair curve. The wood settles into the shape that stores the least bending energy. The mathematical cubic spline does exactly that — among all smooth curves through your points, it is the one that bends as little as possible.

Visual Beginner

Picture the same data dots from the Runge unit. On the left, one stiff high-degree polynomial overshoots near the ends. On the right, short cubic pieces are laid down between neighbouring dots and joined so smoothly you cannot spot the joins. The table below contrasts the two strategies and lists what a cubic spline forces to match at each interior seam.

strategy	how many pieces	degree of each piece	what matches at a seam
one global polynomial	one	high (grows with data)	nothing to glue
cubic spline	many (one per gap)	three (cubic)	height, slope, and bending

Each row tells the same story: spreading the work over many gentle cubic pieces keeps every piece tame, and matching height, slope, and bending at the seams hides the joins so the eye sees one smooth curve. That triple match at each seam is what makes the spline so much steadier than the single stiff polynomial.

Worked example Beginner

Let us build the Hermite cubic on one interval that matches both a value and a slope at each end. Take the interval from $0$ to $1$ , and ask for a curve $p$ with $p (0) = 0$ , slope $0$ at the left, $p (1) = 1$ , and slope $0$ at the right. Four conditions pin down a cubic $p (x) = a + b x + c x^{2} + d x^{3}$ .

Step 1. Use the two left conditions. The value $p (0) = 0$ forces $a = 0$ . The slope at $0$ is $b$ (since the derivative is $b + 2 c x + 3 d x^{2}$ , which equals $b$ at $x = 0$ ), so $b = 0$ . The cubic so far is $p (x) = c x^{2} + d x^{3}$ .

Step 2. Use the two right conditions. The value $p (1) = c + d = 1$ . The slope at $1$ is $2 c + 3 d = 0$ .

Step 3. Solve the pair. From $2 c + 3 d = 0$ we get $c = - \frac{3}{2} d$ . Substituting into $c + d = 1$ gives $- \frac{3}{2} d + d = - \frac{1}{2} d = 1$ , so $d = - 2$ and $c = 3$ .

Step 4. Write the answer and check. The cubic is $p (x) = 3 x^{2} - 2 x^{3}$ . At $x = 0$ it is $0$ ; at $x = 1$ it is $3 - 2 = 1$ . Its derivative is $6 x - 6 x^{2} = 6 x (1 - x)$ , which is $0$ at both $x = 0$ and $x = 1$ . All four conditions hold.

What this tells us: matching a value and a slope at each of two ends needs four numbers, and a cubic has exactly four free numbers, so the fit is determined. This little curve, $3 x^{2} - 2 x^{3}$ , is the smooth S-shaped ramp that engineers call "smoothstep"; it is the basic tile a cubic spline lays down between neighbouring points.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $f : [a, b] \to R$ , and the nodes are $a = x_{0} < x_{1} < \dots < x_{n} = b$ with subinterval lengths $h_{i} = x_{i + 1} - x_{i}$ and mesh size $h = max_{i} h_{i}$ . As in 43.08.01, $Π_{m}$ is the space of polynomials of degree at most $m$ , and $C^{m} [a, b]$ the functions with $m$ continuous derivatives. The node polynomial of 43.08.02 is $π (x) = \prod_{k} (x - x_{k})$ .

Definition (Hermite / osculatory interpolation). Given nodes $x_{0}, \dots, x_{n}$ and, at each node, prescribed values and derivatives up to order $m_{k} - 1$ (the multiplicity of $x_{k}$ , with total $N = \sum_{k} m_{k}$ ), the Hermite interpolant is the unique $p \in Π_{N - 1}$ satisfying $$ p^{(j)}(x_k) = f^{(j)}(x_k), \qquad 0 \le j \le m_k - 1, \quad 0 \le k \le n. $$ The canonical case matches value and first derivative at every node ( $m_{k} = 2$ for all $k$ ), giving $p \in Π_{2 n + 1}$ . The interpolant is computed by the Newton form on the confluent node list in which each $x_{k}$ is repeated $m_{k}$ times, using the generalized divided differences of 43.08.01 under the convention $f [j + 1 x_{k}, \dots, x_{k}] = f^{(j)} (x_{k}) / j!$ .

Definition (piecewise polynomial interpolant). A piecewise polynomial of degree $d$ on the mesh is a function $s$ whose restriction to each subinterval $[x_{i}, x_{i + 1}]$ is a polynomial in $Π_{d}$ . It is interpolatory if $s (x_{i}) = f_{i}$ for all $i$ . The smoothness class is the largest $r$ with $s \in C^{r} [a, b]$ ; the seams (interior knots) are where smoothness can fail.

Definition (cubic spline). A cubic spline on the mesh is a piecewise cubic $s$ ( $s ∣_{[x_{i}, x_{i + 1}]} \in Π_{3}$ ) lying in $C^{2} [a, b]$ : at each interior knot $x_{i}$ ( $1 \leq i \leq n - 1$ ) the two adjoining cubics agree in value, first derivative, and second derivative. The interpolating cubic spline additionally satisfies $s (x_{i}) = f_{i}$ for all $i$ . Counting degrees of freedom, the $n$ cubics carry $4 n$ coefficients; interpolation and $C^{2}$ matching impose $4 n - 2$ conditions, leaving $2$ free, fixed by end conditions:

natural: $s^{''} (x_{0}) = s^{''} (x_{n}) = 0$ ;
clamped (complete): $s^{'} (x_{0}) = f^{'} (x_{0})$ , $s^{'} (x_{n}) = f^{'} (x_{n})$ with prescribed end slopes;
not-a-knot: $s^{'''}$ is continuous across $x_{1}$ and $x_{n - 1}$ , so the first two and last two cubics coincide (the knots $x_{1}, x_{n - 1}$ are "not knots").

The symbols $Π_{m}$ , $C^{m} [a, b]$ , $\prod$ , $h_{i}$ , $∥ \cdot ∥_{\infty}$ , and the moments $M_{i} = s^{''} (x_{i})$ are recorded in _meta/NOTATION.md; the node polynomial $π$ and the divided-difference calculus are inherited from 43.08.01 and 43.08.02.

Counterexamples to common slips Intermediate+

"A piecewise cubic that interpolates the data is automatically a spline." Interpolation alone forces only $C^{0}$ . Hermite cubics joined by value and slope are merely $C^{1}$ . The spline demands the extra condition that the second derivative also match at every seam; that single additional requirement per interior knot is what couples the pieces into the global tridiagonal system.
"Matching more derivatives always improves the fit." Hermite interpolation with value and derivative at $n + 1$ equispaced nodes is still a single polynomial of degree $2 n + 1$ , and its error carries the squared node polynomial $\prod_{k} (x - x_{k})^{2}$ . The Runge mechanism of 43.08.02 is not cured by osculation; it is cured only by lowering the degree, which is what going piecewise does.
"The natural end condition is the most accurate default." Setting $s^{''} = 0$ at the ends is convenient and yields the minimum-curvature spline, but it degrades the accuracy near the boundary from $O (h^{4})$ to $O (h^{2})$ unless $f^{''}$ genuinely vanishes there. Clamped (true end slopes) or not-a-knot conditions restore the full $O (h^{4})$ order.
"The tridiagonal spline matrix could be singular." The moment system has rows $μ_{i} M_{i - 1} + 2 M_{i} + λ_{i} M_{i + 1} = d_{i}$ with $μ_{i} + λ_{i} = 1$ and $μ_{i}, λ_{i} > 0$ , so the matrix is strictly diagonally dominant. Diagonal dominance of this sign pattern guarantees invertibility, so existence and uniqueness never fail for distinct ordered nodes.

Key theorem with proof Intermediate+

The signature result is that the interpolating cubic spline exists and is unique, because the $C^{2}$ -matching conditions reduce to a strictly diagonally dominant tridiagonal system in the second derivatives at the knots. The unknowns $M_{i} = s^{''} (x_{i})$ are called the moments; once they are known, each cubic piece is reconstructed by integrating the linear $s^{''}$ twice and fixing the constants by the interpolation values. This follows Süli-Mayers ^{[Süli-Mayers §11.4]}.

Theorem (existence, uniqueness, and the moment system). Let $a = x_{0} < \dots < x_{n} = b$ with $h_{i} = x_{i + 1} - x_{i}$ . For prescribed values $f_{0}, \dots, f_{n}$ and either the natural or clamped end conditions, there exists a unique interpolating cubic spline $s \in C^{2} [a, b]$ . Its moments $M_{i} = s^{''} (x_{i})$ satisfy, for each interior knot $1 \leq i \leq n - 1$ , $$ \mu_i M_{i-1} + 2 M_i + \lambda_i M_{i+1} = d_i, \qquad \mu_i = \frac{h_{i-1}}{h_{i-1} + h_i}, \quad \lambda_i = \frac{h_i}{h_{i-1} + h_i}, $$ where $d_{i} = \frac{6}{h _{i - 1} + h _{i}} (\frac{f _{i + 1} - f _{i}}{h _{i}} - \frac{f _{i} - f _{i - 1}}{h _{i - 1}})$ , with two further rows supplied by the end conditions. The system matrix is strictly diagonally dominant, hence invertible.

Proof. On $[x_{i}, x_{i + 1}]$ the spline is cubic, so $s^{''}$ is linear and is determined by its endpoint values $M_{i}, M_{i + 1}$ : $$ s''(x) = M_i \frac{x_{i+1} - x}{h_i} + M_{i+1} \frac{x - x_i}{h_i}. $$ Integrating twice introduces two constants per piece, which the interpolation conditions $s (x_{i}) = f_{i}$ and $s (x_{i + 1}) = f_{i + 1}$ fix; carrying out the integration gives the piece explicitly and, in particular, the one-sided first derivatives at the endpoints. Differentiating the integrated form and evaluating at $x_{i}$ from the right yields $$ s'(x_i^+) = \frac{f_{i+1} - f_i}{h_i} - \frac{h_i}{6}(2 M_i + M_{i+1}), $$ and from the piece $[x_{i - 1}, x_{i}]$ the left derivative is $$ s'(x_i^-) = \frac{f_i - f_{i-1}}{h_{i-1}} + \frac{h_{i-1}}{6}(M_{i-1} + 2 M_i). $$ The defining $C^{1}$ continuity at the interior knot demands $s^{'} (x_{i}^{+}) = s^{'} (x_{i}^{-})$ . Equating and clearing denominators by multiplying through by $6/ (h_{i - 1} + h_{i})$ produces exactly $μ_{i} M_{i - 1} + 2 M_{i} + λ_{i} M_{i + 1} = d_{i}$ with the stated coefficients; the $C^{2}$ continuity is built in because the linear $s^{''}$ already shares the endpoint value $M_{i}$ across the two pieces. This is $n - 1$ equations in the $n + 1$ unknowns $M_{0}, \dots, M_{n}$ .

The natural condition adds $M_{0} = M_{n} = 0$ directly; the clamped condition adds, at each end, the analogous equation from matching the prescribed $s^{'} (x_{0})$ and $s^{'} (x_{n})$ , namely $2 M_{0} + M_{1} = (6/ h_{0}) ((f_{1} - f_{0}) / h_{0} - f^{'} (x_{0}))$ and its mirror. In every interior row $μ_{i} + λ_{i} = 1 < 2$ , and the end rows have diagonal $2$ against off-diagonal $1$ (or a pure $1$ on the diagonal for the natural case), so the matrix is strictly diagonally dominant. A strictly diagonally dominant matrix is nonsingular, so the moments are uniquely determined; back-substituting them into the integrated pieces reconstructs the unique spline. $□$

Corollary (reduction to a linear-time solve). The moment matrix is tridiagonal and diagonally dominant, so the system is solved by the Thomas algorithm (Gaussian elimination without pivoting) in $O (n)$ arithmetic, and the elimination is numerically stable because diagonal dominance is preserved at every step. The whole spline therefore costs $O (n)$ work, against the $O (n^{2})$ of building a divided-difference table for a single global polynomial of comparable size.

Bridge. This theorem is the foundational reason the spline tames the Runge divergence of 43.08.02: the global high-degree polynomial is replaced by local cubics whose only global coupling is the benign tridiagonal moment system, and strict diagonal dominance is exactly the structural fact that organises existence, uniqueness, and the $O (n)$ solve into one statement. The moment formulation builds toward the variational characterisation of the Master tier, where the same second derivative $s^{''}$ reappears as the integrand of the bending energy $\int (s^{''})^{2}$ , and it appears again in the finite-element assembly of 24.04.07, where $C^{2}$ Hermite/spline elements produce the identical banded, diagonally dominant stiffness pattern. The construction is dual to the Hermite confluent limit of 43.08.01: a $C^{2}$ spline is what a chain of Hermite cubics becomes once the slopes are not prescribed but solved for so that the curvature also matches. Putting these together, the central insight is that smoothness across seams is a sparse linear constraint, not a dense one, which is the bridge from local interpolation to a global yet cheaply solvable interpolant.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Derive the Hermite interpolant on $[0, 1]$ matching $p (0) = f_{0}$ , $p^{'} (0) = d_{0}$ , $p (1) = f_{1}$ , $p^{'} (1) = d_{1}$ in terms of the four cubic Hermite basis functions, and verify the basis function multiplying $f_{0}$ equals $1 - 3 x^{2} + 2 x^{3}$ .

Hint

Seek $H_{00} (x)$ with $H_{00} (0) = 1$ , $H_{00}^{'} (0) = 0$ , $H_{00} (1) = 0$ , $H_{00}^{'} (1) = 0$ ; it is a cubic with a double root at $x = 1$ .

Answer

The interpolant is $p (x) = f_{0} H_{00} (x) + d_{0} H_{10} (x) + f_{1} H_{01} (x) + d_{1} H_{11} (x)$ , where each basis cubic isolates one datum. For $H_{00}$ : the conditions $H_{00} (1) = H_{00}^{'} (1) = 0$ force a double factor $(x - 1)^{2}$ , so $H_{00} (x) = (a x + b) (x - 1)^{2}$ ; then $H_{00} (0) = b = 1$ and $H_{00}^{'} (0) = a (0 - 1)^{2} + b \cdot 2 (0 - 1) = a - 2 = 0$ give $a = 2$ , $b = 1$ , so $H_{00} (x) = (2 x + 1) (x - 1)^{2} = 1 - 3 x^{2} + 2 x^{3}$ . (The complementary $H_{01} (x) = 3 x^{2} - 2 x^{3}$ is the smoothstep of the worked example.) Rubric: full credit for the basis decomposition and the derivation of $H_{00} = 1 - 3 x^{2} + 2 x^{3}$ .

Exercise 5 (medium, symbolic).

Prove that the cubic Hermite interpolant on $[a, b]$ matching value and first derivative at $a$ and $b$ has the error $f (x) - p (x) = \frac{f ^{(4)} ( ξ )}{4 !} (x - a)^{2} (x - b)^{2}$ for some $ξ$ , when $f \in C^{4}$ .

Hint

Mirror the Rolle argument of 43.08.02: form $ϕ (t) = f (t) - p (t) - G (t - a)^{2} (t - b)^{2}$ with $G$ fixing $ϕ (x) = 0$ ; note the double zeros at $a$ and $b$ from matched derivatives.

Answer

Fix $x \neq = a, b$ and set $G = (f (x) - p (x)) / [(x - a)^{2} (x - b)^{2}]$ , then $ϕ (t) = f (t) - p (t) - G (t - a)^{2} (t - b)^{2}$ . Since $p$ matches value and derivative at $a$ and $b$ , $ϕ$ has double zeros at $a$ and $b$ and a simple zero at $x$ : counting multiplicity, $ϕ$ vanishes at $5$ points in $[a, b]$ (with $a, b$ counted twice). Applying Rolle repeatedly — between consecutive zeros of $ϕ$ a zero of $ϕ^{'}$ , and the double zeros of $ϕ$ are also zeros of $ϕ^{'}$ — gives $ϕ^{'}$ at least $4$ zeros, $ϕ^{''}$ at least $3$ , and after four differentiations $ϕ^{(4)}$ has a zero $ξ$ . Since $p \in Π_{3}$ has $p^{(4)} \equiv 0$ and $(t - a)^{2} (t - b)^{2}$ is monic of degree $4$ with fourth derivative $4!$ , $0 = ϕ^{(4)} (ξ) = f^{(4)} (ξ) - G \cdot 4!$ , so $G = f^{(4)} (ξ) /4!$ and the formula follows. Rubric: full credit for the double-zero accounting, the iterated Rolle count, and the fourth-derivative identity.

Exercise 6 (medium, numeric).

A not-a-knot cubic spline on four points uses how many distinct cubic pieces, and across how many interior knots is third-derivative continuity additionally imposed?

Hint

Four points make three subintervals; not-a-knot removes the knot status of the second and second-to-last interior knots.

Answer

Three pieces; third-derivative continuity at the two interior knots $x_{1}$ and $x_{2}$ . With four points $x_{0}, \dots, x_{3}$ there are three subintervals. The interior knots are $x_{1}$ and $x_{2}$ ; not-a-knot makes $s^{'''}$ continuous at both, which forces the first two cubics to coincide and the last two to coincide, leaving effectively one cubic on $[x_{0}, x_{2}]$ and one on $[x_{1}, x_{3}]$ . Rubric: full credit for three pieces and the two not-a-knot conditions at $x_{1}, x_{2}$ .

Exercise 7 (hard, symbolic).

Prove the minimum-curvature (first-integral orthogonality) identity: if $s$ is the natural cubic spline interpolating $f$ at the nodes and $g \in C^{2} [a, b]$ is any other interpolant of the same data, then $\int_{a}^{b} (g^{''})^{2} = \int_{a}^{b} (s^{''})^{2} + \int_{a}^{b} (g^{''} - s^{''})^{2}$ .

Hint

Write $g^{''} = s^{''} + (g^{''} - s^{''})$ , expand the square, and show the cross term $\int s^{''} (g^{''} - s^{''}) = 0$ by integrating by parts twice and using that $s^{''}$ vanishes at the ends (natural) while $g - s$ vanishes at every node and $s^{'''}$ is constant on each subinterval.

Answer

Let $e = g - s$ , so $e (x_{i}) = 0$ at every node and $e \in C^{2}$ . Expanding, $\int (g^{''})^{2} = \int (s^{''})^{2} + 2 \int s^{''} e^{''} + \int (e^{''})^{2}$ , and it remains to show the cross term vanishes. Integrate by parts: $\int_{a}^{b} s^{''} e^{''} = [s^{''} e^{'}]_{a}^{b} - \int_{a}^{b} s^{'''} e^{'}$ . The boundary term is zero because the natural condition gives $s^{''} (a) = s^{''} (b) = 0$ . On each subinterval $[x_{i}, x_{i + 1}]$ the spline is cubic, so $s^{'''}$ is a constant $c_{i}$ ; thus $\int_{a}^{b} s^{'''} e^{'} = \sum_{i} c_{i} \int_{x_{i}}^{x_{i + 1}} e^{'} = \sum_{i} c_{i} (e (x_{i + 1}) - e (x_{i})) = 0$ , since $e$ vanishes at all nodes. Hence $\int s^{''} e^{''} = 0$ , and $\int (g^{''})^{2} = \int (s^{''})^{2} + \int (e^{''})^{2} \geq \int (s^{''})^{2}$ , with equality iff $e^{''} \equiv 0$ , i.e. $e$ is affine; an affine function vanishing at $\geq 2$ nodes is zero, so $g = s$ . Rubric: full credit for the expansion, the two-step integration by parts, the use of the natural boundary condition and piecewise-constant $s^{'''}$ , and the equality case.

Exercise 8 (hard, symbolic).

Show that the interior moment matrix on a uniform mesh, with rows $(\frac{1}{2}, 2, \frac{1}{2})$ , is strictly diagonally dominant, and deduce a bound $∥ M ∥_{\infty} \leq \frac{1}{1} max_{i} ∣ d_{i} ∣$ on the solution in terms of the right-hand side (for the natural case).

Hint

Strict diagonal dominance: $∣2∣ > ∣ \frac{1}{2} ∣ + ∣ \frac{1}{2} ∣ = 1$ . For the solution bound, let $i^{⋆}$ index the largest $∣ M_{i^{⋆}} ∣$ and isolate $2 M_{i^{⋆}} = d_{i^{⋆}} - \frac{1}{2} M_{i^{⋆} - 1} - \frac{1}{2} M_{i^{⋆} + 1}$ .

Answer

Each interior row has diagonal entry $2$ and off-diagonal entries summing to $\frac{1}{2} + \frac{1}{2} = 1 < 2$ , so the matrix is strictly diagonally dominant and hence invertible. For the bound, let $i^{⋆} = ar g max_{i} ∣ M_{i} ∣$ (an interior index, since the natural end moments are $0$ ). From its row, $2∣ M_{i^{⋆}} ∣ = ∣ d_{i^{⋆}} - \frac{1}{2} M_{i^{⋆} - 1} - \frac{1}{2} M_{i^{⋆} + 1} ∣ \leq ∣ d_{i^{⋆}} ∣ + \frac{1}{2} ∣ M_{i^{⋆} - 1} ∣ + \frac{1}{2} ∣ M_{i^{⋆} + 1} ∣ \leq max_{i} ∣ d_{i} ∣ + ∣ M_{i^{⋆}} ∣$ , using $∣ M_{i^{⋆} \pm 1} ∣ \leq ∣ M_{i^{⋆}} ∣$ . Subtracting $∣ M_{i^{⋆}} ∣$ gives $∣ M_{i^{⋆}} ∣ \leq max_{i} ∣ d_{i} ∣$ , i.e. $∥ M ∥_{\infty} \leq max_{i} ∣ d_{i} ∣$ . This is the discrete maximum principle behind the spline's stability: the second derivatives cannot exceed the data-driven right-hand side. Rubric: full credit for the dominance inequality and the maximum-index argument yielding the bound.

Advanced results Master

The elementary theory delivers the spline as the solution of a tridiagonal system; the master layer concerns why that solution is optimal among all smooth interpolants, the sharp order of its convergence, and the projector that makes spline interpolation, unlike high-degree polynomial interpolation, uniformly bounded.

Theorem 1 (minimum bending energy / variational characterisation). Among all $g \in C^{2} [a, b]$ interpolating the data $f_{0}, \dots, f_{n}$ , the natural cubic spline $s$ is the unique minimiser of the bending energy $\int_{a}^{b} (g^{''})^{2} d x$ , with the Pythagorean identity $\int (g^{''})^{2} = \int (s^{''})^{2} + \int (g^{''} - s^{''})^{2}$ ^{[Holladay 1957]}. The clamped spline is the corresponding minimiser within the class additionally matching the prescribed end slopes. The identity is the function-space statement that $s$ is the orthogonal projection of any interpolant onto the spline subspace in the seminorm $g \mapsto (\int (g^{''})^{2})^{1/2}$ : the cross term $\int s^{''} (g^{''} - s^{''})$ vanishes because $s^{'''}$ is piecewise constant and $g - s$ vanishes at the nodes, while the boundary term is killed by $s^{''} = 0$ at the ends. This is the precise mathematical content of the draughtsman's flexible strip, whose stored elastic energy is, to leading order, $\int (y^{''})^{2}$ .

Theorem 2 (sharp $O (h^{4})$ convergence). For $f \in C^{4} [a, b]$ and the clamped (or not-a-knot) interpolating cubic spline $s$ on a mesh of size $h$ , $$ \lVert f - s \rVert_\infty \le \frac{5}{384}, h^4 \lVert f^{(4)} \rVert_\infty, $$ with the constant $5/384$ best possible, and the derivatives converge at the graded orders $∥ f^{(j)} - s^{(j)} ∥_{\infty} = O (h^{4 - j})$ for $j = 1, 2, 3$ ^{[Hall-Meyer 1976]}. Thus the spline recovers not only the function to fourth order but its slope, curvature, and even its third derivative to orders $h^{3}, h^{2}, h$ — the superconvergence picture that makes cubic splines the default for smooth interpolation and for the construction of $C^{2}$ geometric curves. The natural end condition degrades the boundary rate to $O (h^{2})$ where $f^{''} \neq = 0$ at the ends, a localised loss that does not propagate into the interior.

Theorem 3 (boundedness of the spline projector). The map $P_{h} : C [a, b] \to S_{h}$ sending $f$ to its interpolating (not-a-knot or complete) cubic spline is a linear projection onto the spline space $S_{h}$ , and its $sup$ -norm operator norm — the spline analogue of the Lebesgue constant of 43.08.02 — is bounded by a constant independent of the mesh: $∥ P_{h} ∥_{\infty} \leq 3$ for complete splines on any mesh, and is uniformly bounded for not-a-knot on quasi-uniform meshes ^{[de Boor 2001]}. This mesh-independent bound is the structural reason the spline cannot exhibit the Runge phenomenon: where the polynomial interpolation projector has Lebesgue constant growing like $2^{n}$ for equispaced nodes, the spline projector stays bounded, so $∥ f - P_{h} f ∥_{\infty} \leq (1 + ∥ P_{h} ∥_{\infty}) dist (f, S_{h})$ converges whenever the best spline approximation does.

Theorem 4 (Hermite error and the confluent remainder). The Hermite interpolant matching value and first derivative at $n + 1$ distinct nodes has, for $f \in C^{2 n + 2}$ , the error $$ f(x) - p(x) = \frac{f^{(2n+2)}(\xi(x))}{(2n+2)!}\prod_{k=0}^{n}(x - x_k)^2, $$ the confluent limit of the Lagrange remainder of 43.08.02 in which each node is doubled and the node polynomial acquires squared factors ^{[Süli-Mayers §6.5]}. The squared product is everywhere non-negative, so a value-and-slope Hermite fit at a single pair of nodes has one-signed error, but for many equispaced nodes $\prod_{k} (x - x_{k})^{2}$ inherits the endpoint swelling of 43.08.02 amplified by the square: osculation does not defeat Runge. The cubic spline is the resolution — it is, on each subinterval, exactly a cubic Hermite piece whose interior slopes are solved for rather than prescribed, so the degree stays fixed at three while $C^{2}$ smoothness is achieved globally.

Synthesis. The variational theorem is the foundational reason the cubic spline is the canonical smooth interpolant: it is the unique minimiser of $\int (s^{''})^{2}$ among $C^{2}$ interpolants, and the orthogonality $\int s^{''} (g^{''} - s^{''}) = 0$ is exactly the projection statement that organises the whole subject — the same second-derivative object that appears as the moment $M_{i}$ in the tridiagonal system, as the integrand of the bending energy, and as the curvature whose continuity defines the spline. The central insight is that going piecewise trades one global high-degree polynomial for many fixed-degree cubics coupled only through a sparse, diagonally dominant, $O (n)$ -solvable system, and this is precisely what bounds the spline projector independently of the mesh where the polynomial Lebesgue constant of 43.08.02 diverges. Putting these together, the cubic spline is dual to the Hermite confluent limit of 43.08.01: a chain of Hermite cubics whose slopes are not given but determined by demanding the curvature also match, which generalises local osculatory interpolation into a global interpolant with the sharp $5/384 h^{4}$ accuracy of Hall-Meyer. The bridge is that smoothness across seams is a sparse linear constraint and minimum energy is its variational shadow, so the spline at once defeats the Runge divergence that 43.08.02 diagnosed, supplies the $C^{2}$ shape primitives used in 24.04.07 and in computer-aided geometry, and prefigures the finite-element philosophy of replacing global polynomials by assembled local ones — one principle, piecewise low degree plus matched derivatives, behind interpolation, approximation, and the numerical solution of differential equations alike.

Full proof set Master

Proposition 1 (moment system and existence-uniqueness). For distinct ordered nodes and natural or clamped end conditions, the interpolating cubic spline exists and is unique, its moments solving a strictly diagonally dominant tridiagonal system.

Proof. On $[x_{i}, x_{i + 1}]$ , $s^{''}$ is linear with endpoint values $M_{i}, M_{i + 1}$ , so $s^{''} (x) = M_{i} (x_{i + 1} - x) / h_{i} + M_{i + 1} (x - x_{i}) / h_{i}$ . Integrating twice and imposing $s (x_{i}) = f_{i}$ , $s (x_{i + 1}) = f_{i + 1}$ determines the two constants, giving $$ s(x) = M_i\frac{(x_{i+1}-x)^3}{6h_i} + M_{i+1}\frac{(x-x_i)^3}{6h_i} + \Big(f_i - \frac{M_i h_i^2}{6}\Big)\frac{x_{i+1}-x}{h_i} + \Big(f_{i+1} - \frac{M_{i+1}h_i^2}{6}\Big)\frac{x-x_i}{h_i}. $$ Differentiating gives the one-sided slopes $s^{'} (x_{i}^{+}) = (f_{i + 1} - f_{i}) / h_{i} - h_{i} (2 M_{i} + M_{i + 1}) /6$ and, from the left piece, $s^{'} (x_{i}^{-}) = (f_{i} - f_{i - 1}) / h_{i - 1} + h_{i - 1} (M_{i - 1} + 2 M_{i}) /6$ . Imposing $C^{1}$ continuity $s^{'} (x_{i}^{+}) = s^{'} (x_{i}^{-})$ and multiplying by $6/ (h_{i - 1} + h_{i})$ yields $μ_{i} M_{i - 1} + 2 M_{i} + λ_{i} M_{i + 1} = d_{i}$ with $μ_{i} = h_{i - 1} / (h_{i - 1} + h_{i})$ , $λ_{i} = h_{i} / (h_{i - 1} + h_{i})$ , $d_{i} = \frac{6}{h _{i - 1} + h _{i}} ((f_{i + 1} - f_{i}) / h_{i} - (f_{i} - f_{i - 1}) / h_{i - 1})$ ; $C^{2}$ continuity holds automatically as $s^{''}$ shares $M_{i}$ across pieces. The natural condition closes the system with $M_{0} = M_{n} = 0$ ; the clamped condition adds the end rows from matching $s^{'} (x_{0}), s^{'} (x_{n})$ . Interior rows satisfy $∣2∣ > μ_{i} + λ_{i} = 1$ and the clamped end rows $∣2∣ > 1$ , so the matrix is strictly diagonally dominant, hence nonsingular (a nonzero kernel vector would, at its largest-modulus entry, violate dominance). The moments are unique; substitution reconstructs $s$ . $□$

Proposition 2 (minimum-bending-energy theorem). Let $s$ be the natural cubic spline interpolating the data and $g \in C^{2} [a, b]$ any interpolant of the same data. Then $\int_{a}^{b} (g^{''})^{2} = \int_{a}^{b} (s^{''})^{2} + \int_{a}^{b} (g^{''} - s^{''})^{2}$ , so $s$ uniquely minimises $\int (g^{''})^{2}$ .

Proof. Put $e = g - s \in C^{2} [a, b]$ with $e (x_{i}) = 0$ for all $i$ . Then $\int (g^{''})^{2} = \int (s^{''})^{2} + 2 \int s^{''} e^{''} + \int (e^{''})^{2}$ . Integrate the cross term by parts: $\int_{a}^{b} s^{''} e^{''} = [s^{''} e^{'}]_{a}^{b} - \int_{a}^{b} s^{'''} e^{'}$ . The natural condition $s^{''} (a) = s^{''} (b) = 0$ annihilates the boundary term. On each $[x_{i}, x_{i + 1}]$ the cubic spline has constant $s^{'''} = c_{i}$ , so $\int_{a}^{b} s^{'''} e^{'} = \sum_{i} c_{i} \int_{x_{i}}^{x_{i + 1}} e^{'} = \sum_{i} c_{i} (e (x_{i + 1}) - e (x_{i})) = 0$ because $e$ vanishes at every node. Hence $\int s^{''} e^{''} = 0$ and the Pythagorean identity holds, giving $\int (g^{''})^{2} \geq \int (s^{''})^{2}$ . Equality forces $\int (e^{''})^{2} = 0$ , so $e^{''} \equiv 0$ and $e$ is affine; an affine function vanishing at the $n + 1 \geq 2$ nodes is identically zero, so $g = s$ . $□$

Proposition 3 (Hermite confluent remainder). For $f \in C^{2 n + 2}$ and $p$ the Hermite interpolant matching $f$ and $f^{'}$ at distinct $x_{0}, \dots, x_{n}$ , every $x$ admits $ξ$ with $f (x) - p (x) = f^{(2 n + 2)} (ξ) / (2 n + 2)! \prod_{k} (x - x_{k})^{2}$ .

Proof. Set $w (t) = \prod_{k} (t - x_{k})^{2}$ , monic of degree $2 n + 2$ . For $x$ off the node set put $G = (f (x) - p (x)) / w (x)$ and $ϕ (t) = f (t) - p (t) - Gw (t)$ . Since $p$ matches value and first derivative at each $x_{k}$ , both $f - p$ and $w$ have double zeros at every $x_{k}$ ; thus $ϕ$ has a double zero at each of the $n + 1$ nodes and a simple zero at $x$ , accounting for $2 (n + 1) + 1 = 2 n + 3$ zeros counted with multiplicity. Iterated Rolle (a double zero of $ϕ$ is also a zero of $ϕ^{'}$ , and between consecutive zeros of $ϕ$ lies a zero of $ϕ^{'}$ ) gives $ϕ^{'}$ at least $2 n + 2$ zeros, and after $2 n + 2$ differentiations $ϕ^{(2 n + 2)}$ has a zero $ξ$ . As $p \in Π_{2 n + 1}$ has $p^{(2 n + 2)} \equiv 0$ and $w$ is monic of degree $2 n + 2$ with $w^{(2 n + 2)} \equiv (2 n + 2)!$ , $0 = ϕ^{(2 n + 2)} (ξ) = f^{(2 n + 2)} (ξ) - G (2 n + 2)!$ , so $G = f^{(2 n + 2)} (ξ) / (2 n + 2)!$ and the claim follows. $□$

Proposition 4 ( $O (h^{4})$ uniform convergence, complete spline). For $f \in C^{4} [a, b]$ the complete cubic spline interpolant $s$ on a uniform mesh of size $h$ satisfies $∥ f - s ∥_{\infty} \leq C h^{4} ∥ f^{(4)} ∥_{\infty}$ for an absolute constant $C$ (the sharp value being $5/384$ ).

Proof (order, with the constant cited). Let $e = f - s$ . The moments $M_{i} = s^{''} (x_{i})$ solve the tridiagonal system whose right-hand side is the centred second difference $d_{i}$ of $f$ , which equals $f^{''} (x_{i}) + O (h^{2})$ by Taylor expansion of $f$ to fourth order ^{[Süli-Mayers §11.5]}. Subtracting the exact relation satisfied by $f^{''} (x_{i})$ (with the consistency error from the difference approximation) gives a tridiagonal system for $M_{i} - f^{''} (x_{i})$ with right-hand side $O (h^{2}) ∥ f^{(4)} ∥_{\infty}$ ; strict diagonal dominance bounds the solution by its right-hand side (Exercise 8's maximum-index argument), so $∥ s^{''} - f^{''} ∥_{\infty, nodes} = O (h^{2}) ∥ f^{(4)} ∥_{\infty}$ . On each subinterval $s$ is the cubic Hermite interpolant of the nodal values and the (spline-determined) slopes; writing $e$ via the Hermite/Peano representation on $[x_{i}, x_{i + 1}]$ and inserting the nodal second-derivative error propagates the bound to $∥ e ∥_{\infty} = O (h^{4}) ∥ f^{(4)} ∥_{\infty}$ . The sharp constant $5/384$ , attained in the limit by the worst-case quartic, is established by the Peano-kernel optimisation of Hall-Meyer ^{[Hall-Meyer 1976]}; the graded derivative bounds $∥ f^{(j)} - s^{(j)} ∥_{\infty} = O (h^{4 - j})$ follow from differentiating the same representation. $□$

Connections Master

The Hermite interpolant of this unit is the confluent (repeated-node) limit of the Lagrange/Newton construction of 43.08.01: the generalized divided differences $f [x_{k}, \dots, x_{k}] = f^{(j)} (x_{k}) / j!$ are the Hermite-Newton coefficients, and the squared node polynomial $\prod_{k} (x - x_{k})^{2}$ in the error is the doubled form of the node polynomial of 43.08.02; the cubic spline assembles local Hermite cubics into a $C^{2}$ global interpolant by solving for the interior slopes rather than prescribing them.
The cubic spline directly resolves the Runge phenomenon diagnosed in 43.08.02: where equispaced high-degree polynomial interpolation has a Lebesgue constant growing like $2^{n}$ , the spline interpolation projector is bounded independently of the mesh, so the spline converges at the sharp $O (h^{4})$ rate for smooth functions instead of diverging; the same node polynomial that swelled at the endpoints there is here confined to a single short subinterval where it cannot grow.
The tridiagonal moment system and its strict diagonal dominance prefigure the assembly of $C^{1}$ / $C^{2}$ finite elements in 24.04.07, where Hermite and spline elements produce the identical banded, diagonally dominant stiffness matrices solved in $O (n)$ ; the minimum-bending-energy variational characterisation is the interpolation-theoretic shadow of the energy-minimisation (Ritz) principle on which the finite-element method rests, and the spline is the one-dimensional thin-beam element.
The minimum-curvature theorem realises the spline as an orthogonal projection in the second-derivative seminorm, the same projection structure that governs the best $L^{2}$ approximation by orthogonal polynomials in 43.08.05; both are instances of a Hilbert-space best-approximation theorem, with the spline projecting onto a piecewise-polynomial subspace under the bending-energy inner product and the least-squares approximant projecting onto $Π_{n}$ under a weighted $L^{2}$ inner product.

Historical & philosophical context Master

The word spline names a thin flexible strip of wood or metal that shipwrights and draughtsmen bent through fixed pegs (ducks) to trace a fair curve; the strip settles into the shape minimising its stored bending energy, which to leading order is $\int (y^{''})^{2}$ . The transfer of this physical object to a mathematical interpolation scheme is due to Isaac Jacob Schoenberg, whose 1946 Quarterly of Applied Mathematics papers introduced spline functions and the contributions to their approximation theory, founding the subject during his wartime work at the Aberdeen Proving Ground. The variational characterisation — that the natural cubic spline uniquely minimises $\int (g^{''})^{2}$ among $C^{2}$ interpolants — was proved by John C. Holladay in 1957, giving the precise sense in which the mathematical spline reproduces the physical one ^{[Holladay 1957]}.

Hermite interpolation carries the name of Charles Hermite, whose work on the confluent form of the divided-difference calculus in the 1870s, with Angelo Genocchi, supplied the integral representation that handles repeated nodes and derivative matching. The sharp error analysis of cubic spline interpolation, including the best-possible constant $5/384$ in the $C^{4}$ uniform bound and the graded convergence of the derivatives, was settled by Charles A. Hall and W. Weston Meyer in 1976 ^{[Hall-Meyer 1976]}. The comprehensive computational and approximation-theoretic treatment, including the B-spline basis that makes splines numerically robust and the boundedness of the spline projector, is Carl de Boor's 1978 A Practical Guide to Splines, for which he received the 2003 National Medal of Science ^{[de Boor 2001]}.

Bibliography Master

@book{sulimayers2003,
  author    = {S\"{u}li, Endre and Mayers, David F.},
  title     = {An Introduction to Numerical Analysis},
  publisher = {Cambridge University Press},
  year      = {2003}
}

@book{deboor2001,
  author    = {de Boor, Carl},
  title     = {A Practical Guide to Splines},
  edition   = {Revised},
  series    = {Applied Mathematical Sciences},
  volume    = {27},
  publisher = {Springer},
  year      = {2001}
}

@article{holladay1957,
  author  = {Holladay, John C.},
  title   = {A smoothest curve approximation},
  journal = {Mathematical Tables and Other Aids to Computation},
  volume  = {11},
  number  = {60},
  year    = {1957},
  pages   = {233--243}
}

@article{hallmeyer1976,
  author  = {Hall, Charles A. and Meyer, W. Weston},
  title   = {Optimal error bounds for cubic spline interpolation},
  journal = {Journal of Approximation Theory},
  volume  = {16},
  number  = {2},
  year    = {1976},
  pages   = {105--122}
}

@article{schoenberg1946,
  author  = {Schoenberg, Isaac J.},
  title   = {Contributions to the problem of approximation of equidistant data by analytic functions},
  journal = {Quarterly of Applied Mathematics},
  volume  = {4},
  year    = {1946},
  pages   = {45--99 and 112--141}
}

Prerequisites

43.08.01
43.08.02

Tier anchors

beginner: Süli-Mayers 2003 *An Introduction to Numerical Analysis* (Cambridge) §6.5, §11.2-11.5 (Hermite interpolation, and splines as the smooth piecewise alternative to high-degree fits, at the level of a first numerical-methods course); Burden-Faires 2011 *Numerical Analysis* 9e (Brooks/Cole) §3.4-3.5 for the Hermite divided-difference table and the cubic-spline bookkeeping on worked nodes
intermediate: Süli-Mayers 2003 *An Introduction to Numerical Analysis* (Cambridge) §6.5 (Hermite interpolation and its error term with the squared node polynomial) and §11.2-11.5 (the cubic spline, the tridiagonal moment system, natural and clamped end conditions, existence-uniqueness); de Boor 2001 *A Practical Guide to Splines* revised ed. (Springer) Ch. IV-V (the B-spline view, the tridiagonal system, and not-a-knot)
master: de Boor 2001 *A Practical Guide to Splines* revised ed. (Springer) Ch. V (the minimum-curvature variational characterisation, the $O(h^4)$ convergence, and the spline projector) and Ch. IX-X; Hall-Meyer 1976 *J. Approx. Theory* 16(2) (the sharp $5/384$ constant in the clamped cubic-spline error bound); Holladay 1957 *Quart. Appl. Math.* 15 (the original minimum-bending-energy theorem for the natural cubic spline)

References

Süli, E. & Mayers, D. F. — An Introduction to Numerical Analysis · Cambridge University Press (2003). §6.5 builds Hermite (osculatory) interpolation as the confluent limit of Lagrange interpolation in which nodes are repeated according to the number of derivative conditions matched, gives the generalized divided-difference table with repeated arguments, and proves the error term f(x)-p(x) = f^{(2n+2)}(xi)/(2n+2)! prod_k (x-x_k)^2 for the two-point-per-node case. Chapter 11 (§11.2-11.5) develops piecewise polynomial and spline interpolation: the natural and complete (clamped) cubic spline, the derivation of the tridiagonal linear system for the second derivatives (moments) M_i = s''(x_i) via continuity of s' across interior knots, strict diagonal dominance of the tridiagonal matrix giving existence and uniqueness, and the order-h^4 error bound for the interpolation of a smooth function by the cubic spline on a quasi-uniform mesh.
de Boor, C. — A Practical Guide to Splines · Springer, revised edition (2001). Chapters IV-V develop the cubic spline as the central example of the B-spline theory: the interpolation problem with the various end conditions (natural, complete/clamped, not-a-knot, periodic), the tridiagonal system in the slopes or the moments and its diagonal dominance, and the minimum-curvature (minimum bending energy) variational theorem stating that among all C^2 interpolants of the data the natural cubic spline uniquely minimises int (g'')^2; Chapter V also gives the O(h^4) convergence of the clamped spline to a C^4 function and the boundedness of the spline interpolation projector, the property that distinguishes splines from high-degree polynomial interpolation whose Lebesgue constants diverge.
Holladay, J. C. — A smoothest curve approximation · Mathematical Tables and Other Aids to Computation (later Quart. Appl. Math.) 11 (1957). The original proof that, among all twice-continuously-differentiable functions interpolating a prescribed table, the natural cubic spline is the unique minimiser of the integral of the square of the second derivative — the discrete analogue of the bending energy of a thin elastic beam (the draughtsman's spline), giving the variational characterisation that names the object.
Hall, C. A. & Meyer, W. W. — Optimal error bounds for cubic spline interpolation · Journal of Approximation Theory 16(2) (1976), 105-122. Proves the sharp pointwise and uniform error bounds for clamped cubic spline interpolation of a function in C^4: ||f - s||_infty <= (5/384) h^4 ||f^{(4)}||_infty with the constant 5/384 best possible, together with the order-h^{4-j} bounds on the j-th derivatives s^{(j)} for j=1,2,3, establishing the full superconvergence picture and that the cubic spline recovers the function and its first three derivatives to the orders h^4, h^3, h^2, h respectively.

Estimated time

beginner: 20m
intermediate: 50m
master: 90m