A natural-spline tridiagonal residual bounds the spline value error

Let $x_1<\cdots <x_r$, $r\ge 3$, and let $s$ be the natural cubic interpolant of nodal values $y_i$. Write $h_i=x_{i+1}-x_i$ and $H={\max }_i{h}_i$. Let $A$ be the $(r-2)\times (r-2)$ interior coefficient matrix in the system of Natural cubic interpolation at distinct knots, and let $b$ be its right-hand side. Put $\delta ={\min }_{2\le i\le r-1}(h_{i-1}+h_i)>0.$ For any proposed nodal second derivatives ${\widehat{M}}_1={\widehat{M}}_r=0$, let $\widehat{m}$ be their interior vector, let $\rho =A\widehat{m}-b$ be the exact residual, and let $\widehat{s}$ be the piecewise cubic given by the natural piece formula with these second derivatives and the same values $y_i$. Then, on the knot hull, $\mathrm{\parallel }\widehat{s}-s{\mathrm{\parallel }}_{L^{\mathrm{\infty }}([x_1,x_r])}\le \frac{H^2}{8\delta }\mathrm{\parallel }\rho {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ This is a residual-to-function transfer statement; it does not assert that a particular floating-point solver returns a residual certificate.