Comparison of the trapezoidal rule error with the spline rule error
The main result of the paper is the following theorem. \textbf{Theorem.} Let \(\Delta = \{ x_j : j = 0,1,\dots,n \}\) is a partition of the interval \([a,b]\) by knots \(a= x_0 < x_1 <\dots < x_n = b\), and \(h_{j + 1} = x_{j + 1} - x_j , j = 0,1,\dots,n -1\). For the function \(f \in C^2[a,b]\) we set \(y_j = f(x_j ),\;j = 0,1,\dots,n\), and denote by \(S_\Delta (x)\) the cubic spline from the class \(C^2[a,b]\) which interpolates the values of the function \(f\) at knots \(x_0 ,x_1 ,\dots,x_n \) and satisfies the conditions \({S}''_\Delta (a) = {S}''_\Delta (b) = 0\). Then we obtain the following equation for the error of integration of \(f\) by cubic spline: \[ \int_a^b {S_\Delta (x)\,dx - \int_a^b {f(x)\,dx} = \frac{1}{12}\left\{ {\sum_{j = 0}^{n - 1} {h_{j + 1}^3 f''(\eta _j ) - \sum_{j = 0}^{n - 1} {h_{j + 1}^3 \left( {\frac{M_j + M_{j + 1} }{2}} \right)} } } \right\}} , \] where \(\eta _j \in [x_j ,x_{j + 1} ], M_j = {S}''_\Delta (x_j )\) for \(j = 0,1,\dots,n\). For uniform partition \(\Delta = \{x_j: x_j = a + jh,\,j = 0,1,\dots,n \}\) and \(h = \frac{b - a}{n}\) we have \[ \int_a^b {S_\Delta (x)\,dx - \int_a^b {f(x)\,dx} = \frac{(b - a)}{12}h^2f''(\eta ) - \frac{h^3}{12}\sum_{j = 1}^{n - 1} {M_j ,\quad \eta \in [a,b]} } . \]
scientific article; zbMATH DE number 5128310