Skip to content

Business Mathematics and Statistics · Ch 5 — Numerical Methods (Finite Differences, Interpolation)

Newton's Backward Interpolation Formula

3

Newton's Backward Interpolation Formula

When to use it

Newton's backward formula estimates yy for an xx-value near the END of an equally-spaced table, using the LAST row's own backward differences.

The formula

With p=x−xnhp=\dfrac{x-x_n}{h} (xnx_n the LAST tabulated point, hh the common spacing):

y=yn+p ∇yn+p(p+1)2! ∇2yn+p(p+1)(p+2)3! ∇3yn+⋯y=y_n+p\,\nabla y_n+\frac{p(p+1)}{2!}\,\nabla^2y_n+\frac{p(p+1)(p+2)}{3!}\,\nabla^3y_n+\cdots

Note

The sign pattern flips between forward and backward

Forward uses p(p−1)p(p-1), p(p−1)(p−2)…p(p-1)(p-2)\ldots; backward uses p(p+1)p(p+1), p(p+1)(p+2)…p(p+1)(p+2)\ldots — a genuinely common setup mistake is copying the forward formula's (p−1)(p-1) factors into a backward calculation, or vice versa.

Worked reasoning …

Definition 1Newton's Backward Interpolation Formula

y=yn+p∇yn+p(p+1)2!∇2yn+⋯y=y_n+p\nabla y_n+\frac{p(p+1)}{2!}\nabla^2y_n+\cdots, where p=x−xnhp=\frac{x-x_n}{h}; used for interpolating near the END of a …

Start your 14-day free trial to unlock the full solution →
← Previous · 2Newton's Forward Interpolation FormulaNext · 4 →Lagrange's Interpolation Formula