Finite Difference Stencil & Accuracy Calculator
Choose the derivative, target order, stencil direction, and grid spacing. The calculator returns the smallest supported stencil that meets the requested formal accuracy, with exact normalized weights and the leading truncation-error term.
Required sample points3
- Sample offsets
- x0 - h, x0, x0 + h
- Normalized weights
- -1/2, 0, 1/2
- Scaled coefficients
- -0.5, 0, 0.5
- Finite difference formula
- f^(1)(x0) ~= (-1/2 f(x0 - h) + 1/2 f(x0 + h)) / h
- Achieved accuracy order
- 2
- Leading truncation error
- 1/6 h^2 f^(3)(x0)
Stencil weights
| Point | Offset | Normalized weight | Scaled coefficient |
|---|---|---|---|
| 1 | x0 - h | -1/2 | -0.5 |
| 2 | x0 | 0 | 0 |
| 3 | x0 + h | 1/2 | 0.5 |
Normalized weights are exact rational coefficients before division by h^m.
Large high-order stencils can amplify floating-point roundoff and data noise.
How to use this calculator
- Enter the derivative order you need, such as 1 for f prime or 2 for f double prime.
- Enter the target accuracy order p for a truncation error of O(h^p) or better.
- Choose centered for an interior point, or forward or backward for a boundary point.
- Enter the uniform grid spacing h and read the selected offsets, weights, and error term.
How the stencil is calculated
A finite difference stencil approximates a derivative by combining nearby sampled values of a smooth function. For derivative order m, grid spacing h, and integer offsets j, the approximation has the form:
f^(m)(x0) ~= h^(-m) sum w_j f(x0 + jh)
The normalized weights w_j are chosen so that the Taylor-series moments match the requested derivative and cancel the lower terms:
sum w_j j^k = m! when k = m, and sum w_j j^k = 0 for the other matched powers.
The implementation uses Fornberg's recursive finite-difference weight algorithm. That avoids directly inverting a Vandermonde matrix, which is a common source of unnecessary numerical sensitivity. Because the offsets are integers, the displayed normalized weights are kept as rational numbers. The scaled coefficients are those same weights divided by h^m.
How the smallest stencil is chosen
The calculator tries candidate stencils in increasing size. A forward stencil uses offsets 0, 1, ..., N-1. A backward stencil uses -(N-1), ..., -1, 0. A centered stencil uses symmetric odd widths -q, ..., 0, ..., q. Candidates with fewer than m + 1 sample points are skipped.
After weights are found, the calculator scans the Taylor moments beyond the requested derivative. The first nonzero moment M_k = sum w_j j^k gives the achieved accuracy p = k - m and the leading truncation term:
(M_k / k!) h^p f^(k)(x0)
Centered stencils often cancel an extra moment by symmetry, so they can exceed the requested order. For example, the three-point centered first-derivative stencil has second-order accuracy even though only three values are used.
What changes the coefficients most
The derivative order and stencil direction have the largest effect on the weights. Higher derivatives and one-sided boundary formulas usually need more points for the same accuracy and can create larger coefficients. Grid spacing does not change the normalized weights, but the usable coefficients scale as 1 / h^m, so a small h can strongly amplify roundoff, cancellation, and measurement noise.
What this calculator leaves out
The formulas assume one spatial dimension, equally spaced points, and a sufficiently smooth function near x0. They do not build formulas for irregular grids, multidimensional mixed derivatives, boundary conditions, filtering, or time-dependent stability. Formal truncation order also does not choose the best h; in floating-point work, making h smaller eventually increases roundoff error even while the Taylor truncation term shrinks.
Worked example
With the default inputs, m = 1, target order 2, centered stencil, and h = 1. The smallest centered candidate that works has offsets -1, 0, 1.
Fornberg's weights are -1/2, 0, 1/2, so the formula is f^(1)(x0) ~= (-1/2 f(x0 - h) + 1/2 f(x0 + h)) / h. The first uncanceled Taylor moment after the first derivative is at k = 3, giving achieved order 2 and leading error h^2 f^(3)(x0) / 6.
Common questions
What is a finite difference stencil?
A finite difference stencil is the set of sample locations used to approximate a derivative from function values. On an equally spaced grid, the locations can be written as offsets from x0, such as x0 - h, x0, and x0 + h.
How many points are needed for a requested accuracy?
For one-sided stencils, each added point usually raises the formal order by one once there are enough points for the derivative. Centered stencils use symmetric odd widths and may gain an extra order because odd or even Taylor moments cancel by symmetry.
When should I use forward, backward, or centered differences?
Use centered differences when x0 has usable sample points on both sides. Use forward differences near a left boundary and backward differences near a right boundary, where values on one side of x0 are unavailable or outside the domain.
Why can a centered stencil exceed the requested order?
A symmetric centered stencil cancels Taylor terms with the wrong parity for the requested derivative. That symmetry can make the first nonzero omitted term farther away than the minimum target order you entered.
How do grid spacing and derivative order scale the coefficients?
The normalized weights depend on the offsets and derivative order, not on h itself. The coefficients applied to function values are the normalized weights divided by h^m, so higher derivatives and smaller spacing make the applied coefficients larger.
Why can making h extremely small reduce numerical accuracy?
The truncation term shrinks with h, but the weighted sum can subtract nearly equal floating-point values. Once cancellation and rounding dominate, reducing h further can make the numerical derivative worse even though the formal stencil order is unchanged.