Enter three consecutive real sequence values, s0, s1 and s2, in order. You can also enter three consecutive partial sums of a series. This calculator applies one Aitken delta-squared transformation and shows the differences used in the calculation. Inputs are dimensionless; no known limit or function expression is required. It does not generate sequence terms or iterate automatically.
Aitken’s process targets linearly convergent sequences, whose errors approach a fixed geometric ratio in magnitude below one. It can be useful when your consecutive approximations behave approximately this way, without requiring you to know the limit. The transformation and its convergence setting are described in NIST DLMF §3.9(iii).
Three values alone cannot establish convergence or guarantee that the transformed estimate is more accurate. A finite result is an estimate, not a convergence test or an error bound.
If s2 − 2s1 + s0 is exactly zero, the transformed estimate is unavailable because its formula divides by zero. This also applies when all three values are equal: the expression becomes 0/0, even though the supplied values are identical.
A second difference close to zero relative to the input values can make the estimate highly sensitive to rounding or small input errors. Subtracting nearly equal numbers can lose significant digits, and division by a small denominator can magnify those errors. Retain as many reliable input digits as possible. A very small nonzero difference may display as zero after rounding to eight decimal places; the displayed digits do not certify accuracy.
| s0 | s1 | s2 | s1 − s0 | s2 − s1 | Second difference | Transformed estimate |
|---|---|---|---|---|---|---|
| 1 | 1.5 | 1.75 | 0.5 | 0.25 | −0.25 | 2 |
| 1 | 2 | 3 | 1 | 1 | 0 | Unavailable |
| 2 | 2 | 2 | 0 | 0 | 0 | Unavailable |
The default example uses the first three partial sums of 1 + 1/2 + 1/4 + ⋯. The correction is 0.5 × 0.5 ÷ (−0.25) = −1, so the estimate is 1 − (−1) = 2. For this particular geometric series, 2 is the exact sum. That success does not establish the limit of an arbitrary sequence with the same first three values.
Use the Series Convergence Calculator to investigate convergence from an nth-term expression. Use the Richardson Extrapolation Calculator when your estimates follow its known second-order step-size error model. Aitken’s calculator instead transforms three consecutive sequence values without requiring that prescribed refinement model.