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LANDAU Function (Gnumeric)

Statistical Intermediate Gnumeric

Evaluate the approximate standard Landau probability density in Gnumeric, with a checked value at zero and guidance on density, scaling, and numerical limits.

Verification status: See the evidence and test scope described in this guide. How formula examples are checked

LANDAU evaluates an approximation to the probability density of the Landau distribution at a supplied x. The returned height is a density, not the probability that a continuous random variable equals that exact value, and not a cumulative probability up to x. This distinction matters when interpreting small numbers from a distribution function.

Syntax and parameterization

=LANDAU(x)

x is one required numeric argument. There are no optional location, scale, cumulative-mode or precision arguments. The inspected implementation uses a fixed standard parameterization and piecewise approximations derived from GNU Scientific Library code. A differently parameterized Landau distribution in another package may require a transformation before values can be compared.

Do not replace LANDAU with a normal density because the plotted peak looks familiar. Its asymmetric tail behavior is part of a different model, and a density function does not generate random observations.

Reproducible value at zero

=LANDAU(0) has expected Gnumeric approximation 0.1788541609. In the source’s interval containing zero, the rational approximation reduces at x=0 to its leading numerator coefficient divided by its leading denominator coefficient,0.1788541609/1.

A separate numerical quadrature of the standard Landau density integral at zero gives approximately 0.178854160675249. The difference is about 2.25×10^-10, consistent with this being an approximation rather than an exact symbolic value. The comparison uses an independent integral calculation, not a reimplementation of the same rational polynomial. Use a practical tolerance such as 10^-8 for this single check rather than demanding identical final digits.

For a small worksheet, put0 in A2, the formula =LANDAU(A2) in B2, and label B1 Density. Keep the word density visible so a reader does not interpret0.17885 as a cumulative 17.885% probability. Obtaining a probability over an interval would require integrating the density over that interval; LANDAU has no interval endpoints or cumulative switch.

Advanced use: comparing parameterizations

If a model explicitly defines Y = location + scale×X with positive scale and X following this exact standard parameterization, a transformed density is LANDAU((y−location)/scale)/scale. The division by scale is essential for a density transformation. This algebraic construction does not establish that an external library uses the same standard Landau convention; verify that first.

Avoid interpreting a grid of sampled density heights as probabilities unless the integration weights and approximation are handled correctly. A plot is useful for shape inspection, but the sum of raw density heights depends on the chosen grid spacing.

Errors, numerical limits and version history

Supply numeric values and check any upstream error. The implementation has piecewise branches for different x ranges and floating-point tail calculations. The zero fixture does not establish accurate behavior for huge magnitudes, underflow, overflow or all branch boundaries. A displayed zero in an extreme tail is not proof of mathematically zero density.

NEWS records LANDAU among new statistical functions in Gnumeric 1.1.7, integrated from R/GSL material. The retained implementation explicitly describes an approximate density. This guide therefore does not claim an exact distribution evaluator, a quantile function, a random generator, or a retirement date.

Native fixture results

The native 1.12.57 run returned exactly 0.1788541609 for LANDAU(0). The independent quadrature differs in the tenth decimal place, within the stated tolerance.

Availability and verification scope

The function is present in the inspected official source. Bounded examples are independently calculated; native Gnumeric ssconvert 1.12.57 results are recorded separately below. Retained statistical source 1.12.62 was originally obtained October 13–14, 2026 and inspected October 17; HPFILTER source 1.12.57 was freshly retrieved October 17. No historical-release execution, GUI operation, screenshot, or native 1.12.62 result is implied. The last functioning release is not established. English formulas use commas, with locale-specific separators adjusted as needed.

Array-returning functions require an appropriate result area. The INDEX formulas shown inspect individual elements without promising automatic dynamic-array spilling in the GUI.