solveVandermonde: Numerically stable non-confluent Vandermonde system solver

In pnd: Parallel Numerical Derivatives, Gradients, Jacobians, and Hessians of Arbitrary Accuracy Order

Numerically stable non-confluent Vandermonde system solver

Description

Numerically stable non-confluent Vandermonde system solver

Usage

solveVandermonde(s, b)

Arguments

s	Numeric vector of stencil points defining the Vandermonde matrix on the left-hand side, where each element S_{i, j} is calculated as s[j]^(i-1).
b	Numeric vector of the right-hand side of the equation. This vector must be the same length as s.

Details

This function utilises the \insertCitebjorck1970solutionpnd algorithm for an accurate solution to non-confluent Vandermonde systems, which are known for their numerical instability. Unlike Gaussian elimination, which suffers from ill conditioning, this algorithm achieves numerical stability through exploiting the ordering of the stencil. An unsorted stencils will trigger a warning. Additionally, the stencil must contain unique points, as repeated values make the Vandermonde matrix confluent and therefore non-invertible.

This implementation is a verbatim translation of Algorithm 4.6.2 from \insertCitegolub2013matrixpnd, which is robust against the issues typically associated with Vandermonde systems.

See \insertCitehigham1987errorpnd for an in-depth error analysis of this algorithm.

Value

A numeric vector of coefficients solving the Vandermonde system, matching the length of s.

References

\insertAllCited

Examples

# Approximate the 4th derivatives on a non-negative stencil
solveVandermonde(s = 0:5, b = c(0, 0, 0, 0, 24, 0))

# Small numerical inaccuracies: note the 6.66e-15 in the 4th position --
# it should be rounded towards zero:
solveVandermonde(s = -3:3, b = c(0, 1, rep(0, 5))) * 60

pnd documentation built on Sept. 9, 2025, 5:44 p.m.