Implementation of a 1D Interpolation Module

[srinjoy933]

To implement a high-performance 1D interpolation module in fortran_stdlib, I propose a design centered on a unified stdlib_interpolation interface that leverages the fypp preprocessor to ensure type-safety across all supported real kinds (sp, dp, xdp, qp). The core architecture will utilize an $O(\log n)$ bisection-based interval hunting strategy to efficiently locate target coordinates within discretized datasets, preventing the performance degradation typical of linear scans in large-scale scientific simulations. This module will integrate two primary mathematical backends: a piecewise linear routine for rapid $f(x)$ lookups and a more sophisticated cubic spline implementation that ensures $C^2$ continuity across all knots.
The cubic spline implementation will require the internal integration of a robust tridiagonal matrix solver, specifically the Thomas Algorithm, to compute the second derivatives at each knot point under specified boundary conditions (e.g., natural or clamped). By embedding this numerical logic directly into stdlib, we provide a "Fortran-native" solution that maintains a clear separation of responsibilities while eliminating the overhead of linking against external libraries like LAPACK for common fitting tasks. This feature directly supports the stdlib mission by providing a scalable, preprocessed framework that can eventually be extended to bilinear and bicubic interpolation for multi-dimensional data arrays.

[srinjoy933]

@jalvesz what's your opinion on this module? Looking for your feedback.

[jalvesz]

Hi @srinjoy933, the idea is appealing but the text looks like the summary of a first prompt to an LLM. Nothing inherently bad about it but it is rough and needs digging:

What's your current understanding of the state of the art for this in other ecosystems? Numpy, SciPy, scikit-learn, Julia, R, C++? This is important to check common patterns in the consumer facing API, as well as efficient internal implementations. Have you searched for packages in the Fortran ecosystem which propose this to see how they do it?

The text talks about not needing lapack, but stdlib already embeddes a modernized version of blas\lapack as well as allowing linking against optimized libraries such as MKL, OpenBLAS, etc. You might want to ponder this in the proposal. @loiseaujc also established the base to enable tridiagonal matrices and solvers around them. So you might want to try and build on top of that.

[loiseaujc]

Tridiagonal matrices and associated solvers are indeed already provided by the stdlib_specialmatrices module so you could build on top of that. Moreover, as I mentioned here, @jacobwilliams has already modernized a whole bunch of interpolation softwares in Fortran. These include: bspline-fortran, splpak, pchip, or finterp.

Along with doing a survey of what's available in other standard languages/libraries (e.g. numpy, scipy, Julia, etc) to get a feel for what a good API may look like, you could also look at how @jacobwilliams implemented things in Fortran to get some ideas. I don't know exactly which licences @jacobwilliams is using. Maybe you could also discuss with him to see if some parts of his modernized implementations could be re-used directly into stdlib.