TSSQ: Translated Singularity Swap Quadrature

tssq-matlab provides a MATLAB implementation of the quadrature proposed in the publication Stabilizing the singularity swap quadrature for near-singular line integrals by D. Krantz, A. H. Barnett and A.-K. Tornberg (BIT Numerical Mathematics, 2026).

Translated singularity swap quadrature (TSSQ) improves the standard SSQ method with translated bases that avoid catastrophic cancellation and achieve high accuracy for very close evaluation.

Dependencies

This repository builds on two external packages:

• ludvigak/linequad
  Original SSQ code containing core quadrature routines.
• taiya/kdtree
  Minimalist kd-tree implementation for neighbor searches.

Optional: chebfun (only used in some test scripts)

Setup

After cloning, cd to the root of this repo and initialize the submodules by running

git submodule update --init --recursive

Compilation

linequad

From the root of this repo within MATLAB:

cd external/linequad/matlab
make

kdtree

From the root of this repo within MATLAB:

cd external/kdtree/toolbox
kdtree_compile

NOTE: To compile kdtree on macOS, replace line 154 in external/kdtree/toolbox/MyHeaps.h

return std::__is_heap(heap.begin(), heap.end() );

with

return std::is_heap(heap.begin(), heap.end());

Usage

1. Start MATLAB in the root of this repo.
2. Run init to add paths.
3. Try the examples:
   • Basic examples:
     • matlab/examples/example_open_curve_monomial.m
     • matlab/examples/example_closed_curve_fourier.m
   • Paper examples:
     • matlab/examples-paper/long_filament.m
     • matlab/examples-paper/deformed_starfish.m
     • matlab/examples-paper/bases_comp_flat_panel.m

• Comparison of SSQ and TSSQ for close evaluation of the slender-body kernel with error tolerance $10^{-6}$.

Main functions

Core entry points for users are:

• ssq_sbt(curve,density,targets,opts)
  Standard SSQ for the slender-body kernel.

• tssq_sbt(curve,density,targets,opts)
  Translated SSQ (TSSQ) for the slender-body kernel. Automatically switches between standard and modified bases.

Supporting routines

These are lower-level routines used internally by SSQ/TSSQ, but may be useful for advanced users.

Monomial (open curves)

• Standard basis integrals:
  external/linequad/matlab/src/recursions/rsqrt_pow_integrals.m

• Translated basis integrals:
  matlab/src/recursions/rsqrt_pow_integrals_shift.m

• SSQ weights:
  external/linequad/matlab/src/line3_near_weights.m

• TSSQ weights:
  matlab/src/near_weights_monomial.m

• Quadrature driver:
  matlab/src/interpolatory_quadrature_monomial.m

Fourier (closed curves)

• Standard basis integrals:
  matlab/src/recursions/rsqrt_pow_integrals_fourier.m

• TSSQ weights:
  matlab/src/near_weights_fourier.m

• Adjoint helper for modified Fourier basis:
  matlab/src/adjoint_modified_fourier.m

• Modified Fourier coefficients:
  matlab/src/fourier2modcoeffs.m

• Quadrature driver:
  matlab/src/interpolatory_quadrature_fourier.m

Code structure

matlab/
├── examples/        % simpler demos
├── examples-paper/  % scripts for figures in the paper
├── src/             % main TSSQ and SSQ routines
├── test/            % tests and exploratory scripts
├── utils/           % helper routines (interpolation, Vandermonde, etc.)

External dependencies are in external/.

Citation

If you find this code useful in your research, please cite our publication:

• D. Krantz, A. H. Barnett and A.-K. Tornberg, Stabilizing the singularity swap quadrature for near-singular line integrals, BIT Numerical Mathematics, 66 (2026), p. 39, https://doi.org/10.1007/s10543-026-01132-w.

The software itself is also archived on Zenodo and can be cited as:

• D. Krantz, davidkrantz/tssq-matlab: v1.1.0, Mar. 2026, https://doi.org/10.5281/zenodo.17962659.