## Special pointsets

- For special pointsets over intervals, triangles (simplex), disk, square, and polygons, see the following url.

## Recent codes

**Matlab codes for computing cubature rules over general polygons.**

Object: this software computes cubature rules over general polygons (convex or not convex, connected or disconnected, simply or not simply connected). It needs polyshape environment.

Applications to optics are given.

» Paper: Compressed cubature over polygons with applications to optical design

» Source: [.zip]

» Last update: May 12, 2019.**Matlab codes for computing Gaussian Rules for Symmetric weight functions.**

» Source: [ZIP]

» Last update: July 27, 2012.**Matlab codes for computing Fejer and Clenshaw-Curtis rules for general weight functions.**

» Source: [ZIP]

» Last update: July 27, 2012 (Tested August 22, 2016)]**Matlab codes for computing good interpolation points or quadrature rules on geographic subregions of the sphere.**

» Object: In this version we determine cubature rules, hyper interpolation and wam on spherical rectangles of the 2-sphere. The novelty is the usage of new cubature rules that are numerically exact but with inferior cardinality. The zip file contains also the new version of the routine trigauss with some tests.

» Source:- New version: [MATLAB CODES (zip file)]
- Old version: [MATLAB CODES (zip file), interpolation], [MATLAB CODES (zip file), cubature]

**Matlab codes for computing good cubature points on polygons.**

» Object: The polygauss.m routine allows the computation of cubature rules with a certain degree of exactness on a polygon (without self-intersections). The new routine works via quadrangles panelling or triangularization. Some new demos show its usage. We point out that these rules can be compressed, providing a rule with less nodes/weights but the same algebraic degree of exactness (see comprexcub.m).

» Source:- New version: [POLYGAUSS_230816.zip]
- Old version: [POLYGAUSS_2013_300413.zip]

**Matlab codes for computing Orthogonal polynomials and hyper interpolation over general regions.**

» Object: The MULTIVOP_220816.zip suite allows to compute hyperinterpolants over general regions. Here we show examples on some circular regions or polygons. The more important routine is multivop.m (see the paper http://www.math.unipd.it/~marcov/pdf/multivop.pdf) that allows the numerical computation of orthogonal polynomials over general regions.

» Source: Matlab code for Gauss-like cubature over polygons [MULTIVOP_220816]

» Last update: August 25, 2016.**Trigauss: quadrature rules for trigonometric polynomials..**

» Object: The routine*trigauss.m*computes a quadrature rule over an interval, exact for trigonometric polynomial degree N. Several versions, including the Gauss-Kronrod version (it is not known if they always exist!), are available.

» Source:- New version: [MATLAB CODES (zip file)]
- Old version: [ZIP] [MATLAB CODES (zip file), cubature]

**HYPERTRIG and HYPERPOL suites.**

» Object: Software for 1D trigonometric hyperinterpolation and hyperinterpolation over spherical rectangles. HYPERTRIG is based on trigonometric tensorial hyperinterpolation, while HYPERPOL consists in polynomial hyperinterpolation over spherical rectangles (of total degree). For details, see the paper G. Da Fies, A. Sommariva and M. Vianello,*Subperiodic trigonometric hyperinterpolation*.

» Source:- HYPERTRIG: [MATLAB CODES (zip file)]
- HYPERPOL: [MATLAB CODES (zip file)]

**CATCH suite.**

» Object: Demo of least squares on a 2D mesh and least squares on a compressed mesh. Examples on*union of disks*and*polygons*. For details, see the paper F. Piazzon, A. Sommariva and M. Vianello, Caratheodory-Tchakaloff Subsampling.

» Source:- CATCH: [MATLAB CODES (zip file)]

**Optimal mesh suite.**

» Object: Computation of optimal meshes on sphere, ball and torus. For details, see the paper A. Sommariva and M. Vianello, Discrete norming inequalities on sections of sphere, ball and torus.

» Source:- Actual version: [MATLAB CODES (zip file)]

**Sensors suite.**

» Object: Computation of nested cubature rules on polygons, from bivariate data. For details, see the paper A. Sommariva and M. Vianello, Nearly optimal nested sensors location for polynomial regression on complex geometriess.

» Source:- Actual version: [MATLAB CODES (zip file)]

## Old versions of our software

**Polygauss**

Matlab code for Gauss-like cubature over polygons [POLYGAUSS_2013_300413.zip]

Previous version: Matlab code for Gauss-like cubature over polygons [GAUSSCUB_2013.zip]. The main routine is the m. file polygauss.m. As demo, please use demo_gausscub.m. A different version, that uses less points but is more time expensive is polygaussj.m. We point out that the routines work in the GNU Octave version 2.1.73 (i686-pc-cygwin) environment. For some numerical experiments in Matlab/Octave see [readme_polygauss.pdf].

**GreenDisk**

Meshless cubature from scattered data over disks and annular sectors by TPS and Green's formula [GreenDisk_v1_0a.zip]. The demo file is demoRBFsimple.m. As alternative, You can use the more complicated demoRBFcells_multiple.m that allows to subdivide the disk or annular sector into certain equal area annular sectors.

Status: correction of minor mistakes in quadrature_rules_1D.m and cubature_rules_1D.m: some indexes

*i*in a 1D have been modified into

*index*in the gaussian rule (that however was working in Matlab 6.1). [Last update: May 07, 2007]

**Splinegauss**

Matlab code for Gauss-like cubature over spline curvilinear polygons [SPLINEGAUSS_2009.zip]. The main routine is the m. file splinegauss_2009b that performs cubature over domains with spline curvilinear boundaries.. As demo, please use demo.m for computing cubature over spline curvilinear polygons. In addition, try demo_splinegauss_moments.m for tests on computing moments w.r.t. some specific bases on such domains. All numerical experiments have been done on MAC Book, with MAC OS X, Matlab 7.6.0.324 Release 2008a. [Latest update: November 19, 2009]. [Last update: March 27, 2009]

**Padua_CC (Cubature using Padua points)**

Matlab code for cubature over a square [Padua_CC.zip] (using non tensorial Padua and Morrow-Patterson Xu points). The main routine is the m. file main.m that performs cubature over a square using different formulas. All numerical experiments have been tested on Intel Centrino Duo T2400, with Windows XP, Matlab 6.1.0.450 Release 12.1 and GNU Octave 2.1.73. [Last update: October 09, 2007]

**Padua2DM**

Software for Padua points interpolation and cubature. [Padua2DM.tar.gz].

**ChebfunGauss**

*ChebfunGauss: a Matlab code for Gauss-Green cubature by the Chebfun package*: Matlab code that uses a suitable version of Splinegauss for bivariate domains defined analytically and approximated by Chebfuns. The files are stored in the compressed folder [CHEBFUNGAUSS_2017.zip]. For a demo use the file demo.m, for plotting cubature points plot_pts.m.

» Source:

- New Version: [CHEBFUNGAUSS_2017.zip]
- Old Version: [CHEBFUNGAUSS_2011.zip]

» Last update: November 26, 2017.

**WAM**

Some routines that can be used to make experiments with WAMs can be found in the following compressed files:

- [WAM_2D_280210.zip]: a complete version that works on several bivariate domains, not easily adaptable to add own WAMs but right for experiments on general 2D domains.
- [WAM_2De_010310.zip]: a simple version that does not work on many bivariate domains, but that is easily adaptable to add 2D WAMs.
- [WAM_3D_010310.zip]: a simple version that does not work on many trivariate domains, but that is easily adaptable to add 3D WAMs.

**Polygint**

[POLYGINT_010411.zip] contains several Matlab files for cubature/interpolation over polygons based on AFP (Approximate Fekete Points) and DLP (Discrete Leja Points). The zip file contains also Matlab routines for a minimal triangulation of a polygon and a quadrangulation of a polygon using few quadrangles and triangles. [Last Update: April 01, 2011]

**LEBFEK**

*LEBFEK: Matlab codes and data sets for bivariate Lebesgue and Fekete interpolation*.

In

*M. Briani, A. Sommariva, M. Vianello, Computing Fekete and Lebesgue points: simplex, square, disk*we have described some algorithms for computing (quasi-)Lebesgue and (quasi-)Fekete points on the unit simplex, a square and on the unit disk. The respective codes and sets for these domains can be found in the links below. [Last Update: February 14, 2011]

**Computation of (almost-)minimal rules on the square.**

Matlab codes for computing (almost-)minimal cubature rules on the square. [Last Update: December 11, 2011]

**Gaussian Rules for Symmetric weight functions.**

Matlab codes for computing Gaussian Rules for Symmetric weight functions. [Last Update: July 27, 2012]

**Fejer and Clenshaw-Curtis rules for general weight functions.**

Matlab codes for computing Fejer and Clenshaw-Curtis rules for general weight functions. [Last Update: July 27, 2012]

**Matlab codes for computing trigonometric quadrature rules.**

Matlab codes for computing trigonometric quadrature rules. A comparison. [Last Update: February 27, 2013]

Matlab codes for computing good interpolation points or quadrature rules on geographic subregions of the sphere. [Last Update: November 20, 2013]