Alvise Sommariva

Computation of Marcinkiewicz-Zygmund constants (shortened as MZ) and least squares/hyperinterpolation comparisons. It contains many rules of interest.

In particular it contains demos for

a) computation of MZ constants on unit-interval, unit-disk, unit-simplex, unit-square, unit-sphere, unit-cube;

b) comparisons between unfettered hyperinterpolation, classical hyperinterpolation and least-squares on unit-interval (based on Clenshaw-Curtis rule), unit-square (based on Padua-points rule), unit-cube (based on QMC rule).

There are many rules of interest:

a) interval: Gauss-Legendre, Clenshaw-Curtis, Fejer

b) disk    : product-rules, almost minimal rules

c) simplex : Stroud conical rules, almost minimal rules

d) square  : product rules, Padua points rules, Morrow-Patterson-Xu rules, QMC rules

e) sphere  : product rules, spherical designs (low degree of precision, see R. Womersley homepage for more)

f) cube    : product rules, QMC rules.

Furthermore we include orthornormal basis and their Vandermonde matrices.

Papers :

On the role of Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases  
C.; An, A. Sommariva and M. Vianello

» PDF: 

Codes:

• Last versions:
  » MATLAB:  MZ.zip (A. Sommariva homepage), GitHub (A. Sommariva homepage) 

• » PYTHON:   not available

Presentations:

• -

>> First update: December 27, 2025. 

>> Last update:  December 27, 2025.

Object:

ORTHOCUB: integral and differential cubature rules by orthogonal moments.

In what follows, we mention the routines that are stored in this project.

• cheap_startup: given the algebraic degree of precision ade, it computes a low cardinality cubature rule rule_ref on the reference square [−1, 1]^2 or cube [−1, 1]^3 w.r.t. tensorial-Chebyshev weight (17) or (18). Next, calling the subroutine dCHEBVAND_orthn, it evaluates the tensorial-orthonormal Chebyshev basis {ψ_j}_j of total degree ade at the nodes, storing the results in the Van- dermonde matrix V_ref. The sequence of indices (h, k) or (h, k, l), defining the ordering of the basis elements, is listed in the matrix basis_indices. We stress that this routine in independent of the functional L to be used and is only related to the ADE of the reference cubature rule.

• cheap_rule: given the matrix dbox of dimension 2 × d in which the i column con- tains the extrema of the bounding box w.r.t. the i-th variable, the moments m storing L(ψ j({Λ−1}(· − C))), as well as rule_ref, V_ref computed by cheap_startup, it determines the nodes X and the weights w of the ORTHOCUB cubature rule.

• dCHEBVAND_orthn evaluates the Vandermonde matrix on the set X ⊂ Ω ⊆ B, relatively to the scaled tensorial-Chebyshev {φ_j}_j. As input, it is requires the variable dbox describing B and the sequence of degree indices basis_indices.

• vandermonde_jacobi allows the evaluation of the tensorial-Jacobi polynomials. This general purpose function takes into account a specific normalization of the basis, e.g. monic, orthonormal as well classical ones depending on the weight.

• cub_square_mpx, cub_cube_mpx are low cardinality cubature rules, respectively on the unit-square [−1, 1]^2 and on the cube [−1, 1]^3 relatively to the tensorial-Chebyshev measure.

• mom_derivative_2D, mom_derivative_3D produce the derivatives of order with re- spect to a variable specified by the user of the polynomial basis {φ_k} on a specific 2D or 3D mesh. Depending on the mesh, it is possible to define a suitable bounding box B via the variable dbox.

As for the examples regarding numerical cubature, the algorithms require some additional functions.

• compute_spline_boundary is useful in the case of the spline-curvilinear domains. It provides the description of its boundary, given as inputs the abscissae of the vertices XV and the relative ordinates YV , a variable spline_parms storing the order of the piecewise splines and the vertices involved. In case the spline is cubic, spline_type fixes the additional conditions (e.g. natural, periodic). As output, the routine supplies the vector of structures Sx and Sy that determine the piecewise splines x = x(t), y = y(t) parametrizing the boundary.

• spline_chebmom, given the polynomial degree n, the vectors of structures Sx, Sy, the basis ordering basis_indices, computes the moments of the basis {ψ_j}_j and, if required, of {φ_j}_j. This purpose is obtained by applying Green theorem, requiring the numerical integration of a certain piecewise polynomial function on the boundary.

• QMC_union_balls, given the vectors of centers and radii defining the balls Bi, i = 1, 2, . . ., computes card Halton points XB in the bounding box B of the integration domain Ω = ∪_i B_i. Next, after the application of an in-domain function, provides as output the nodes and the weights of a QMC rule in Ω.

5.2 Description of the demos

In order to replicate the examples, we have implemented the following demos, used for the numerical experiments above.

• demo_cubature_splines illustrates the basic usage of the ORTHOCUB technique to approximate definite integrals on a spline-curvilinear domain Ω1 (see Figure 1). • demo_cubature_ade_splines, fixed a degree of precision n equal to 2, 4, . . . , 16, approximates via the rules proposed in this work 100 integrals of random polynomials of the form p_n(x, y) = (c_0 + c_1 x + c_2 y)^n. Each exact integral is computed via Green theorem. Finally it plots the single relative errors and the value of their geometric mean.

• demo_cubature_sumweights_spline first computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for integration on Ω1 and then their stability ratios as in Table 2.

• demo_cubature_weights_spline, first computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for integration on Ω1, and then sorts the weights, illustrating their distribution as in Figure 1.

• demo_cubature_QMC shows the basic usage of the ORTHOCUB technique to approximate definite integrals on the domain Ω2 that is union of 5 balls (see Figure 2).

• demo_cubature_ade_QMC determines via QMC_union_balls two QMC rules SL, SH on Ω2, based respectively on 105 and 106 in the bounding box B. Next, fixed a degree of precision n equal to 2, 4, . . . , 16, approximates via the rules proposed in this work 100 integrals of random polynomials of the form p_n(x, y, z) = (c_0 + c_1 x + c_2 y * c_3 z)^n. In particular, takes into account SH (p_n) as reference value, while as approximation of the moments m uses the vector whose i-th component is SL(φ_i). Finally it plots the single relative errors and the value of their geometric mean.

• demo_cubature_sumweights_QMC computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for QMC integration on Ω1 and their stability ratios as in Table 2.

• demo_cubature_weights_QMC computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for QMC integration on Ω2, sorts the weights and illustrates their distribution as in Figure 2.

• demo_derivative_ade_2D computes by the ORTHOCUB technique described above the first and second order derivatives of random polynomials of the form p_n(x, y) = (c_0 + c_1 x + c_2 y)^n on a mesh of the first 10000 Halton points {P_k} in [−1, 1]^2, for n = 2, 4, . . . , 16. For any exponent n and point of mesh Pk, we determine by an ORTHOCUB rule on [−1, 1]^2 defined by mom_derivative_2D the numerical approximation of these derivatives on P_k. Since the exact value is known analytically, we can easily establish the relative error in norm 2. Integration and differentiation by orthogonal moments

• demo_derivative_ade_3D computes by the ORTHOCUB technique described above the first and second order derivatives of random polynomials of the form p_n(x, y, z) = (c_0 + c_1 x + c_2 y + c_3 z)^n on a mesh of the first 10000 Halton points {P_k} in [−1, 1]^3, for n = 2, 4, . . . , 16. For any exponent n and point of mesh P_k, we determine by an ORTHOCUB rule on [−1, 1]^3 defined by mom_derivative_3D the numerical approximation of these derivatives on P_k. Since the exact value is known analytically, we can easily establish the relative error in norm 2.

For more details see the paper:

"ORTHOCUB: integral and differential cubature rules by orthogonal moments", 2025, by L. Rinaldi, A. Sommariva, M. Vianello. matlab codes https://github.com/alvisesommariva/ORTHOCUB

Papers :

ORTHOCUB: integral and differential cubature rules by orthogonal moments
L. Rinaldi, A. Sommariva and M. Vianello

» PDF: 2510.23069

Codes:

• Last versions:
  » MATLAB:  ORTHOCUB.zip (A. Sommariva homepage), GitHub (A. Sommariva homepage) 

• » PYTHON:   GitHub (L. Rinaldi homepage)

Presentations:

• -

>> First update: December 05, 2025. 

>> Last update:  December 05, 2025.

Object:

Matlab files for cheap quadrature over spline curvilinear polygons and 3D domains (w.r.t. QMC based measures).

The .zip files contain two folder with demos that perform experiments respectively over spline curvilinear polygons and 3D complex domains.

Each folder contains in particular

• demo_ADE: quadrature of polynomial integrands (to establish numerically the properties regarding the degree of exactness), 

• demo_CUB: quadrature of polynomial integrands,

• demo_CPU: average cputimes of the rules,

• demo_WEIGHTS: distribution of the weights,

• demo_SUMWEIGHTS: stability of the rules.

Furthermore, a routine shows the geometry of the domain and the distribution of the nodes in a parallelepiped containing the integration domain.

Papers :

Effective numerical integration on complex shaped elements by discrete signed measures,
L. Rinaldi, A. Sommariva and M. Vianello

» Arxiv repository: 2510.23069

Codes:

• Last versions:
  » MATLAB:  CHEAP3.zip (A. Sommariva homepage), GitHub (A. Sommariva homepage) 

• » PYTHON:   GitHub (L. Rinaldi homepage)

Presentations:

• -

>> First update: October 28, 2025. 

>> Last update: October 28, 2025.

Adaptive quadrature over spherical polygons on scattered data, via TPS and Radial Powers.

The main function of this repository is SCATCUB_SRBF.m which consists in the algorithm described in the paper

Roberto Cavoretto, Alessandra De Rossi, Giacomo Elefante and Alvise Sommariva - "Adaptive RBF cubature by scattered data on spherical polygons"

Moreover there are some demos to reproduce the various numerical experiments included in the paper.

It is necessary to use the package CHEBFUN to run the code.

Papers :

• R. Cavoretto, A. De Rossi, G. Elefante, A. Sommariva, Adaptive RBF cubature by scattered data on spherical polygons

Codes:

• Last versions:
  » MATLAB: SCATCUB_SRBF.zip (.zip file, G. Elefante's GitHub).

Presentations:

• -

>> First update: August 12, 2024. 

>> Last update: October 15, 2025.

The main demo "demo_driver.m" performs battery of experiments in the paper:

  "Unisolvence of randomized MultiQuadric Kansa collocation for convection-diffusion with mixed boundary conditions" by M. Mohammadi, A. Sommariva and M. Vianello.

The demo "demo_driver_02.m" makes a battery of tests for discussing the relationship between separation distance, fill distance, conditioning and quality of the method.

The demo "demo_tests.m" performs single experiments for the demo above in the paper:

  "Unisolvence of randomized MultiQuadric Kansa collocation for convection-diffusion with mixed boundary conditions" by M. Mohammadi, A. Sommariva and M. Vianello.

It consists in defining a mixed Dirichlet-Neumann problem on the square  [0,1] x [0,1] of which the solution "sol" is known, and approximate numerically "sol" at collocation points "pts_coll" via linear combination of RBF centered at "pts_centers".

The differential problem has the form

           lapl(u)+<grad(u),b>=f    in (0,1) x (0,1),

           u=g                      on the vertical sides, incl. vertices

           grad(u)*out_normal=g     on the horiz. sides (no vertices).

 The collocation points are defined as tensor product of "N" equispacedpoints in [0,1].

 The RBF centers are choosen randomly, one for each collocation point P, n the disk B(P,delta).

 The RBF are 

  1. TPS of the form "r^k*log(r)", 

  2. Radial Powers of the form "r^k", 

  3. Multiquadrics of the form "sqrt(1+(ep*r)^2)".

  The paper above is focussed on Multiquadrics, but all the codes may work with other RBF as TPS.

Papers :

• M. Mohammadi, A. Sommariva and M. Vianello, Unisolvence of randomized MultiQuadric Kansa collocation for convection-diffusion with mixed boundary conditions,

Codes:

• Last versions:
  » MATLAB: Kansa_MQ.zip (.zip file).

Presentations:

• -

>> First update: August 12, 2024. 

>> Last update: March 02, 2025.

Usage notes:

a) The zip file contains 4 demos.

The files demo_01 and demo_02 illustrate the numerical examples performed in the section "Implementation and examples" of the paper "Cheap and stable quadrature on polyhedral elements" by Alvise Sommariva, Marco Vianello.

The files demo_03 and demo_04 show how to use the software on integrating functions and to perform comparisons between the available methods, also using cubature compression

b) The cheap main routines are

• cubature_polyhedron_cheap.m

• cheap_startup.m

Subroutines used are

• cubature_triangle.m

• dCHEBVAND.m

• mono_next_grlex.m

• polygauss_2013 / polygauss_2018 .m

• tenscheb_norm2sq.m

Papers:

• A. Sommariva and M. Vianello, Cheap and stable quadrature on polyhedral elements

Codes:

• CheapQ (at github),

• CheapQ.zip (.zip file).

Presentations:

» Last update: February 5, 2025.

The zip file contains Matlab codes for performing the examples in the numerical section of the paper "Polynomial meshes on algebraic sets". The latter can be run by selecting proper parameters in demo_driver.m.

Papers :

• L. Bialas-Ciez, A. Kowalska, A. Sommariva, Polynomial meshes on algebraic sets

Codes:

• Last versions:
  » MATLAB: , MeshAlgebraicSets.zip (.zip file).

Presentations:

• -

>> Last update: January 25, 2025.

This demo performs battery of experiments named as "test 1" and "test 2"  in the paper:  "Unisolvence of Kansa collocation for elliptic equations by polyharmonic splines with random fictitious centers", by M. Mohammadi, A. Sommariva and M. Vianello.

 ............................... Test 1  .................................

 It consists in defining a Dirichlet problem on the square [0,1] x [0,1] of which the solution "sol" is known, and approximate numerically "sol" at collocation points "pts_coll" via linear combination of RBF centered at "pts_centers".

  The differential problem has the form

•   lapl(u)=f    in (0,1) x (0,1),

•     u=g          at the boundary of [0,1] x [0,1]

The collocation points are defined as tensor product of "N" equispaced points in [0,1]. The RBF centers are choosen randomly, one for each collocation point P, in the disk B(P,delta). The RBF are TPS of the form "r^k*log(r)" or Radial Powers of the form "r^k".  In output we provides the RMSE (vector whose i-th component is the RMS  for the i-th test).

  ............................... Test 2  .................................

 It consists in defining a mixed Dirichlet-Neumann problem on the square  [0,1] x [0,1] of which the solution "sol" is known, and approximate numerically "sol" at collocation points "pts_coll" via linear combination of RBF centered at "pts_centers".

The differential problem has the form

•            lapl(u)+<grad(u),b>=f    in (0,1) x (0,1),

•            u=g                      on the vertical sides, incl. vertices

•            grad(u)*out_normal=g     on the horiz. sides (no vertices).

The collocation points are defined as tensor product of "N" equispaced points in [0,1]. The RBF centers are choosen randomly, one for each collocation point P  in the disk B(P,delta). The RBF are TPS of the form "r^k*log(r)" or Radial Powers of the form  "r^k".

Note: a more general code can be found in Kansa_MQ.zip where RBF can be more general. The usage is the same. In particular in the demo "demo_driver_02.m" makes a battery of tests for discussing the relationship between separation distance, fill distance, conditioning and quality of the method.

Papers :

• M. Mohammadi, A. Sommariva and M. Vianello, Unisolvence of Kansa collocation for elliptic equations by polyharmonic splines with random fictitious centers,

Codes:

• Last versions:
  » MATLAB: Kansa_TPS.zip (.zip file).
  » MATLAB: Kansa_MQ.zip (.zip file).

Presentations:

• -

>> First update: August 12, 2024. 

>> Last update: February 18, 2025.

Usage notes:

Matlab version of CPOLYMESH The Matlab package includes two demos, that we briefly comment.

1. demo_cdes_1: by this routine we show
   
   

   • how to define the complex domain (several ways),

   • compute an admissible mesh (AM) of a fixed degree,

   • extract extremal sets,

   • compute a certified Lebesgue constant.

2.  demo_cdes_2: by this routine we perform all batteries of numerical tests that are described above. In particular, varying the degrees, we

   • compute an admissible mesh (AM) of a fixed degree;

   • extract the AFP, DLP, PLP extremal sets;

   • for each of them we compute a certified Lebesgue constant;

   • plot Lebesgue contants, domain and extremal points.
     
     Note: Changing the value of the variable domain_type, from 1 to 22, one can test our routines also on other complex geometries, as hypocycloids, epicycloids, epitrochoids, limacons, rhodoneas, eggs, bifoliums, Talbot curves, tricuspoids, torpedos, ellipses and heart-shaped domain.

Observe that there is also a Python version whose author is Dimitri J. Kenne.

» Sources:

Papers:

• Dimitri J. Kenne, Alvise Sommariva and Marco Vianello, CPOLYMESH: Matlab and Python codes for complex polynomial approximation by Chebyshev admissible meshes.

• Leokadia Bialas-Ciez, Dimitri Jordan Kenne, Alvise Sommariva, Marco Vianello, Chebyshev admissible meshes and Lebesgue constants of complex polynomial projections.

Codes:

• Last versions:
  » MATLAB: CPOLYMESH at Github, CPOLYMESH (.zip file).

Presentations:

» First update: June 28, 2024. Last update: June 28, 2024.

» Object: Near-algebraic quadrature rules on spherical polygons.

1. Construction of rules thar have (numerically) fixed algebraic degree of precision "n". 

2. Tchakaloff formulas, with a cardinality at most "(n+1)^2" can be extracted by means of a new implementation of Lawson-Hanson algorithm.

3. Polynomial hyperinterpolation at degree "n" is determined by means of the rules above of degree "2n" and discrete orthogonal polynomials.

The suite is subdivided in two folders, EASY and NUMERICAL_EXPERIMENTS.

In EASY there are two subfolders CUB and HYP in which we provide the codes for cubature and hyperinterpolation (respectively).

In CUB there is the demo demo_cub_sphpgon that shows how to use the main routine cub_sphpgon and a basic adaptive code adaptive_cub_sphpgon.

In HYP there is the demo demo_hyp_sphpgon that shows how to compute hyperinterpolants on spherical polygons.

In NUMERICAL_EXPERIMENTS there are three subfolders CUB01 and HYP01, HYP02 in which we provide the codes used for experiments about cubature and hyperinterpolation and its variants (respectively).

» Sources:

• Papers:
  
  

  1. https://www.math.unipd.it/~alvise/PAPERS_2023/SPHPOLY/sphpoly.pdf .

• Last version: [MATLAB CODES (zip file)]

» Last update: March 22, 2024.

Usage notes:

The main folder contains several subfolders that lead to the wanted domain. In particular we examine the Lebesgue constants for some interpolation sets in the

• interval,

• simplex,

• square,

• disk,

• cube,

• ball,

as well as we analyse the Lebesgue constant for Least-Squares on

• square,

• disk.

Each folder contains:

• demo_basic.m that is a basic demo for understanding how the Lebesgue costant estimator lebconst_eval_*.m works;

• demo_leb_*.m makes all the experiments cited in the paper L. Bialas-Ciez, D. Kenne, A. Sommariva and M. Vianello, Evaluating Lebesgue constants by Chebyshev polynomial meshes on cube, simplex and ball.

• The folder _DATA contains the plots of the papers, a file "numerical_experiments.txt" with the statistics of the experiments, and if it is useful, the files set_AFP_disk, set_DLP_disk containing respectively Approximate Fekete Points and Discrete Leja Points for mild degrees.

» Sources:

• 
  Papers:Leokadia Bialas-Ciez, Dimitri Jordan Kenne, Alvise Sommariva, Marco Vianello, Evaluating Lebesgue constants by Chebyshev polynomial meshes on cube, simplex and ball.

• 
  Codes:Last versions:
  » MATLAB: LEBEVAL at Github, LEBEVAL (.zip file).

• 
  Presentations:-

» First update: November 23, 2023. Last update: November 23, 2023.

Usage notes:

• The main Matlab code used in our numerical tests, can be found in the folder "General" and is named "demo_scattcub_driver_01.m". There You can change the domain, the algebraic degrees of exactness of the rule, cardinali- ty and type of the scattered data.
  
  

• For a simple usage of the algorithm, we have added the routine "demo_scattcub", where one can use one of the algorithms to compute the wanted integrals. It makes comparisons with the algebraic rule used in the integral evaluation knowing the function evaluations at the nodes.
  
  

• IMPORTANT: The directory "EXTERNAL_ROUTINES" contains methods and cubature rules used in the paper.

» Sources:

Papers:

• R. Cavoretto, F. Dell'Accio, A. De Rossi, F. Di Tommaso, N. Siar, A. Sommariva, and M. Vianello, Numerical cubature on scattered data by adaptive interpolation.

Codes:

• SCATTCUB at Github, 

• SCATTCUB (.zip file).

Presentations:-

» First update: July 07, 2023. Last update: July 07, 2023.

Papers:

• Congpei An, Jia-Shu Ran, Alvise Sommariva, Hybrid hyperinterpolation over general regions.

Codes:

• Hyper23.zip at Github, 

• Hyper23.zip.

Presentations:-

» First update: June 02, 2023. Last update: August 27, 2023.

» Description

The main folder contains the following files:

• demo_CQMC_sphpoly: In this demo we compute a cubature rule over spherical polygons via QMC rules. Once the domain is available, we compute a full rule QMC and a compressed one, say CQMC, sharing the same integrals for polynomials up to a total degree.

• demo_CQMC_torus: same as above on a subdomain of a toroidal surface;

• demo_CQMC_franke: same as above on a subdomain of a surface given by Franke's function.

• pts_domain: corresponds to items (i) and (ii) of Algorithm Qsurf (see attached paper);

• cqmc_v2: corresponds to items (iii) and (vi) of Algorithm Qsurf (see attached paper);

• dCHEBVAND_v2: basis selection in item (iii) of Algorithm Qsurf (see attached paper);

• lawsonhanson, LHDM: NNLS implementation;

• dCATCH, implements Caratheodory-like compression via NNLS.

» Sources:

Papers:

G. Elefante, A. Sommariva and M. Vianello, PDF Qsurf: compressed QMC integration on parametric surfaces, (submitted),

Codes:

• Qsurf.zip [MATLAB CODES (Personal homepage)]

• Qsurf.zip [MATLAB CODES (GitHub)]

Presentations:

» First update: April 13, 2023. Last update: September 5, 2023.

» Object: 

The main folder contains the following files:

• demo_CQMC: In this demo we compute a cubature rule on a union of balls or on its surface via QMC rules. Once the domain is available, we compute a full rule QMC and a compressed one, say CQMC, sharing the same integrals for polynomials up to a total degree.

• demo_CQMC_battery: numerical cubature over volumes/surfaces defined by union of balls/spheres. The routine tests the quality of quadrature rules with degree of precision "deg" over polynomials of the form f=@(x,y,z) (a1+b1*x+c1*y+d1*z).^deg; where "a1", "b1", "c1", "d1" are random numbers. Finally it plots, if requires, the relative errors of each test, for some "deg" and logarithmic averages.

• cqmc_v2: the routine computes a Compressed Quasi-Monte Carlo formula from a large low-discrepancy sequence on a low-dimensional compact set preserving the QMC polynomial moments up to a given degree, via Tchakaloff sets and NNLS moment-matching.

» Sources:

Papers:

• G. Elefante, A. Sommariva and M. Vianello, Compressed QMC volume and surface integration on union of balls, (submitted),

Codes:

• qBubble_040323.zip [MATLAB CODES],

• qBubble at GitHub [MATLAB CODES],

Presentations:

• Constructive approximation of functions 3 (Krakow, Poland, 22-24 February 2023)
  » [PDF] ,

» First update: November 29, 2022. Last update: November 29, 2022.

» Object: TetraFreeQ: a software package for Tchakaloff-like quadrature on polyhedral elements.

• we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape;

• if a tetrahedralization is available, we also provide an algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape;

The main folder contains the following files:

• demo_01: battery of tests will be performed, on different polyhedra, for the methods working with or without triangulation; the software compares the performance of the method, on random polynomials of total degree "ade".

• demo_02: single cubature test over polyhedra (convex or concave, even with odd shapes); the demo expains the subroutines usage.

• cubature_polyhedron_free: cubature over polyhedral elements without tetrahedralization;

• cubature_polyhedron_tri: cubature over polyhedral elements via tetrahedralization;

» Sources:

Papers:

• A. Sommariva and M. Vianello, TetraFreeQ: Tetrahedra-free quadrature on polyhedral elements.

Codes:

• TETRAFREEQ.zip [MATLAB CODES],

Presentations:

• FAATNA 20-> 22 (Matera, Italy, July 5-8, 2022)
  » [PDF] ,

» First update: November 29, 2022. Last update: November 29, 2022.

» Object: UNIONOFDISKS: a software package for determining cubature rules over union of disks.

• detects the connected components of the union;

• detects the outer and possible inner boundaries of each connected component (i.e. detects also the possible holes providing their boundary);

• constructs nodes and weights of a PI-formula exact for polynomials of a given total-degree, with cardinality increasing proportionally to the overall number of disks times the degree squared;

• allows then to compute immediately at machine precision some relevant features in applications, such as the area of the union and its first and second monomial moments (i.e. its ``center of mass'' and ``moment of inertia'' for a constant density);

• finally, provides a compressed PI-formula with cardinality not exceeding the dimension of the exactness polynomial space, irrespectively of the overall number of disks.

The main folder contains the following compressed folders:

• COMPARISONS, contains all the subroutines used for our numerical experiments (including previous software that is compared with the new approach); the main demo is the file demo make experiments that performs the numerical experiments presented in the paper Cubature rules with positive weights on union of disks, comparing the new rules determined by cub_uniondisks, including the compressed version, and the old routine achieved by gqmultibubble;

  1. demo_make_experiments: it performs the experiments presented in the paper;

  2. demo_cubature_uniondisk: it compares the results of new rules (including the compresse one) on integrating some functions on certain test domains;

• ROUTINES, contains all the routines necessary for cubature over disks; the main routines are

  1. cub_uniondiscs: implements the algorithm for determining a cubature formula on the union of disks, including the Tchakaloff rules (i.e. their compressed version);

  2. demo_cub_uniondiscs: demo explaining how to use the subroutines and comparing the new rules on some domains.

» Sources:

Papers:

• A. Sommariva and M. Vianello, Cubature rules with positive weights on union of disks.

      Codes:

•  UNIONOFDISKS.zip [MATLAB CODES],

• 
  

» First update: October 18, 2022. Last update: October 18, 2022.

» Matlab codes for determining cubature rules over intersection of disks (October 2022).

• computation of algebraic cubature rules over the intersection of disks (formula with positive weights and internal nodes).

• finally, provides a compressed PI-formula with cardinality not exceeding the dimension of the exactness polynomial space, irrespectively of the overall number of disks.

» Object: INTERSECTIONOFDISKS: a software package for determining cubature rules over intersection of disks (2017-2022).

• The main routine for computing the intersection of disks is gqintdisk.m that requires additional routines saved in the folder;

• the demo to test the performance of the previous routine is demo_intersection.m.

Papers:[1] A. Sommariva and M. Vianello, Numerical quadrature on the intersection of planar disks, FILOMAT 31:13 (2017), 4105--4115 (open access).

Codes:Last versions:
» INTERSECTIONOFDISKS_2022.zip [MATLAB CODES],

» First update: October 24, 2022. Last update: October 24, 2022.

» Object: 

Improved version of Compressed Quasi-MonteCarlo cubature (CQMC) on large low-discrepancy samples is implemented, on 2D and 3D regions with complex shape. The algorithms rests on the concept of Tchakaloff set and on NNLS solution of a sequence of small moment-matching systems.

The main folder contains the following compressed folders:

• ROUTINES_2D.zip, that include many routines for computation of integrals via QMC/CQMC methods on domains obtained by set operation from two bivariate domains whose boundary is defined by piecewise NURBS;

• ROUTINES_3D.zip, that include many routines for computation of integrals via QMC/CQMC methods on domains obtained by set operation from a trivariate polyhedron and a sphere;

• README_FIRST.txt, that explains how to replicate the tests in the paper.

In both cases, the main routines are:

• make_rule, that defines a QMC rule over the domain and then extracts the CQMC rule via cqmc_01;

• cqmc_01, routine that from a sufficiently dense QMC rule computes a low cardinality CQMC rule that matches the QMC moments up to a certain total polynomial degree.

» Sources:

Papers:

• G. Elefante, A. Sommariva and M. Vianello, CQMC: an improved code for low-dimensional Compressed Quasi-MonteCarlo cubature, (submitted),

Codes:

• ROUTINES_2D.zip [MATLAB CODES],

• ROUTINES_3D.zip [MATLAB CODES]

        Presentation:

• Presentation at York (January 2023): [PDF]

» Last update: January 05, 2022.

» Object: 

The main folder IN_RS contains:

» inRS.m, a new routine for indomain over NURBS-shaped domains;

» inRS_21.m, an old routine for indomain over NURBS-shaped domains;

» inRSpoly2.m, a routine for indomain over NURBS-shaped domains (based on approximating the domain with a polygon, it requires inpoly2 by D. Engwirda);

» demo_nurbs_indomain_01.m, demo_nurbs_indomain_02.m, demo_nurbs_indomain_03.m are demos, useful to define the domains and to understand how to use the indomain routine.

» demo_nurbs_indomain_04.m, evaluates the performance of our two indomain routines for NURBS presented in [1] and [2], comparing them to one in which the NURBS boundary is approximated by a polygon, finally using inpoly2 by D. Engwirda. To make that comparison, please download the routine inpoly2 and save it in the IN_RS folder.

» the routine demo_nurbs_indomain_05.m shows how to define the problem when only some vertices of the curved polygon are available;

» the routine demo_nurbs_indomain_06.m considers the case of a domain whose boundary is defined only by a point cloud of possibly inexact data; this routine requires spcsp.m (cubic smoothing periodic spline), by M. Zanetti, available at the Matlab Central File Exchange;

» the routine demo_nurbs_indomain_07.m shows how to define the problem when the whole domain is defined via point cloud and NURBS use the vertices, provided by alphashape polygonal approximation of the boundary, as control points;

» the routine demo_nurbs_indomain_08.m shows how to define the problem when the whole domain is defined via point cloud and parametric cubic smoothing periodic spline are used on the data provided by alphashape polygonal approximation of the boundary; this routine requires spcsp.m, by M. Zanetti, available at the Matlab Central File Exchange;

» Sources:

Papers:

• A. Sommariva and M. Vianello, Low-cardinality Positive Interior cubature on NURBS-shaped domains, [PDF]

• A. Sommariva and M. Vianello, inRS: implementing the indicator function for NURBS-shaped planar domains [PDF].

Codes:

• [MATLAB CODES (zip file)],

• [MATLAB CODES (zip file) at github.com],

Presentation:

•  Presentation at SA 2022: [PDF (Beamer)]» First update: January 07, 2022. 

>> Last update: February 22, 2022.

» Object: 

Software for computing algebraic cubature rules of degree n on domains defined parametrically by rational splines. It is suitable for domains whose boundary are defined by arcs that are NURBS, Bezier curves as well as general parametric splines. These rules are of PI type, i.e. with nodes that are in the integration domain and positive weights. They do not require a triangulation of the domain and have cardinality at most equal to (n+1)(n+2)/2.

In this toolbox we also develop an indomain routine suitable to establish what points in a bivariate set are inside these domains.

The main folder CUB_RS contains:

• PAPER_EXPERIMENTS that includes all the Matlab routines useful for reproducing the experiments described in the work [1] and [2] (see references below);

• BEZIER_ROUTINES that includes all the Matlab routines useful for integration over domains whose boundary is defined by composite Bezier arcs;

• NURBS_ROUTINES that includes all the Matlab routines useful for integration over domains whose boundary is defined by NURBS;

• SPLINE_ROUTINES that includes all the Matlab routines useful for integration over domains whose boundary is defined by general parametric splines.

All these folders contain several demos, useful to define the domains, to understand how to use the indomain routine as well as the cubature routine. Some examples on hyperinterpolation are also available.

The main routines are:

• inRS that implements a fast indomain algorithm for these instances; the last update presents a faster indomain algorithm inRS, presented in [2]; the older version introduced in [1] is present in the file inRS_21;

• cubRS that determines an algebraic cubature formula of the desired degree of exactness, with low cardinality, internal nodes and positive weights.

» Sources:

Papers:

•  A. Sommariva and M. Vianello, Low-cardinality Positive Interior cubature on NURBS-shaped domains, [PDF];

•  A. Sommariva and M. Vianello, inRS: implementing the indicator function for NURBS-shaped planar domains.

Codes:

Last version: [MATLAB CODES (zip file)],

Presentation:

• Presentation at SA 2022: [PDF (Beamer)]

» Last update: January 05, 2022.

» Object: 

The main routine ispolygcirc_2021.m that computes an algebraic cubature rule on a domain defined by the union/difference of a disk with a polygon.

The software includes the demo file:

demo_plot that tests the cubature results on some domains.

» Sources:

Paper:

• E. Artioli, A. Sommariva and M. Vianello, Algebraic cubature on polygonal elements with a circular edge, [PDF],

Codes:

•  [MATLAB CODES (zip file)],

» Last update: December 02, 2021.

Description: 

The main routine is cub_polygon_adaptive.m that computes adaptive quadrature of an integrand over polygonal regions.
The software includes the demos file demo_adaptive_polygons.m that tests the cubature results on some domains and integrands.

Sources:

Codes:

• Last version: [MATLAB CODES (zip file)],

» Last update: July 13, 2021.

Description

Matlab codes for computing moments and cubature rules on polygonal regions (via triangulation or alternatively by Gauss-Green theorem in polar coordinates). The regions can be general, i.e. polygons that are not simply connected or even disconnected. It requires Matlab built-in environment polyshape. With respect to a previous version, depending on the values of the integrand at the centers, we optimize the shape parameter so to achieve better cubature results. 

The main routines are:

• RBF_cub_polygon_OPT.m that determines the RBF based cubature rule on the polygonal region;

• RBF_moms_polar.m and RBF_moms_tri.m that determines the RBF moments on the polygonal region;

• RBF_optimal_scale.m that chooses the optimal RBF scale parameter;

The software includes several demos:

• demo_RBF_cub_OPT.m that performs a battery of tests, allowing different integrands, polygonal regions, RBF, number of centers and choices of the RBF shape parameter.

• demo_RBF_cub_OPT_basic that performs a simple example, on a certain integrand, polygonal region, RBF, number of centers and with almost optimal RBF shape parameter (useful to adapt the code to user experiments).
  
  

For the demos of the previous version published in RBF moment computation and meshless cubature on general polygonal regions, see:

RBF_cub_polygon.m that determines the RBF based cubature rule on the polygonal region (not optimized shape parameter);

Source 

Papers:

• • R. Cavoretto, A. De Rossi, A. Sommariva and M. Vianello, RBFCUB: a numerical package for near-optimal meshless cubature on general polygons;

  • A. Sommariva and M. Vianello, RBF moment computation and meshless cubature on general polygonal regions

Codes:

• [MATLAB CODES (zip file)]

•  [MATLAB CODES (zip file at GitHub)]

» Last update: July 9, 2021.

Description.

Matlab codes for determining "norming meshes" and "good interpolation points" on spherical triangles.The meshes, if a numerically verified conjecture holds, are norming meshes and Approximate Fekete Points, useful for polynomial interpolation, can be determined for mild degrees.

The zip file contains three folders:

• BASIC_USAGE: here we have introduced a demo demo_am_00.m on the norming mesh and its properties as well as the main routine am_sphtri.m for the determination of the norming mesh, depending on a spherical triangle contained in a certain hemisphere.

• ADVANCED_USAGE: here there are two demos for the computation of norming meshes on spherical triangle T, as well as a set of Approximate Fekete Points, useful for polynomial interpolation.

  • demo_am_01.m demo on using "dPOLYFIT", "dPOLYVAL", "dLEB" for algebraic polynomial interpolation/least squares approximation on spherical triangle. The approximation points are taken from AMs on the spherical triangle; the new routine introduced here is "am_sphtri" that determines an Admissible Mesh over a spherical triangle.

  • demo_am_02.m this routine computes a AM on a spherical triangle and extracts AFEK pointset; next it determines its main features and plots AM and AFEK.

• FIGURES: it contains all the Matlab files useful for determining the figures and their experiments, as cited in the reference paper Chebyshev-Dubiner norming grids and Fekete-like interpolation on spherical triangles.

Papers:

• • Near-optimal polynomial interpolation on spherical triangles

Codes:

• • [MATLAB CODES (zip file)],

  • [MATLAB CODES (zip file), at Github]

» Last update: February 18, 2021.

Description:

The rules have (numerically) fixed algebraic degree of precision "n". Tchakaloff formulas, with a cardinality at most "(n+1)^2" can be extracted by means of a new implementation of Lawson-Hanson algorithm.

Polynomial hyperinterpolation at degree "n" is determined by means of the rules above of degree "2n" and discrete orthogonal polynomials.

The main routine iscub_sphtri.m that determines full and compressed quadrature rules with fixed (numerical) algebraic degree of precision on spherical triangles; it requires dCATCHsph.m, dCHEBVAND.m, gqcircsect.m, LHDM.m, trigauss.m.

The software includes the demos files:

• demo_01.m that tests the cubature rules on spherical harmonics (it requires sphharmVAND.m and plot_s2.m),

• demo_02.m that tests the cubature rules on a battery of tests functions.
  
  

• demo_hyperinterpolation_01.m that tests the hyperinterpolants of several functions on some spherical triangles,

• demo_hyperinterpolation_02.m that shows how to determine the hyperinterpolant on a single example.

• demo_hyperinterpolation_03.m that shows estimates of the Lesbegue constant based on Christoffel function.
  
  

Papers:

• • Near-algebraic Tchakaloff-like quadrature on spherical triangles

  • Numerical hyperinterpolation over spherical triangles

Codes:

• Last version: [MATLAB CODES (zip file)], [MATLAB CODES (zip file), at Github]

• Older version: [MATLAB CODES (zip file)]

» Last update: March 22, 2021.

Description:

Matlab codes for computing moments and cubature rules on polygonal regions (via triangulation or alternatively by Gauss-Green theorem in polar coordinates). The regions can be general, i.e. polygons that are not simply connected or even disconnected. It requires Matlab built-in environment polyshape. See the codes in the work "RBFCUB: Matlab package for near-optimal meshless cubature on general polygons (2021)" for more accurate results with an almost optimal shape parameter.

The main routines are

• RBF_moms_polar.m and RBF_moms_tri.m that determine the RBF moments on the polygonal region;

• RBF_cub_polygon.m that determines the RBF based cubature rule on the polygonal region;
  
  

The software includes several demos that test moments as well as the cubature rules. In particular the tests in the paper can be obtained by means of demo_RBF_moms.m and demo_RBF_cub.m.

Sources:

Paper: 

• RBF moment computation and meshless cubature on general polygonal regions

Codes:

• [MATLAB CODES (zip file)]

» Last update: December 28, 2020.

Description:

Matlab codes for computing cubature rules on geographic subregions of the sphere (spherical rectangles of longitude-latitude type). It includes cubature over spherical caps.

The main routine is cub_sphrect.m that computes rules with a fixed algebraic degree of precision on a spherical rectangle.

Another relevant function is cub_sphrect_adaptive_2020.m that is a basic adaptive code for cubature on a spherical rectangle.

The software includes two demos, on integration of a general function as well as test on the algebraic degree of precision of the rule.

Sources:

Papers:

• Polynomial approximation and quadrature on geographic rectangles

Codes:

• New version: [MATLAB CODES (zip file)]

• Old version: [MATLAB CODES (zip file), interpolation], [MATLAB CODES (zip file), cubature]

» Last update: December 20, 2020.

Description:

Incurvpolygon_cheb: indomain routine on bivariate Jordan domains whose boundary is defined by chebfuns. The files are stored in the compressed folder, that contains:

» incurvpolygon_cheb.m: it is an indomain routine on domains defined whose boundary is defined by chebfuns;

» in the folder there is a demo demo_incurvpolygon_cheb.m, that tests the provided software.

Sources:

Codes:

• Last Version: [INCURVPOLYGON_CHEB_2020.zip]

» Last update: October 10, 2020.

Description:

ChebfunGauss: a Matlab code for cubature on Jordan bivariate domains defined analytically and approximated by Chebfuns. The files are stored in the compressed folder, that contains:

» chebcub_2019.m: it is a more stable version of the previous software and quickly produces a rule with many points, of the choosen degree of precision;

» chebfuncub_compression.m: by this routine one can compress the previous rule to one with the same degree of precision but fewer nodes; first it tries to compute one with internal nodes and positive weights, then one with internal nodes and possible negative weights, then one with possible external nodes and positive weights and finally one with possible external nodes and possible negative weights;

» chebcub_2020.m: it is a version that produces a rule with few points, of the choosen degree of precision (the rule has internal nodes and usually positive weights);

» incurvpolygon_cheb.m: it is an indomain routine on bivariate Jordan domains defined whose boundary is defined by chebfuns;

» in the folder there are several demo,

• demo_2019.m,

• demo_2020.m,

• demo_random_polynomials.m,

• demo_19vs20.m,

• demo_incurvpolygon_cheb.m,

that test the provided software.

Sources:

Codes:

• New Version: [CHEBFUNGAUSS_2020.zip]

• Old Version: [CHEBFUNGAUSS_2011.zip]

» Last update: October 10, 2020.

Description:

In this software, we define a Matlab code for establishing numerically if a point is inside or on the boundary of a spline-curvilinear polygon (the so called "in domain" algorithms). The main routine is incurvpolygon.m. Several demos are available, that show how to use incurvpolygon.m.

Sources:

Papers:

• Tchakaloff-like cubature rules on spline curvilinear polygons with M. Vianello.

Codes:

• [.zip],

» First Version: April 2019, Last update: September 29, 2020.

Description:

Fast Matlab codes for computing algebraic cubature rules on spline-curvilinear polygons (rules with possible negative weights and nodes outside the domain). By compression with splinegauss_compression.m the code tries to achieve low cardinality rules with positive weights and nodes inside the domain.

The main routine is the m. file splinegauss_2020.m that performs cubature over domains with spline curvilinear boundaries. In case one wants to recover rules with less points (possibly with points in the domain and positive weights) one can try to achieve such rule by the routine splinegauss_compression.m.

Several demos are available, that show how to use splinegauss_2020.m and splinegauss_compression.m to produce cubature rules over spline-curvilinear polygons.

Sources:

Paper:

• Gauss-Green cubature and moment computation over arbitrary geometries, with M. Vianello.

Codes:

• [.zip].

» First Version: March 2009, Last update: September 20, 2020.

Description:

This software computes algebraic cubature rules with fixed degree of precision over spline-curvilinear polygons. It includes an in-domain routine incurvpolygon.m that determines if a point is inside one of such regions.

The cubature formula has all points inside the domain as well as positive weights.

Sources:

Paper:

• Tchakaloff-like cubature rules on spline curvilinear polygons with M. Vianello.

Codes:

• [.zip]

» First Version: May 2020, Last update: May 2020.

Description:

This software computes algebraic cubature rules with fixed degree of precision over general polygons (convex or not convex, connected or disconnected, simply or not simply connected). It needs Matlab-built in polyshape environment.

Sources:

Paper:

• Compressed cubature over polygons with applications to optical design

Codes:

• [.zip]

» Last update: May 12, 2019.

Description:

This software computes cubature rules over the sphere, via Radial Basis Functions and Spherical Radial Basis Functions. We point out that as described in demoRBFweights.m and demoSRBFweights.m , cubature weights are obtained, without the need of the RBF interpolant (though via moments and transpose of Vandermonde matrix).

Sources:

Paper:

•  Integration by RBF over the sphere

Codes:

• [.zip],

» First Version: May 2005, Last update: September 26, 2019.

Description:

This software computes cubature rules over disks and annular sectors, via Radial Basis Functions. We point out that as described in demoRBFweights.m, cubature weights are obtained, without the need of the RBF interpolant (though via moments and transpose of Vandermonde matrix).

Sources:

Paper:

• Meshless cubature over the disk by Thin-Plates Splines

Codes:

•  [.zip]

• Old sources: [GreenDisk_v1_0.zip], [GreenDisk_v1_0a.zip]

» First Version: 2008, Last update: September 24, 2019.

Description:

This software computes cubature rules over rectangles, via Radial Basis Functions. The first .zip file considers RBF with the same scale for all the centers, the second zip file manages the variable scale case. We point out that as described in demo_2019_weights.m, cubature weights can be obtained, without the need of the RBF interpolant.

Sources:

Paper:

• Numerical cubature on scattered datas by Radial Basis Functions

Codes:

• [.zip], 

• [.zip]

» First Version: May 2004, Last update: September 23, 2019.

Description:

This software computes cubature rules over polygons (convex or not convex), via Radial Basis Functions (Thin Plate Splines)

Sources:

Paper:

• Meshless cubature by Green's formula.

Codes:

•  [.zip]

» First Version: May 2006, Last update: September 11, 2019.

Description:

Matlab codes for computing Gaussian Rules for Symmetric weight functions.

Sources:

Paper:

Codes:

•  [ZIP]

» Last update: July 27, 2012.

Description:

Matlab codes for computing Fejer and Clenshaw-Curtis rules for general weight functions.

Sources:

Paper:

Codes:

• [ZIP]

» Last update: July 27, 2012 (Tested August 22, 2016)]

Description:

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

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 that allows the numerical computation of orthogonal polynomials over general regions.

Sources:

Paper:

Codes:

• [MULTIVOP_220816]

» Last update: August 25, 2016.

Description:

Trigonometric Gaussian rules with fixed degree of precision

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.

Sources:

Paper:

Codes:

• New version: [MATLAB CODES (zip file)]

• Old version: [ZIP] [MATLAB CODES (zip file), cubature]

Last update:

•  September 13, 2016.

Description:

1D trigonometric hyperinterpolation and hyperinterpolation over spherical rectangles

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).

Sources:

Paper:

• G. Da Fies, A. Sommariva and M. Vianello, Subperiodic trigonometric hyperinterpolation.

Codes:

• HYPERTRIG: [MATLAB CODES (zip file)]

• HYPERPOL: [MATLAB CODES (zip file)]

Last update:

• October 24, 2016.

Description:

Least squares on a 2D mesh and least squares on a compressed mesh

Demo of least squares on a 2D mesh and least squares on a compressed mesh. Examples on union of disks and polygons.

Sources:

Paper:

• F. Piazzon, A. Sommariva and M. Vianello, Caratheodory-Tchakaloff Subsampling.

Codes:

• CATCH: [MATLAB CODES (zip file)]

Last update:

• November 20, 2016.

Description:

Computation of optimal meshes on sphere, ball and torus.

Sources:

Paper:

• A. Sommariva and M. Vianello, Discrete norming inequalities on sections of sphere, ball and torus.

Codes:

• [MATLAB CODES (zip file)]

Last update:

• April 24, 2017.

Description:

Computation of nested cubature rules on polygons, from bivariate data.

Sources:

Paper:

• A. Sommariva and M. Vianello, Nearly optimal nested sensors location for polynomial regression on complex geometries

Codes:

• [MATLAB CODES (zip file)]

Last update:

• May 23, 2017.

Description:

Matlab code for cubature over a square (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. 

Sources:

Paper:

Codes:

•  [Padua_CC.zip]

Last update:

• October 09, 2007]

Description:

Software for interpolation and cubature via Padua points.

Sources:

Paper:

Codes:

• [Padua2DM.tar.gz].

Last update:

Description:

Matlab code for WAM computation (good meshes over domains).

Sources:

Paper:

Codes:

• [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.

Last update:

• March 01, 2010