CC_DISPLAY
Display 2D Quadrature Rules

CC_DISPLAY is a MATLAB program which displays the abscissas used in a 2D quadrature rule.

The current version only considers a single Clenshaw Curtis grid, in which the orders of the rule in the X and Y directions may differ.

Eventually, some other variations will be added, in which the points associated with an isotropic Smolyak rule will be displayed, or in which several sets of Clenshaw Curtis grids will be superimposed, or a Gauss-Legendre rule will be displayed.

Related Data and Programs:

BOX_PLOT is a MATLAB program which can color in specified entries of a checkerboard, corresponding to pairs of integer data.

CLENSHAW_CURTIS is a MATLAB library of routines for defining a Clenshaw Curtis quadrature rule.

FEKETE is a MATLAB library of routines for defining a Fekete rule for quadrature or interpolation over a triangle.

GL_DISPLAY is a MATLAB library of routines for displaying a single Gauss Legendre product rule quadrature grid in 2D.

INTLIB is a FORTRAN90 library containing a variety of routines for numerical estimation of integrals in 1D.

NINTLIB is a MATLAB library containing a variety of routines for numerical estimation of integrals in multiple dimensions.

QUADPACK is a FORTRAN90 library containing a variety of routines for numerical estimation of integrals in 1D.

QUADRULE is a MATLAB library containing quadrature rules.

SPARSE_GRID_DISPLAY is a library of MATLAB routines which can display a sparse grid in 2D or 3D.

STROUD is a MATLAB library containing quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_INT is a FORTRAN90 library containing a number of functions that may be used as test integrands for quadrature rules in 1D.

TEST_NINT is a MATLAB library containing a number of functions that may be used as test integrands for quadrature rules in multiple dimensions.

TOMS351 is a FORTRAN77 library which estimates an integral using Romberg integration.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
2. Charles Clenshaw, Alan Curtis,
   A Method for Numerical Integration on an Automatic Computer,
   Numerische Mathematik,
   Volume 2, Number 1, December 1960, pages 197-205.
3. W Morven Gentleman,
   Algorithm 424: Clenshaw-Curtis Quadrature,
   Communications of the ACM,
   Volume 15, Number 5, May 1972, pages 353-355.
4. Lloyd Trefethen,
   Is Gauss Quadrature Better than Clenshaw-Curtis?
5. Joerg Waldvogel,
   Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
   BIT Numerical Mathematics,
   Volume 43, Number 1, 2003, pages 1-18.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• cc_display_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• cc_display_m_src.tar.gz the GZIP'ed TAR file.

Source Code:

• cc_abscissa.m returns the I-th Clenshaw Curtis abscissa.
• cc_grid.m returns a multidimensional Clenshaw-Curtis grid.
• cc_grid_display.m displays a multidimensional Clenshaw-Curtis grid.
• cc_grids_animate.m displays a sequence of merged CC grids from order [1,1] to [A,B].
• cc_grids_constrained.m computes CC orders and grids satisfying a constraint.
• cc_grids_constrained_display.m displays grids generated by CC_GRIDS_CONSTRAINED.
• cc_grids_constrained_size.m computes CC orders and grids with a constraint.
• cc_grids_minmax.m computes CC orders and grids with Q_MIN <= Q <= Q_MAX.
• cc_grids_minmax_display.m displays grids generated by CC_GRIDS_MINMAX.
• cc_grids_minmax_size.m returns sizes for the CC_GRIDS_MINMAX.
• cc_level_compose_animate.m displays the individual CC grids that compose a given level.
• cc_level_to_order.m converts CC nesting levels to CC orders.
• cc_levels_constrained.m computes CC levels and grids satisfying a constraint.
• cc_levels_constrained_display.m displays grids generated by CC_LEVELS_CONSTRAINED.
• cc_levels_constrained_animate.m displays a sequence of grids generated by CC_LEVELS_CONSTRAINED.
• cc_levels_constrained_size.m computes sizes for CC_LEVELS_CONSTRAINED.
• cc_levels_minmax_animate.m displays a sequence of merged grids generated by CC_LEVELS_MINMAX.
• cc_levels_minmax_display.m displays grids generated by CC_LEVELS_MINMAX.
• cc_levels_minmax.m computes CC grids with LEVEL_MIN <= LEVEL <= LEVEL_MAX.
• cc_levels_minmax_size.m returns sizes for CC_LEVELS_MINMAX.
• comp_next.m computes the compositions of the integer N into K parts.
• compnz_next.m computes the compositions of the integer N into K nonzero parts.
• timestamp.m prints the current YMDHMS date as a time stamp.
• vec_next_gray.m computes the elements of a product space.
• vector_constrained_next4.m returns the "next" constrained vector.

Examples and Tests:

CC_GRID displays a single Clenshaw Curtis grid of orders M by N:

• cc_grid_2x16.png, a PNG image of a 2 by 16 Clenshaw Curtis grid.
• cc_grid_5x9.png, a PNG image of a 5 by 9 Clenshaw Curtis grid.

CC_GRIDS_MINMAX displays all Clenshaw Curtis grids of orders M by N such that M+N is between Q_MIN and Q_MAX:

• cc_grids_minmax_2to2.png, a PNG image of Clenshaw Curtis grids with Q_MIN = Q_MAX = 2.
• cc_grids_minmax_3to3.png, a PNG image of Clenshaw Curtis grids with Q_MIN = Q_MAX = 3.
• cc_grids_minmax_4to4.png, a PNG image of Clenshaw Curtis grids with Q_MIN = Q_MAX = 4.
• cc_grids_minmax_5to5.png, a PNG image of Clenshaw Curtis grids with Q_MIN = Q_MAX = 5.
• cc_grids_minmax_6to6.png, a PNG image of Clenshaw Curtis grids with Q_MIN = Q_MAX = 6.

CC_LEVELS_MINMAX displays all Clenshaw Curtis grids of levels M by N such that M+N is between LEVEL_MIN and LEVEL_MAX. Note that the order, or number of points, in the one dimensional rules, is related to the one dimensional levels by ORDER=2**LEVEL+1 (except that the LEVEL=0 corresponds to ORDER=1):

• cc_levels_minmax_0to0.png, a PNG image of Clenshaw Curtis grids with LEVEL_MIN = LEVEL_MAX = 0.
• cc_levels_minmax_1to1.png, a PNG image of Clenshaw Curtis grids with LEVEL_MIN = LEVEL_MAX = 1.
• cc_levels_minmax_2to2.png, a PNG image of Clenshaw Curtis grids with LEVEL_MIN = LEVEL_MAX = 2.
• cc_levels_minmax_3to3.png, a PNG image of Clenshaw Curtis grids with LEVEL_MIN = LEVEL_MAX = 3.
• cc_levels_minmax_4to4.png, a PNG image of Clenshaw Curtis grids with LEVEL_MIN = LEVEL_MAX = 4.
• cc_levels_minmax_5to5.png, a PNG image of Clenshaw Curtis grids with LEVEL_MIN = LEVEL_MAX = 5.
• cc_levels_minmax_6to6.png, a PNG image of Clenshaw Curtis grids with LEVEL_MIN = LEVEL_MAX = 6.

CC_GRIDS_CONSTRAINED displays all Clenshaw Curtis grids of orders M by N such that M and N lie between ORDER_MIN and ORDER_MAX, and ALPHA1*M+ALPHA2*N<=Q_MAX:

• cc_grids_constrained_plot1.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 13, ALPHA=[2,3], ORDER_MIN=[1,1], ORDER_MAX=[5,5].
• cc_grids_constrained_plot2.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 16, ALPHA=[2,3], ORDER_MIN=[1,1], ORDER_MAX=[5,5].
• cc_grids_constrained_plot3.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 20, ALPHA=[2,3], ORDER_MIN=[1,1], ORDER_MAX=[5,5].
• cc_grids_constrained_plot4.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 20, ALPHA=[2,3], ORDER_MIN=[1,1], ORDER_MAX=[8,8].
• cc_grids_constrained_plot5.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 20, ALPHA=[5,2], ORDER_MIN=[1,1], ORDER_MAX=[8,8].
• cc_grids_constrained_plot6.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 30, ALPHA=[5,2], ORDER_MIN=[1,1], ORDER_MAX=[8,8].

CC_LEVELS_CONSTRAINED displays all Clenshaw Curtis grids of levels M by N such that M and N lie between LEVELS_MIN and LEVELS_MAX, and ALPHA1*M+ALPHA2*N<=Q_MAX:

• cc_levels_constrained_plot1.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 13, ALPHA=[2,3], LEVEL_MIN=[1,1], LEVEL_MAX=[5,5].
• cc_levels_constrained_plot2.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 16, ALPHA=[2,3], LEVEL_MIN=[1,1], LEVEL_MAX=[5,5].
• cc_levels_constrained_plot3.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 20, ALPHA=[2,3], LEVEL_MIN=[1,1], LEVEL_MAX=[5,5].
• cc_levels_constrained_plot4.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 20, ALPHA=[2,3], LEVEL_MIN=[1,1], LEVEL_MAX=[8,8].
• cc_levels_constrained_plot5.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 20, ALPHA=[5,2], LEVEL_MIN=[1,1], LEVEL_MAX=[8,8].
• cc_levels_constrained_plot6.png, a PNG image of Clenshaw Curtis grids with Q_MAX = 30, ALPHA=[5,2], LEVEL_MIN=[1,1], LEVEL_MAX=[8,8].

You can go up one level to the MATLAB source codes.