# TRIANGLE_NCO_RULE Newton-Cotes Open Quadrature Rules for the Triangle

TRIANGLE_NCO_RULE is a C library which defines the weights and abscisass for a sequence of 9 Newton-Cotes open quadrature rules over the interior of a triangle in 2D.

Newton-Cotes rules have the characteristic that the abscissas are equally spaced. For a triangle, this refers to spacing in the unit reference triangle, or in the barycentric coordinate system. These rules may be mapped to an arbitrary triangle, and will still be valid.

The rules are said to be "open" when they do not include points on the boundary of the triangle.

The use of equally spaced abscissas may be important for your application. That may how your data was collected, for instance. On the other hand, the use of equally spaced abscissas carries a few costs. In particular, for a given degree of polynomial accuracy, there will be rules that achieve this accuracy, but use fewer abscissas than Newton-Cotes. Moreover, the Newton-Cotes approach almost always results in negative weights for some abscissas. This is generally an undesirable feature, particularly when higher order quadrature rules are being used.

### Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

### Languages:

TRIANGLE_NCO_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version

### Related Data and Programs:

ANNULUS_RULE, a C library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

CUBE_FELIPPA_RULE, a C library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

LINE_NCO_RULE, a C library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

PYRAMID_FELIPPA_RULE, a C library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a C library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_FELIPPA_RULE, a C library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

STROUD, a C library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TETRAHEDRON_FELIPPA_RULE, a C library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a C library which defines Newton-Cotes open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_EXACTNESS, a C program which investigates the monomial exactness quadrature rule over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a C library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a C library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_MONTE_CARLO, a C program which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

TRIANGLE_NCC_RULE, a C library which defines Newton-Cotes closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_SYMQ_RULE, a C library which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2D, by Hong Xiao and Zydrunas Gimbutas.

WEDGE_FELIPPA_RULE, a C library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

### Reference:

1. Peter Silvester,
Symmetric Quadrature Formulae for Simplexes,
Mathematics of Computation,
Volume 24, Number 109, January 1970, pages 95-100.

### List of Routines:

• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• R8_HUGE returns a "huge" R8.
• R8_NINT returns the nearest integer to an R8.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_AREA computes the area of a triangle.
• TRIANGLE_NCO_DEGREE returns the degree of an NCO rule for the triangle.
• TRIANGLE_NCO_ORDER_NUM returns the order of an NCO rule for the triangle.
• TRIANGLE_NCO_RULE returns the points and weights of an NCO rule.
• TRIANGLE_NCO_RULE_NUM returns the number of NCO rules available.
• TRIANGLE_NCO_SUBORDER returns the suborders for an NCO rule.
• TRIANGLE_NCO_SUBORDER_NUM returns the number of suborders for an NCO rule.
• TRIANGLE_NCO_SUBRULE returns a compressed NCO rule.
• TRIANGLE_NCO_SUBRULE_01 returns a compressed NCO rule 1.
• TRIANGLE_NCO_SUBRULE_02 returns a compressed NCO rule 2.
• TRIANGLE_NCO_SUBRULE_03 returns a compressed NCO rule 3.
• TRIANGLE_NCO_SUBRULE_04 returns a compressed NCO rule 4.
• TRIANGLE_NCO_SUBRULE_05 returns a compressed NCO rule 5.
• TRIANGLE_NCO_SUBRULE_06 returns a compressed NCO rule 6.
• TRIANGLE_NCO_SUBRULE_07 returns a compressed NCO rule 7.
• TRIANGLE_NCO_SUBRULE_08 returns a compressed NCO rule 8.
• TRIANGLE_NCO_SUBRULE_09 returns a compressed NCO rule 9.

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

Last revised on 15 June 2014.