<html> <head> <title> PYRAMID_RULE - Generate a Quadrature Rule for a Pyramid </title> </head> <body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055"> <h1 align = "center"> PYRAMID_RULE <br> Generate a Quadrature Rule for a Pyramid </h1> <hr> <p> <b>PYRAMID_RULE</b> is a MATLAB program which generates a quadrature rule for a pyramid. </p> <p> The quadrature rules generated by <b>PYRAMID_RULE</b> are all examples of <i>conical product rules</i>, and involve a kind of direct product of the form: <blockquote> Legendre rule in X * Legendre rule in Y * Jacobi rule in Z </blockquote> where the Jacobi rule includes a factor of (1-Z)^2. </p> <p> The integration region is: <pre> - ( 1 - Z ) <= X <= 1 - Z - ( 1 - Z ) <= Y <= 1 - Z 0 <= Z <= 1. </pre> When Z is zero, the integration region is a square lying in the (X,Y) plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the radius of the square diminishes, and when Z reaches 1, the square has contracted to the single point (0,0,1). </p> <h3 align = "center"> Usage: </h3> <p> <blockquote> <b>pyramid_rule</b> ( <i>legendre_order</i>, <i>jacobi_order</I>, <i>'filename'</i> ) </blockquote> where <ul> <li> <i>legendre_order</i> is the order of the 1D Legendre quadrature rule to be used. The X and Y dimensions will use a product of this rule. </li> <li> <i>jacobi_order</i> is the order of the 1D Jacobi quadrature rule to be used. The Z dimension will use this rule which will include a factor of (1-X)^2 which accounts for the narrowing of the pyramid. </li> <li> <i>'filename'</i> is the common prefix for the files containing the region, weight and abscissa information of the quadrature rule; </li> </ul> </p> <h3 align = "center"> Licensing: </h3> <p> The computer code and data files described and made available on this web page are distributed under <a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a> </p> <h3 align = "center"> Languages: </h3> <p> <b>PYRAMID_RULE</b> is available in <a href = "../../cpp_src/pyramid_rule/pyramid_rule.html">a C++ version</a> and <a href = "../../f90_src/pyramid_rule/pyramid_rule.html">a FORTRAN90 version</a> and <a href = "../../m_src/pyramid_rule/pyramid_rule.html">a MATLAB version.</a> </p> <h3 align = "center"> Related Data and Programs: </h3> <p> <a href = "../../m_src/felippa/felippa.html"> FELIPPA</a>, a MATLAB library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons. </p> <p> <a href = "../../m_src/geometry/geometry.html"> GEOMETRY</a>, a MATLAB library which performs geometric calculations in 2, 3 and N dimensional space. </p> <p> <a href = "../../m_src/jacobi_rule/jacobi_rule.html"> JACOBI_RULE</a>, a MATLAB program which can compute and print a Gauss-Jacobi quadrature rule. </p> <p> <a href = "../../m_src/legendre_rule/legendre_rule.html"> LEGENDRE_RULE</a>, a MATLAB program which can compute and print a Gauss-Legendre quadrature rule. </p> <p> <a href = "../../m_src/pyramid_exactness/pyramid_exactness.html"> PYRAMID_EXACTNESS</a>, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the pyramid. </p> <p> <a href = "../../datasets/quadrature_rules_pyramid/quadrature_rules_pyramid.html"> QUADRATURE_RULES_PYRAMID</a>, a dataset directory which contains quadrature rules for a pyramid with a square base. </p> <p> <a href = "../../m_src/quadrule/quadrule.html"> QUADRULE</a>, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions. </p> <p> <a href = "../../m_src/stroud/stroud.html"> STROUD</a>, a MATLAB library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions. </p> <h3 align = "center"> Reference: </h3> <p> <ol> <li> Carlos Felippa,<br> A compendium of FEM integration formulas for symbolic work,<br> Engineering Computation,<br> Volume 21, Number 8, 2004, pages 867-890. </li> <li> Arthur Stroud,<br> Approximate Calculation of Multiple Integrals,<br> Prentice Hall, 1971,<br> ISBN: 0130438936,<br> LC: QA311.S85. </li> </ol> </p> <h3 align = "center"> Source Code: </h3> <p> <ul> <li> <a href = "pyramid_rule.m">pyramid_rule.m</a>, the source code. </li> </ul> </p> <p> You can go up one level to <a href = "../m_src.html"> the MATLAB source codes</a>. </p> <hr> <i> Last revised on 23 July 2009. </i> <!-- John Burkardt --> </body> <!-- Initial HTML skeleton created by HTMLINDEX. --> </html>