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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra nqip.as}
+\author{Michael Richardson}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{NagQuadratureInterfacePackage}
+<<NagQuadratureInterfacePackage>>=
++++ Author: M.G. Richardson
++++ Date Created: 1995 Dec. 07
++++ Date Last Updated:
++++ Basic Functions:
++++ Related Constructors:
++++ Also See:
++++ AMS Classifications:
++++ Keywords:
++++ References:
++++ Description:
++++ This package provides Axiom-like interfaces to some of the NAG
++++ quadrature (numerical integration) routines in the NAGlink.
+
+NagQuadratureInterfacePackage: with {
+
+  nagPolygonIntegrate : (LDF, LDF) ->
+			        RCD(integral : DF, errorEstimate : DF) ;
+
+  ++ nagPolygonIntegrate(xlist,ylist) evaluates the definite integral
+#if saturn
+  ++ $\int_{x_{1}}^{x_{n}}y(x) \, dx$
+#else
+  ++ integrate(y(x), x=x[1]..x[n])
+#endif
+  ++ where the numerical value of the function \spad{y} is specified at
+  ++ the \spad{n} distinct points
+#if saturn
+  ++ $x_{1}, x_{2}, \ldots , x_{n}$.
+#else
+  ++ x[1], x[2] ... x[n].
+#endif
+  ++ The \spad{x} and \spad{y} values are specified in the lists
+  ++ \spad{xlist} and \spad{ylist}, respectively; the \spad{xlist}
+  ++ values must form a strictly monotonic sequence of four or more
+  ++ points.
+  ++ The calculation is performed by the NAG routine D01GAF.
+  ++
+  ++ An estimate of the numerical error in the calculation is also
+  ++ returned; however, by choosing unrepresentative data points to
+  ++ approximate the function it is possible to achieve an arbitrarily
+  ++ large difference between the true integral and the value
+  ++ calculated.
+  ++ For more detailed information, please consult the NAG
+  ++ manual via the Browser page for the operation d01gaf.
+
+  nagPolygonIntegrate : MDF -> RCD(integral : DF, errorEstimate : DF) ;
+
+
+} == add {
+
+  import from NagIntegrationPackage ;
+  import from NagResultChecks ;
+  import from AnyFunctions1 DF ;
+  import from STRG ;
+  import from List STRG ;
+  import from Symbol ;
+  import from LLDF ;
+  import from VDF ;
+  import from MDF ;
+  import from ErrorFunctions ;
+
+  local ipIfail : INT := -1 ;
+  local d01gafError : DF := 0 ;
+
+  nagPolygonIntegrate(xlist : LDF, ylist : LDF)
+		     : RCD(integral : DF, errorEstimate : DF) == {
+ 
+    local lx, ly : INT ;
+    local d01gafResult : RSLT ;
+
+    lx := (# xlist) pretend INT ;
+    ly := (# ylist) pretend INT ;
+    if lx ~= ly
+    then error ["The lists supplied to nagPolygonIntegrate are of ",
+		"different lengths: ",
+		string(lx),
+		" and ",
+		string(ly),
+		"."]
+    else {
+      d01gafResult := d01gaf(matrix [xlist],matrix [ylist],lx,ipIfail) ;
+      [checkResult(d01gafResult,"ans","D01GAF"),
+       retract(d01gafResult."er") @ DF]
+    }
+  }
+
+  nagPolygonIntegrate(coords : MDF)
+		     : RCD(integral : DF, errorEstimate : DF) ==
+    if (ncols(coords) pretend INT) ~= 2
+    then error ["Please supply the coordinate matrix in ",
+		"nagPolygonIntegrate with each row consisting of ",
+		"a single x-y pair."]
+    else nagPolygonIntegrate(members column(coords,1),
+		             members column(coords,2)) ;
+
+}
+
+#if NeverAssertThis
+
+-- Note that the conversions of results from DoubleFloat to Float
+-- will become unnecessary if outputGeneral is extended to apply to
+-- DoubleFloat quantities.
+
+)lib nrc
+)lib nqip
+
+outputGeneral 5
+
+xvals := [0.00,0.04,0.08,0.12,0.22,0.26,0.30,0.38,0.39,0.42,0.45, 
+               0.46,0.60,0.68,0.72,0.73,0.83,0.85,0.88,0.90,1.00];
+
+yvals := [4.0000,3.9936,3.9746,3.9432,3.8135,3.7467,3.6697,3.4943,
+                 3.4719,3.4002,3.3264,3.3017,2.9412,2.7352,2.6344,
+                        2.6094,2.3684,2.3222,2.2543,2.2099,2.0000];
+
+result := nagPolygonIntegrate(xvals,yvals);
+result.integral :: Float             
+
+--       3.1414
+
+result.errorEstimate :: Float        
+
+--       - 0.000025627
+
+coords := transpose matrix [xvals, yvals];
+result := nagPolygonIntegrate coords;
+result.integral :: Float             
+
+--       3.1414
+
+result.errorEstimate :: Float        
+
+--       - 0.000025627
+
+nagPolygonIntegrate([1,2,3],[1,2,3,4])
+ 
+--   Error signalled from user code:
+--      The lists supplied to nagPolygonIntegrate are of different 
+--      lengths: 3 and 4.
+
+nagPolygonIntegrate([[1,2,3],[4,5,6]])
+
+--   Error signalled from user code:
+--      Please supply the coordinate matrix in nagPolygonIntegrate with
+--      each row consisting of single a x-y pair.
+
+outputGeneral()
+
+output "End of tests"
+
+#endif
+
+@
+\section{License}
+<<license>>=
+--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+--All rights reserved.
+--
+--Redistribution and use in source and binary forms, with or without
+--modification, are permitted provided that the following conditions are
+--met:
+--
+--    - Redistributions of source code must retain the above copyright
+--      notice, this list of conditions and the following disclaimer.
+--
+--    - Redistributions in binary form must reproduce the above copyright
+--      notice, this list of conditions and the following disclaimer in
+--      the documentation and/or other materials provided with the
+--      distribution.
+--
+--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
+--      names of its contributors may be used to endorse or promote products
+--      derived from this software without specific prior written permission.
+--
+--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
+--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+@
+<<*>>=
+<<license>>
+
+-- NagQuadratureInterfacePackage
+
+-- To test:
+-- sed -ne '1,/^#if NeverAssertThis/d;/#endif/d;p' < nqip.as > nqip.input
+-- axiom
+-- )set nag host <some machine running nagd>
+-- )r nqip.input
+ 
+#unassert saturn
+
+#include "axiom.as"
+
+DF   ==> DoubleFloat ;
+LDF  ==> List DoubleFloat ;
+LLDF ==> List LDF ;
+VDF  ==> Vector DoubleFloat ;
+MDF  ==> Matrix DoubleFloat ;
+INT  ==> Integer ;
+RCD  ==> Record ;
+RSLT ==> Result ;
+STRG ==> String ;
+
+<<NagQuadratureInterfacePackage>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}