* algebra/axtimer.as.pamphlet: Remove.

	* algebra/ffrac.as.pamphlet: Likewise.
	* algebra/herm.as.pamphlet: Likewise.
	* algebra/interval.as.pamphlet: Likewise.
	* algebra/invnode.as.pamphlet: Likewise.
	* algebra/invrender.as.pamphlet: Likewise.
	* algebra/invtypes.as.pamphlet: Likewise.
	* algebra/invutils.as.pamphlet: Likewise.
	* algebra/iviews.as.pamphlet: Likewise.
	* algebra/ndftip.as.pamphlet: Likewise.
	* algebra/nepip.as.pamphlet: Likewise.
	* algebra/noptip.as.pamphlet: Likewise.
	* algebra/nqip.as.pamphlet: Likewise.
	* algebra/nrc.as.pamphlet: Likewise.
	* algebra/nsfip.as.pamphlet: Likewise.

1 files changed, 0 insertions, 1174 deletions

diff --git a/src/algebra/ndftip.as.pamphlet b/src/algebra/ndftip.as.pamphlet
deleted file mode 100644
index 4c186a67..00000000
--- a/src/algebra/ndftip.as.pamphlet
+++ /dev/null
@@ -1,1174 +0,0 @@
-\documentclass{article}
-\usepackage{open-axiom}
-\begin{document}
-\title{\$SPAD/src/algebra ndftip.as}
-\author{Michael Richardson}
-\maketitle
-\begin{abstract}
-\end{abstract}
-\eject
-\tableofcontents
-\eject
-\section{NagDiscreteFourierTransformInterfacePackage}
-<<NagDiscreteFourierTransformInterfacePackage>>=
-+++ Author: M.G. Richardson
-+++ Date Created: 1995 Dec. 08
-+++ Date Last Updated:
-+++ Basic Functions:
-+++ Related Constructors:
-+++ Also See:
-+++ AMS Classifications:
-+++ Keywords:
-+++ References:
-+++ Description:
-+++ This package provides Axiom-like interfaces to the NAG
-+++ Finite Fourier Transform routines in the NAGlink.
-
-NagDiscreteFourierTransformInterfacePackage: with {
-
-  nagDFT : VDF -> VCDF ;                                --  test  1
-
-++ nagDFT(seq) calculates the discrete Fourier transform of a sequence
-++ of real data values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the vector seq.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06EAF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06eaf.
-
-  nagDFT : VCDF -> VCDF ;                               --  test  3
-
-++ nagDFT(seq) calculates the discrete Fourier transform of a sequence
-++ of complex data values 
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the vector seq.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06ECF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06ecf.
-
-  nagDFT : PHSDF -> VDF ;                               --  test  7
-
-++ nagDFT(seq) calculates the discrete Fourier transform of a Hermitian
-++ sequence of complex data values,
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the PackedHermitianSequence seq.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06EBF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06ebf.
-
-  nagDFT : LVDF -> LVCDF ;                               --  test 10, 19
-
-++ nagDFT(seqs) calculates the discrete Fourier transform of each of a
-++ list of sequences of real data values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the list of vectors, seqs.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FPF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06fpf.
-
-  nagDFT : LVCDF -> LVCDF ;                             --  test 16
-
-++ nagDFT(seqs) calculates the discrete Fourier transform of each of a
-++ list of sequences of complex data values 
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the list of vectors, seqs.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FRF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06frf.
-
-  nagDFT : LPHSDF -> LVDF ;                              --  test 12, 21
-
-++ nagDFT(seq) calculates the discrete Fourier transform of a each of a
-++ list of Hermitian sequences of complex data values,
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the List PackedHermitianSequence, seq.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FQF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06fqf.
-
-  nagInverseDFT : VDF -> VCDF ;                         --  test  8
-
-++ nagInverseDFT(seq) calculates the inverse discrete Fourier
-++ transform of a sequence of real data values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the vector seq.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06EAF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06eaf.
-
-  nagInverseDFT : VCDF -> VCDF ;                         --  test  2,  4
-
-++ nagInverseDFT(seq) calculates the inverse discrete Fourier
-++ transform of a sequence of complex data values 
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the vector seq.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06ECF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06ecf.
-
-  nagInverseDFT : PHSDF -> VDF ;                        --  test  6
-
-++ nagInverseDFT(seq) calculates the inverse discrete Fourier transform
-++ of a Hermitian sequence of complex data values 
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the PackedHermitianSequence seq.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06EBF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06ebf.
-
-  nagInverseDFT : LVDF -> LVCDF ;                       --  test 13
-
-++ nagInverseDFT(seqs) calculates the inverse discrete Fourier
-++ transform of each of a list of sequences of real data values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the list of vectors, seqs.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FPF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06fpf.
-
-  nagInverseDFT : LVCDF -> LVCDF ;                       --  test 11, 17
-
-++ nagInverseDFT(seqs) calculates the inverse discrete Fourier
-++ transform of each of a list of sequences of complex data values 
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the list of vectors, seqs.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FRF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06frf.
-
-  nagInverseDFT : LPHSDF -> LVDF ;                      --  test 15
-
-++ nagInverseDFT(seqs) calculates the inverse discrete Fourier transform
-++ of each of a list of Hermitian sequences of complex data values 
-#if saturn
-++ $z_{1} \ldots z_{n}$
-#else
-++ \spad{z[1] .. z[n]}
-#endif
-++ supplied in the List PackedHermitianSequence, seqs.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} z_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(z[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FQF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06fqf.
-
-  nagHermitianDFT : VDF -> PHSDF ;                      --  test  5
-
-++ nagHermitianDFT(seq) calculates the discrete Fourier transform, in
-++ packed Hermitian form, of a sequence of real data values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the vector seq.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06EAF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06eaf.
-
-  nagHermitianDFT : LVDF -> LPHSDF ;                     --  test 14, 20
-
-++ nagHermitianDFT(seqs) calculates the discrete Fourier transform, in
-++ packed Hermitian form, of each of a list of sequences of real data
-++ values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the list of vectors, seqs.
-++ Note that the definition used for the discrete Fourier transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{-i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(-i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FPF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06fpf.
-
-  nagHermitianInverseDFT : VDF -> PHSDF ;               --  test  9
-
-++ nagHermitianInverseDFT(seq) calculates the inverse discrete Fourier
-++ transform, in packed Hermitian form, of a sequence of real data
-++ values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the vector seq.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06EAF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06eaf.
-
-  nagHermitianInverseDFT : LVDF -> LPHSDF ;             --  test 18
-
-++ nagHermitianInverseDFT(seqs) calculates the inverse discrete Fourier
-++ transform, in packed Hermitian form, of each of a list of sequences
-++ of real data values 
-#if saturn
-++ $x_{1} \ldots x_{n}$
-#else
-++ \spad{x[1] .. x[n]}
-#endif
-++ supplied in the list of vectors, seqs.
-++ Note that the definition used for the inverse discrete Fourier
-++ transform is
-#if saturn
-++ \[ \frac{1}{\sqrt{n} \sum_{j=0}^{n-1} x_{j} e^{i \frac{2 \pi j k}{n}
-++ \qquad k = 0 \ldots  n - 1 \]
-#else
-++ \spad{1/sqrt(n)*sum(x[j]*%e^(i*2*%pi*j*k/n), j=0..(n-1))} for
-++ \spad{k=0..(n-1)}.
-#endif
-++ The numerical calculation is performed by the NAG routine C06FPF.
-++
-++ For more detailed information, please consult the NAG
-++ manual via the Browser page for the operation c06fpf.
-
-} == add {
-
-  import from AnyFunctions1 MDF ;
-  import from CDF;
-  import from ErrorFunctions ;
-  import from LLDF ;
-  import from MCDF ;
-  import from MDF ;
-  import from NagResultChecks ;
-  import from NagSeriesSummationPackage ;
-  import from PHSDF;
-  import from STRG ;
-  import from List STRG ;
-  import from Symbol ;
-  import from VDF ;
-
-  local (..)(a:INT,b:INT):Generator INT == {
-    generate {
-      t := a ;
-      while (t <= b) repeat {
-        yield t ;
-        t := t + 1 ;
-        }
-      }
-    }
-
-  local ipIfail : INT := -1 ;
-
--- First, the functions corresponding to single NAGlink calls of C06E
--- routines (single vector transforms):
-
--- c06eaf:
-
-  nagHermitianDFT(seq : VDF) : PHSDF ; == {
-    local lseq : INT ;
-
-    lseq := ((# seq)@NNI) pretend INT ; -- @ to eliminate SI possibility
-    row(checkMxDF(c06eaf(lseq,matrix [members seq],ipIfail),
-		   "x",
-		    "C06EAF"),
-			      1)
-				 pretend PHSDF
-  }
-
--- c06ebf:
-
-  nagDFT(seq : PHSDF) : VDF == {
-    local lseq : INT ;
-
-    lseq := ((# seq)@NNI) pretend INT ; -- @ to eliminate SI possibility
-    row(checkMxDF(c06ebf(lseq,matrix [members seq],ipIfail),
-		   "x",
-		    "C06EBF"),
-			      1)
-  }
-
--- c06ecf:
-
-  nagDFT(seq : VCDF) : VCDF == {
-    local nseq : NNI ;
-    local lseq : INT ;
-    local rvec, ivec : VDF ;
-    local cvec : VCDF ;
-    local c06ecfResult : RSLT ;
-
-    nseq := # seq ;
-    lseq := nseq pretend INT ;
-    rvec := new(nseq,0) ;
-    ivec := new(nseq,0) ;
-    for i in 1..lseq repeat {
-      rvec(i) := real seq(i) ;
-      ivec(i) := imag seq(i) ;
-    }
-    c06ecfResult := c06ecf(lseq,
-			    matrix [members rvec],
-			     matrix [members ivec],
-			      ipIfail) ;
-    rvec := row(checkMxDF(c06ecfResult,"x","C06ECF"),1) ;
-    ivec := row((retract(c06ecfResult."y") @ MDF),1) ;
-    cvec := new(nseq,0) ;
-    for i in 1..lseq repeat cvec(i) := complex(rvec(i),ivec(i)) ;
-    cvec
-  }
-
--- inverse transforms, in terms of these and functions from PHS:
-
-  nagInverseDFT(seq : PHSDF) : VDF == nagDFT conjHerm seq ;
-
-  nagHermitianInverseDFT(seq : VDF) : PHSDF
-			== conjHerm nagHermitianDFT seq ;
-
-  nagInverseDFT(seq : VCDF) : VCDF == {
-    local nseq : NNI ;
-    local lseq : INT ;
-    local rvec, ivec : VDF ;
-    local cvec : VCDF ;
-    local c06ecfResult : RSLT ;
-
-    nseq := # seq ;
-    lseq := nseq pretend INT ;
-    rvec := new(nseq,0) ;
-    ivec := new(nseq,0) ;
-    for i in 1..lseq repeat {
-      rvec(i) := real seq(i) ;
-      ivec(i) := - imag seq(i) ;
-    }
-    c06ecfResult := c06ecf(lseq,
-			    matrix [members rvec],
-			     matrix [members ivec],
-			      ipIfail) ;
-    rvec := row(checkMxDF(c06ecfResult,"x","C06ECF"),1) ;
-    ivec := row((retract(c06ecfResult."y") @ MDF),1) ;
-    cvec := new(nseq,0) ;
-    for i in 1..lseq repeat cvec(i) := complex(rvec(i), - ivec(i)) ;
-    cvec
-  }
-
--- "Full form" equivalents of c06eaf and inverse:
-
-  nagDFT(seq : VDF) : VCDF == expand nagHermitianDFT seq ;
-
-  nagInverseDFT(seq : VDF) : VCDF == expand nagHermitianInverseDFT seq ;
-
-
--- Next, the functions corresponding to single NAGlink calls of C06F
--- routines (multiple vector transforms):
-
--- basic routines:
-
--- c06fpf
-
-  nagHermitianDFT(seqs : LVDF) : LPHSDF ; == {
-
-    local nr, nc : NNI ;
-    local inr, inc : INT ;
-    local seqMat, trig, result : MDF ;
-    local nextSeq : PHSDF ;
-    local hermDFTs : LPHSDF ;
-
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    nc := # (seqs.1) ;
-    inc := nc pretend INT ;
-    seqMat := new(nr,nc,0) ;
-    for j in 1 .. inc repeat seqMat(1,j) := (seqs.1).j ;
-    for i in 2 .. inr repeat
-      if (# seqs.i) ~= nc 
-      then error ["The data sequences in nagHermitianDFT must all",
-		  " have the same length.  ",
-		  "The length of sequence 1 is ",
-		  string(inc),
-		  "that of sequence ",
-		  string(i pretend INT),
-		  " is ",
-		  string((# seqs.i)@NNI pretend INT), -- @ avoids SI
-		  "."]
-      else for j in 1 .. inc repeat seqMat(i,j) := (seqs.i).j ;
-    trig := new(1@NNI,2*nc,0) ;
-    result :=
-      checkMxDF(c06fpf(inr,inc,"i",seqMat,trig,ipIfail),"x","C06FPF") ;
-    hermDFTs := [] ;
-    for i in inr .. 1  by -1 repeat {
-      nextSeq := new(nc,0) ;
-      for j in 1 .. inc repeat nextSeq(j) := result(1,(j-1)*inr + i) ;
-      hermDFTs := cons(nextSeq,hermDFTs) ;
-    }
-    hermDFTs
-  }
-
--- c06fqf
-
-  nagDFT(seqs : LPHSDF) : LVDF == {
-    
-    local nr, nc : NNI ;
-    local inr, inc : INT ;
-    local seqMat, trig, result : MDF ;
-    local nextSeq : VDF ;
-    local dfts : LVDF ;
-  
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    nc := # (seqs.1) ;
-    inc := nc pretend INT ;
-    seqMat := new(nr,nc,0) ;
-    for j in 1 .. inc repeat seqMat(1,j) := (seqs.1).j ;
-    for i in 2 .. inr repeat
-      if (# seqs.i) ~= nc 
-      then error ["The data sequences in nagDFT must all",
-                  " have the same length.  ",
-                  "The length of sequence 1 is ",
-                  string(inc),
-                  "that of sequence ",
-                  string(i pretend INT),
-                  " is ",
-                  string((# seqs.i)@NNI pretend INT), -- @ avoids SI
-                  "."]
-      else for j in 1 .. inc repeat seqMat(i,j) := (seqs.i).j ;
-    trig := new(1@NNI,2*nc,0) ;
-    result :=
-      checkMxDF(c06fqf(inr,inc,"i",seqMat,trig,ipIfail),"x","C06FQF") ;
-    dfts := [] ;
-    for i in inr .. 1 by -1 repeat {
-      nextSeq := new(nc,0) ;
-      for j in 1 .. inc repeat nextSeq(j) := result(1,(j-1)*inr + i) ;
-      dfts := cons(nextSeq,dfts) ;
-    }
-    dfts
-  }
-
--- c06frf
-
-  nagDFT(seqs : LVCDF) : LVCDF == {
-
-    local nr, nc : NNI ;
-    local inr, inc : INT ;
-    local trig, rMat, iMat : MDF ;
-    local result : RSLT ;
-    local nextSeq : VCDF ;
-    local dfts : LVCDF ;
-
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    nc := # (seqs.1) ;
-    inc := nc pretend INT ;
-    rMat := new(nr,nc,0) ;
-    iMat := new(nr,nc,0) ;
-    for j in 1 .. inc repeat {
-      rMat(1,j) := real((seqs.1).j) ;
-      iMat(1,j) := imag((seqs.1).j) ;
-    }
-    for i in 2 .. inr repeat {
-      if (# seqs.i) ~= nc
-      then error ["The data sequences in nagDFT must all",
-		  " have the same length.  ",
-		  "The length of sequence 1 is ",
-		  string(inc),
-		  "that of sequence ",
-		  string(i pretend INT),
-		  " is ",
-		  string((# seqs.i)@NNI pretend INT), -- @ avoids SI
-		  "."]
-      else for j in 1 .. inc repeat {
-	rMat(i,j) := real((seqs.i).j) ;
-	iMat(i,j) := imag((seqs.i).j) ;
-      }
-    }
-    trig := new(1@NNI,2*nc,0) ;
-    result := c06frf(inr,inc,"i",rMat,iMat,trig,ipIfail) ;
-    rMat := checkMxDF(result, "x", "C06FRF") ;
-    iMat := retract(result."y") @ MDF ;
-    dfts := [] ;
-    for i in inr .. 1 by -1 repeat {
-      nextSeq := new(nc,0) ;
-      for j in 1 .. inc repeat
-	nextSeq(j) := complex(rMat(1,(j-1)*inr+i),iMat(1,(j-1)*inr+i)) ;
-      dfts := cons(nextSeq,dfts) ;
-    }
-    dfts
-  }
-
--- inverse transforms, in terms of these and functions from PHS:
-
-  nagInverseDFT(seqs : LVCDF) : LVCDF == {
-
-    local nr, nc : NNI ;
-    local inr, inc : INT ;
-    local conjSeq : VCDF ;
-    local temp, invdfts : LVCDF ;
-
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    temp := [] ;
-    for i in inr .. 1 by -1 repeat {
-      nc := #(seqs.i) ;
-      inc := nc pretend INT ;
-      conjSeq := new(nc,0) ;
-      for j in 1 .. inc repeat
-        conjSeq(j) := conjugate((seqs.i).j) ;
-      temp := cons(conjSeq,temp) ;
-    }
-    temp := nagDFT temp ;
-    invdfts := [] ;
-    for i in inr .. 1 by -1 repeat {
-      conjSeq := new(nc,0) ;
-      for j in 1 .. inc repeat -- know inc is constant after nagDFT call
-	conjSeq(j) := conjugate((temp.i).j) ;
-      invdfts := cons(conjSeq,invdfts) ;
-    }
-    invdfts
-  }
-
-  nagInverseDFT(seqs : LPHSDF) : LVDF == {
-    local nr : NNI ;
-    local inr : INT ;
-    local conjSeqs : LPHSDF ;
-  
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    conjSeqs := [] ;
-    for i in inr .. 1 by -1 repeat
-      conjSeqs := cons(conjHerm(seqs.i),conjSeqs) ;
-    nagDFT conjSeqs ;
-  }
-
-  nagHermitianInverseDFT(seqs : LVDF) : LPHSDF == {
-    local nr : NNI ;
-    local inr : INT ;
-    local conjSeqs, invSeqs : LPHSDF ;
-  
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    conjSeqs := nagHermitianDFT seqs ;
-    invSeqs := [] ;
-    for i in inr .. 1 by -1 repeat
-      invSeqs := cons(conjHerm(conjSeqs.i),invSeqs) ;
-    invSeqs
-  }
-
--- "Full form" equivalents of c06fpf and inverse:
-
-  nagDFT(seqs : LVDF) : LVCDF == {
-  
-    local nr : NNI ;
-    local inr : INT ;
-    local hermdfts : LPHSDF ;
-    local dfts : LVCDF ;
-
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    hermdfts := nagHermitianDFT seqs ;
-    dfts := [] ;
-    for i in inr .. 1 by -1 repeat
-      dfts := cons(expand(hermdfts.i),dfts) ;
-    dfts
-  }
-
-  nagInverseDFT(seqs : LVDF) : LVCDF == {
-    local nr : NNI ;
-    local inr : INT ;
-    local hermdfts : LPHSDF ;
-    local invdfts : LVCDF ;
-
-    nr := # seqs ;
-    inr := nr pretend INT ;
-    hermdfts := nagHermitianDFT seqs ;
-    invdfts := [] ;
-    for i in inr .. 1 by -1 repeat
-      invdfts := cons(expand conjHerm(hermdfts.i),invdfts) ;
-    invdfts
-  }
-    
-}
-
-#if NeverAssertThis
-
--- Note that the conversions of results from DoubleFloat to Float
--- will become unnecessary if outputGeneral is extended to apply to
--- DoubleFloat quantities.  Those results not converted will, of
--- course, then be displayed to 6 s.f.
-
-)lib nrc
-)lib herm
-)lib ndftip
-
-outputGeneral 6
-
-seqA := [0.34907,0.54890,0.74776,0.94459,1.1385,1.3285,1.5137];
-
-seqB := [0.34907 - 0.37168*%i,  _
-         0.54890 - 0.35669*%i,  _
-         0.74776 - 0.31175*%i,  _
-         0.94459 - 0.23702*%i,  _
-         1.13850 - 0.13274*%i,  _
-         1.32850 + 0.00074*%i,  _
-         1.51370 + 0.16298*%i];
-
-hseqC : PackedHermitianSequence DoubleFloat
-hseqC := packHS [0.34907,        _
-                 0.54890 + %i*1.51370,  _
-                 0.74776 + %i*1.32850,  _
-                 0.94459 + %i*1.13850,  _
-                 0.94459 - %i*1.13850,  _
-                 0.74776 - %i*1.32850,  _
-                 0.54890 - %i*1.51370];
-
-seqsD : List Vector DoubleFloat;
-seqsD := [vector [0.3854, 0.6772, 0.1138, 0.6751, 0.6362, 0.1424], _
-          vector [0.5417, 0.2983, 0.1181, 0.7255, 0.8638, 0.8723], _
-          vector [0.9172, 0.0644, 0.6037, 0.6430, 0.0428, 0.4815]];
-
-seqsE : List PackedHermitianSequence DoubleFloat;
-seqsE := [pHS [0.3854, 0.6772, 0.1138, 0.6751, 0.6362, 0.1424], _
-          pHS [0.5417, 0.2983, 0.1181, 0.7255, 0.8638, 0.8723], _
-          pHS [0.9172, 0.0644, 0.6037, 0.6430, 0.0428, 0.4815]];
-
-seqsF : List Vector Complex DoubleFloat
-seqsF := [vector [0.3854 + 0.5417*%i, 0.6772 + 0.2983*%i,   _
-                  0.1138 + 0.1181*%i, 0.6751 + 0.7255*%i,   _
-                  0.6362 + 0.8638*%i, 0.1424 + 0.8723*%i],  _
-          vector [0.9172 + 0.9089*%i, 0.0644 + 0.3118*%i,   _
-                  0.6037 + 0.3465*%i, 0.6430 + 0.6198*%i,   _
-                  0.0428 + 0.2668*%i, 0.4815 + 0.1614*%i],  _
-          vector [0.1156 + 0.6214*%i, 0.0685 + 0.8681*%i,   _
-                  0.2060 + 0.7060*%i, 0.8630 + 0.8652*%i,   _
-                  0.6967 + 0.9190*%i, 0.2792 + 0.3355*%i]];
-
--- test  1
-
-dftA := nagDFT seqA;
-dftA :: Vector Complex Float :: Matrix Complex Float
-                             -- Matrix to force display as a column,
-                             -- Float to allow outputGeneral to work.
-
---       +         2.48361         +
---       |                         |
---       |- 0.265985 + 0.530898 %i |
---       |                         |
---       |- 0.257682 + 0.202979 %i |
---       |                         |
---       |- 0.256363 + 0.0580623 %i|
---       |                         |
---       |- 0.256363 - 0.0580623 %i|
---       |                         |
---       |- 0.257682 - 0.202979 %i |
---       |                         |
---       +- 0.265985 - 0.530898 %i +
-
--- test  2
-
-nagInverseDFT dftA :: Vector Float
- 
---       [0.34907,0.5489,0.74776,0.94459,1.1385,1.3285,1.5137]
-
--- test  3
-
-dftB := nagDFT seqB;
-dftB :: Vector Complex Float :: Matrix Complex Float
-
---       +  2.48361 - 0.471004 %i  +
---       |                         |
---       | - 0.5518 + 0.496841 %i  |
---       |                         |
---       |- 0.367113 + 0.0975621 %i|
---       |                         |
---       |- 0.287669 - 0.0586476 %i|
---       |                         |
---       |- 0.225057 - 0.174772 %i |
---       |                         |
---       |- 0.148251 - 0.308396 %i |
---       |                         |
---       + 0.0198297 - 0.564956 %i +
- 
--- test  4
- 
-(nagInverseDFT dftB) :: Vector Complex Float :: Matrix Complex Float
- 
---       +0.34907 - 0.37168 %i+
---       |                    |
---       |0.5489 - 0.35669 %i |
---       |                    |
---       |0.74776 - 0.31175 %i|
---       |                    |
---       |0.94459 - 0.23702 %i|
---       |                    |
---       |1.1385 - 0.13274 %i |
---       |                    |
---       |1.3285 + 0.00074 %i |
---       |                    |
---       +1.5137 + 0.16298 %i +
-
--- test  5
-
-hdftA := nagHermitianDFT seqA;
-(expand hdftA) :: Vector Complex Float :: Matrix Complex Float
-
---       +         2.48361         +
---       |                         |
---       |- 0.265985 + 0.530898 %i |
---       |                         |
---       |- 0.257682 + 0.202979 %i |
---       |                         |
---       |- 0.256363 + 0.0580623 %i|
---       |                         |
---       |- 0.256363 - 0.0580623 %i|
---       |                         |
---       |- 0.257682 - 0.202979 %i |
---       |                         |
---       +- 0.265985 - 0.530898 %i +
- 
--- test  6
- 
-(nagInverseDFT hdftA) :: Vector Float
-
---       [0.34907,0.5489,0.74776,0.94459,1.1385,1.3285,1.5137]
-
--- test  7
-
-dftC := nagDFT hseqC;
-dftC :: Vector Float
- 
--- [1.82616,1.86862,- 0.017503,0.502001,- 0.598725,- 0.0314404,- 2.62557]
-
--- test  8
-
-(nagInverseDFT dftC) :: Vector Complex Float
- 
--- [0.34907, 0.5489 + 1.5137 %i, 0.74776 + 1.3285 %i, 0.94459 + 1.1385 %i,
---  0.94459 - 1.1385 %i, 0.74776 - 1.3285 %i, 0.5489 - 1.5137 %i]
-
--- test  9
-
-nagHermitianInverseDFT dftC
- 
--- [0.34907000000000005, 0.54889999999999983, 0.74775999999999987,
---  0.94459000000000004, 1.1385000000000003, 1.3284999999999998,
---  1.5136999999999998]
-
--- test 10:
-
-dftsD := nagDFT seqsD;
-
-dftsD :: List Vector Complex Float
- 
--- [
---   [1.07373, - 0.104062 - 0.00438406 %i, 0.112554 - 0.373777 %i, - 0.146684,
---    0.112554 + 0.373777 %i, - 0.104062 + 0.00438406 %i]
---   ,
-
---   [1.39609, - 0.0365178 + 0.466584 %i, 0.077955 - 0.0607051 %i, - 0.152072,
---    0.077955 + 0.0607051 %i, - 0.0365178 - 0.466584 %i]
---   ,
-
---   [1.12374, 0.0914068 - 0.050841 %i, 0.393551 + 0.345775 %i, 0.153011,
---    0.393551 - 0.345775 %i, 0.0914068 + 0.050841 %i]
---   ]
-
--- test 11:
-
-invdftsD := nagInverseDFT dftsD ;
-invdftsD :: List Vector Complex Float
- 
--- [[0.3854,0.6772,0.1138,0.6751,0.6362,0.1424],
---  [0.5417,0.2983,0.1181,0.7255,0.8638,0.8723],
---  [0.9172,0.0644,0.6037,0.643,0.0428,0.4815]]
-
--- test 12:
-
-dftsE := nagDFT seqsE;
-dftsE :: List Vector Float
- 
--- [[1.0788,0.662291,- 0.239146,- 0.578284,0.459192,- 0.438816],
---  [0.857321,1.22614,0.353348,- 0.222169,0.341327,- 1.22908],
---  [1.18245,0.262509,0.674406,0.552278,0.0539906,- 0.478963]]
-
--- test 13:
-
-invdftsE := nagInverseDFT dftsE;
-invdftsE :: List Vector Complex Float
- 
--- [
---   [0.3854, 0.6772 + 0.1424 %i, 0.1138 + 0.6362 %i, 0.6751,
---    0.1138 - 0.6362 %i, 0.6772 - 0.1424 %i]
---   ,
-
---   [0.5417, 0.2983 + 0.8723 %i, 0.1181 + 0.8638 %i, 0.7255,
---    0.1181 - 0.8638 %i, 0.2983 - 0.8723 %i]
---   ,
-
---   [0.9172, 0.0644 + 0.4815 %i, 0.6037 + 0.0428 %i, 0.643,
---    0.6037 - 0.0428 %i, 0.0644 - 0.4815 %i]
---   ]
-
--- test 14:
-
-hdftsD := nagHermitianDFT seqsD;
-map(expand,hdftsD) :: List Vector Complex Float
- 
--- [
---   [1.07373, - 0.104062 - 0.00438406 %i, 0.112554 - 0.373777 %i, - 0.146684,
---    0.112554 + 0.373777 %i, - 0.104062 + 0.00438406 %i]
---   ,
-
---   [1.39609, - 0.0365178 + 0.466584 %i, 0.077955 - 0.0607051 %i, - 0.152072,
---    0.077955 + 0.0607051 %i, - 0.0365178 - 0.466584 %i]
---   ,
-
---   [1.12374, 0.0914068 - 0.050841 %i, 0.393551 + 0.345775 %i, 0.153011,
---    0.393551 - 0.345775 %i, 0.0914068 + 0.050841 %i]
---   ]
-
--- test 15:
-
-(nagInverseDFT hdftsD) :: List Vector Float
- 
--- [[0.3854,0.6772,0.1138,0.6751,0.6362,0.1424],
---  [0.5417,0.2983,0.1181,0.7255,0.8638,0.8723],
---  [0.9172,0.0644,0.6037,0.643,0.0428,0.4815]]
-
--- test 16:
-
-dftsF := nagDFT seqsF;
-dftsF :: List Vector Complex Float
- 
--- [
---   [1.07373 + 1.39609 %i, - 0.570647 - 0.0409019 %i, 0.173259 - 0.295822 %i,
---    - 0.146684 - 0.152072 %i, 0.0518489 + 0.451732 %i,
---    0.362522 - 0.0321337 %i]
---   ,
-
---   [1.12374 + 1.06765 %i, 0.172759 + 0.0385858 %i, 0.418548 + 0.748083 %i,
---    0.153011 + 0.17522 %i, 0.368555 + 0.0565331 %i, 0.0100542 + 0.140268 %i]
---   ,
-
---   [0.909985 + 1.76167 %i, - 0.305418 + 0.0624335 %i,
---    0.407884 - 0.0694786 %i, - 0.078547 + 0.0725049 %i,
---    - 0.119334 + 0.128511 %i, - 0.531409 - 0.433531 %i]
---   ]
-
--- test 17:
-
-invdftsF := nagInverseDFT dftsF ;
-invdftsF :: List Vector Complex Float
- 
--- [
---   [0.3854 + 0.5417 %i, 0.6772 + 0.2983 %i, 0.1138 + 0.1181 %i,
---    0.6751 + 0.7255 %i, 0.6362 + 0.8638 %i, 0.1424 + 0.8723 %i]
---   ,
-
---   [0.9172 + 0.9089 %i, 0.0644 + 0.3118 %i, 0.6037 + 0.3465 %i,
---    0.643 + 0.6198 %i, 0.0428 + 0.2668 %i, 0.4815 + 0.1614 %i]
---   ,
-
---   [0.1156 + 0.6214 %i, 0.0685 + 0.8681 %i, 0.206 + 0.706 %i,
---    0.863 + 0.8652 %i, 0.6967 + 0.919 %i, 0.2792 + 0.3355 %i]
---   ]
-
--- test 18:
-
-nagHermitianInverseDFT dftsE
- 
--- [
---   [0.38540000000000013, 0.67720000000000025, 0.11380000000000001,
---    0.67510000000000014, 0.63620000000000021, 0.14240000000000003]
---   ,
-
---   [0.54170000000000018, 0.29830000000000012, 0.1181, 0.72550000000000014,
---    0.86380000000000023, 0.87230000000000019]
---   ,
-
---   [0.91720000000000035, 0.064399999999999999, 0.60370000000000024,
---    0.64300000000000013, 0.042799999999999991, 0.48150000000000015]
---   ]
-
--- error tests:
-
--- test 19:
-
-nagDFT [vector [0.3854 + 0.5417*%i, 0.6772 + 0.2983*%i,   _
-                0.1138 + 0.1181*%i, 0.6751 + 0.7255*%i,   _
-                0.6362 + 0.8638*%i, 0.1424 + 0.8723*%i],  _
-        vector [0.1156 + 0.6214*%i, 0.0685 + 0.8681*%i,   _
-                0.6967 + 0.9190*%i, 0.2792 + 0.3355*%i]]
- 
--- Error signalled from user code:
---    The data sequences in nagDFT must all have the same length. The 
---    length of sequence 1 is 6 that of sequence 2 is 4.
-
--- test 20:
-
-nagHermitianDFT [vector [0.3854, 0.6751, 0.6362, 0.1424], _
-                 vector [0.5417, 0.7255, 0.8638, 0.8723], _
-                 vector [0.9172, 0.0428, 0.4815]]
- 
--- Error signalled from user code:
---    The data sequences in nagHermitianDFT must all have the same 
---    length. The length of sequence 1 is 4 that of sequence 3 is 3.
-
--- test 21:
-
-badSeqs : List PackedHermitianSequence DoubleFloat
-badSeqs := [pHS [0.3854, 0.1138, 0.6751, 0.6362, 0.1424],         _
-            pHS [0.5417, 0.2983, 0.1181, 0.7255, 0.8638, 0.8723], _
-            pHS [0.9172, 0.0644, 0.6037, 0.6430, 0.0428, 0.4815]];
- 
-nagDFT badSeqs
- 
--- Error signalled from user code:
---    The data sequences in nagDFT must all have the same length. The 
---    length of sequence 1 is 5 that of sequence 2 is 6.
-
-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>>
-
--- To test:
--- sed -ne '1,/^#if NeverAssertThis/d;/#endif/d;p' < ndftip.as > ndftip.input
--- axiom
--- )set nag host <some machine running nagd>
--- )r ndftip.input
-
-#unassert saturn
-
-#include "axiom.as"
-
-DF     ==> DoubleFloat ;
-CDF    ==> Complex DoubleFloat ;
-LDF    ==> List DoubleFloat ;
-LLDF   ==> List LDF ;
-VDF    ==> Vector DoubleFloat ;
-LVDF   ==> List VDF ;
-VCDF   ==> Vector Complex DoubleFloat ;
-LVCDF  ==> List VCDF ;
-MDF    ==> Matrix DoubleFloat ;
-MCDF   ==> Matrix Complex DoubleFloat ;
-INT    ==> Integer ;
-NNI    ==> NonNegativeInteger ;
-RSLT   ==> Result ;
-STRG   ==> String ;
-PHSDF  ==> PackedHermitianSequence DF;
-LPHSDF ==> List PackedHermitianSequence DF;
-
-<<NagDiscreteFourierTransformInterfacePackage>>
-@
-\eject
-\begin{thebibliography}{99}
-\bibitem{1} nothing
-\end{thebibliography}
-\end{document}