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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} |