From f6f645092daa7054e3c2ed695c7f0c77acd69497 Mon Sep 17 00:00:00 2001
From: Tim Daly
Date: Sun, 20 Mar 2016 23:32:49 -0400
Subject: [PATCH] books/bookvolbib Add Brow01, the McCallum CAD projection
Goal: Axiom Algebra
@article{Brow01,
author="Brown, Christopher W.",
title="The McCallum projection, lifting, and order-invariance",
year="2001",
url="http://www.usna.edu/Users/cs/wcbrown/research/MOTS2001.1.ps.gz",
paper = "Brow01.pdf",
abstract =
"The McCallum Projection for Cylindrical Algebraic Decomposition (CAD)
produces a smaller projection factor set than previous projections,
however it does not always produce a sign-invariant CAD for the set of
input polynomials. Problems may arise when a ($k+1$)-level projection
factor vanishes identically over a k-level cell. According to
McCallum's paper, when this happens (and $k+1$ is not the highest
level in the CAD) we do not know whether the projection is valid,
i.e. whether or not a sign-invariant CAD for the set of input
polynomials will be produced when lifting is performed in the usual
way. When the $k$-level cell in question has dimension 0, McCallum
suggests a modification of the lifting method that will ensure the
validity of his projection, although to my knowledge this has never
been implemented.
In this paper we give easily computable criteria that often allow us
to conclude that McCallum's projection is valid even though a
projection factor vanishes identically over a cell. We also improve on
McCallum's modified lifting method.
We've incorporated the ideas contained in the paper into QEPCAD, the
most complete implementation of CAD. When McCallum's projection is
invalid because of a projection factor not being order-invariant over a
region on which it vanishes identically, at least a warning message
ought to be issued. Currently, QEPCAD may print warning messages that
are not needed, and may fail to print warning messages when they are
needed. Our implementation in QEPCAD ensures that warning messages are
printed when needed, and reduces the number of times warning messages
are printed when not needed. Neither McCallum's modified lifting
method nor our improvement of it have been implemented in QEPCAD. The
design of the system would make implementing such a feature quite
difficult."
}
---
books/bookvolbib.pamphlet | 45 +++++++++++++++++++++++++++++
changelog | 2 +
patch | 62 ++++++++++++++++++++++++---------------
src/axiom-website/patches.html | 2 +
4 files changed, 87 insertions(+), 24 deletions(-)
diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index 7156519..22fa31f 100644
--- a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ -6779,6 +6779,51 @@ Proc ISSAC 97 pp172-175 (1997)
\end{chunk}
+\index{Brown, Christopher W.}
+\begin{chunk}{axiom.bib}
+@article{Brow01,
+ author="Brown, Christopher W.",
+ title="The McCallum projection, lifting, and order-invariance",
+ year="2001",
+ url="http://www.usna.edu/Users/cs/wcbrown/research/MOTS2001.1.ps.gz",
+ paper = "Brow01.pdf",
+ abstract =
+ "The McCallum Projection for Cylindrical Algebraic Decomposition (CAD)
+ produces a smaller projection factor set than previous projections,
+ however it does not always produce a sign-invariant CAD for the set of
+ input polynomials. Problems may arise when a ($k+1$)-level projection
+ factor vanishes identically over a k-level cell. According to
+ McCallum's paper, when this happens (and $k+1$ is not the highest
+ level in the CAD) we do not know whether the projection is valid,
+ i.e. whether or not a sign-invariant CAD for the set of input
+ polynomials will be produced when lifting is performed in the usual
+ way. When the $k$-level cell in question has dimension 0, McCallum
+ suggests a modification of the lifting method that will ensure the
+ validity of his projection, although to my knowledge this has never
+ been implemented.
+
+ In this paper we give easily computable criteria that often allow us
+ to conclude that McCallum's projection is valid even though a
+ projection factor vanishes identically over a cell. We also improve on
+ McCallum's modified lifting method.
+
+ We've incorporated the ideas contained in the paper into QEPCAD, the
+ most complete implementation of CAD. When McCallum's projection is
+ invalid because of a projection factor not being order-invariant over a
+ region on which it vanishes identically, at least a warning message
+ ought to be issued. Currently, QEPCAD may print warning messages that
+ are not needed, and may fail to print warning messages when they are
+ needed. Our implementation in QEPCAD ensures that warning messages are
+ printed when needed, and reduces the number of times warning messages
+ are printed when not needed. Neither McCallum's modified lifting
+ method nor our improvement of it have been implemented in QEPCAD. The
+ design of the system would make implementing such a feature quite
+ difficult."
+
+}
+
+\end{chunk}
+
\index{Chyzak, Fr\'ed\'eric}
\index{Davenport, James H.}
\index{Koutschan, Christoph}
diff --git a/changelog b/changelog
index ce5bcda..8ddfd0d 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20160320 tpd src/axiom-website/patches.html 20160320.04.tpd.patch
+20160320 tpd books/bookvolbib Add Brow01, the McCallum CAD projection
20160320 tpd src/axiom-website/patches.html 20160320.03.tpd.patch
20160320 tpd books/bookvolbib Add Fern92 Accurate Singular Values
20160320 tpd src/axiom-website/patches.html 20160320.02.tpd.patch
diff --git a/patch b/patch
index e14b1c8..0e4da59 100644
--- a/patch
+++ b/patch
@@ -1,29 +1,43 @@
-books/bookvolbib Add Fern92 Accurate Singular Values
+books/bookvolbib Add Brow01, the McCallum CAD projection
-Goal: Axiom Numerics
+Goal: Axiom Algebra
-@techreport{Fern92,
- author = "Fernando, K. Vince and Parlett, Beresford N.",
- title = "Accurate Singular Values and Differential qd Algorithms",
- year = "1992",
- institution = "University of California, Berkeley",
- type = "Technical Report",
- number = "PAM-554",
- paper = "Fern92.pdf",
- url = "http://www.dtic.mil/dtic/tr/fulltext/u2/a256582.pdf",
+@article{Brow01,
+ author="Brown, Christopher W.",
+ title="The McCallum projection, lifting, and order-invariance",
+ year="2001",
+ url="http://www.usna.edu/Users/cs/wcbrown/research/MOTS2001.1.ps.gz",
+ paper = "Brow01.pdf",
abstract =
- "We have discovered a new implementaiton of the qd algorithm that has
- a far wider domain of stability than Rutishauser's version. Our
- algorithm was developed from an examination of the LR-Cholesky
- transformation and can be adapted to parallel computation in stark
- contrast to traditional qd. Our algorithm also yields useful a
- posteriori upper and lower bounds on the smallest singular value of a
- bidiagonal matrix.
+ "The McCallum Projection for Cylindrical Algebraic Decomposition (CAD)
+ produces a smaller projection factor set than previous projections,
+ however it does not always produce a sign-invariant CAD for the set of
+ input polynomials. Problems may arise when a ($k+1$)-level projection
+ factor vanishes identically over a k-level cell. According to
+ McCallum's paper, when this happens (and $k+1$ is not the highest
+ level in the CAD) we do not know whether the projection is valid,
+ i.e. whether or not a sign-invariant CAD for the set of input
+ polynomials will be produced when lifting is performed in the usual
+ way. When the $k$-level cell in question has dimension 0, McCallum
+ suggests a modification of the lifting method that will ensure the
+ validity of his projection, although to my knowledge this has never
+ been implemented.
- The zero-shift bidiagonal QR of Demmel and Kahan computes the smallest
- singlar values to maximal relative accuracy and the others to maximal
- absolute accuracy with little or no degradation in efficiency when
- compared with the LINPACK code. Our algorithm obtains maximal relative
- accuracy for all the singluar values and runs at least four times
- faster than the LINPACK code."
+ In this paper we give easily computable criteria that often allow us
+ to conclude that McCallum's projection is valid even though a
+ projection factor vanishes identically over a cell. We also improve on
+ McCallum's modified lifting method.
+
+ We've incorporated the ideas contained in the paper into QEPCAD, the
+ most complete implementation of CAD. When McCallum's projection is
+ invalid because of a projection factor not being order-invariant over a
+ region on which it vanishes identically, at least a warning message
+ ought to be issued. Currently, QEPCAD may print warning messages that
+ are not needed, and may fail to print warning messages when they are
+ needed. Our implementation in QEPCAD ensures that warning messages are
+ printed when needed, and reduces the number of times warning messages
+ are printed when not needed. Neither McCallum's modified lifting
+ method nor our improvement of it have been implemented in QEPCAD. The
+ design of the system would make implementing such a feature quite
+ difficult."
}
diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html
index 677da42..bff2cd6 100644
--- a/src/axiom-website/patches.html
+++ b/src/axiom-website/patches.html
@@ -5242,6 +5242,8 @@ books/bookvolbib Sutt13, Computing the Complete CS Decomposition

books/bookvolbib Add Demm88 Computing Small Singular Values

20160320.03.tpd.patch
books/bookvolbib Add Fern92 Accurate Singular Values

+20160320.04.tpd.patch
+books/bookvolbib Add Brow01, the McCallum CAD projection

--
1.7.5.4