Changeset 1555 for trunk/doc/pslib/psLibADD.tex

Timestamp:

Aug 16, 2004, 5:09:16 PM (22 years ago)

Author:

Paul Price

Message:

Added Powell's method.

File:

• trunk/doc/pslib/psLibADD.tex (modified) (9 diffs)

trunk/doc/pslib/psLibADD.tex

@@ -1 @@
-%%% $Id: psLibADD.tex,v 1.25 2004-08-17 01:56:41 eugene Exp $
+%%% $Id: psLibADD.tex,v 1.26 2004-08-17 03:09:16 price Exp $
 \documentclass[panstarrs]{panstarrs}
 
@@ -450 +450 @@
 
 \paragraph{Non-linear Fitting: Levenberg-Marquardt Method}
-
-\TBD{describe LMM for psMinimize and psMinimizeChi2}
 
 For models in which the system of equations defined by the partial
@@ -523 +521 @@
 \end{center}
 
+
+\paragraph{Non-linear fitting: Powell's method}
+
+Powell's method is a type of ``Direction Set'' methods in
+multi-dimensions for finding a local minimum.  Given a starting point
+(the ``best guess'' for the minimum) and a set of direction vectors, a
+direction set method advances in the direction of the vectors,
+determines a new direction vector by some method, and proceeds in this
+manner until the advances along the vectors are smaller than some
+pre-defined tolerance.  Such direction set methods, including Powell's
+Quadratically Convergent method are discussed in NR\S10.5.
+
+We will use for our algorithm the modified version of Powell's
+Quadratically Convergent Method, which is described below, adapted
+from NR.
+
+\begin{enumerate}
+\item Given a function in $N$ dimensions to minimize, $f$, and a best
+  guess for the minimum, point $P$ in $N$ dimensions, take an initial
+  set of $N$ vectors, $v_i$, to be the unit vectors.
+\item Set point $Q = P$.
+\item For each dimension in turn, move $Q$ \textit{only} in the
+  direction $v_i$ to minimize the function of interest.
+\item Set vector $u = Q - P$.
+\item Move $Q$ \textit{only} in the direction $u$
+\item Replace the vector along which the largest minimization was
+  made, $v_{i,\rm max}$, with $u$, except under either of the
+  following circumstances:
+  \begin{itemize}
+  \item If $f_QP \ge f_P$, then there is no point in keeping the new
+    vector, because there is no further minimization to be made in
+    that direction.
+  \item If $2 ( f_P - 2f_Q + f_{QP} ) \left[ ( f_P - f_Q ) -
+  \Delta_{\rm max} \right]^2 \ge ( f_P - f_{QP} )^2 \Delta_{\rm max}$,
+  then either the decrease in the function was not due to any single
+  direction, or we are close to the minimum.
+  \end{itemize}
+  where $f_P = f(P)$, $f_Q = f(Q)$, $f_{QP} = f(2Q - P)$, and
+  $\Delta_{\rm max} \ge 0$ is the magnitude of the minimization made
+  along $v_{i,\rm max}$.
+\item Set $P$ to $Q$.
+\item Return to step 3 until the change in this last move is less
+  than some specified tolerance, or a maximum number of iterations
+  has been reached.
+\end{enumerate}
+
+In regards to minimizing the function only in a particular direction,
+we shall adopt, as NR recommends, bracketing the minimum before
+applying Brent's method, \tbd{which will be specified in detail
+later}.
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -644 +694 @@
 
 Correct time representation is critical in astronomical software.
-PSLib uses the \code{psTime} structure to represent all time values.
-This structure represents a time which is consists of seconds and
-fractions of seconds in a time system defined by the \code{psTimeType}
-element \code{type}.  Two possible time systems are currently
-available: TAI and UTC.  Both are defined in terms of the reference
-epoch 1970-01-01T00:00:00Z, but with minor modifications for
-leap-seconds as needed.  The first represenatation, TAI (International
-Atomic Time), has seconds of uniform length and no leap seconds.  The
-exact zero reference is 1970/01/01,00:00:10 UTC.  The second
-representations is UTC, which has seconds of uniform length and
-leap-seconds as needed to adjust it to remain within 0.9 seconds of
-the Earth's rotation.  It has a zero-point of exactly
-1970/01/01,00:00:00 UTC.
+PSLib uses the \code{psTime} structure to represent all time
+values.  This structure represents a time which is equivalent to TAI
+(International Atomic Time) and has the following properties:
+\begin{itemize}
+\item it represents both seconds and microseconds
+\item the seconds are continuous (no leap seconds)
+\item the zero reference point is \tbd{1970/01/01,00:00:10} UTC.
+\end{itemize}
 
 Julian Day (JD) and Modified Julian Day (MJD) are both continuous time
@@ -713 +758 @@
 \end{verbatim}
 
-The conversion from a time and longitude to local mean sidereal time
-is performed using the SLA Lib function \code{sla_GMST}.  This
-function requires the value $\Delta$ UT1 = UTC - UT1.  The value of
-$\Delta$ UT1 may be determined from the following site in real time:
-
-\code{ftp://maia.usno.navy.mil/ser7/ser7.dat}
-
-In addition, the long-term values may be determined from the table
-found at: \code{ftp://maia.usno.navy.mil/ser7/finals.all}.  See also
-the web page \code{http://maia.usno.navy.mil/}.  The most significant
-accuracy requirements are for the current value when calculating the
-LST.  For this purpose, the table above (\code{ser7.dat}), which
-provides predictions over a 2 month period, must be made available
-locally to PSLib and updated regularly. 
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -862 +892 @@
 \subsubsection{Projections}
 
-We implement three types of projections: {\em zenithal}, {\em
-cylindrical} and {\em pseudocylindrical}, each requiring slightly
-different handling.  Our representations are based on the treatment of
-projections presented by
+The following information is from
 \href{http://www.cv.nrao.edu/fits/documents/wcs/wcs.all.ps}{Greisen \&
-Calabretta (1995, ADASS, 4, 233)}.  In all of these projections, we
-are converting from a spherical coordinate $\alpha,\delta$ to a linear
-(2-D) coordinate $x_p,y_p$.  The projection is defined by the
-projection type, the projection center ($\alpha_p, \delta_p$) and the
-the plate scales in the $x_p$ and $y_p$ directions ($\rho_x,\rho_y$).
-
-In the structure, \code{psProjection}, the projection type is defined
-by the element \code{type}, the projection center $\alpha_p,\delta_p$
-is defined by the elements \code{R,D}, and the plate scales,
-$\rho_x,\rho_y$, are defined by the elements \code{Xs,Ys}.  The plate
-scales are applied independently to the $x$ and $y$ coordinates to
-convert them to the corresponding linear units (ie, pixels):
-%
-\begin{eqnarray}
-x_p & = & \rho_x x \\
-y_p & = & \rho_y y \\
-\end{eqnarray}
-% 
-In the discussions below, we ignore this last step (or first step,
-depending on the direction of the conversion).
-
-\paragraph{Zenithal Projections}
-
-The {\em zenithal} projections are defined relative to a set of
-spherical coordinates with pole at the center of the projection
-($\alpha_p, \delta_p$), and which thus represents a coordinate system
-rotated relative to the coordinate system of $\alpha, \delta$.  In
-this spherical coordinate system, the coordinate of longitude is
-labeled $\phi$, and has domain of $-\pi < \phi \le \pi$, while the
-latitude, measured from the pole, is labeled $\theta$ and has domain
-$0 \le \theta \le \pi$.
-
-For an arbitrary projection center, it is necessary to convert the
-spherical coordinates to be projected ($\alpha,\delta$) to the
-projection spherical coordinate system coordinates ($\phi, \theta$).
-In practice, we construct the following useful trigonometric
-relationships between $\phi$ and $\theta$ which may be employed in the
-equations of $x,y$ below:
-%
-\begin{eqnarray}
-\sin \theta           & = & \sin \delta \sin \delta_p + \cos \delta \cos \delta_p \cos (\alpha - \alpha_p) \\
-\cos \theta \cos \phi & = & \sin \delta \cos \delta_p - \cos \delta \sin \delta_p \cos (\alpha - \alpha_p) \\
-\cos \theta \sin \phi & = & - \cos \delta \sin (\alpha - \alpha_p)
-\end{eqnarray}
-%
-For the inverse transformations, the equivalent relationships are:
-%
-\begin{eqnarray}
-\sin \delta                          & = & \sin \theta \sin \delta_p + \cos \theta \cos \delta_p \cos \phi \\
-\cos \delta \cos (\alpha - \alpha_p) & = & \sin \theta \cos \delta_p - \cos \theta \sin \delta_p \cos \phi \\
-\cos \delta \sin (\alpha - \alpha_p) & = & - \cos \theta \sin (\phi - \phi_p)
-\end{eqnarray}
-%
-For zenithal projections, the linear coordinates are related to
-$\phi,\theta$ by:
-%
-\begin{eqnarray}
-x & = & R_\theta \sin \phi \\
-y & = & -R_\theta \cos \phi
-\end{eqnarray}
-%
-and the inverse:
-%
-\begin{eqnarray}
-R_\theta & = & \sqrt{x^2 + y^2} \\
-\phi     & = & {\rm arg} (-y,x)
-\end{eqnarray}
-%
-The coordinates $x,y$ above are defined to be in angular units (ie,
-radians).  
-
-From these relationships, we can calculate $\alpha, \delta$ as:
-%
-\begin{eqnarray}
-\alpha - \alpha_p & = & \arctan (\sin \alpha, \cos \alpha) \\
-\delta            & = & \arcsin (\sin \delta) \\
-\end{eqnarray}
-%
-Note that if $(x,y) = (0,0)$, then $\alpha = \alpha_p, \delta = \delta_p$.
-
-\subparagraph{Gnomonic}
-
-The Gnomonic projection (``TAN'') is a zenithal projection with
-$R_\theta = \cot \theta$.  The resulting relationships for $(x,y)$ and
-for $\sin \theta, \cos \theta$ are:
-
-\begin{eqnarray}
-x           & = & \frac{\cos \theta \sin \phi}{\sin \theta} \\
-y           & = & \frac{-\cos \theta \cos \phi}{\sin \theta} \\
-\sin \theta & = & \zeta / \sqrt{1 + \zeta^2} \\
-\cos \theta & = & 1 / \sqrt{1 + \zeta^2} \\
-\end{eqnarray}
-
-where $\zeta = 1 / R_\theta$.
-
-\subparagraph{Orthographic}
-
-The Orthographic projection (``SIN'') is a zenithal projection with
-$R_\theta = \cos \theta$.  The resulting relationships for $(x,y)$ and
-for $\sin \theta, \cos \theta$ are:
-
-\begin{eqnarray}
-x           & = & \cos \theta \sin \phi \\
-y           & = & -\cos \theta \cos \phi \\
-\sin \theta & = & \sqrt{1 - R_\theta^2} \\
-\cos \theta & = & R_\theta \\
-\end{eqnarray}
-
-\paragraph{Cylindrical and Pseudocylindrical Projections}
-
-The {\em cylindrical} and {\em pseudocylindrical} projections are
-defined relative to a set of cylindrical coordinates whose pole is
-coincident with the pole of the spherical coordinates.  These
-projections are particularly used for full-sky representations, and
-are only defined for projection centers with $\delta_p = 0$.  In this
-spherical coordinate system, the coordinate of longitude is labeled
-$\phi$, and has domain of $-\pi < \phi \le \pi$, while the latitude,
-measured from the pole, is labeled $\theta$ and has domain $0 \le
-\theta \le \pi$.  The projection center longitude, $\alpha_p$
-corresponds to $\phi = 0$, thus the value of $\phi$ is determined as
-$\alpha - \alpha_p$ for all such projections.
-
-\subparagraph{Cartesian}
-
-The Cartesian projection (``CAR'') is a very simple cylindrical
-projection with the following relationships between $x,y$ and
-$\phi,\theta$:
+Calabretta (1995, ADASS, 4, 233)}.
+
+Let the latitude be $\phi$ and the longitude $\theta$.  The domains of
+these are $-\pi < \phi \le \pi$ and $-\pi/2 \le \theta \le \pi/2$.
+
+For zenithal projections (e.g.\ Gnomonic and Orthographic) the
+following hold:
+
+\begin{eqnarray}
+x & = & R \sin (\phi) \\
+y & = & -R \cos (\phi)
+\end{eqnarray}
+
+and
+
+\begin{eqnarray}
+R & = & \sqrt{x^2 + y^2} \\
+\phi & = & {\rm arg} (-y,x)
+\end{eqnarray}
+
+\paragraph{Gnomonic}
+
+The Gnomonic projection (``TAN'') is a zenithal projection.
+
+\begin{eqnarray}
+R & = & \cot (\theta) 180^\circ/\pi \\
+\theta & = & \arctan (180^\circ/(\pi R))
+\end{eqnarray}
+
+\paragraph{Orthographic}
+
+The Orthographic projection (``SIN'') is a zenithal projection.
+
+\begin{eqnarray}
+R & = & \cos (\theta) 180^\circ/\pi \\
+\theta & = & \arccos (\pi R / 180^\circ)
+\end{eqnarray}
+
+\paragraph{Cartesian}
+
+The Cartesian projection (``CAR'') is a very simple cylindrical projection.
 
 \begin{eqnarray}
@@ -1000 +941 @@
 \end{eqnarray}
 
-\subparagraph{Mercator}
+\paragraph{Mercator}
 
 The Mercator projection (``MER'') is a cylindrical projection.
@@ -1006 +947 @@
 \begin{eqnarray}
 x & = & \phi \\
-y & = & \ln \left( \tan (\pi/4 + \theta/2) \right) \\
-{\rm and}\hspace{1cm} \theta & = & 2 \arctan \left( e^y \right) - \pi/2
-\end{eqnarray}
-
-\subparagraph{Hammer-Aitoff}
-
-The Hammer-Aitoff projection(``AIT'') is a pseudocylindrical projection, and is defined:
-
-\begin{eqnarray}
-x & = & 2 \zeta \cos \theta \sin \frac{\phi}{2} \\
-y & = & \zeta \sin \theta \\
-{\rm where}\hspace{1cm} \zeta^{-1} & \equiv & \sqrt{\frac{1}{2}\left(1 + \cos \theta \cos \frac{\phi}{2} \right)}
+y & = & \ln \left( \tan (45^\circ + \theta/2) \right) 180^\circ/\pi \\
+{\rm and}\hspace{1cm} \theta & = & 2 \arctan \left( e^{y\pi/180^\circ} \right) - 90^\circ
+\end{eqnarray}
+
+\paragraph{Hammer-Aitoff}
+
+The Hammer-Aitoff projection is a general projection, and is defined:
+
+\begin{eqnarray}
+x & = & 2 \alpha \cos (\theta) \sin (\phi/2) \\
+y & = & \alpha \sin \theta \\
+{\rm where}\hspace{1cm} \alpha^{-1} & \equiv & (180^\circ/\pi) \sqrt{\left(1 + \cos (\theta) \cos (\phi/2) \right) / 2}
 \end{eqnarray}
 
@@ -1023 +964 @@
 
 \begin{eqnarray}
-\phi & = & 2 {\rm \arctan} (2z^2 - 1, x z) \\
-\theta & = & \arcsin (yz) \\
-{\rm where}\hspace{1cm} z & \equiv & \sqrt{1 - (x/2)^2 - y^2}
-\end{eqnarray}
-
-\subparagraph{Parabolic}
-
-The Parabolic projection (``PAR'') is a pseudocylindrical projection, and is defined:
-
-\begin{eqnarray}
-x & = & \phi \left( 2 \cos \frac{2 \theta}{3} - 1 \right) \\
-y & = & \pi \sin \frac{\theta}{3} \\
-\end{eqnarray}
-
-And in reverse:
-
-\begin{eqnarray}
-\theta & = & 3 \sin^{-1} \rho \\
-\phi   & = & \frac{x}{1 - 4\rho^2} \\
-{\rm where}\hspace{1cm} \rho & \equiv & y/\pi \\
+\phi & = & 2 {\rm arg} (2z^2 - 1, xz \pi/360^\circ) \\
+\theta & = & \arcsin (yz\pi/180^\circ) \\
+{\rm where}\hspace{1cm} z & \equiv & \sqrt{1 - (x\pi/720^\circ)^2 - (y\pi/360^\circ)^2}
 \end{eqnarray}