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revision 1.18 by afe, Thu May 6 15:21:01 2004 UTC revision 1.24 by molod, Mon Jul 18 20:45:27 2005 UTC
# Line 10  Line 10 
10  %%    o automatically inserted at \section{Reference}  %%    o automatically inserted at \section{Reference}
11    
12    
13  \section{exch2: Extended Cubed Sphere \mbox{Topology}}  \subsection{exch2: Extended Cubed Sphere \mbox{Topology}}
14  \label{sec:exch2}  \label{sec:exch2}
15    
16    
17  \subsection{Introduction}  \subsubsection{Introduction}
18    
19  The \texttt{exch2} package extends the original cubed sphere topology  The \texttt{exch2} package extends the original cubed sphere topology
20  configuration to allow more flexible domain decomposition and  configuration to allow more flexible domain decomposition and
# Line 41  of these files with alternate topologies Line 41  of these files with alternate topologies
41  \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}  \file{utils/exch2/code-mods} along with the appropriate \file{SIZE.h}
42  file for single-processor execution.  file for single-processor execution.
43    
44  \subsection{Invoking exch2}  \subsubsection{Invoking exch2}
45    
46  To use exch2 with the cubed sphere, the following conditions must be  To use exch2 with the cubed sphere, the following conditions must be
47  met: \\  met: \\
# Line 54  $\bullet$ The exch2 package is included Line 54  $\bullet$ The exch2 package is included
54    
55  $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and  $\bullet$ An example of \file{W2\_EXCH2\_TOPOLOGY.h} and
56    \file{w2\_e2setup.F} must reside in a directory containing files    \file{w2\_e2setup.F} must reside in a directory containing files
57    symbolically linked when \file{genmake2} runs.  The safest place to    symbolically linked by the \file{genmake2} script.  The safest place to
58    put these is the directory indicated in the \code{-mods=DIR} command    put these is the directory indicated in the \code{-mods=DIR} command
59    line modifier (typically \file{../code}), or the build directory.    line modifier (typically \file{../code}), or the build directory.
60    The default versions of these files reside in \file{pkg/exch2} and    The default versions of these files reside in \file{pkg/exch2} and
# Line 77  $\bullet$ As always when compiling MITgc Line 77  $\bullet$ As always when compiling MITgc
77    \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain    \file{W2\_EXCH2\_TOPOLOGY.h} and \file{w2\_e2setup.F}.  Domain
78    decomposition issues particular to exch2 are addressed in Section    decomposition issues particular to exch2 are addressed in Section
79    \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}    \ref{sec:topogen} \sectiontitle{Generating Topology Files for exch2}
80    and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and MPI}; a more    and \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and Multiprocessing}; a more
81    general background on the subject relevant to MITgcm is presented in    general background on the subject relevant to MITgcm is presented in
82    Section \ref{sect:specifying_a_decomposition}    Section \ref{sect:specifying_a_decomposition}
83    \sectiontitle{Specifying a decomposition}.\\    \sectiontitle{Specifying a decomposition}.\\
# Line 96  verification/hs94.cs-32x32x5 Line 96  verification/hs94.cs-32x32x5
96    
97    
98    
99  \subsection{Generating Topology Files for exch2}  \subsubsection{Generating Topology Files for exch2}
100  \label{sec:topogen}  \label{sec:topogen}
101    
102  Alternate cubed sphere topologies may be created using the Matlab  Alternate cubed sphere topologies may be created using the Matlab
# Line 107  from the Matlab prompt (there are no par Line 107  from the Matlab prompt (there are no par
107  exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and  exch2 topology files \file{W2\_EXCH2\_TOPOLOGY.h} and
108  \file{w2\_e2setup.F} in the working directory and displays a figure of  \file{w2\_e2setup.F} in the working directory and displays a figure of
109  the topology via Matlab -- figures \ref{fig:6tile}, \ref{fig:12tile},  the topology via Matlab -- figures \ref{fig:6tile}, \ref{fig:12tile},
110  and \ref{fig:24tile} are examples.  The other m-files in the directory are  and \ref{fig:24tile} are examples of the generated diagrams.  The other
111  subroutines of \file{driver.m} and should not be run ``bare'' except  m-files in the directory are
112    subroutines called from \file{driver.m} and should not be run ``bare'' except
113  for development purposes. \\  for development purposes. \\
114    
115  The parameters that determine the dimensions and topology of the  The parameters that determine the dimensions and topology of the
116  generated configuration are \code{nr}, \code{nb}, \code{ng},  generated configuration are \code{nr}, \code{nb}, \code{ng},
117  \code{tnx} and \code{tny}, and all are assigned early in the script. \\  \code{tnx} and \code{tny}, and all are assigned early in the script. \\
118    
119  The first three determine the size of the subdomains and  The first three determine the height and width of the subdomains and
120  hence the size of the overall domain.  Each one determines the number  hence the size of the overall domain.  Each one determines the number
121  of grid points, and therefore the resolution, along the subdomain  of grid points, and therefore the resolution, along the subdomain
122  sides in a ``great circle'' around each the three spatial axes of the cube.  At the time  sides in a ``great circle'' around each the three spatial axes of the cube.  At the time
123  of this writing MITgcm requires these three parameters to be equal,  of this writing MITgcm requires these three parameters to be equal,
124  but they provide for future releases  to accomodate different  but they provide for future releases  to accomodate different
125  resolutions around the axes to allow (for example) greater resolution  resolutions around the axes to allow subdomains with differing resolutions.\\
 around the equator.\\  
126    
127  The parameters \code{tnx} and \code{tny} determine the width and height of  The parameters \code{tnx} and \code{tny} determine the width and height of
128  the tiles into which the subdomains are decomposed, and must evenly  the tiles into which the subdomains are decomposed, and must evenly
# Line 182  by their tile number in the topology, se Line 182  by their tile number in the topology, se
182    
183    
184    
185  \subsection{exch2, SIZE.h, and multiprocessing}  \subsubsection{exch2, SIZE.h, and Multiprocessing}
186  \label{sec:exch2mpi}  \label{sec:exch2mpi}
187    
188  Once the topology configuration files are created, the Fortran  Once the topology configuration files are created, the Fortran
# Line 190  Once the topology configuration files ar Line 190  Once the topology configuration files ar
190  Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying  Section \ref{sect:specifying_a_decomposition} \sectiontitle{Specifying
191  a decomposition} provides a general description of domain  a decomposition} provides a general description of domain
192  decomposition within MITgcm and its relation to \file{SIZE.h}. The  decomposition within MITgcm and its relation to \file{SIZE.h}. The
193  current section specifies certain constraints the exch2 package  current section specifies constraints that the exch2 package
194  imposes as well as describes how to enable parallel execution with  imposes and describes how to enable parallel execution with
195  MPI. \\  MPI. \\
196    
197  As in the general case, the parameters \varlink{sNx}{sNx} and  As in the general case, the parameters \varlink{sNx}{sNx} and
# Line 207  levels in the model.\\ Line 207  levels in the model.\\
207  The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},  The parameters \varlink{nSx}{nSx}, \varlink{nSy}{nSy},
208  \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of  \varlink{nPx}{nPx}, and \varlink{nPy}{nPy} relate to the number of
209  tiles and how they are distributed on processors.  When using exch2,  tiles and how they are distributed on processors.  When using exch2,
210  the tiles are stored in a single dimension, and so  the tiles are stored in the $x$ dimension, and so
211  \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as  \code{\varlink{nSy}{nSy}=1} in all cases.  Since the tiles as
212  configured by exch2 cannot be split up accross processors without  configured by exch2 cannot be split up accross processors without
213  regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well. \\  regenerating the topology, \code{\varlink{nPy}{nPy}=1} as well. \\
# Line 215  regenerating the topology, \code{\varlin Line 215  regenerating the topology, \code{\varlin
215  The number of tiles MITgcm allocates and how they are distributed  The number of tiles MITgcm allocates and how they are distributed
216  between processors depends on \varlink{nPx}{nPx} and  between processors depends on \varlink{nPx}{nPx} and
217  \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per  \varlink{nSx}{nSx}.  \varlink{nSx}{nSx} is the number of tiles per
218  processor and \varlink{nPx}{nPx} the number of processors.  The total  processor and \varlink{nPx}{nPx} is the number of processors.  The
219  number of tiles in the topology minus those listed in  total number of tiles in the topology minus those listed in
220  \file{blanklist.txt} must equal \code{nSx*nPx}. \\  \file{blanklist.txt} must equal \code{nSx*nPx}.  Note that in order to
221    obtain maximum usage from a given number of processors in some cases,
222    this restriction might entail sharing a processor with a tile that
223    would otherwise be excluded because it is topographically outside of
224    the domain and therefore in \file{blanklist.txt}.  For example,
225    suppose you have five processors and a domain decomposition of
226    thirty-six tiles that allows you to exclude seven tiles.  To evenly
227    distribute the remaining twenty-nine tiles among five processors, you
228    would have to run one ``dummy'' tile to make an even six tiles per
229    processor.  Such dummy tiles are \emph{not} listed in
230    \file{blanklist.txt}.\\
231    
232    
233  The following is an example of \file{SIZE.h} for the twelve-tile  The following is an example of \file{SIZE.h} for the twelve-tile
234  configuration illustrated in figure \ref{fig:12tile} running on  configuration illustrated in figure \ref{fig:12tile} running on
# Line 260  figure \ref{fig:24tile} running on six p Line 271  figure \ref{fig:24tile} running on six p
271    
272    
273    
274  \subsection{Key Variables}  \subsubsection{Key Variables}
275    
276  The descriptions of the variables are divided up into scalars,  The descriptions of the variables are divided up into scalars,
277  one-dimensional arrays indexed to the tile number, and two and  one-dimensional arrays indexed to the tile number, and two and
# Line 270  scalars are common to every part of the Line 281  scalars are common to every part of the
281  arrays to individual tiles, and the arrays indexed by tile and  arrays to individual tiles, and the arrays indexed by tile and
282  neighbor to relationships between tiles and their neighbors. \\  neighbor to relationships between tiles and their neighbors. \\
283    
284  \subsubsection{Scalars}  Scalars:
285    
286  The number of tiles in a particular topology is set with the parameter  The number of tiles in a particular topology is set with the parameter
287  \code{NTILES}, and the maximum number of neighbors of any tiles by  \code{NTILES}, and the maximum number of neighbors of any tiles by
# Line 287  setup of six tiles (Fig. \ref{fig:6tile} Line 298  setup of six tiles (Fig. \ref{fig:6tile}
298  topology of twenty-four square tiles, four per subdomain (as in figure  topology of twenty-four square tiles, four per subdomain (as in figure
299  \ref{fig:24tile}), will have \code{exch2\_domain\_nxt=12} and  \ref{fig:24tile}), will have \code{exch2\_domain\_nxt=12} and
300  \code{exch2\_domain\_nyt=2}.  Note that these parameters express the  \code{exch2\_domain\_nyt=2}.  Note that these parameters express the
301  tile layout to allow global data files that are tile-layout-neutral  tile layout in order to allow global data files that are tile-layout-neutral.
302  and have no bearing on the internal storage of the arrays.  The tiles  They have no bearing on the internal storage of the arrays.  The tiles
303  are stored internally in a range from \code{(1:\varlink{bi}{bi})} the  are stored internally in a range from \code{\varlink{bi}{bi}=(1:NTILES)} in the
304  $x$ axis, and the $y$ axis variable \varlink{bj}{bj} is assumed to  $x$ axis, and the $y$ axis variable \varlink{bj}{bj} is assumed to
305  equal \code{1} throughout the package. \\  equal \code{1} throughout the package. \\
306    
307  \subsubsection{Arrays Indexed to Tile Number}  Arrays indexed to tile number:
308    
309  The following arrays are of length \code{NTILES} and are indexed to  The following arrays are of length \code{NTILES} and are indexed to
310  the tile number, which is indicated in the diagrams with the notation  the tile number, which is indicated in the diagrams with the notation
# Line 303  The arrays \varlink{exch2\_tnx}{exch2_tn Line 314  The arrays \varlink{exch2\_tnx}{exch2_tn
314  \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of  \varlink{exch2\_tny}{exch2_tny} express the $x$ and $y$ dimensions of
315  each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and  each tile.  At present for each tile \texttt{exch2\_tnx=sNx} and
316  \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in  \texttt{exch2\_tny=sNy}, as assigned in \file{SIZE.h} and described in
317  section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and  Section \ref{sec:exch2mpi} \sectiontitle{exch2, SIZE.h, and
318  multiprocessing}.  Future releases of MITgcm may allow varying tile  Multiprocessing}.  Future releases of MITgcm may allow varying tile
319  sizes. \\  sizes. \\
320    
321  The location of the tiles' Cartesian origin within a subdomain are  The arrays \varlink{exch2\_tbasex}{exch2_tbasex} and
322  determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and  \varlink{exch2\_tbasey}{exch2_tbasey} determine the tiles'
323  \varlink{exch2\_tbasey}{exch2_tbasey}.  These variables are used to  Cartesian origin within a subdomain  
324  relate the location of the edges of different tiles to each other.  As  and locate the edges of different tiles relative to each other.  As
325  an example, in the default six-tile topology (Fig. \ref{fig:6tile})  an example, in the default six-tile topology (Fig. \ref{fig:6tile})
326  each index in these arrays is set to \code{0} since a tile occupies  each index in these arrays is set to \code{0} since a tile occupies
327  its entire subdomain.  The twenty-four-tile case discussed above will  its entire subdomain.  The twenty-four-tile case discussed above will
328  have values of \code{0} or \code{16}, depending on the quadrant the  have values of \code{0} or \code{16}, depending on the quadrant of the
329  tile falls within the subdomain.  The elements of the arrays  tile within the subdomain.  The elements of the arrays
330  \varlink{exch2\_txglobalo}{exch2_txglobalo} and  \varlink{exch2\_txglobalo}{exch2_txglobalo} and
331  \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to  \varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to
332  \varlink{exch2\_tbasex}{exch2_tbasex} and  \varlink{exch2\_tbasex}{exch2_tbasex} and
333  \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the  \varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tile edges within the
334  global address space, similar to that used by global output and input  global address space, similar to that used by global output and input
335  files. \\  files. \\
336    
# Line 328  the subdomain of each tile, in a range \ Line 339  the subdomain of each tile, in a range \
339  standard cube topology and indicated by \textbf{\textsf{f}}$n$ in  standard cube topology and indicated by \textbf{\textsf{f}}$n$ in
340  figures \ref{fig:12tile} and  figures \ref{fig:12tile} and
341  \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}  \ref{fig:24tile}. \varlink{exch2\_nNeighbours}{exch2_nNeighbours}
342  contains a count of the neighboring tiles each tile has, and is used  contains a count of the neighboring tiles each tile has, and sets
343  for setting bounds for looping over neighboring tiles.  the bounds for looping over neighboring tiles.
344  \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each  \varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each
345  tile, and is used in interprocess communication.  \\  tile, and is used in interprocess communication.  \\
346    
# Line 338  The arrays \varlink{exch2\_isWedge}{exch Line 349  The arrays \varlink{exch2\_isWedge}{exch
349  \varlink{exch2\_isEedge}{exch2_isEedge},  \varlink{exch2\_isEedge}{exch2_isEedge},
350  \varlink{exch2\_isSedge}{exch2_isSedge}, and  \varlink{exch2\_isSedge}{exch2_isSedge}, and
351  \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the  \varlink{exch2\_isNedge}{exch2_isNedge} are set to \code{1} if the
352  indexed tile lies on the respective edge of a subdomain, \code{0} if  indexed tile lies on the edge of its subdomain, \code{0} if
353  not.  The values are used within the topology generator to determine  not.  The values are used within the topology generator to determine
354  the orientation of neighboring tiles, and to indicate whether a tile  the orientation of neighboring tiles, and to indicate whether a tile
355  lies on the corner of a subdomain.  The latter case requires special  lies on the corner of a subdomain.  The latter case requires special
# Line 346  exchange and numerical handling for the Line 357  exchange and numerical handling for the
357  corners of the cube. \\  corners of the cube. \\
358    
359    
360  \subsubsection{Arrays Indexed to Tile Number and Neighbor}  Arrays Indexed to Tile Number and Neighbor:
361    
362  The following arrays have vectors of length \code{MAX\_NEIGHBOURS} and  The following arrays have vectors of length \code{MAX\_NEIGHBOURS} and
363  \code{NTILES} and describe the orientations between the the tiles. \\  \code{NTILES} and describe the orientations between the the tiles. \\
# Line 371  This provides a back-reference from the Line 382  This provides a back-reference from the
382  The arrays \varlink{exch2\_pi}{exch2_pi} and  The arrays \varlink{exch2\_pi}{exch2_pi} and
383  \varlink{exch2\_pj}{exch2_pj} specify the transformations of indices  \varlink{exch2\_pj}{exch2_pj} specify the transformations of indices
384  in exchanges between the neighboring tiles.  These transformations are  in exchanges between the neighboring tiles.  These transformations are
385  necessary in exchanges between subdomains because the array index in  necessary in exchanges between subdomains because a horizontal dimension
386  one dimension may map to the other index in an adjacent subdomain, and  in one subdomain
387  may be have its indexing reversed. This swapping arises from the  may map to other horizonal dimension in an adjacent subdomain, and
388    may also have its indexing reversed. This swapping arises from the
389  ``folding'' of two-dimensional arrays into a three-dimensional  ``folding'' of two-dimensional arrays into a three-dimensional
390  cube. \\  cube. \\
391    
# Line 381  The dimensions of \code{exch2\_pi(t,N,T) Line 393  The dimensions of \code{exch2\_pi(t,N,T)
393  are the neighbor ID \code{N} and the tile number \code{T} as explained  are the neighbor ID \code{N} and the tile number \code{T} as explained
394  above, plus a vector of length \code{2} containing transformation  above, plus a vector of length \code{2} containing transformation
395  factors \code{t}.  The first element of the transformation vector  factors \code{t}.  The first element of the transformation vector
396  holds the factor to multiply the index in the same axis, and the  holds the factor to multiply the index in the same dimension, and the
397  second element holds the the same for the orthogonal index.  To  second element holds the the same for the orthogonal dimension.  To
398  clarify, \code{exch2\_pi(1,N,T)} holds the mapping of the $x$ axis  clarify, \code{exch2\_pi(1,N,T)} holds the mapping of the $x$ axis
399  index of tile \code{T} to the $x$ axis of tile \code{T}'s neighbor  index of tile \code{T} to the $x$ axis of tile \code{T}'s neighbor
400  \code{N}, and \code{exch2\_pi(2,N,T)} holds the mapping of \code{T}'s  \code{N}, and \code{exch2\_pi(2,N,T)} holds the mapping of \code{T}'s
# Line 397  the arrays for all tile neighbors on the Line 409  the arrays for all tile neighbors on the
409  \code{(1,0)}, since all tiles on the same subdomain are oriented  \code{(1,0)}, since all tiles on the same subdomain are oriented
410  identically.  An axis that corresponds to the orthogonal dimension  identically.  An axis that corresponds to the orthogonal dimension
411  with the same index direction in a particular tile-neighbor  with the same index direction in a particular tile-neighbor
412  orientation will have \code{(0,1)}.  Those in the opposite index  orientation will have \code{(0,1)}.  Those with the opposite index
413  direction will have \code{(0,-1)} in order to reverse the ordering. \\  direction will have \code{(0,-1)} in order to reverse the ordering. \\
414    
415  The arrays \varlink{exch2\_oi}{exch2_oi},  The arrays \varlink{exch2\_oi}{exch2_oi},
# Line 483  the neighbor tile is \code{Tn=5}: Line 495  the neighbor tile is \code{Tn=5}:
495  \code{Tn}'s $y$ axis corresponds to \code{T}'s $x$ axis, \code{T}'s  \code{Tn}'s $y$ axis corresponds to \code{T}'s $x$ axis, \code{T}'s
496  northern edge exchanges with \code{Tn}'s western edge.  The western  northern edge exchanges with \code{Tn}'s western edge.  The western
497  edge of the tiles corresponds to the lower bound of the $x$ axis, so  edge of the tiles corresponds to the lower bound of the $x$ axis, so
498  \code{exch2\_itlo\_c} \code{exch2\_ithi\_c} are \code{0}. The range of  \code{exch2\_itlo\_c} and \code{exch2\_ithi\_c} are \code{0}, in the
499    western halo region of \code{Tn}. The range of
500  \code{exch2\_jtlo\_c} and \code{exch2\_jthi\_c} correspond to the  \code{exch2\_jtlo\_c} and \code{exch2\_jthi\_c} correspond to the
501  width of \code{T}'s northern edge, plus the halo. \\  width of \code{T}'s northern edge, expanded by one into the halo. \\
502    
503    
504  \subsection{Key Routines}  \subsubsection{Key Routines}
505    
506  Most of the subroutines particular to exch2 handle the exchanges  Most of the subroutines particular to exch2 handle the exchanges
507  themselves and are of the same format as those described in  themselves and are of the same format as those described in
508  \ref{sect:cube_sphere_communication} \sectiontitle{Cube sphere  \ref{sect:cube_sphere_communication} \sectiontitle{Cube sphere
509  communication}.  Like the original routines, they are written as  communication}.  Like the original routines, they are written as
510  templates which the local Makefile converts from RX into RL and RS  templates which the local Makefile converts from \code{RX} into
511  forms. \\  \code{RL} and \code{RS} forms. \\
512    
513  The interfaces with the core model subroutines are  The interfaces with the core model subroutines are
514  \code{EXCH\_UV\_XY\_RX}, \code{EXCH\_UV\_XYZ\_RX} and  \code{EXCH\_UV\_XY\_RX}, \code{EXCH\_UV\_XYZ\_RX} and
# Line 510  for scalars and \code{EXCH2\_RX2\_CUBE} Line 523  for scalars and \code{EXCH2\_RX2\_CUBE}
523  the singularities at the cube corners. \\  the singularities at the cube corners. \\
524    
525  The separate scalar and vector forms of \code{EXCH2\_RX1\_CUBE} and  The separate scalar and vector forms of \code{EXCH2\_RX1\_CUBE} and
526  \code{EXCH2\_RX2\_CUBE} reflect that the vector-handling subrouine  \code{EXCH2\_RX2\_CUBE} reflect that the vector-handling subroutine
527  needs to pass both the $u$ and $v$ components of the phsical vectors.  needs to pass both the $u$ and $v$ components of the physical vectors.
528  This arises from the topological folding discussed above, where the  This swapping arises from the topological folding discussed above, where the
529  $x$ and $y$ axes get swapped in some cases.  This swapping is not an  $x$ and $y$ axes get swapped in some cases, and is not an
530  issue with the scalar version. These subroutines call  issue with the scalar case. These subroutines call
531  \code{EXCH2\_SEND\_RX1} and \code{EXCH2\_SEND\_RX2}, which do most of  \code{EXCH2\_SEND\_RX1} and \code{EXCH2\_SEND\_RX2}, which do most of
532  the work using the variables discussed above. \\  the work using the variables discussed above. \\
533    
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