/[escript]/trunk/doc/user/escript.tex
ViewVC logotype

Contents of /trunk/doc/user/escript.tex

Parent Directory Parent Directory | Revision Log Revision Log


Revision 804 - (show annotations)
Thu Aug 10 01:12:16 2006 UTC (12 years, 8 months ago) by gross
File MIME type: application/x-tex
File size: 44264 byte(s)
the new function swap_axes + tests added. (It replaces swap).


1 % $Id$
2 %
3 % Copyright © 2006 by ACcESS MNRF
4 % http://www.access.edu.au
5 % Primary Business: Queensland, Australia.
6 % Licensed under the Open Software License version 3.0
7 % http://www.opensource.org/licenses/osl-3.0.php
8 %
9
10
11 \chapter{The module \escript}
12 \label{ESCRIPT CHAP}
13
14 \declaremodule{extension}{escript}
15 \modulesynopsis{Data manipulation}
16
17 \begin{figure}
18 \includegraphics[width=\textwidth]{figures/EscriptDiagram1.eps}
19 \caption{\label{ESCRIPT DEP}Dependency of Function Spaces. An arrow indicates that a function in the
20 function space at the starting point can be interpreted as a function in the function space of the arrow target.}
21 \end{figure}
22
23 \escript is an extension of Python to handle functions represented by their values on
24 \DataSamplePoints for the geometrical region on which
25 the function is defined. The region as well as the method which is used
26 to interpolate value on the \DataSamplePoints is defined by
27 \Domain class objects. For instance when using
28 the finite element method (FEM) \index{finite element method}
29 \Domain object holds the information about the FEM mesh, eg.
30 a table of nodes and a table of elements. Although \Domain contains
31 the discretization method to be used \escript does not use this information directly.
32 \Domain objects are created from a module which want to make use
33 \escript, e.g. \finley.
34
35 The solution of a PDE is a function of its location in the domain of interest $\Omega$.
36 When solving a partial differential equation \index{partial differential equation} (PDE) using FEM
37 the solution is (piecewise) differentiable but, in general, its gradient
38 is discontinuous. To reflect these different degrees of smoothness different
39 representations of the functions are used. For instance; in FEM
40 the displacement field is represented by its values at the nodes of the mesh, while the
41 strain, which is the symmetric part of the gradient of the displacement field, is stored on the
42 element centers. To be able to classify functions with respect to their smoothness, \escript has the
43 concept of the "function space". A function space is described by a \FunctionSpace object.
44 The following statement generates the object \var{solution_space} which is
45 a \FunctionSpace object and provides access to the function space of
46 PDE solutions on the \Domain \var{mydomain}:
47 \begin{python}
48 solution_space=Solution(mydomain)
49 \end{python}
50 The following generators for function spaces on a \Domain \var{mydomain} are available:
51 \begin{itemize}
52 \item \var{Solution(mydomain)}: solutions of a PDE.
53 \item \var{ReducedSolution(mydomain)}: solutions of a PDE with a reduced smoothness requirement.
54 \item \var{ContinuousFunction(mydomain)}: continuous functions, eg. a temperature distribution.
55 \item \var{Function(mydomain)}: general functions which are not necessarily continuous, eg. a stress field.
56 \item \var{FunctionOnBoundary(mydomain)}: functions on the boundary of the domain, eg. a surface pressure.
57 \item \var{FunctionOnContact0(mydomain)}: functions on side $0$ of the discontinuity.
58 \item \var{FunctionOnContact1(mydomain)}: functions on side $1$ of the discontinuity.
59 \end{itemize}
60 The reduced smoothness for PDE solution is often used to fulfill the Ladyzhenskaya–-Babuska–-Brezzi condition \cite{LBB} when
61 solving saddle point problems \index{saddle point problems}, eg. the Stokes equation.
62 A discontinuity \index{discontinuity} is a region within the domain across which functions may be discontinuous.
63 The location of discontinuity is defined in the \Domain object.
64 \fig{ESCRIPT DEP} shows the dependency between the types of function spaces.
65 The solution of a PDE is a continuous function. Any continuous function can be seen as a general function
66 on the domain and can be restricted to the boundary as well as to any side of the
67 discontinuity (the result will be different depending on
68 which side is chosen). Functions on any side of the
69 discontinuity can be seen as a function on the corresponding other side.
70 A function on the boundary or on one side of
71 the discontinuity cannot be seen as a general function on the domain as there are no values
72 defined for the interior. For most PDE solver libraries
73 the space of the solution and continuous functions is identical, however in some cases, eg.
74 when periodic boundary conditions are used in \finley, a solution
75 fulfils periodic boundary conditions while a continuous function does not have to be periodic.
76
77 The concept of function spaces describes the properties of
78 functions and allows abstraction from the actual representation
79 of the function in the context of a particular application. For instance,
80 in the FEM context a
81 function in the \Function function space
82 is typically represented by its values at the element center,
83 but in a finite difference scheme the edge midpoint of cells is preferred.
84 Using the concept of function spaces
85 allows the user to run the same script on different
86 PDE solver libraries by just changing the creator of the \Domain object.
87 Changing the function space of a particular function
88 will typically lead to a change of its representation.
89 So, when seen as a general function,
90 a continuous function which is typically represented by its values
91 on the node of the FEM mesh or finite difference grid
92 must be interpolated to the element centers or the cell edges,
93 respectively.
94
95 \Data class objects store functions of the location in a domain.
96 The function is represented through its values on \DataSamplePoints where
97 the \DataSamplePoints are chosen according to the function space
98 of the function.
99 \Data class objects are used to define the coefficients
100 of the PDEs to be solved by a PDE solver library
101 and to store the returned solutions.
102
103 The values of the function have a rank which gives the
104 number of indices, and a \Shape defining the range of each index.
105 The rank in \escript is limited to the range $0$ through $4$ and
106 it is assumed that the rank and \Shape is the same for all \DataSamplePoints.
107 The \Shape of a \Data object is a tuple \var{s} of integers. The length
108 of \var{s} is the rank of the \Data object and \var{s[i]} is the maximum
109 value for the \var{i}-th index.
110 For instance, a stress field has rank $2$ and
111 \Shape $(d,d)$ where $d$ is the spatial dimension.
112 The following statement creates the \Data object
113 \var{mydat} representing a
114 continuous function with values
115 of \Shape $(2,3)$ and rank $2$:
116 \begin{python}
117 mydat=Data(value=1,what=ContinuousFunction(myDomain),shape=(2,3))
118 \end{python}
119 The initial value is the constant $1$ for all \DataSamplePoints and
120 all components.
121
122 \Data objects can also be created from any \numarray
123 array or any object, such as a list of floating point numbers,
124 that can be converted into a \numarray.NumArray \Ref{NUMARRAY}.
125 The following two statements
126 create objects which are equivalent to \var{mydat}:
127 \begin{python}
128 mydat1=Data(value=numarray.ones((2,3)),what=ContinuousFunction(myDomain))
129 mydat2=Data(value=[[1,1],[1,1],[1,1]],what=ContinuousFunction(myDomain))
130 \end{python}
131 In the first case the initial value is \var{numarray.ones((2,3))}
132 which generates a $2 \times 3$ matrix as a \numarray.NumArray
133 filled with ones. The \Shape of the created \Data object
134 it taken from the \Shape of the array. In the second
135 case, the creator converts the initial value, which is a list of lists,
136 and converts it into a \numarray.NumArray before creating the actual
137 \Data object.
138
139 For convenience \escript provides creators for the most common types
140 of \Data objects in the following forms (\var{d} defines the
141 spatial dimension):
142 \begin{itemize}
143 \item \var{Scalar(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(,))},
144 e.g a temperature field.
145 \item \var{Vector(0,Function(mydomain))}is the same as \var{Data(0,Function(myDomain),(d))}, e.g
146 a velocity field.
147 \item \var{Tensor(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d))},
148 eg. a stress field.
149 \item \var{Tensor4(0,Function(mydomain))} is the same as \var{Data(0,Function(myDomain),(d,d,d,d))}
150 eg. a Hook tensor field.
151 \end{itemize}
152 Here the initial value is $0$ but any object that can be converted into a \numarray.NumArray and whose \Shape
153 is consistent with \Shape of the \Data object to be created can be used as the initial value.
154
155 \Data objects can be manipulated by applying unitary operations (eg. cos, sin, log)
156 and can be combined by applying binary operations (eg. +, - ,* , /).
157 It is to be emphasized that \escript itself does not handle any spatial dependencies as
158 it does not know how values are interpreted by the processing PDE solver library.
159 However \escript invokes interpolation if this is needed during data manipulations.
160 Typically, this occurs in binary operation when both arguments belong to different
161 function spaces or when data are handed over to a PDE solver library
162 which requires functions to be represented in a particular way.
163
164 The following example shows the usage of {\tt Data} objects: Assume we have a
165 displacement field $u$ and we want to calculate the corresponding stress field
166 $\sigma$ using the linear--elastic isotropic material model
167 \begin{eqnarray}\label{eq: linear elastic stress}
168 \sigma\hackscore {ij}=\lambda u\hackscore {k,k} \delta\hackscore {ij} + \mu ( u\hackscore {i,j} + u\hackscore {j,i})
169 \end{eqnarray}
170 where $\delta\hackscore {ij}$ is the Kronecker symbol and
171 $\lambda$ and $\mu$ are the Lame coefficients. The following function
172 takes the displacement {\tt u} and the Lame coefficients
173 \var{lam} and \var{mu} as arguments and returns the corresponding stress:
174 \begin{python}
175 from esys.escript import *
176 def getStress(u,lam,mu):
177 d=u.getDomain().getDim()
178 g=grad(u)
179 stress=lam*trace(g)*kronecker(d)+mu*(g+transpose(g))
180 return stress
181 \end{python}
182 The variable
183 \var{d} gives the spatial dimension of the
184 domain on which the displacements are defined.
185 \var{kronecker} returns the Kronecker symbol with indexes
186 $i$ and $j$ running from $0$ to \var{d}-1. The call \var{grad(u)} requires
187 the displacement field \var{u} to be in the \var{Solution} or \ContinuousFunction
188 function space. The result \var{g} as well as the returned stress will be in the \Function function space.
189 If \var{u} is available, eg. by solving a PDE, \var{getStress} might be called
190 in the following way:
191 \begin{python}
192 s=getStress(u,1.,2.)
193 \end{python}
194 However \var{getStress} can also be called with \Data objects as values for
195 \var{lam} and \var{mu} which,
196 for instance in the case of a temperature dependency, are calculated by an expression.
197 The following call is equivalent to the previous example:
198 \begin{python}
199 lam=Scalar(1.,ContinuousFunction(mydomain))
200 mu=Scalar(2.,Function(mydomain))
201 s=getStress(u,lam,mu)
202 \end{python}
203 The function \var{lam} belongs to the \ContinuousFunction function space
204 but with \var{g} the function \var{trace(g)} is in the \Function function space.
205 Therefore the evaluation of the product \var{lam*trace(g)} in the stress calculation
206 produces a problem, as both functions are represented differently, eg. in FEM
207 \var{lam} by its values on the node, and in \var{trace(g)} by its values at the element centers.
208 In the case of inconsistent function spaces of arguments in a binary operation, \escript
209 interprets the arguments in the appropriate function space according to the inclusion
210 defined in Table~\ref{ESCRIPT DEP}. In this example that means
211 \escript sees \var{lam} as a function of the \Function function space.
212 In the context of FEM this means the nodal values of
213 \var{lam} are interpolated to the element centers. Behind the scenes
214 \escript calls the appropriate function from the PDE solver library.
215
216 \begin{figure}
217 \includegraphics[width=\textwidth]{figures/EscriptDiagram2.eps}
218 \caption{\label{Figure: tag}Element Tagging. A rectangular mesh over a region with two rock types {\it white} and {\it gray}.
219 The number in each cell refers to the major rock type present in the cell ($1$ for {\it white} and $2$ for {\it gray}).
220 }
221 \end{figure}
222
223 Material parameters such as the Lame coefficients are typically dependent on rock types present in the
224 area of interest. A common technique to handle these kinds of material parameters is "tagging". \fig{Figure: tag}
225 shows an example. In this case two rock types {\it white} and {\it gray} can be found in the domain. The domain
226 is subdivided into triangular shaped cells. Each
227 cell has a tag indicating the rock type predominately found in this cell. Here $1$ is used to indicate
228 rock type {\it white} and $2$ for rock type {\it gray}. The tags are assigned at the time when the cells are generated
229 and stored in the \Domain class object. The following statements show how for the
230 example of \fig{Figure: tag} and the stress calculation discussed before tagged values are used for
231 \var{lam}:
232 \begin{python}
233 lam=Scalar(value=2.,what=Function(mydomain))
234 lam.setTaggedValue(1,30.)
235 lam.setTaggedValue(2,5000.)
236 s=getStress(u,lam,2.)
237 \end{python}
238 In this example \var{lam} is set to $30$ for those cells with tag $1$ and to $5000.$ for those cells
239 with tag $2$. The initial value $2$ of \var{lam} is used as a default value for the case when a tag
240 is encountered which has not been linked with a value. Note that the \var{getStress} method
241 is called without modification. \escript resolves the tags when \var{lam*trace(g)} is calculated.
242
243 The \Data class provides a transparent interface to various data representations and the
244 translations between them. As shown in the example of stress calculation, this allows the user to
245 develop and test algorithms for a simple case (for instance with the Lame coefficients as constants)
246 and then without further modifications of the program code to apply the algorithm in a
247 more complex application (for instance a definition of the Lame coefficients using tags).
248 As described here, there are three ways in which \Data objects are represented internally, constant,
249 tagged, and expanded (other representations will become available in later versions of \escript):
250 In the constant case, if the same value is used at each sample point a single value is stored to save memory and compute time.
251 Any operation on this constant data will only be performed on the single value.
252 In the expanded case, each sample point has an individual value, eg. the solution of a PDE,
253 and the values are stored as a complete array. The tagged case has already been discussed above.
254
255 Values are accessed through a sample reference number. Operations on expanded \Data
256 objects have to be performed for each sample point individually. If tagged values are used values are
257 held in a dictionary. Operations on tagged data require processing the set of tagged values only, rather than
258 processing the value for each individual sample point.
259 \escript allows use of constant, tagged and expanded data in a single expression.
260
261 \section{\Domain class}
262 \begin{classdesc}{Domain}{}
263 A \Domain object is used to describe a geometrical region together with
264 a way of representing functions over this region.
265 The \Domain class provides an abstract access to the domain of \FunctionSpace and \Data objects.
266 \Domain itself has no initialization but implementations of \Domain are
267 instantiated by numerical libraries making use of \Data objects.
268 \end{classdesc}
269 The following methds are available:
270 \begin{methoddesc}[Domain]{getDim}{}
271 returns the spatial dimension of the \Domain.
272 \end{methoddesc}
273
274 \begin{methoddesc}[Domain]{getX}{}
275 returns the locations in the \Domain. The \FunctionSpace of the returned
276 \Data object is chosen by the \Domain implementation. Typically it will be
277 in the \Function.
278 \end{methoddesc}
279
280 \begin{methoddesc}[Domain]{setX}{newX}
281 assigns a new location to the \Domain. \var{newX} has to have \Shape $(d,)$
282 where $d$ is the spatial dimension of the domain. Typically \var{newX} must be
283 in the \ContinuousFunction but the space actually to be used depends on the \Domain implementation.
284 \end{methoddesc}
285
286 \begin{methoddesc}[Domain]{getNormal}{}
287 returns the surface normals on the boundary of the \Domain as \Data object.
288 \end{methoddesc}
289
290 \begin{methoddesc}[Domain]{getSize}{}
291 returns the local sample size, e.g. the element diameter, as \Data object.
292 \end{methoddesc}
293
294 \begin{methoddesc}[Domain]{__eq__}{arg}
295 returns \True of the \Domain \var{arg} describes the same domain. Otherwise
296 \False is returned.
297 \end{methoddesc}
298
299 \begin{methoddesc}[Domain]{__ne__}{arg}
300 returns \True of the \Domain \var{arg} does not describe the same domain.
301 Otherwise \False is returned.
302 \end{methoddesc}
303
304 \begin{methoddesc}[Domain]{__str__}{g}
305 returns string represention of the \Domain.
306 \end{methoddesc}
307
308 \section{\FunctionSpace class}
309 \begin{classdesc}{FunctionSpace}{}
310 \FunctionSpace objects are used to define properties of \Data objects, such as continuity. \FunctionSpace objects
311 are instantiated by generator functions. \Data objects in particular \FunctionSpace are
312 represented by their values at \DataSamplePoints which are defined by the type and the \Domain of the
313 \FunctionSpace.
314 \end{classdesc}
315 The following methds are available:
316 \begin{methoddesc}[FunctionSpace]{getDim}{}
317 returns the spatial dimension of the \Domain of the \FunctionSpace.
318 \end{methoddesc}
319
320 \begin{methoddesc}[FunctionSpace]{getX}{}
321 returns the location of the \DataSamplePoints.
322 \end{methoddesc}
323
324 \begin{methoddesc}[FunctionSpace]{getNormal}{}
325 If the domain of functions in the \FunctionSpace
326 is a hypermanifold (e.g. the boundary of a domain)
327 the method returns the outer normal at each of the
328 \DataSamplePoints. Otherwise an exception is raised.
329 \end{methoddesc}
330
331 \begin{methoddesc}[FunctionSpace]{getSize}{}
332 returns a \Data objects measuring the spacing of the \DataSamplePoints.
333 The size may be zero.
334 \end{methoddesc}
335
336 \begin{methoddesc}[FunctionSpace]{getDomain}{}
337 returns the \Domain of the \FunctionSpace.
338 \end{methoddesc}
339
340 \begin{methoddesc}[FunctionSpace]{__eq__}{arg}
341 returns \True of the \Domain \var{arg} describes the same domain. Otherwise
342 \False is returned.
343 \end{methoddesc}
344
345 \begin{methoddesc}[FunctionSpace]{__ne__}{arg}
346 returns \True of the \Domain \var{arg} describes the note same domain.
347 Otherwise \False is returned.
348 \end{methoddesc}
349
350 \begin{methoddesc}[Domain]{__str__}{g}
351 returns string represention of the \Domain.
352 \end{methoddesc}
353
354 The following function provide generators for \FunctionSpace objects:
355 \begin{funcdesc}{Function}{domain}
356 returns the \Function on the \Domain \var{domain}. \Data objects in this type of \Function
357 are defined over the whole geometrical region defined by \var{domain}.
358 \end{funcdesc}
359
360 \begin{funcdesc}{ContinuousFunction}{domain}
361 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function
362 are defined over the whole geometrical region defined by \var{domain} and assumed to represent
363 a continuous function.
364 \end{funcdesc}
365
366 \begin{funcdesc}{FunctionOnBoundary}{domain}
367 returns the \ContinuousFunction on the \Domain domain. \Data objects in this type of \Function
368 are defined on the boundary of the geometrical region defined by \var{domain}.
369 \end{funcdesc}
370
371 \begin{funcdesc}{FunctionOnContactZero}{domain}
372 returns the \FunctionOnContactZero the \Domain domain. \Data objects in this type of \Function
373 are defined on side 0 of a discontinutiy within the geometrical region defined by \var{domain}.
374 The discontinutiy is defined when \var{domain} is instantiated.
375 \end{funcdesc}
376
377 \begin{funcdesc}{FunctionOnContactOne}{domain}
378 returns the \FunctionOnContactOne on the \Domain domain.
379 \Data objects in this type of \Function
380 are defined on side 1 of a discontinutiy within the geometrical region defined by \var{domain}.
381 The discontinutiy is defined when \var{domain} is instantiated.
382 \end{funcdesc}
383
384 \begin{funcdesc}{Solution}{domain}
385 returns the \SolutionFS on the \Domain domain. \Data objects in this type of \Function
386 are defined on geometrical region defined by \var{domain} and are solutions of
387 partial differential equations \index{partial differential equation}.
388 \end{funcdesc}
389
390 \begin{funcdesc}{ReducedSolution}{domain}
391 returns the \ReducedSolutionFS on the \Domain domain. \Data objects in this type of \Function
392 are defined on geometrical region defined by \var{domain} and are solutions of
393 partial differential equations \index{partial differential equation} with a reduced smoothness
394 for the solution approximation.
395 \end{funcdesc}
396
397 \section{\Data Class}
398 \label{SEC ESCRIPT DATA}
399
400 The following table shows binary and unitary operations that can be applied to
401 \Data objects:
402 \begin{tableii}{l|l}{textrm}{expression}{Description}
403 \lineii{+\var{arg0}} {just \var{arg} \index{+}}
404 \lineii{-\var{arg0}} {swapping the sign\index{-}}
405 \lineii{\var{arg0}+\var{arg1}} {adds \var{arg0} and \var{arg1} \index{+}}
406 \lineii{\var{arg0}*\var{arg1}} {multiplies \var{arg0} and \var{arg1} \index{*}}
407 \lineii{\var{arg0}-\var{arg1}} {difference \var{arg1} from\var{arg1} \index{-}}
408 \lineii{\var{arg0}/\var{arg1}} {ratio \var{arg0} by \var{arg1} \index{/}}
409 \lineii{\var{arg0}**\var{arg1}} {raises \var{arg0} to the power of \var{arg1} \index{**}}
410 \end{tableii}
411 At least one of the arguments \var{arg0} or \var{arg1} must be a
412 \Data object. One of the arguments may be an object that can be
413 converted into a \Data object. If \var{arg0} or \var{arg1} are
414 defined on different \FunctionSpace an attempt is made to embed \var{arg0}
415 into the \FunctionSpace of \var{arg1} or to embed \var{arg1} into
416 the \FunctionSpace of \var{arg0}. Boths arguments must have the same
417 \Shape or one of the arguments my be of rank 0. In the
418 latter case it is assumed that the particular argument is of the same
419 \Shape as the other argument but constant over all components.
420
421 The returned \Data object has the same \Shape and is defined on
422 the \DataSamplePoints as \var{arg0} or \var{arg1}.
423
424 The following table shows the update operations that can be applied to
425 \Data objects:
426 \begin{tableii}{l|l}{textrm}{expression}{Description}
427 \lineii{\var{arg0}+=\var{arg2}} {adds \var{arg0} to \var{arg2} \index{+}}
428 \lineii{\var{arg0}*=\var{arg2}} {multiplies \var{arg0} with \var{arg2} \index{*}}
429 \lineii{\var{arg0}-=\var{arg2}} {subtracts \var{arg2} from\var{arg2} \index{-}}
430 \lineii{\var{arg0}/=\var{arg2}} {divides \var{arg0} by \var{arg2} \index{/}}
431 \lineii{\var{arg0}**=\var{arg2}} {raises \var{arg0} by \var{arg2} \index{**}}
432 \end{tableii}
433 \var{arg0} must be a \Data object. \var{arg1} must be a
434 \Data object or an object that can be converted into a
435 \Data object. \var{arg1} must have the same \Shape like
436 \var{arg1} or has rank 0. In the latter case it is
437 assumed that the values of \var{arg1} are constant for all
438 components. \var{arg1} must be defined in the same \FunctionSpace as
439 \var{arg0} or it must be possible to interpolate \var{arg1} onto the
440 \FunctionSpace of \var{arg1}.
441
442 The \Data class supports getting slices as well as assigning new values to components in an existing
443 \Data object. \index{slicing}
444 The following expression for getting (expression on the right hand side of the
445 equal sign) and setting slices (expression on the left hand side of the
446 equal sign) are valid:
447 \begin{tableiii}{l|ll}{textrm}{rank of \var{arg}}{slicing expression}{\Shape of returned and assigned object}
448 \lineiii{0}{ no slicing } {-}
449 \lineiii{1}{\var{arg[l0:u0]}} {(\var{u0}-\var{l0},)}
450 \lineiii{2}{\var{arg[l0:u0,l1:u1]}} {(\var{u0}-\var{l0},\var{u1}-\var{l1})}
451 \lineiii{3}{\var{arg[l0:u0,l1:u1,l2:u2]} } {(\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2})}
452 \lineiii{4}{\var{arg[l0:u0,l1:u1,l2:u2,l3:u3]}} {(\var{u0}-\var{l0},\var{u1}-\var{l1},\var{u2}-\var{l2},\var{u3}-\var{l3})}
453 \end{tableiii}
454 where
455 $0 \le \var{l0} \le \var{u0} \le \var{s[0]}$,
456 $0 \le \var{l1} \le \var{u1} \le \var{s[1]}$,
457 $0 \le \var{l2} \le \var{u2} \le \var{s[2]}$,
458 $0 \le \var{l3} \le \var{u3} \le \var{s[3]}$ and \var{s} the \Shape if \var{arg}.
459 Any of the lower indexes \var{l0}, \var{l1}, \var{l2} and \var{l3} may not be present in which case
460 $0$ is assumed.
461 Any of the upper indexes \var{u0}, \var{u1}, \var{u2} and \var{u3} may not be present in which case
462 \var{s} is assumed. The lower and upper index may be identical, in which case the column and the lower or upper
463 index may be dropped. In the returned or in the object assigned to a slice the corresponding component is dropped,
464 i.e. the rank is reduced by one in comparison to \var{arg}.
465 The following examples show slicing usage:
466 \begin{python}
467 t=Data(1.,(4,4,6,6),Function(mydomain))
468 t[1,1,1,0]=9.
469 s=t[:2,:,2:6,5] # s has rank 3
470 s[:,:,1]=1.
471 t[:2,:2,5,5]=s[2:4,1,:2]
472 \end{python}
473
474 \subsection{Generation of \Data class objects}
475 \begin{classdesc}{Data}{value=0,shape=(,),what=FunctionSpace(),expand=\False}
476 creates a \Data object with \Shape \var{shape} in the \FunctionSpace \var{what}.
477 The values at all \DataSamplePoints are set to the double value \var{value}. If \var{expanded} is \True
478 the \Data object is represented in expanded from.
479 \end{classdesc}
480
481 \begin{classdesc}{Data}{value,what=FunctionSpace(),expand=\False}
482 creates a \Data object in the \FunctionSpace \var{what}.
483 The value for each \DataSamplePoints is set to \numarray, \Data object \var{value} or a dictionary of
484 \numarray or floating point numbers. In the latter case the keys muts be integers and are used
485 as tags.
486 The \Shape of the returned object is equal to the \Shape of \var{value}. If \var{expanded} is \True
487 the \Data object is represented in expanded from.
488 \end{classdesc}
489
490 \begin{classdesc}{Data}{}
491 creates an \EmptyData object. The \EmptyData object is used to indicate that an argument is not present
492 where a \Data object is required.
493 \end{classdesc}
494
495 \begin{funcdesc}{Scalar}{value=0.,what=escript::FunctionSpace(),expand=\False}
496 returns a \Data object of rank 0 in the \FunctionSpace \var{what}.
497 Values are initialed with the double \var{value}. If \var{expanded} is \True
498 the \Data object is represented in expanded from.
499 \end{funcdesc}
500
501 \begin{funcdesc}{Vector}{value=0.,what=escript::FunctionSpace(),expand=\False}
502 returns a \Data object of \Shape \var{(d,)} in the \FunctionSpace \var{what}
503 where \var{d} is the spatial dimension of the \Domain of \var{what}.
504 Values are initialed with the double \var{value}. If \var{expanded} is \True
505 the \Data object is represented in expanded from.
506 \end{funcdesc}
507
508 \begin{funcdesc}{Tensor}{value=0.,what=escript::FunctionSpace(),expand=\False}
509 returns a \Data object of \Shape \var{(d,d)} in the \FunctionSpace \var{what}
510 where \var{d} is the spatial dimension of the \Domain of \var{what}.
511 Values are initialed with the double \var{value}. If \var{expanded} is \True
512 the \Data object is represented in expanded from.
513 \end{funcdesc}
514
515 \begin{funcdesc}{Tensor3}{value=0.,what=escript::FunctionSpace(),expand=\False}
516 returns a \Data object of \Shape \var{(d,d,d)} in the \FunctionSpace \var{what}
517 where \var{d} is the spatial dimension of the \Domain of \var{what}.
518 Values are initialed with the double \var{value}. If \var{expanded} is \True
519 the \Data object is re\var{arg}presented in expanded from.
520 \end{funcdesc}
521
522 \begin{funcdesc}{Tensor4}{value=0.,what=escript::FunctionSpace(),expand=\False}
523 returns a \Data object of \Shape \var{(d,d,d,d)} in the \FunctionSpace \var{what}
524 where \var{d} is the spatial dimension of the \Domain of \var{what}.
525 Values are initialed with the double \var{value}. If \var{expanded} is \True
526 the \Data object is represented in expanded from.
527 \end{funcdesc}
528
529 \subsection{\Data class methods}
530 This is a list of frequently used methods of the
531 \Data class. A complete list can be fond on \ReferenceGuide.
532 \begin{methoddesc}[Data]{getFunctionSpace}{}
533 returns the \FunctionSpace of the object.
534 \end{methoddesc}
535
536 \begin{methoddesc}[Data]{getDomain}{}
537 returns the \Domain of the object.
538 \end{methoddesc}
539
540 \begin{methoddesc}[Data]{getShape}{}
541 returns the \Shape of the object as a \class{tuple} of
542 integers.
543 \end{methoddesc}
544
545 \begin{methoddesc}[Data]{getRank}{}
546 returns the rank of the data on each data point. \index{rank}
547 \end{methoddesc}
548
549 \begin{methoddesc}[Data]{isEmpty}{}
550 returns \True id the \Data object is the \EmptyData object.
551 Otherwise \False is returned.
552 \end{methoddesc}
553
554 \begin{methoddesc}[Data]{setTaggedValue}{tag,value}
555 assigns the \var{value} to all \DataSamplePoints which have the tag
556 \var{tag}. \var{value} must be an object of class
557 \class{numarray.NumArray} or must be convertible into a
558 \class{numarray.NumArray} object. \var{value} (or the corresponding
559 \class{numarray.NumArray} object) must be of rank $0$ or must have the
560 same rank like the object.
561 If a value has already be defined for tag \var{tag} within the object
562 it is overwritten by the new \var{value}. If the object is expanded,
563 the value assigned to \DataSamplePoints with tag \var{tag} is replaced by
564 \var{value}.
565 \end{methoddesc}
566
567 \begin{methoddesc}[Data]{__str__}{}
568 returns a string representation of the object.
569 \end{methoddesc}
570
571 \section{Functions of \Data class objects}
572 This section lists the most important functions for \Data class objects \var{a}.
573 A complete list and a more detailed description of the functionality can be fond on \ReferenceGuide.
574 \begin{funcdesc}{saveVTK}{filename,**kwdata}
575 writes \Data defined by keywords in the file with \var{filename} using the
576 vtk file format \VTK file format. The key word is used as an identifier. The statement
577 \begin{python}
578 saveVTK("out.xml",temperature=T,velocity=v)
579 \end{python}
580 will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
581 file \file{out.xml}. Restrictions on the allowed combinations of \FunctionSpace apply.
582 \end{funcdesc}
583 \begin{funcdesc}{saveDX}{filename,**kwdata}
584 writes \Data defined by keywords in the file with \var{filename} using the
585 vtk file format \OpenDX file format. The key word is used as an identifier. The statement
586 \begin{python}
587 saveDX("out.dx",temperature=T,velocity=v)
588 \end{python}
589 will write the scalar \var{T} as \var{temperature} and the vector \var{v} as \var{velocity} into the
590 file \file{out.dx}. Restrictions on the allowed combinations of \FunctionSpace apply.
591 \end{funcdesc}
592 \begin{funcdesc}{kronecker}{d}
593 returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
594 \begin{equation}
595 \code{kronecker(d)}\left[ i,j\right] = \left\{
596 \begin{array}{cc}
597 1 & \mbox{ if } i=j \\
598 0 & \mbox{ otherwise }
599 \end{array}
600 \right.
601 \end{equation}
602 If \var{d} is an integer a $(d,d)$ \numarray array is returned.
603 \end{funcdesc}
604 \begin{funcdesc}{identityTensor}{d}
605 returns a \RankTwo \Data object in \FunctionSpace \var{d} such that
606 \begin{equation}
607 \code{identityTensor(d)}\left[ i,j\right] = \left\{
608 \begin{array}{cc}
609 1 & \mbox{ if } i=j \\
610 0 & \mbox{ otherwise }
611 \end{array}
612 \right.
613 \end{equation}
614 If \var{d} is an integer a $(d,d)$ \numarray array is returned.
615 \end{funcdesc}
616 \begin{funcdesc}{identityTensor4}{d}
617 returns a \RankFour \Data object in \FunctionSpace \var{d} such that
618 \begin{equation}
619 \code{identityTensor(d)}\left[ i,j,k,l\right] = \left\{
620 \begin{array}{cc}
621 1 & \mbox{ if } i=k \mbox{ and } j=l\\
622 0 & \mbox{ otherwise }
623 \end{array}
624 \right.
625 \end{equation}
626 If \var{d} is an integer a $(d,d,d,d)$ \numarray array is returned.
627 \end{funcdesc}
628 \begin{funcdesc}{unitVector}{i,d}
629 returns a \RankOne \Data object in \FunctionSpace \var{d} such that
630 \begin{equation}
631 \code{identityTensor(d)}\left[ j \right] = \left\{
632 \begin{array}{cc}
633 1 & \mbox{ if } j=i\\
634 0 & \mbox{ otherwise }
635 \end{array}
636 \right.
637 \end{equation}
638 If \var{d} is an integer a $(d,)$ \numarray array is returned.
639
640 \end{funcdesc}
641
642 \begin{funcdesc}{Lsup}{a}
643 returns the $L^{sup}$ norm of \var{arg}. This is the maximum of the absolute values
644 over all components and all \DataSamplePoints of \var{a}.
645 \end{funcdesc}
646
647 \begin{funcdesc}{sup}{a}
648 returns the maximum value over all components and all \DataSamplePoints of \var{a}.
649 \end{funcdesc}
650
651 \begin{funcdesc}{inf}{a}
652 returns the minimum value over all components and all \DataSamplePoints of \var{a}
653 \end{funcdesc}
654
655 \begin{funcdesc}{sin}{a}
656 applies sine function to \var{a}.
657 \end{funcdesc}
658
659 \begin{funcdesc}{cos}{a}
660 applies cosine function to \var{a}.
661 \end{funcdesc}
662
663 \begin{funcdesc}{tan}{a}
664 applies tangent function to \var{a}.
665 \end{funcdesc}
666
667 \begin{funcdesc}{asin}{a}
668 applies arc (inverse) sine function to \var{a}.
669 \end{funcdesc}
670
671 \begin{funcdesc}{acos}{a}
672 applies arc (inverse) cosine function to \var{a}.
673 \end{funcdesc}
674
675 \begin{funcdesc}{atan}{a}
676 applies arc (inverse) tangent function to \var{a}.
677 \end{funcdesc}
678
679 \begin{funcdesc}{sinh}{a}
680 applies hyperbolic sine function to \var{a}.
681 \end{funcdesc}
682
683 \begin{funcdesc}{cosh}{a}
684 applies hyperbolic cosine function to \var{a}.
685 \end{funcdesc}
686
687 \begin{funcdesc}{tanh}{a}
688 applies hyperbolic tangent function to \var{a}.
689 \end{funcdesc}
690
691 \begin{funcdesc}{asinh}{a}
692 applies arc (inverse) hyperbolic sine function to \var{a}.
693 \end{funcdesc}
694
695 \begin{funcdesc}{acosh}{a}
696 applies arc (inverse) hyperbolic cosine function to \var{a}.
697 \end{funcdesc}
698
699 \begin{funcdesc}{atanh}{a}
700 applies arc (inverse) hyperbolic tangent function to \var{a}.
701 \end{funcdesc}
702
703 \begin{funcdesc}{exp}{a}
704 applies exponential function to \var{a}.
705 \end{funcdesc}
706
707 \begin{funcdesc}{sqrt}{a}
708 applies square root function to \var{a}.
709 \end{funcdesc}
710
711 \begin{funcdesc}{log}{a}
712 applies the natural logarithm to \var{a}.
713 \end{funcdesc}
714
715 \begin{funcdesc}{log10}{a}
716 applies the base-$10$ logarithm to \var{a}.
717 \end{funcdesc}
718
719 \begin{funcdesc}{sign}{a}
720 applies the sign function to \var{a}, that is $1$ where \var{a} is positive,
721 $-1$ where \var{a} is negative and $0$ otherwise.
722 \end{funcdesc}
723
724 \begin{funcdesc}{wherePositive}{a}
725 returns a function which is $1$ where \var{a} is positive and $0$ otherwise.
726 \end{funcdesc}
727
728 \begin{funcdesc}{whereNegative}{a}
729 returns a function which is $1$ where \var{a} is negative and $0$ otherwise.
730 \end{funcdesc}
731
732 \begin{funcdesc}{whereNonNegative}{a}
733 returns a function which is $1$ where \var{a} is non--negative and $0$ otherwise.
734 \end{funcdesc}
735
736 \begin{funcdesc}{whereNonPositive}{a}
737 returns a function which is $1$ where \var{a} is non--positive and $0$ otherwise.
738 \end{funcdesc}
739
740 \begin{funcdesc}{whereZero}{a\optional{, tol=0.}}
741 returns a function which is $1$ where \var{a} equals zero with tolerance \var{tol} and $0$ otherwise.
742 \end{funcdesc}
743
744 \begin{funcdesc}{whereNonZero}{a\optional{, tol=0.}}
745 returns a function which is $1$ where \var{a} different from zero with tolerance \var{tol} and $0$ otherwise.
746 \end{funcdesc}
747
748 \begin{funcdesc}{minval}{a}
749 returns at each \DataSamplePoints the minumum value over all components.
750 \end{funcdesc}
751
752 \begin{funcdesc}{maxval}{a}
753 returns at each \DataSamplePoints the maximum value over all components.
754 \end{funcdesc}
755
756 \begin{funcdesc}{length}{a}
757 returns at Euclidean norm at each \DataSamplePoints. For a \RankFour function \var{a} this is
758 \begin{equation}
759 \code{length(a)}=\sqrt{\sum\hackscore{ijkl} \var{a} \left[i,j,k,l\right]^2}
760 \end{equation}
761 \end{funcdesc}
762 \begin{funcdesc}{trace}{a\optional{,axis_offset=0}}
763 returns the trace of \var{a}. This is the sum over components \var{axis_offset} and \var{axis_offset+1} with the same index. For instance in the
764 case of a \RankTwo function and this is
765 \begin{equation}
766 \code{trace(a)}=\sum\hackscore{i} \var{a} \left[i,i\right]
767 \end{equation}
768 and for a \RankFour function and \code{axis_offset=1} this is
769 \begin{equation}
770 \code{trace(a,1)}\left[i,j\right]=\sum\hackscore{k} \var{a} \left[i,k,k,j\right]
771 \end{equation}
772 \end{funcdesc}
773
774 \begin{funcdesc}{transpose}{a\optional{, axis_offset=None}}
775 returns the transpose of \var{a}. This swaps the first \var{axis_offset} components of \var{a} with the rest. If \var{axis_offset} is not
776 present \code{int(r/2)} is used where \var{r} is the rank of \var{a}.
777 the sum over components \var{axis_offset} and \var{axis_offset+1} with the same index. For instance in the
778 case of a \RankTwo function and this is
779 \begin{equation}
780 \code{transpose(a)}\left[i,j\right]=\var{a} \left[j,i\right]
781 \end{equation}
782 and for a \RankFour function and \code{axis_offset=1} this is
783 \begin{equation}
784 \code{transpose(a,1)}\left[i,j,k,l\right]=\var{a} \left[j,k,l,i\right]
785 \end{equation}
786 \end{funcdesc}
787
788 \begin{funcdesc}{swap_axes}{a\optional{, axis0=0 \optional{, axis1=1 }}}
789 returns \var{a} but with swapped componets \var{axis0} and \var{axis1}. The argument \var{a} must be
790 at least of \RankTwo. For instance in the
791 for a \RankFour argument, \code{axis0=1} and \code{axis1=2} this is
792 \begin{equation}
793 \code{swap_axes(a,1,2)}\left[i,j,k,l\right]=\var{a} \left[i,k,j,l\right]
794 \end{equation}
795 \end{funcdesc}
796
797 \begin{funcdesc}{symmetric}{a}
798 returns the symmetric part of \var{a}. This is \code{(a+transpose(a))/2}.
799 \end{funcdesc}
800 \begin{funcdesc}{nonsymmetric}{a}
801 returns the non--symmetric part of \var{a}. This is \code{(a-transpose(a))/2}.
802 \end{funcdesc}
803 \begin{funcdesc}{inverse}{a}
804 return the inverse of \var{a}. This is
805 \begin{equation}
806 \code{matrixmult(inverse(a),a)=kronecker(d)}
807 \end{equation}
808 if \var{a} has shape \code{(d,d)}. The current implementation is restricted to arguments of shape
809 \code{(2,2)} and \code{(3,3)}.
810 \end{funcdesc}
811 \begin{funcdesc}{eigenvalues}{a}
812 return the eigenvalues of \var{a}. This is
813 \begin{equation}
814 \code{matrixmult(a,V)=e[i]*V}
815 \end{equation}
816 where \code{e=eigenvalues(a)} and \var{V} is suitable non--zero vector \var{V}.
817 The eigenvalues are ordered in increasing size.
818 The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
819 The current implementation is restricted to arguments of shape
820 \code{(2,2)} and \code{(3,3)}.
821 \end{funcdesc}
822 \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}
823 return the eigenvalues and eigenvectors of \var{a}. This is
824 \begin{equation}
825 \code{matrixmult(a,V[:,i])=e[i]*V[:,i]}
826 \end{equation}
827 where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are orthogonal and normalized, ie.
828 \begin{equation}
829 \code{matrixmult(transpose(V),V)=kronecker(d)}
830 \end{equation}
831 if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing size.
832 The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.
833 The current implementation is restricted to arguments of shape
834 \code{(2,2)} and \code{(3,3)}.
835 \end{funcdesc}
836 \begin{funcdesc}{maximum}{*a}
837 returns the maximum value over all arguments at all \DataSamplePoints and for each component.
838 For instance
839 \begin{equation}
840 \code{maximum(a0,a1)}\left[i,j\right]=max(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
841 \end{equation}
842 at all \DataSamplePoints.
843 \end{funcdesc}
844 \begin{funcdesc}{minimum}{*a}
845 returns the minimum value over all arguments at all \DataSamplePoints and for each component.
846 For instance
847 \begin{equation}
848 \code{minimum(a0,a1)}\left[i,j\right]=min(\var{a0} \left[i,j\right],\var{a1} \left[i,j\right])
849 \end{equation}
850 at all \DataSamplePoints.
851 \end{funcdesc}
852
853 \begin{funcdesc}{clip}{a\optional{, minval=0.}\optional{, maxval=1.}}
854 cuts back \var{a} into the range between \var{minval} and \var{maxval}. A value in the returned object equals
855 \var{minval} if the corresponding value of \var{a} is less than \var{minval}, equals \var{maxval} if the
856 corresponding value of \var{a} is greater than \var{maxval}
857 or corresponding value of \var{a} otherwise.
858 \end{funcdesc}
859 \begin{funcdesc}{inner}{a0,a1}
860 returns the inner product of \var{a0} and \var{a1}. For instance in the
861 case of \RankTwo arguments and this is
862 \begin{equation}
863 \code{inner(a)}=\sum\hackscore{ij}\var{a0} \left[j,i\right] \cdot \var{a1} \left[j,i\right]
864 \end{equation}
865 and for a \RankFour arguments this is
866 \begin{equation}
867 \code{inner(a)}=\sum\hackscore{ijkl}\var{a0} \left[i,j,k,l\right] \cdot \var{a1} \left[j,i,k,l\right]
868 \end{equation}
869 \end{funcdesc}
870 \begin{funcdesc}{matrixmult}{a0,a1}
871 returns the matrix product of \var{a0} and \var{a1}. If \var{a1} is \RankOne this is
872 \begin{equation}
873 \code{matrixmult(a)}\left[i\right]=\sum\hackscore{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k\right]
874 \end{equation}
875 and if \var{a1} is \RankTwo this is
876 \begin{equation}
877 \code{matrixmult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0} \cdot \left[i,k\right]\var{a1} \left[k,j\right]
878 \end{equation}
879 \end{funcdesc}
880 \begin{funcdesc}{outer}{a0,a1}
881 returns the outer product of \var{a0} and \var{a1}. For instance if \var{a0} and \var{a1} both are \RankOne then
882 \begin{equation}
883 \code{outer(a)}\left[i,j\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j\right]
884 \end{equation}
885 and if \var{a0} is \RankOne and \var{a1} is \RankThree
886 \begin{equation}
887 \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]
888 \end{equation}
889 \end{funcdesc}
890 \begin{funcdesc}{tensormult}{a0,a1}
891 returns the tensor product of \var{a0} and \var{a1}. If \var{a1} is \RankTwo this is
892 \begin{equation}
893 \code{tensormult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[k,l\right]
894 \end{equation}
895 and if \var{a1} is \RankFour this is
896 \begin{equation}
897 \code{tensormult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[m,n,k,l\right]
898 \end{equation}
899 \end{funcdesc}
900 \begin{funcdesc}{grad}{a\optional{, where=None}}
901 returns the gradient of \var{a}. If \var{where} is present the gradient will be calculated in \FunctionSpace \var{where} otherwise a
902 default \FunctionSpace is used. In case that \var{a} has \RankTwo one has
903 \begin{equation}
904 \code{grad(a)}\left[i,j,k\right]=\frac{\partial \var{a} \left[i,j\right]}{\partial x\hackscore{k}}
905 \end{equation}
906 \end{funcdesc}
907 \begin{funcdesc}{integrate}{a\optional{ ,where=None}}
908 returns the integral of \var{a} where the domain of integration is defined by the \FunctionSpace of \var{a}. If \var{where} is
909 present the argument is interpolated into \FunctionSpace \var{where} before integration. For instance in the case of
910 a \RankTwo argument in \ContinuousFunction it is
911 \begin{equation}
912 \code{integrate(a)}\left[i,j\right]=\int\hackscore{\Omega}\var{a} \left[i,j\right] \; d\Omega
913 \end{equation}
914 where $\Omega$ is the spatial domain and $d\Omega$ volume integration. To integrate over the boundary of the domain one uses
915 \begin{equation}
916 \code{integrate(a,where=FunctionOnBoundary(a.getDomain))}\left[i,j\right]=\int\hackscore{\partial \Omega} a\left[i,j\right] \; ds
917 \end{equation}
918 where $\partial \Omega$ is the surface of the spatial domain and $ds$ area or line integration.
919 \end{funcdesc}
920 \begin{funcdesc}{interpolate}{a,where}
921 interpolates argument \var{a} into the \FunctionSpace \var{where}.
922 \end{funcdesc}
923 \begin{funcdesc}{div}{a\optional{ ,where=None}}
924 returns the divergence of \var{a}. This
925 \begin{equation}
926 \code{div(a)}=trace(grad(a),where)
927 \end{equation}
928 \end{funcdesc}
929 \begin{funcdesc}{jump}{a\optional{ ,domain=None}}
930 returns the jump of \var{a} over the discontinuity in its domain or if \Domain \var{domain} is present
931 in \var{domain}.
932 \begin{equation}
933 \code{jump(a)}=interpolate(a,FunctionOnContactOne(domain))-interpolate(a,FunctionOnContactZero(domain))
934 \end{equation}
935 \end{funcdesc}
936 \begin{funcdesc}{L2}{a}
937 returns the $L^2$-norm of \var{a} in its function space. This is
938 \begin{equation}
939 \code{L2(a)}=integrate(length(a)^2) \; .
940 \end{equation}
941 \end{funcdesc}
942
943 \section{\Operator Class}
944 The \Operator class provides an abstract access to operators build
945 within the \LinearPDE class. \Operator objects are created
946 when a PDE is handed over to a PDE solver library and handled
947 by the \LinearPDE class defining the PDE. The user can gain access
948 to the \Operator of a \LinearPDE object through the \var{getOperator}
949 method.
950
951 \begin{classdesc}{Operator}{}
952 creates an empty \Operator object.
953 \end{classdesc}
954
955 \begin{methoddesc}[Operator]{isEmpty}{fileName}
956 returns \True is the object is empty. Otherwise \True is returned.
957 \end{methoddesc}
958
959 \begin{methoddesc}[Operator]{setValue}{value}
960 resets all entires in the obeject representation to \var{value}
961 \end{methoddesc}
962
963 \begin{methoddesc}[Operator]{solves}{rhs}
964 solves the operator equation with right hand side \var{rhs}
965 \end{methoddesc}
966
967 \begin{methoddesc}[Operator]{of}{u}
968 applies the operator to the \Data object \var{u}
969 \end{methoddesc}
970
971 \begin{methoddesc}[Operator]{saveMM}{fileName}
972 saves the object to a matrix market format file of name
973 \var{fileName}, see
974 \ulink{maths.nist.gov/MatrixMarket}{\url{http://maths.nist.gov/MatrixMarket}}.
975 \index{Matrix Market}
976 \end{methoddesc}
977

Properties

Name Value
svn:eol-style native
svn:keywords Author Date Id Revision

  ViewVC Help
Powered by ViewVC 1.1.26