(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.2'

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Cell["ModelReduction`: Manual", "Section",
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Cell[TextData[{
  "The goal of the package is to provide a framework for prototyping new \
model reduction algorithms. I find it much more convenient first to make \
research at the ",
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  " level and only then plan changes in ",
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  ".\nThe design of the package was to reuse functions from Post4MOR as much \
as possible. As such, understanding of Post4MOR is required.\nAn example on \
how functions from the package can be used to perform parametric model \
reduction can be found at the  ",
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            StyleBox[\(gives\ transposed\ projection\ matrix\ V\ with\ dim\ \
columns\ generated\ by\ the\ Arnoldi\ process\ with\ the\ starting\ vector\ \
startVec\ and\ the\ function\ nextVec\ to\ generate\ the\ next\ vector\ as\ a\
\ function\ of\ a\ previous\ \(\(vector\)\(.\)\)\),
              "DefinitionBox"]},
          {\(ProjectSystem[sys, 
              V]\), \(gives\ a\ reduced\ dynamic\ system\ projected\ on\ the\ \
basis\ V . \ ProjectSystem[psys, 
                  V]\ gives\ a\ reduced\ parametric\ system . \ 
                V\ must\ be\ the\ transposed\ projection\ matrix\ as\ \
returned\ by\ \(\(ArnoldiProcess\)\(.\)\)\)},
          {\(Orthogonalize[Vini, V, 
              tol]\), \(gives\ a\ joint\ orthogonalized\ basis\ of\ Vini\ and\
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1*^-12)\)\(.\)\)\)}
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  "ArnoldiProcess is the main function in the package. It implements the \
Arnoldi process and expects that a user specifies a function to compute the \
next vector from the previous vector. It returns the Arnoldi vectors (the \
projection basis) in the transposed form. The reason for this decision is \
that ",
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  " stores matrices by rows and this way allows us to choose the desired \
number of vectors by using Take[V, num].\nProjectSystem takes a DynamicSystem \
from Post4MOR and projects it on the low-dimensional basis found by \
ArnoldiProcess. It can also project ParametricSystem described belows.\n\
ArnoldiProcess and ProjectSystem support a complex-valued DynamicSystem.\n\
Orthogonalize merges two bases. The first basis is supposed to be \
orthonormal."
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\)\)\),
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Cell["\<\
ModelReduction is a simple shell to ArnolidProcess and \
ProjectSystem to make model reduction for DynamicSystem. In the case of a \
second-order system, it uses the default strategy of MOR for ANSYS when \
MatrixE is ignored during model reduction but then projected afterward with \
the basis found from MatrixM and MatrixE. See papers on the MOR for ANSYS \
website for more detail. The expansion point by default is zero.\
\>", "Text"]
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Cell["\<\
This is a pretty primitive container that was made to support \
research on parametric model reduction. Its goal is to keep an arbitrary \
number of system matrices and three function to convert the matrices to that \
required for DynamicSystem (MatrixM, MatrixE, MatrixK).\
\>", "Text"],

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                  mats\ is\ a\ list\ of\ matrices\ and\ funs\ is\ a\ list\ of\
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              matE\ and\ matK\ of\ DynamicSystem . \ For\ example, 
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              "DefinitionBox"]},
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= \(\(\(MatrixKFunction[psys]\)[mats, \ par]\)\(.\)\)\)}
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Cell["\<\
One can use parametric system by converting it to DynamicSystem for \
particular parameter values and then using StationarySolution, \
TransientSolution or HarmonicSolution.\
\>", "Text"],

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          },
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      "DefinitionBox"]], "Text"],

Cell["\<\
ProjectSystem describe above can be used for ParametricSystem as \
well. In this case, it returns projected (low-dimensional) \
ParametricSystem.\
\>", "Text"]
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