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433 lines
12 KiB
TeX
433 lines
12 KiB
TeX
%
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% part of numlib docs. In time this won't be a standalone pdf anymore, but part of a larger part.
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% for now I keep it directly compliable. Uses fpc.sty from fpc-docs pkg.
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%
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\documentclass{report}
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\usepackage{fpc}
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\lstset{%
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basicstyle=\small,
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language=delphi,
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commentstyle=\itshape,
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keywordstyle=\bfseries,
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showstringspaces=false,
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frame=
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}
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\makeindex
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\newcommand{\FunctionDescription}{\item[Description]\rmfamily}
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\newcommand{\Dataorganisation}{\item[Data Struct]\rmfamily}
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\newcommand{\DeclarationandParams}{\item[Declaration]\rmfamily}
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\newcommand{\References}{\item[References]\rmfamily}
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\newcommand{\Method}{\item[Method]\rmfamily}
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\newcommand{\Precision}{\item[Precision]\rmfamily}
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\newcommand{\Remarks}{\item[Remarks]\rmfamily}
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\newcommand{\Example}{\item[Example]\rmfamily}
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\newcommand{\ProgramData}{\item[Example Data]\rmfamily}
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\newcommand{\ProgramResults}{\item[Example Result]\rmfamily}
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\makeatletter
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\@ifpackageloaded{tex4ht}{%
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\newcommand{\NUMLIBexample}[1]{
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\par \file{\textbf{Listing:} \exampledir/#1.pas}%
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\HCode{<HR/>}%
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\listinginput[9999]{5000}{\exampledir/#1.pas}%
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\HCode{<HR/>}%
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}%
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}{% else ifpackageloaded
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\newcommand{\NUMLIBexample}[1]{%
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\par \file{\textbf{Listing:} \exampledir/#1.pas}%
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\lstinputlisting{\exampledir/#1.pas}%
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}% End of FPCExample
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}% End of ifpackageloaded.
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\makeatother
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%
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\begin{document}
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\FPCexampledir{../examples}
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\section{general comments}
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\textbf{Original comments:} \\
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The used floating point type \textbf{real} depends on the used version,
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see the general introduction for more information. You'll need to USE
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units typ an inv to use these routines.
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\textbf{MvdV notes:} \\
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Integers used for parameters are of type "ArbInt" to avoid problems with
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systems that define integer differently depending on mode.
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Floating point values are of type "ArbFloat" to allow writing code
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that is independent of the exact real type. (Contrary to directly
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specifying single, real, double or extended in library and
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examples). Typ.pas and the central includefile have some conditional
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code to switch between floating point types.
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These changes were already prepared somewhat when I got the lib, but weren't
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consequently applied. I did that while porting to FPC.
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\section{Unit inv}
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\begin{procedure}{invgen}
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\FunctionDescription
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Procedure to calculate the inverse of a matrix.
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\Dataorganisation
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The procedure assumes that the calling program has declared a two
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dimensional matrix containing the matrix elements in a square
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partial block.
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\DeclarationandParams
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\lstinline|procedure invgen(n, rwidth: ArbInt; var ai: ArbFloat;var term: ArbInt);|
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\begin{description}
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\item[n: integer] \mbox{ } \\
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The parameter {\bf n} describes the size of the matrix
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\item[rwidth: integer] \mbox{} \\
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The parameter {\bf rwidth} describes the declared rowlength of the two dimensional
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matrix.
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\item[var ai: real] \mbox{} \\
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The parameter {\bf ai} must contain to the two dimensional array element
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that is top-left in the matrix.
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After the function has ended successfully (\textbf{term=1}) then
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the input matrix has been changed into its inverse, otherwise the contents
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of the input matrix are undefined.
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\item[var term: integer] \mbox{} \\
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After the procedure has run, this variable contains the reason for
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the termination of the procedure:\\
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{\bf term}=1, successful termination, the inverse has been calculated
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{\bf term}=2, inverse matrix could not be calculated because the matrix
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is (very close to) being singular.
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{\bf term}=3, wrong input n$<$1
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\end{description}
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\Remarks
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This procedure does not do array range checks. When called with invalid
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parameters, invalid/nonsense responses or even crashes may be the result.
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\Example
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Calculate the inverse of
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\[
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A=
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\left(
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\begin{array}{rrrr}
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4 & 2 & 4 & 1 \\
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30 & 20 & 45 & 12 \\
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20 & 15 & 36 & 10 \\
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35 & 28 & 70 & 20
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\end{array}
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\right)
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.
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\]
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Below is the source of the invgenex demo that demonstrates invgenex,
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some routines of iom were used to construct matrices.
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\NUMLIBexample{invgenex}
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\ProgramData
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The input datafile looks like:
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\begin{verbatim}
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4 2 4 1
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30 20 45 12
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20 15 36 10
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35 28 70 20
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\end{verbatim}
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\ProgramResults
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\begin{verbatim}
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program results invgenex
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A =
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4.0000000000000000E+0000 2.0000000000000000E+0000
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3.0000000000000000E+0001 2.0000000000000000E+0001
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2.0000000000000000E+0001 1.5000000000000000E+0001
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3.5000000000000000E+0001 2.8000000000000000E+0001
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4.0000000000000000E+0000 1.0000000000000000E+0000
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4.5000000000000000E+0001 1.2000000000000000E+0001
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3.6000000000000000E+0001 1.0000000000000000E+0001
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7.0000000000000000E+0001 2.0000000000000000E+0001
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term= 1
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inverse of A =
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4.0000000000000000E+0000 -2.0000000000000000E+0000
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-3.0000000000000000E+0001 2.0000000000000000E+0001
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2.0000000000000000E+0001 -1.5000000000000000E+0001
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-3.4999999999999999E+0001 2.7999999999999999E+0001
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3.9999999999999999E+0000 -1.0000000000000000E+0000
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-4.4999999999999999E+0001 1.2000000000000000E+0001
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3.5999999999999999E+0001 -1.0000000000000000E+0001
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-6.9999999999999999E+0001 2.0000000000000000E+0001
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\end{verbatim}
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\Precision
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The procedure doesn't supply information about the precision of the
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operation after termination.
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\Method
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The calculation of the inverse is based on LU decomposition with partial
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pivoting.
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\References
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Wilkinson, J.H. and Reinsch, C.\\
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Handbook for Automatic Computation.\\
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Volume II, Linear Algebra.\\
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Springer Verlag, Contribution I/7, 1971.
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\end{procedure}
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\begin{procedure}{invgpd}
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\FunctionDescription
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Procedure to calculate the inverse of a symmetrical, positive
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definitive matrix
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\Dataorganisation The procedure assumes that the calling program has
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declared a two dimensional matrix containing the matrix elements in
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a square partial block.
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\DeclarationandParams
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\lstinline|procedure invgpd(n, rwidth: ArbInt; var ai: ArbFloat; var term: ArbInt);|
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\begin{description}
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\item[n: integer] \mbox{ } \\
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The parameter {\bf n} describes the size of the matrix
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\item[rwidth: integer] \mbox{} \\
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The parameter {\bf rwidth} describes the declared row length of the two dimensional
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matrix.
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\item[var ai: real] \mbox{} \\
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The parameter {\bf ai} must contain to the two dimensional array element
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that is top-left in the matrix.
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After the function has ended successfully (\textbf{term=1}) then
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the input matrix has been changed into its inverse, otherwise the contents
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of the input matrix are undefined.
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\item[var term: integer] \mbox{} \\
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After the procedure has run, this variable contains the reason for
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the termination of the procedure:\\
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{\bf term}=1, successful termination, the inverse has been calculated
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{\bf term}=2, inverse matrix could not be calculated because the matrix
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is (very close to) being singular.
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{\bf term}=3, wrong input n$<$1
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\end{description}
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\Remarks
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\begin{itemize}
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\item Only the bottom left part of the matrix $A$ needs to be passed.
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\item \textbf{Warning} This procedure does not do array range checks. When called with invalid
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parameters, invalid/nonsense responses or even crashes may be the result.
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\end{itemize}
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\Example
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Calculate the inverse of
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\[
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A=
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\left(
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\begin{array}{rrrr}
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5 & 7 & 6 & 5 \\
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7 & 10 & 8 & 7 \\
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6 & 8 & 10 & 9 \\
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5 & 7 & 9 & 10
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\end{array}
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\right)
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.
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\]
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\NUMLIBexample{invgpdex}
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\ProgramData
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\begin{verbatim}
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5
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7 10
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6 8 10
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5 7 9 10
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\end{verbatim}
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\ProgramResults
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\begin{verbatim}
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program results invgpdex
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A =
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5.0000000000000000E+0000 7.0000000000000000E+0000
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7.0000000000000000E+0000 1.0000000000000000E+0001
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6.0000000000000000E+0000 8.0000000000000000E+0000
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5.0000000000000000E+0000 7.0000000000000000E+0000
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6.0000000000000000E+0000 5.0000000000000000E+0000
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8.0000000000000000E+0000 7.0000000000000000E+0000
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1.0000000000000000E+0001 9.0000000000000000E+0000
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9.0000000000000000E+0000 1.0000000000000000E+0001
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term= 1
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inverse of A =
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6.8000000000000000E+0001 -4.1000000000000000E+0001
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-4.1000000000000000E+0001 2.5000000000000000E+0001
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-1.7000000000000000E+0001 1.0000000000000000E+0001
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1.0000000000000000E+0001 -6.0000000000000000E+0000
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-1.7000000000000000E+0001 1.0000000000000000E+0001
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1.0000000000000000E+0001 -6.0000000000000000E+0000
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5.0000000000000000E+0000 -3.0000000000000000E+0000
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-3.0000000000000000E+0000 2.0000000000000000E+0000
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\end{verbatim}
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\Precision
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The procedure doesn't supply information about the precision of the
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operation after termination.
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\Method
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The calculation of the inverse is based on Cholesky decomposition
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for a symmetrical positive definitive matrix.
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\References
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Wilkinson, J.H. and Reinsch, C.\\ Handbook for Automatic Computation.\\
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Volume II, Linear Algebra.\\ Springer Verlag, Contribution I/7, 1971.
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\end{procedure}
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\begin{procedure}{invgsy}
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\FunctionDescription
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Procedure to calculate the inverse of a symmetrical matrix.
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\Dataorganisation
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The procedure assumes that the calling program has declared a two
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dimensional matrix containing the matrix elements in (the bottom
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left part of) a square partial block.
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\DeclarationandParams
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\lstinline|procedure invgsy(n, rwidth: ArbInt; var ai: ArbFloat;var term:ArbInt);|
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\begin{description}
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\item[n: integer] \mbox{ } \\
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The parameter {\bf n} describes the size of the matrix
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\item[rwidth: integer] \mbox{} \\
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The parameter {\bf rwidth} describes the declared row length of the two dimensional
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matrix.
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\item[var ai: real] \mbox{} \\
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The parameter {\bf ai} must contain to the two dimensional array element
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that is top-left in the matrix.
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After the function has ended successfully (\textbf{term=1}) then
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the input matrix has been changed into its inverse, otherwise the contents
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of the input matrix are undefined.
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\item[var term: integer] \mbox{} \\
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After the procedure has run, this variable contains the reason for
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the termination of the procedure:\\
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{\bf term}=1, successful termination, the inverse has been calculated
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{\bf term}=2, inverse matrix could not be calculated because the matrix
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is (very close to) being singular.
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{\bf term}=3, wrong input n$<$1
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\end{description}
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\Remarks
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\begin{itemize}
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\item Only the bottom left part of the matrix $A$ needs to be passed.
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\item \textbf{Warning} This procedure does not do array range checks. When called with invalid
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parameters, invalid/nonsense responses or even crashes may be the result.
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\end{itemize}
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\Example
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Calculating the inverse of
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\[
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A=
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\left(
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\begin{array}{rrrr}
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5 & 7 & 6 & 5 \\
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7 & 10 & 8 & 7 \\
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6 & 8 & 10 & 9 \\
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5 & 7 & 9 & 10
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\end{array}
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\right)
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.
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\]
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\NUMLIBexample{invgsyex}
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\ProgramData
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\begin{verbatim}
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5
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7 10
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6 8 10
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5 7 9 10
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\end{verbatim}
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\ProgramResults
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\begin{verbatim}
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program results invgsyex
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A =
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5.0000000000000000E+0000 7.0000000000000000E+0000
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7.0000000000000000E+0000 1.0000000000000000E+0001
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6.0000000000000000E+0000 8.0000000000000000E+0000
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5.0000000000000000E+0000 7.0000000000000000E+0000
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6.0000000000000000E+0000 5.0000000000000000E+0000
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8.0000000000000000E+0000 7.0000000000000000E+0000
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1.0000000000000000E+0001 9.0000000000000000E+0000
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9.0000000000000000E+0000 1.0000000000000000E+0001
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term= 1
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inverse of A =
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6.8000000000000001E+0001 -4.1000000000000001E+0001
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-4.1000000000000001E+0001 2.5000000000000000E+0001
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-1.7000000000000000E+0001 1.0000000000000000E+0001
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1.0000000000000000E+0001 -6.0000000000000001E+0000
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-1.7000000000000000E+0001 1.0000000000000000E+0001
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1.0000000000000000E+0001 -6.0000000000000001E+0000
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5.0000000000000001E+0000 -3.0000000000000000E+0000
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-3.0000000000000000E+0000 2.0000000000000000E+0000
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\end{verbatim}
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\Precision
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The procedure doesn't supply information about the precision of the
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operation after termination.
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\Method
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The calculation of the inverse is based on reduction of a symmetrical
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matrix to a tridiagonal form.
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\References
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Aasen, J. O. \\
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On the reduction of a symmetric matrix to tridiagonal form. \\
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BIT, 11, (1971), pag. 223-242.
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\end{procedure}
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\end{document}
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