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packages/extra/numlib/doc/inv.tex
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packages/extra/numlib/doc/inv.tex
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%
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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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\begin{document}
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\section{Unit inv}
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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 that is
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independant of the exact real type. (Contrary to directly specifying single,
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real, double or extended in library and examples). Typ.pas and the central
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includefile have some conditional code to switch between floating point
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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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\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 dimensional
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matrix containing the maxtrixelements in a square 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 twodimensional
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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 twodimensional arrayelement
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that is top-left in the matrix.
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After the function has ended succesfully (\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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ongedefinieerd.
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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, succesfull 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 demontrates invgenex, some
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routines of iom were used to construct matrices.
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\begin{lstlisting}
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program invgenex;
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uses typ, iom, inv;
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const n = 4;
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var term : ArbInt;
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A : array[1..n,1..n] of ArbFloat;
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begin
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assign(input, paramstr(1)); reset(input);
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assign(output, paramstr(2)); rewrite(output);
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writeln('program results invgenex');
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{ Read matrix A }
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iomrem(input, A[1,1], n, n, n);
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{ Print matrix A }
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writeln; writeln('A =');
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iomwrm(output, A[1,1], n, n, n, numdig);
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{ Calculation of the inverse of A }
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invgen(n, n, A[1,1], term);
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writeln; writeln('term=', term:2);
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if term=1 then
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{ print inverse inverse of A }
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begin
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writeln; writeln('inverse of A =');
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iomwrm(output, A[1,1], n, n, n, numdig);
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end; {term=1}
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close(input); close(output)
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end.
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\end{lstlisting}
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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, postivive definite
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%van een symmetrische,positief definiete matrix.
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\Dataorganisation
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The procedure assumes that the calling program has declared a two dimensional
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matrix containing the maxtrixelements in 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 rowlength of the twodimensional
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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 twodimensional arrayelement
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that is top-left in the matrix.
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After the function has ended succesfully (\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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ongedefinieerd.
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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, succesfull 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 bottomleft 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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\begin{lstlisting}
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program invgpdex;
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uses typ, iom, inv;
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const n = 4;
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var i, j, term : ArbInt;
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A : array[1..n,1..n] of ArbFloat;
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begin
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assign(input, paramstr(1)); reset(input);
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assign(output, paramstr(2)); rewrite(output);
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writeln('program results invgpdex');
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{ Read bottom leftpart of matrix A}
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for i:=1 to n do iomrev(input, A[i,1], i);
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{ print matrix A }
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writeln; writeln('A =');
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for i:=1 to n do for j:=1 to i-1 do A[j,i]:=A[i,j];
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iomwrm(output, A[1,1], n, n, n, numdig);
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{ Calculate inverse of matrix A}
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invgpd(n, n, A[1,1], term);
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writeln; writeln('term=', term:2);
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if term=1 then
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{ Print inverse of matrix A}
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begin
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writeln; writeln('inverse of A =');
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iomwrm(output, A[1,1], n, n, n, numdig);
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end; {term=1}
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close(input); close(output)
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end.
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\end{lstlisting}
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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 for a
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symmetrical positive definitie 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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|
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The procedure assumes that the calling program has declared a two
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dimensional matrix containing the maxtrixelements in (the bottomleft part
|
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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
|
||||
\item[rwidth: integer] \mbox{} \\
|
||||
The parameter {\bf rwidth} describes the declared rowlength of the twodimensional
|
||||
matrix.
|
||||
\item[var ai: real] \mbox{} \\
|
||||
The parameter {\bf ai} must contain to the twodimensional arrayelement
|
||||
that is top-left in the matrix.
|
||||
After the function has ended succesfully (\textbf{term=1}) then
|
||||
the input matrix has been changed into its inverse, otherwise the contents
|
||||
of the input matrix are undefined.
|
||||
ongedefinieerd.
|
||||
\item[var term: integer] \mbox{} \\
|
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After the procedure has run, this variable contains the reason for
|
||||
the termination of the procedure:\\
|
||||
{\bf term}=1, succesfull termination, the inverse has been calculated
|
||||
{\bf term}=2, inverse matrix could not be calculated because the matrix
|
||||
is (very close to) being singular.
|
||||
{\bf term}=3, wrong input n$<$1
|
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|
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\end{description}
|
||||
|
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\Remarks
|
||||
\begin{itemize}
|
||||
\item Only the bottomleft part of the matrix $A$ needs to be passed.
|
||||
\item \textbf{Warning} This procedure does not do array range checks. When called with invalid
|
||||
parameters, invalid/nonsense responses or even crashes may be the result.
|
||||
\end{itemize}
|
||||
|
||||
\Example
|
||||
|
||||
Het berekenen van de inverse van
|
||||
\[
|
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A=
|
||||
\left(
|
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\begin{array}{rrrr}
|
||||
5 & 7 & 6 & 5 \\
|
||||
7 & 10 & 8 & 7 \\
|
||||
6 & 8 & 10 & 9 \\
|
||||
5 & 7 & 9 & 10
|
||||
\end{array}
|
||||
\right)
|
||||
.
|
||||
\]
|
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\begin{lstlisting}
|
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program invgsyex;
|
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uses typ, iom, inv;
|
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const n = 4;
|
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var i, j, term : ArbInt;
|
||||
A : array[1..n,1..n] of ArbFloat;
|
||||
begin
|
||||
assign(input, paramstr(1)); reset(input);
|
||||
assign(output, paramstr(2)); rewrite(output);
|
||||
writeln('program results invgsyex');
|
||||
{ Read bottomleft part of matrix A}
|
||||
for i:=1 to n do iomrev(input, A[i,1], i);
|
||||
{ print matrix A}
|
||||
writeln; writeln('A =');
|
||||
for i:=1 to n do for j:=1 to i-1 do A[j,i]:=A[i,j];
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
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{ calculate inverse of matrix A}
|
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invgsy(n, n, A[1,1], term);
|
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writeln; writeln('term=', term:2);
|
||||
if term=1 then
|
||||
{ print inverse of matrix A}
|
||||
begin
|
||||
writeln; writeln('inverse of A =');
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
||||
end; {term=1}
|
||||
close(input); close(output)
|
||||
end.
|
||||
\end{lstlisting}
|
||||
|
||||
\ProgramData
|
||||
|
||||
\begin{verbatim}
|
||||
5
|
||||
7 10
|
||||
6 8 10
|
||||
5 7 9 10
|
||||
\end{verbatim}
|
||||
|
||||
\ProgramResults
|
||||
|
||||
\begin{verbatim}
|
||||
program results invgsyex
|
||||
|
||||
A =
|
||||
5.0000000000000000E+0000 7.0000000000000000E+0000
|
||||
7.0000000000000000E+0000 1.0000000000000000E+0001
|
||||
6.0000000000000000E+0000 8.0000000000000000E+0000
|
||||
5.0000000000000000E+0000 7.0000000000000000E+0000
|
||||
|
||||
6.0000000000000000E+0000 5.0000000000000000E+0000
|
||||
8.0000000000000000E+0000 7.0000000000000000E+0000
|
||||
1.0000000000000000E+0001 9.0000000000000000E+0000
|
||||
9.0000000000000000E+0000 1.0000000000000000E+0001
|
||||
|
||||
term= 1
|
||||
|
||||
inverse of A =
|
||||
6.8000000000000001E+0001 -4.1000000000000001E+0001
|
||||
-4.1000000000000001E+0001 2.5000000000000000E+0001
|
||||
-1.7000000000000000E+0001 1.0000000000000000E+0001
|
||||
1.0000000000000000E+0001 -6.0000000000000001E+0000
|
||||
|
||||
-1.7000000000000000E+0001 1.0000000000000000E+0001
|
||||
1.0000000000000000E+0001 -6.0000000000000001E+0000
|
||||
5.0000000000000001E+0000 -3.0000000000000000E+0000
|
||||
-3.0000000000000000E+0000 2.0000000000000000E+0000
|
||||
\end{verbatim}
|
||||
|
||||
\Precision
|
||||
|
||||
The procedure doesn't supply information about the precision of the
|
||||
operation after termination.
|
||||
|
||||
\Method
|
||||
|
||||
The calculation of the inverse is based on reduction of a symmetrical
|
||||
matrix to a tridiagonal form.
|
||||
|
||||
\References
|
||||
|
||||
Aasen, J. O. \\
|
||||
On the reduction of a symmetric matrix to tridiagonal form. \\
|
||||
BIT, 11, (1971), pag. 223-242.
|
||||
|
||||
\end{procedure}
|
||||
|
||||
\end{document}
|
||||
|
||||
4
packages/extra/numlib/examples/invgenex.dat
Normal file
4
packages/extra/numlib/examples/invgenex.dat
Normal file
@ -0,0 +1,4 @@
|
||||
4 2 4 1
|
||||
30 20 45 12
|
||||
20 15 36 10
|
||||
35 28 70 20
|
||||
39
packages/extra/numlib/examples/invgenex.pas
Normal file
39
packages/extra/numlib/examples/invgenex.pas
Normal file
@ -0,0 +1,39 @@
|
||||
{
|
||||
$Id$
|
||||
|
||||
}
|
||||
|
||||
program invgenex;
|
||||
uses typ, iom, inv;
|
||||
const n = 4;
|
||||
var term : arbint;
|
||||
A : array[1..n,1..n] of arbfloat;
|
||||
begin
|
||||
assign(input, paramstr(1)); reset(input);
|
||||
assign(output, paramstr(2)); rewrite(output);
|
||||
writeln('program results invgenex');
|
||||
{ Read matrix A}
|
||||
iomrem(input, A[1,1], n, n, n);
|
||||
{ Print matrix A }
|
||||
writeln; writeln('A =');
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
||||
{ Calculate inverse of A}
|
||||
invgen(n, n, A[1,1], term);
|
||||
writeln; writeln('term=', term:2);
|
||||
if term=1 then
|
||||
{ Print inverse of matrix A}
|
||||
begin
|
||||
writeln; writeln('inverse of A =');
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
||||
end; {term=1}
|
||||
close(input); close(output)
|
||||
end.
|
||||
|
||||
{
|
||||
|
||||
$Log$
|
||||
Revision 1.1 2004-04-18 14:47:11 marco
|
||||
* initial versions
|
||||
|
||||
|
||||
}
|
||||
4
packages/extra/numlib/examples/invgpdex.dat
Normal file
4
packages/extra/numlib/examples/invgpdex.dat
Normal file
@ -0,0 +1,4 @@
|
||||
5
|
||||
7 10
|
||||
6 8 10
|
||||
5 7 9 10
|
||||
37
packages/extra/numlib/examples/invgpdex.pas
Normal file
37
packages/extra/numlib/examples/invgpdex.pas
Normal file
@ -0,0 +1,37 @@
|
||||
{
|
||||
$Id$
|
||||
}
|
||||
|
||||
program invgpdex;
|
||||
uses typ, iom, inv;
|
||||
const n = 4;
|
||||
var i, j, term : ArbInt;
|
||||
A : array[1..n,1..n] of ArbFloat;
|
||||
begin
|
||||
assign(input, paramstr(1)); reset(input);
|
||||
assign(output, paramstr(2)); rewrite(output);
|
||||
writeln('program results invgpdex');
|
||||
{ read bottomleft part of matrix A}
|
||||
for i:=1 to n do iomrev(input, A[i,1], i);
|
||||
{ Print matrix A}
|
||||
writeln; writeln('A =');
|
||||
for i:=1 to n do for j:=1 to i-1 do A[j,i]:=A[i,j];
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
||||
{ Calculate inverse of matrix A}
|
||||
invgpd(n, n, A[1,1], term);
|
||||
writeln; writeln('term=', term:2);
|
||||
if term=1 then
|
||||
{ Print inverse of matrix A}
|
||||
begin
|
||||
writeln; writeln('inverse of A =');
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
||||
end; {term=1}
|
||||
close(input); close(output)
|
||||
end.
|
||||
|
||||
{
|
||||
$Log$
|
||||
Revision 1.1 2004-04-18 14:47:11 marco
|
||||
* initial versions
|
||||
|
||||
}
|
||||
4
packages/extra/numlib/examples/invgsyex.dat
Normal file
4
packages/extra/numlib/examples/invgsyex.dat
Normal file
@ -0,0 +1,4 @@
|
||||
5
|
||||
7 10
|
||||
6 8 10
|
||||
5 7 9 10
|
||||
38
packages/extra/numlib/examples/invgsyex.pas
Normal file
38
packages/extra/numlib/examples/invgsyex.pas
Normal file
@ -0,0 +1,38 @@
|
||||
{
|
||||
$Id $
|
||||
}
|
||||
|
||||
program invgsyex;
|
||||
uses typ, iom, inv;
|
||||
const n = 4;
|
||||
var i, j, term : ArbInt;
|
||||
A : array[1..n,1..n] of ArbFloat;
|
||||
begin
|
||||
assign(input, paramstr(1)); reset(input);
|
||||
assign(output, paramstr(2)); rewrite(output);
|
||||
writeln('program results invgsyex');
|
||||
{ Read bottomleft part of matrix A }
|
||||
for i:=1 to n do iomrev(input, A[i,1], i);
|
||||
{ print matrix A }
|
||||
writeln; writeln('A =');
|
||||
for i:=1 to n do for j:=1 to i-1 do A[j,i]:=A[i,j];
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
||||
{ calculate inverse of matrix A}
|
||||
invgsy(n, n, A[1,1], term);
|
||||
writeln; writeln('term=', term:2);
|
||||
if term=1 then
|
||||
{ print inverse of matrix A }
|
||||
begin
|
||||
writeln; writeln('inverse of A =');
|
||||
iomwrm(output, A[1,1], n, n, n, numdig);
|
||||
end; {term=1}
|
||||
close(input); close(output)
|
||||
end.
|
||||
|
||||
{
|
||||
$Log$
|
||||
Revision 1.1 2004-04-18 14:47:11 marco
|
||||
* initial versions
|
||||
|
||||
|
||||
}
|
||||
Loading…
Reference in New Issue
Block a user