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\usepackage{fpc}
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commentstyle=\itshape,
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frame=
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\newcommand{\Dataorganisation}{\item[Data Struct]\rmfamily}
\newcommand{\DeclarationandParams}{\item[Declaration]\rmfamily}
\newcommand{\References}{\item[References]\rmfamily}
\newcommand{\Method}{\item[Method]\rmfamily}
\newcommand{\Precision}{\item[Precision]\rmfamily}
\newcommand{\Remarks}{\item[Remarks]\rmfamily}
\newcommand{\Example}{\item[Example]\rmfamily}
\newcommand{\ProgramData}{\item[Example Data]\rmfamily}
\newcommand{\ProgramResults}{\item[Example Result]\rmfamily}
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\par \file{\textbf{Listing:} \exampledir/#1.pas}%
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\begin{document}
\FPCexampledir{../examples}
\section{general comments}
\textbf{Original comments:} \\
The used floating point type \textbf{real} depends on the used version,
see the general introduction for more information. You'll need to USE
units typ an inv to use these routines.
\textbf{MvdV notes:} \\
Integers used for parameters are of type "ArbInt" to avoid problems with
systems that define integer differently depending on mode.
Floating point values are of type "ArbFloat" to allow writing code that is
independant of the exact real type. (Contrary to directly specifying single,
real, double or extended in library and examples). Typ.pas and the central
includefile have some conditional code to switch between floating point
types.
These changes were already prepared somewhat when I got the lib, but weren't
consequently applied. I did that while porting to FPC.
\section{Unit inv}
\begin{procedure}{invgen}
\FunctionDescription
Procedure to calculate the inverse of a matrix.
\Dataorganisation
The procedure assumes that the calling program has declared a two dimensional
matrix containing the maxtrixelements in a square partial block.
\DeclarationandParams
\lstinline|procedure invgen(n, rwidth: ArbInt; var ai: ArbFloat;var term: ArbInt);|
\begin{description}
\item[n: integer] \mbox{ } \\
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.
\item[var term: integer] \mbox{} \\
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
\end{description}
\Remarks
This procedure does not do array range checks. When called with invalid
parameters, invalid/nonsense responses or even crashes may be the result.
\Example
Calculate the inverse of
\[
A=
\left(
\begin{array}{rrrr}
4 & 2 & 4 & 1 \\
30 & 20 & 45 & 12 \\
20 & 15 & 36 & 10 \\
35 & 28 & 70 & 20
\end{array}
\right)
.
\]
Below is the source of the invgenex demo that demontrates invgenex, some
routines of iom were used to construct matrices.
\NUMLIBexample{invgenex}
\ProgramData
The input datafile looks like:
\begin{verbatim}
4 2 4 1
30 20 45 12
20 15 36 10
35 28 70 20
\end{verbatim}
\ProgramResults
\begin{verbatim}
program results invgenex
A =
4.0000000000000000E+0000 2.0000000000000000E+0000
3.0000000000000000E+0001 2.0000000000000000E+0001
2.0000000000000000E+0001 1.5000000000000000E+0001
3.5000000000000000E+0001 2.8000000000000000E+0001
4.0000000000000000E+0000 1.0000000000000000E+0000
4.5000000000000000E+0001 1.2000000000000000E+0001
3.6000000000000000E+0001 1.0000000000000000E+0001
7.0000000000000000E+0001 2.0000000000000000E+0001
term= 1
inverse of A =
4.0000000000000000E+0000 -2.0000000000000000E+0000
-3.0000000000000000E+0001 2.0000000000000000E+0001
2.0000000000000000E+0001 -1.5000000000000000E+0001
-3.4999999999999999E+0001 2.7999999999999999E+0001
3.9999999999999999E+0000 -1.0000000000000000E+0000
-4.4999999999999999E+0001 1.2000000000000000E+0001
3.5999999999999999E+0001 -1.0000000000000000E+0001
-6.9999999999999999E+0001 2.0000000000000000E+0001
\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 LU decomposition with partial
pivoting.
\References
Wilkinson, J.H. and Reinsch, C.\\
Handbook for Automatic Computation.\\
Volume II, Linear Algebra.\\
Springer Verlag, Contribution I/7, 1971.
\end{procedure}
\begin{procedure}{invgpd}
\FunctionDescription
Procedure to calculate the inverse of a symmetrical, positive definite
matrix
%van een symmetrische,positief definiete matrix.
\Dataorganisation
The procedure assumes that the calling program has declared a two dimensional
matrix containing the maxtrixelements in a square partial block.
\DeclarationandParams
\lstinline|procedure invgpd(n, rwidth: ArbInt; var ai: ArbFloat; var term: ArbInt);|
\begin{description}
\item[n: integer] \mbox{ } \\
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.
\item[var term: integer] \mbox{} \\
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
\end{description}
\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
Calculate the inverse of
\[
A=
\left(
\begin{array}{rrrr}
5 & 7 & 6 & 5 \\
7 & 10 & 8 & 7 \\
6 & 8 & 10 & 9 \\
5 & 7 & 9 & 10
\end{array}
\right)
.
\]
\NUMLIBexample{invgpdex}
\ProgramData
\begin{verbatim}
5
7 10
6 8 10
5 7 9 10
\end{verbatim}
\ProgramResults
\begin{verbatim}
program results invgpdex
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 =
6.8000000000000000E+0001 -4.1000000000000000E+0001
-4.1000000000000000E+0001 2.5000000000000000E+0001
-1.7000000000000000E+0001 1.0000000000000000E+0001
1.0000000000000000E+0001 -6.0000000000000000E+0000
-1.7000000000000000E+0001 1.0000000000000000E+0001
1.0000000000000000E+0001 -6.0000000000000000E+0000
5.0000000000000000E+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 Cholesky decomposition for a
symmetrical positive definitie matrix.
\References
Wilkinson, J.H. and Reinsch, C.\\ Handbook for Automatic Computation.\\
Volume II, Linear Algebra.\\ Springer Verlag, Contribution I/7, 1971.
\end{procedure}
\begin{procedure}{invgsy}
\FunctionDescription
Procedure to calculate the inverse of a symmetrical matrix.
\Dataorganisation
The procedure assumes that the calling program has declared a two
dimensional matrix containing the maxtrixelements in (the bottomleft part
of) a square partial block.
\DeclarationandParams
\lstinline|procedure invgsy(n, rwidth: ArbInt; var ai: ArbFloat;var term:ArbInt);|
\begin{description}
\item[n: integer] \mbox{ } \\
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.
\item[var term: integer] \mbox{} \\
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
\end{description}
\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
\[
A=
\left(
\begin{array}{rrrr}
5 & 7 & 6 & 5 \\
7 & 10 & 8 & 7 \\
6 & 8 & 10 & 9 \\
5 & 7 & 9 & 10
\end{array}
\right)
.
\]
\NUMLIBexample{invgsyex}
\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}