\input texinfo @c -*-texinfo-*-
@ignore
verifier si on a bien les nouveautes de la 1.02
1/ sensy_to_var
2/ Borel et Boreleig OK
3/ v.p. param OK
4/ Helps et listing divers OK
5/ Revoir Probes avec Pat => et code !
6/ data doivent-elles etre specifique de kalman?
7/ symbolic en premier dans Borel a faire
@end ignore
@setfilename mini_ker.info
@include version.texi

@c @set myversion @value{VERSION}
@set myversion 102

@set myurl @url{http://www.environnement.ens.fr/@/perso/@/dumas/@/mini_ker/@/software.html}

@macro Minik{}
Miniker
@end macro

@settitle @Minik{} @value{myversion} manual

@syncodeindex fn vr

@dircategory Miscellaneous
@direntry
* Miniker: (mini_ker).            The mini_ker modeling tool.
@end direntry


@copying
@noindent Copyright (C) 2004, 2005, 2006, 2007 Alain Lahellec@*
Copyright (C) 2004, 2005, 2006, 2007 Patrice Dumas@*
Copyright (C) 2004, St@'ephane Hallegatte

@quotation
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.1 or
any later version published by the Free Software Foundation; with no
Invariant Sections, with no Front-Cover text and with no Back-Cover Text.  
A copy of the license is included in the section entitled ``GNU Free 
Documentation License.''
@end quotation
@end copying

@titlepage
@title @Minik{} manual
@subtitle for @Minik{} version @value{myversion}, @value{UPDATED}

@author The TEF Collaboration
@c @author Alain Lahellec
@c @author Patrice Dumas
@c @author St@'ephane Hallegatte

@page
@vskip 0pt plus 1filll
@insertcopying
@end titlepage

@ifset texi2html
@node Top
@top @Minik{} @value{myversion} manual


@c @insertcopying
@end ifset

@ifnottex
@node Top
@top @Minik{} @value{myversion} manual

@strong{By: The TEF Collaboration}

@insertcopying

@end ifnottex

@menu
* Introduction::
* TEF overview::
* A model with @Minik{}::
* Advanced programming::
* Dynamic system analysis::
* Advanced use of @Minik{} with make::

Indices

* Concepts index::
* Variables macros and functions index::

Appendices

* Installation::
* Cmz directives reference::
@ignore
* Resolution method::
* @Minik{} macros::
@end ignore
* Copying This Manual::           The GNU Free Documentation License.
@end menu

@contents

@node Introduction
@unnumbered Introduction

@cindex TEF
@cindex cells
@cindex transfers
@cindex ZOOM
@cindex mortran 

   @Minik{} is a modeling tool, built especially in order to implement
models written following the @acronym{TEF,Transfer Evolution Formalism} 
formalism, a mathematical framework for system analysis and
simulation. @Minik{} allows for timewise simulation, system analysis,
adjoint computation, Kalman filtering and more.

@Minik{} uses a fortran preprocessor, @command{mortran}, designed in the 
1970's, to ease model writing using dedicated specific languages.
For example partial derivatives are
symbolicaly determined by @command{mortran} macros in @Minik{}. 
For the selection of 
another compile-time features, another set of preprocessor directives,
the @dfn{cmz directives}, are used. In most cases the user does not need to
know anything about that preprocessing that occurs behind the scene,
he simply writes down the equations of his model and he is done.

@c All partial derivatives needed to solve the TEF system are automatically
@c determined during the pre-compilation stage.
@c Once all models written down and initial conditions
@c given using a pseudo-Fortran type of language, the model is ready to run.

@c The language developed to get automatic symbolic partial derivatives
@c uses the Fortran pre-compiler @command{mortran}, designed in the 1970's.

A comprehensive description
of the @acronym{TEF} formalism in available on 
@url{http://www.lmd.jussieu.fr/ZOOM/doc/tef-GB-partA5.pdf}).
The @Minik{} software is a reduced version of 
@uref{http://www.lmd.jussieu.fr@//zoom,@strong{ZOOM}}, 
that can only handle a hundreds of variables, but is much easier to use.

@menu 
* Intended audience::
* Reading guide::
* Other Manuals and documentation::
@end menu 

@node Intended audience
@unnumberedsec Intended audience

The reader should have notions in system dynamics.
@c and understand the basis of the TEF. 
Moreover a minimal knowledge of programmation and fortran is 
required. What is required is a basic understanding of variable types, 
affectation and fortran expressions.

@node Reading guide
@unnumberedsec Reading guide

The first chapter is a brief overview of the @acronym{TEF}.
The following describes how to write, compile and run a model in @Minik{} 
in its basic and comprehensive syntax.
@c Reading the sections of this chapter up to the section 
@c @emph{Symbolic model description} is required to know the
@c syntax of model description in @Minik{}.
Reading up to the section
@emph{Controlling the run} is required to be able to use @Minik{}. 
In this section it is assumed that @Minik{} is properly setup. The
installation instructions are in the appendix at
@ref{Installation}.

@c 2 programming environment to compile the model are available, with cmz 
@c and make, you can skip the method description you are not interested in. 
@c A reference for the usefull cmz directives is also in the appendix
@c (@pxref{Cmz directives reference}).

@c You should also
@c read the following section, @emph{Symbolic model description} which presents an
@c alternate syntax for model description, such that you can choose what you
@c prefer.

The next chapter describes advanced features, first a general introduction to
features settings and then a description of other model description related
features.

The next chapter describes system analysis tools available with @Minik{}. The
sections are independant and each describes how to use a specific feature. If
you plan on using these features, you should also read 
@ref{Selecting features, , Overview of feature setting}.

A final chapter describes advanced features in a development environment
using make,

In the appendix the instructions for the installation are described
(@pxref{Installation}). 

@node Other Manuals and documentation
@unnumberedsec  Other Manuals and documentation

A programmers'Manual is available (in French), and can be asked for to 
any member of the collabration. See additional documents in 
 @url{http://www.lmd.jussieu.fr/Zoom/doc} or ask for Research 
texts and articles to members.

@node TEF overview
@chapter An overview of the @acronym{TEF} formalism

The @acronym{TEF, Transfer Evolution Formalism} is based on partitionning
and recoupling of model subsystems. It allows the study of the coupling 
between subsystems by the means of linearization and time discretization.

@menu 
* Cell and Transfer::
* Linearization and discretization::
@end menu

@node Cell and Transfer
@section Cell and Transfer equations

In the @acronym{TEF}, a model is
mathematically represented by a set of equations corresponding
to two kinds objects:

@enumerate
@item Cells which are elementary models and correspond to evolution equations
such as:
@ifset latex
@tex
\begin{eqnarray}
\partial_t \eta (t) &=&  g(\eta(t),\varphi(t))
\label{cells}
\end{eqnarray}
@end tex
@end ifset
@ifclear latex
@tex
$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
@end tex
@ifnottex

@noindent @math{d eta(t)/d t = g(eta(t),phi(t))}

@end ifnottex
@end ifclear

Vector @math{\eta} represent the state variables of cells and
the vector @math{\varphi} represent the dependent
boundary conditions, @i{i.e.} the
variables considered as boundary conditions by a cell, but depending upon
the complete model state. This dependent boundary conditions are
required to make the cells correspond to well-posed problems.
@c FIXME acceptable?
These variables are often called state variables, and prognostic 
variables in meteorology.


@item Transfers which are determined by constraint equations such as:
@ifset latex
@tex
\begin{eqnarray}
\varphi(t) &=& f(\eta(t),\varphi(t))
\label{transfer}
\end{eqnarray}
@end tex
@end ifset
@ifclear latex
@tex
$$
\varphi(t) = f(\eta(t),\varphi(t))
$$
@end tex
@ifnottex

@noindent @math{phi(t) = f(eta(t),phi(t))}

@end ifnottex
@end ifclear
These equations are often called algebraic equations, and in meteorology
diagnostic equations.
@end enumerate

@node Linearization and discretization
@section Linearization and discretization in the @acronym{TEF}

The relations between sub-systems is excessively difficult to exhibit when
having to cope with non-linear system. In the @acronym{TEF}, the 
@acronym{TLS, Tangent Linear System} is constructed along the trajectory.
One considers the system over a small portion along the trajectory, say
between @math{t} and @math{t + \delta t}. The variation @math{\delta \eta}
of @math{\eta} and @math{\delta \varphi} of @math{\varphi} is obtained 
through a Pad@'e approximation of the state-transition matrix. The final
form of the algebraic system is closed to the classical Crank-Nicolson scheme:

@c FIXME PAd'e? od Taylor?
@c through a Taylor expansion followed by time integration.
@c A time scheme is then applied to the @acronym{TLS} (a Crank-Nicholson scheme),
@c to obtain an algebraic system describing the relationships between 
@c variations of transfers and cells variables:


@ifset latex
@tex
$$\left(\begin{array}{cc} 
A & B\\
-C^+ & I-D 
\end{array}\right) \left(\begin{array}{c}
\delta \eta\\
\delta \varphi
\end{array}\right) = \left(\begin{array}{c} 
\Gamma\\
\Omega \end{array}\right)$$
@end tex
@end ifset

@ifclear latex
@tex
$$\pmatrix{A & B\cr
-C^+ & I-D\cr} \pmatrix{\delta \eta\cr
\delta \varphi\cr} = \pmatrix{\Gamma\cr
\Omega\cr}$$
@end tex
@end ifclear

The blocks appearing in the Jacobian matrix are constructed with partial derivative
of @math{f} and @math{g}, and with @math{\delta t}. From this system the
elimination of @math{\delta \eta} leads to another formulation giving
the coupling between transfers, and allows for the @math{\delta \varphi}
computation. The @math{\delta \varphi} value is then substitued in 
@math{\delta \eta} to complete the time-step solving process.

@node A model with @Minik{}
@chapter @Minik{} model programming

@cindex sequences

@Minik{} works by combining the model specification code given by 
the user and other source files provided in the package. The code is
assembled, preprocessed, compiled, linked and the resulting program 
can be run to produce the model trajectory and dynamic analysis.

The code provided in the package contains a principal program, some usefull
subroutines and pieces of code called @dfn{sequences} combined with the
different codes. Among these sequences some hold the code describing the model
and are to be written by the user (sequences are similar to
Fortran include files).

@menu
* Structure of the code::
* A model description::
* Setting and running a model::
* Controlling the run::
@end menu

@node Structure of the code
@section General structure of the code

@cindex sequence
@cindex zinit, general

The sequences used to enter model description hold the @c vector dimensions, 
mathematical formulae for each cell and transfer component, dedicated
derived computations, and time-step 
steering. During the code generation stage, 
cmz directives are preprocessed, then the user pseudo-Fortran
instructions are translated by @command{mortran} using macros designed to 
generate in particular all Fortran instructions that compute the Jacobian 
matrices used in @acronym{TEF} modelling.

@c A first users' sequence to program is: @file{dimetaphi} where the model 
@c dimensions are given, for the two vector-array @code{eta(.)} for cells 
@c and @code{ff(.)} for transfers (@pxref{Model size,,Entering model size}).
 
The sequence @file{zinit} contains the mathematical formulation of the model
(@pxref{Model equation and parameters, Entering model equation and parameters}). 
Another sequence, @file{zsteer}, is merged at
the end of the time step advance of the simulation, where the user can 
monitor the time step values and printing levels, and perform particular 
computations etc.
(@pxref{End of time step, ,Executing code at the end of each time step}).

@node A model description
@section @Minik{} programming illustrated

@cindex TEF

The general @acronym{TEF} system writes:
@ifset latex
@tex
\begin{eqnarray}
\partial_t \eta (t) &=&  g(\eta(t),\varphi(t)) \nonumber\\
\varphi(t) &=& f(\eta(t),\varphi(t))
\label{tef}
\end{eqnarray}
@end tex
@end ifset
@ifclear latex
@tex
$$\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
}$$
@end tex
@ifnottex

@noindent @math{d eta(t)/d t = g(eta(t),phi(t))@*
phi(t) = f(eta(t),phi(t))}

@end ifnottex
@end ifclear

To illustrate the model description in @Minik{} a simple predator-prey
model of Lotka-Volterra is used.
This model can be written in the following @acronym{TEF} form:

@ifset latex
@tex
\begin{equation}
\left\{
\begin{array}{cc}
\partial_t \eta _{prey} =&  a \eta _{prey} - a \varphi _{meet} \\
\partial_t \eta _{pred} =&  -c \eta _{pred} + c \varphi _{meet} \nonumber\\
\end{array}
\right.
\end{equation}
\begin{equation}
\varphi _{meet} = \eta _{prey}\eta _{pred}
\label{pred}
\end{equation}
@end tex
@end ifset
@ifclear latex
@tex
$$\left\{\eqalign{\partial_t \eta _{prey} &=  a \eta _{prey} - a \varphi _{meet} \cr
\partial_t \eta _{pred} &=  -c \eta _{pred} + c \varphi _{meet}\cr}\right.$$
@end tex
@tex
$$\varphi _{meet} = \eta _{prey}\eta _{pred}$$
@end tex
@ifnottex
@noindent @math{d eta_prey(t)/d t = a * eta_prey - a * phi_meet@*
d eta_pred(t)/d t = -c * eta_pred +c * phi_meet}

@noindent @math{phi_meet = eta_prey * eta_pred}
@end ifnottex
@end ifclear

with two cell equations, @i{i.e}. state evolution of the prey and predator 
groups, and one transfer accounting for the meeting of individuals of 
different group.

@menu
* A note about mortran and cmz directives::
* Model equation and parameters::
@end menu

@node A note about mortran and cmz directives
@subsection All you need to know about mortran and cmz directives

@cindex mortran 

The first stage of code generation consists in cmz directives preprocessing.
Cmz directives are used for conditional selection of features, and sequence 
inclusion. At that point you don't need to know anything about these
directives. They are only usefull if you want to take advantage of advanced 
features 
(@pxref{Programming with cmz directives}).

The code in sequences is written in Mortran and the second stage of code
generation consists in mortran macro expansion. The mortran language is 
described
in its own manual, here we only explain the very basics which is all you need
to know to use @Minik{}. Mortran basic instructions are almost Fortran,
the differences are the following:

@itemize @bullet
@item The code is free-form, and each statement should end with a semi-colon 
@code{;}.
@item Comments may be introduced by an exclamation mark @code{!} at the 
beginning of a line, or appear within double quotes @code{"} in a single line.
@item It is possible to use blocs, for @code{do} or @code{if} statement 
for example, and they are enclosed within brackets @samp{<} and @samp{>}.
To be in the safe side, a semi-colon @code{;} should be added after a 
closng bracket @code{>}.
@end itemize

The following fictious code is legal mortran:

@example
real 
  param;
param = 3.; ff(1) = ff(3)**eta(1);       "a comment"                 
! a line comment
do inode=1,n_node <eta_move(inode)=0.01; eta_speed(inode)=0.0;>;
@end example

Thanks to mortran the model code is very simply specified, as you'll 
see next.


@node Model equation and parameters
@subsection Entering model equation and parameters

@cindex @file{zinit}
@vindex dt
@vindex time
@vindex nstep
@vindex modzprint

The model equation and parameters and some @Minik{} parameters are entered in
the @file{zinit} sequence. The whole layout of the model is given
before detailing the keywords.

@example
!%%%%%%%%%%%%%%%%%%%%%%
! Parameters           
!%%%%%%%%%%%%%%%%%%%%%%
  real apar,bpar;        "optional Fortran type declaration"

! required parameters
     dt=.01;             "initial time-step"
     nstep=10 000;       "number of iterations along the trajectory"
     time=0.;            "time initialisation "

! model parameters            
     apar = 1.5;             
     cpar = 0.7;          
                                                  
! misceallaneous parameters
     modzprint = 1000;    "printouts frequency" 

print*,'***************************************';
print*,'Lotka-Volterra model with parameters as:';
z_pr: apar,bpar;
print*,'***************************************';

!%%%%%%%%%%%%%%%%%%%%%%
! Transfer definition
!%%%%%%%%%%%%%%%%%%%%%%
! rencontre (meeting)
set_Phi
< var: ff_interact, fun: f_interact = eta_prey*eta_pred;
>;

!%%%%%%%%%%%%%%%%%%%%%%
! Cell definition
!%%%%%%%%%%%%%%%%%%%%%%

set_eta
< var: eta_prey, fun: deta_prey =   apar*eta_prey - apar*ff_interact;
  var: eta_pred, fun: deta_pred = - cpar*eta_pred + cpar*ff_interact;
>;


!%%%%%%%%%%%%%%%%%%%%%%
! Initial states
!%%%%%%%%%%%%%%%%%%%%%%
     eta_prey = 1.;
     eta_pred = 1.;
;
    OPEN(50,FILE='title.tex',STATUS='UNKNOWN');   "title file"
    write(50,5000) apar,cpar;                     
5000;format('Lotka-Volterra par:',2F4.1);
@end example

@subsubheading Variables and model parameters

The following variables are mandatory:

@table @code
@item dt
The time step.
@item time
Model time initialisation.
@item nstep
Number of iterations along the trajectory.
@end table

There are no other mandatory variables. Some optional variables are used
to monitor the printout and ouput of results of the code.
As an example, the variable @code{modzprint} is used to set
the frequency of the printout of the model matrix and vectors during the
run (@pxref{Controlling the printout and data output}).

User's defined variable and Fortran or Mortran instructions can always be
added for intermediate calculus. To avoid conflict with the variables of the
@Minik{} code, the rule is that a users symbol must not have characters 
@samp{o} 
in the first two symbol characters.

In the predator-prey example there are two model parameters. The fortran
variables are called here @code{apar} for @math{a} and @code{cpar} for @math{c}.
If a Fortan type definition is needed, it should be set at the very beginning
of @file{zinit}. The predator-prey code variable initializations finally reads

@example
@group
!%%%%%%%%%%%%%%%%%%%%%%
! Parameters         
!%%%%%%%%%%%%%%%%%%%%%%
  real apar,bpar;        "optional Fortran type declaration"

     dt=.01;           
     nstep=10 000;    
     time=0.;            

! model parameters
     apar = 1.5;           
     cpar = 0.7;                                                            

     modzprint = 1000;  
@end group
@end example

@subsubheading Model equations
@anchor{Model equations}

@findex set_Phi
@findex set_eta
@vindex var:
@vindex fun:
@vindex eqn:

The model equations for cells and model equations for transferts are 
entered in two mortran blocks, one for the transferts, the other for the
cell components.  The model equations for cells are entered into a 
@code{set_eta} block, and the transfer equations are entered into a
@code{set_phi} block.

In each block the couples variable-function are specified. For
transfers the function defines the transfer itself while for cells
the function describes the cell evolution. The variable is specified
with @code{var:}, the function is defined with @code{fun:}.

In the case of the predator-prey model, the transfer variable
associated with @math{\varphi_{meet}} could be called @code{ff_interact}
and the transfer definition would be given by:
@example
set_Phi
< var: ff_interact, fun: f_interact = eta_prey*eta_pred;
>;
@end example

The two cell equations of the predator-prey model, with name
@code{eta_prey} for the prey (@math{\eta_{prey}}) and @code{eta_pred} 
for the predator (@math{\eta_{pred}}) are:

@example
set_eta
< var: eta_prey, fun: deta_prey =   apar*eta_prey - apar*ff_interact;
  var: eta_pred, fun: deta_pred = - cpar*eta_pred + cpar*ff_interact;
>;
@end example

The @samp{;} at the end of the mortran block is important.

@page
The whole model equations are setup with:

@example
@group
!%%%%%%%%%%%%%%%%%%%%%%
! Transfer definition
!%%%%%%%%%%%%%%%%%%%%%%
! rencontre (meeting)
set_Phi
< var: ff_interact, fun: f_interact = eta_prey*eta_pred;
>;

!%%%%%%%%%%%%%%%%%%%%%%
! Cell definition
!%%%%%%%%%%%%%%%%%%%%%%

set_eta
< var: eta_prey, fun: deta_prey =   apar*eta_prey - apar*ff_interact;
  var: eta_pred, fun: deta_pred = - cpar*eta_pred + cpar*ff_interact;
>;
@end group
@end example

Whenever the user is not concerned with giving a specific name to a
function, it is possible to specify the equation only with
@code{eqn:}. Therefore the user may replace an instruction as:
@example
  var: ff_dump,
  fun: f_dump  = - rd*(eta_speed - eta_speed_limiting);
@end example
with:
@example
   eqn: ff_dump = - rd*(eta_speed - eta_speed_limiting);
@end example

In that case, the unnamed function will take the name of the defined
variable preceded by the @samp{$} sign: @code{$ff_dump}.

@subsubheading Starting points

@cindex starting point

The cells equations require state initial conditions. In some case, the 
transfers may also need starting points although they are determined from 
the cell values.

In the predator-prey model the starting points for cells are:
@example
!     initial state
!     -------------
     eta_prey = 1.;
     eta_pred = 1.;
@end example

When there is a non trivial implicit relationship between the transfers
in the model, it may be usefull or even necessary to set some
transfers to non-zero values. This difficulty is only relevant for the very
first step of the simulation and will be used as a 
first guess of @math{\varphi}. The uninitialized transfers having 
a default compiler-dependant (often zero) value, an initialization
to another value may help avoiding singular functions or matrix and
ensure convergence in the Newton algorithm used to solve the transfer implicit
equation. 

@ignore
Indeed a good starting 
point for the transfers may help finding their value at the first time step 
(to help
avoiding a singular matrix during the research of the first transfers and 
ensure convergence), when the implicit equation defining transfers is solved:

@ifset latex
@tex
\begin{eqnarray}
\varphi(t) &=& f(\eta(t),\varphi(t))
\label{transfer}
\end{eqnarray}
@end tex
@end ifset
@ifclear latex
@tex
$$
\varphi(t) = f(\eta(t),\varphi(t))
$$
@end tex
@ifnottex

@noindent @math{phi(t) = f(eta(t),phi(t))}

@end ifnottex
@end ifclear
@end ignore

@subsubheading The cell and transfer arrays

@vindex eta(.)
@vindex ff(.)
@vindex mp
@vindex np

Sometime it is easier to iterate over an array than to use the 
cell or transfer variable name. This is possible because there is a
correspondence between the variable names 
and the fortran array @code{eta(.)} for the cell variables and
the fortran array @code{ff(.)} for the transfer variables@footnote{In fact
the variables names are transformed into fortran array elements
by mortran generated macros, so the symbolic names defined in the
mortran blocks never appears in the generated fortran code, they are
replaced by the fortran arrays.}.

The index of the variable is determined by the order of appearance in 
the variable definition blocks. It is reminded in the output, as
explained later (@pxref{Simulation and output}). 

The number of cells is in the integer @code{np} variable, and the
number of transfer is in the integer @code{mp} variable.

@subsubheading title file

@anchor{Title file}
@cindex title file
@cindex @file{title.tex}

For some graphics generation, a file with name @file{title.tex} is required
which sets the title. The following instructions take care of that:

@example
    OPEN(50,FILE='title.tex',STATUS='UNKNOWN');
    write(50,5000) apar,cpar;                 
5000;format('Lotka-Volterra par:',2F4.1);

    close(50);
@end example

In that case the parameter values are written down, to differenciate between
different runs. This step is in general not needed.

@c The correspondence with basic components are printed out at execution
@c time as explained in @ref{Simulation and output,,
@c Running a simulation and using the output}. Also, a @file{Model.hlp} is
@c generated that recalls the basic names and equations of the model.
@c It may be noted that whenever 
@c the order of variable-functions is the same between indexed declaration and 
@c symbolic, the two generated Fortran code are almost identical.

@node  Setting and running a model
@section Setting and running a model

In this section it is assumed that a programming environment has been
properly setup. This environment may use either cmz or make to drive
the preprocessing and compilation. 
You can skip the part related with the environment you don't intend to use.

For instructions regarding the 
installation, see @ref{Installation}. 


@menu
* Setting up a model with cmz::
* Setting up a model with make::
* Simulation and output::
* Graphics::
@end menu

@node Setting up a model with cmz
@subsection Setup a model and compile with cmz

@cindex @command{mod}
@cindex @file{$zinit}
@cindex @file{$dimetaphi}

The user defined sequences are @samp{KEEP} in the
cmz world. The most common organization is to have a cmz file in a
subdirectory of the directory containing the @file{mini_ker.cmz} 
cmz file. In this
cmz file there should be a @samp{PATCH} called @samp{zinproc}
with the KEEPs within the patch. The KEEP must be called @file{$zinit}.
@c and @file{$dimetaphi}.

From within cmz in the directory of your model the source extraction, 
compilation and linking will be triggered by a @command{mod} command. This macro 
uses the @file{selseq.kumac} information to find the @file{mini_ker.cmz} 
cmz file.
@command{mod} 
shall create a directory with the same name than the cmz file, 
@file{mymodel/} in our example. In this directory there is another 
directory @file{cfs/} containing the sources extracted from the cmz file.

The file @file{mymodel_o.tmp} contains all the mortran code generated 
by cmz with the sequences substituted, including the @file{$zinit}. @c and 
@c @file{$dimetaphi} sequences (assembled code). 
The fortran produced by the preprocessing and
splitting of this file is in files with the traditional @samp{.f} suffix.
The principal program is in @file{principal.f}. An efficient way of getting 
familiar with mini_ker methods is looking at the @file{mymodel_o.tmp} where 
all sequences and main Mortran instructions are gathered. Symbolic derivation 
@c FIXME pas ici la symbolic derivation
is noted as @code{F_D(expression)(/variable)}, and the resulting Fortran code 
is in @file{principal.f}.

@command{mod} also triggers compilation and linking. The object files are in
the same @file{cfs/} directory and the executable is in the @file{mymodel/}
directory, with name @file{mymodel.exe}.

@node Setting up a model with make
@subsection Setup a model and compile with make

@cindex compilation
@c @cindex @file{dimetaphi.mti}
@cindex @file{zinit.mti}
@vindex model_file_name

With make, the sequences are files ending with @samp{.mti} (for
mortran include files),
called, for example, @file{zinit.mti}.
@c  and @file{dimetaphi.mti}. 
They are included by 
@command{mortran} in other source files. You also need a @file{Makefile}
to drive the compilation of the model.

If you don't need additional code or libraries to be linked with your 
model you have two alternatives. 

@enumerate
@item
The simplest alternative is to run
the @command{start_miniker} script with the model file name as argument.
It should copy a @file{zinit.mti} file
ready to be edited and a Makefile ready to compile the model. For
the predator prey model, for example, you could run

@example
$ start_miniker predator
@end example

@item
Otherwise you can copy the Makefile from @file{template/Makefile}
in the directory containing the sequences. You should then change the compiled
model file name, by changing  the value of the
@code{model_file_name} variable to the name of your choice
in the Makefile. It is set to @file{mymodel} in the template. For the 
predator-prey model, it could be set like

@example
model_file_name = predator
@end example

If you want the executable model file to be built in another directory, you could
set

@example
model_file_name = some_dir/predator
@end example

The other items set in the default Makefile should be right. 
@end enumerate

The preprocessing and the compilation are launched with

@example
make all
@end example

The mortran files are generated by the cmz directive preprocessor 
from files found in the package source directories. The mortran files 
end with @samp{.mtn} for the main files and  @samp{.mti} for 
include files. They are output in the current directory.
The mortran preprocessor then preprocess these mortran files and includes
the sequences. The resulting fortran code is also in the current directory, 
in files with a @samp{.f} suffix.
Some fortran files ending with @samp{.F} may also be
created by the cmz directive preprocessor.
The object files resulting from the compilation of all the
fortran files (generated from mortran or directly from fortran files) are
there too. 

In case you want to override the default sequences or a subroutine file 
you just have to create it in your working directory along with the
@file{zinit.mti}. @c and @file{dimetaphi.mti}. 
For example you could want to 
create or modify a @file{zsteer.mti} file (@pxref{End of time step,,
Executing code at the end of each time step}), a @file{zcmd_law.mti} file 
(@pxref{Control laws}), a @file{monitor.f} file 
(@pxref{Turning the model into a subroutine}) to take advantage of 
features presented later in this manual.

More in-depth discussion of using make to run @Minik{} is covered in 
@ref{Advanced use of @Minik{} with make}.
For example it is also possible to create files that are to be 
preprocessed by the cmz directive
preprocessor and separate source files and generated files.
This advanced use is more precisely covered in 
@ref{Programming with cmz directives}.

@page
@node Simulation and output
@subsection Running a simulation and using the output

@cindex running model

Once compiled the model is ready to run, it only has to be executed. On 
standard output informations about the states, transfers, tangent linear
system and other jacobian matrices are printed. 
For example the predator-prey model could be executed with:

@example
./predator > result.lis
@end example

@cindex output file
@vindex dEta(.)
@cindex @file{res.data}
@cindex @file{dres.data}
@cindex @file{tr.data}
@cindex @file{aspha.data}
@cindex @file{Model.hlp}

@c In case of a model entered symbolically 
@c (@pxref{Symbolic model description})
The  correspondance
between the symbolic variables and the basic vectors and functions
are printed at run time:

@example
  ---------------- Informing on Phi definition -----------------
    Var-name,           Function-name,       index in ff vector
             ff_interact              f_interact  1
  ----------------------------------------------------

  --------------- Informing on Eta definition ------------------
   Var-name,           Function-name,       index in eta vector
                eta_prey               deta_prey  1
                eta_pred               deta_pred  2
@end example

A summary of the model equations are in @file{Model.hlp} file.  For
the same example:

@example
======================= set_Phi                                                                
                                                                                         
    1 ff_interact f_interact           eta_pray*eta_pred
======================= set_Eta                                                                
                                                                                         
    1 eta_pray    deta_pray            apar*eta_pray-apar*ff_interact
    2 eta_pred    deta_pred            -cpar*eta_pred+cpar*ff_interact
@end example
@c FIXME never talked about that. Certainly not here
when other general functions are specified with @code{f_set}, it can appear
also in the same help file when replaced by @code{fun_set}.

As far as possible, all data printed in the listing are associated
with a name related to a variable. Here is an extract:

@example
 Gamma :-8.19100E-02-1.42151E-01 3.87150E-02
         eta_courant eta_T_czcx  eta_T_sz   
       ------------------------------------------------
 Omega : 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
         courant_L   T_czcx      Psi_Tczc    Psi_Tsz 
       ------------------------------------------------
@end example
for the two known vectors of the system, and:
@example
 >ker : Matrice de couplage       4 4 4 4
courant_L Raw(1,j=1,4):   1.000     -9.9010E-03  0.000       0.000    
T_czcx    Raw(2,j=1,4): -2.7972E-02   1.000      0.000      9.9900E-04
Psi_Tczcx Raw(3,j=1,4):  0.1605      9.7359E-02  1.000     -5.7321E-03
Psi_Tsz   Raw(4,j=1,4):   0.000     -0.1376     5.7225E-03   1.000    
          Var-Name      courant_L   T_czcx      Psi_Tczc    Psi_Tsz 
          ----------------------------------------------------------
@end example

where the @code{couplage} (coupling matrix) is given that corresponds 
to the matrix coupling the four transfer components after @math{\delta\eta} 
has been eliminated from system. It is computed in the subprogram 
@file{oker} (for kernel) which solves the system.

Basic results are output in a set of @samp{.data} files. 
The first line (or two lines) describes the column with a @samp{#}
character used to mark the lines as comments (for @command{gnuplot} 
for example).
In the @samp{.data} files, the data are simply separated with spaces.
Each data file has the @code{time} variable values as first column.
@footnote{@file{dres.data} has another time related variable as second column:
@cindex @file{dres.data}
@vindex dt
@code{dt}, the time step that can vary in the course of a simulation.}.
Following columns give the values of @code{eta(.)} in @file{res.data}, 
@code{dEta(.)} in @file{dres.data} -- the step by step variation of 
@code{eta(.)} -- and @code{ff(.)} in @file{tr.data}.

Along the simulation the @acronym{TEF} Jacobian matrices are computed. 
A transfer variables elimination process also leads to the definition 
of the classical state advance matrix of the system 
(the corresponding array is @code{aspha(.,.)} in the code).
This matrix is output in the file @file{aspha.data} that is used to
post-run dynamics analyses. The matrix columns are written column wise on each
record.
@xref{Stability of fastest modes,,Stability analysis of fastest modes}. 
@xref{Generalized TLS,,Generalized 
tangent linear system analysis}. It is not used in the solving process.

Other @samp{.data} files will be described later.

@c FIXME already said
@c At the begining of a run, the help file @file{Model.hlp} is generated for
@c global checkiing of the model. In the example, it is:

@c @example
@c ======================= set_Phi                                                                                                                                              
@c     1 ff_interact      f_interact   eta_pray*eta_pred
@c ======================= set_Eta                                           
@c     1 eta_pray         deta_pray    apar*eta_pray-apar*ff_interact
@c     2 eta_pred         deta_pred    -cpar*eta_pred+cpar*ff_interact
@c @end example


@node Graphics
@subsection Doing graphics

@cindex graphics
@cindex graphics with @command{gnuplot}
@cindex graphics with @command{PAW}

@c The format of the @samp{.data} files are coherent with GNU graphics, that is
@c the data are simply separated with spaces. 
Since the data are simply separated with spaces, and comment lines 
begin with @samp{#}, the 
files can be vizualised with many programs. 
With @command{gnuplot}, for example, to plot @code{eta(@var{n})}, 
the @command{gnuplot} statement could be:

@example
plot "res.data" using 1:(@var{n}+1)
@end example

The similar one for @code{ff(@var{n})}:
@example
plot "tr.data" using 1:(@var{n}+1)
@end example

For people using @command{PAW}, the CERN graphical computer code, 
@Minik{} prepares
kumacs that allow to read process the @samp{.data} files in the form of 
@emph{n-tuples} (see the @cite{PAW manual} for more information). 
In that cas, the flag @code{sel paw} has to be gievn in the @file{selsequ.kumac}.
The generated  n-tuples are ready to use only
for vector dimension of at most 10 (including the variable @code{time}).
These kumacs are overwritten each time the model is run. Usaually, gnuplot has
to be preferred, but when using surfaces and histograms, PAW is better.
The @file{gains.f} (and @file{go.xqt}  is provided as an example in the 
@Minik{} files.

@node Controlling the run
@section Controlling the run

@cindex controlling the run

It is possible to add code that will be executed at the end of each time
step. It is also possible to specify which time step leads to a printout on
standard output. For maximal control, the code running te model may be 
turned into a subroutine to be called from another fortran (or C) 
program, this possibility is covered in @ref{Calling the model code}.

@menu
* End of time step::
* Controlling the printout and data output::
@end menu

@node End of time step
@subsection Executing code at the end of each time step

@cindex @file{zsteer}
@cindex @file{zsteer.inc}

The code in the sequence @file{zsteer} is executed at the end of each time
step. It is possible to change the time step length (variable @code{dt})
verify that the non linearity are not too big, or perform 
discontinuous modifications of the states. One available variable @code{res}
might be usefull for time step monitoring. At the end of the time step,
as soon as @math{\varphi} has been computed, a numerical test is applied
on a pseudo relative quadratic residual between 
@math{\varphi=f(\eta(t-dt)+d\varphi} (@code{ ffl}), where @math{d\varphi}
is given by the system resolution in @code{ker},and
@math{\varphi=f(\eta),\varphi)},  Fortran variable (@code{ff}):

@verbatim
! ========================================================
! test linearite ffl - ff
! ========================================================
if (istep.gt.1)
< res=0.; <io=1,m; res = res +(ffl(io)-ff(io))**2/max(one,ff(io)*ff(io)); >;
  if (res .gt. TOL_FFL)
  < print*,'*** pb linearite : res > TOL_FFL a istep',istep,res,' > ',TOL_FFL;
    do io=1,m < z_pr: io,ff(io),ff(io)-ffl(io); >;
  >;
>;
@end verbatim

This test hence applies only for non linearities in tranfer models. Nevertheless,
@code{res} might be usefull to monitor the time step @code{dt} in @code{ZSTEER}
and eventually go backward one step (@code{goto :ReDoStep:}).
This can more appropriatly be coded in the (empty in default case) 
sequence @code{zstep}, inserted just before time-advancing
states and @code{time} variables in @file{principal}.
@vindex ffl(.)
@cindex @code{ffl} (linearity test)
@cindex linearity test

It is also possible to fix the value of the criterium @code{TOL_FFL} in
@file{zinit} different from its default value of @math{10^{-3}} --
independent of the Fortran precision.


Many other variables are available, including
@vtable @code
@item istep
The step number;
@item couplage(.)
The @acronym{TEF} coupling matrix between transfers;
@item H
The Jacobian matrix corresponding with:
@c \varphi(t) &= f(\eta(t),\varphi(t))\cr
@c \frac{\partial g(\eta(t),\varphi(t))}{\partial \eta(t)}
@tex
$$\partial_{\eta} g(\eta(t),\varphi(t));
$$
@end tex
@ifnottex
g_1(eta,phi);
@end ifnottex
@item Bb
The Jacobian matrix corresponding with:
@tex
$$\partial_{\varphi} g(\eta(t),\varphi(t));
$$
@end tex
@ifnottex
g_2(eta,phi);
@end ifnottex
@item Bt
The Jacobian matrix corresponding with:
@tex
$$\partial_{\eta} f(\eta(t),\varphi(t));
$$
@end tex
@ifnottex
f_1(eta,phi);
@end ifnottex
@item D
The Jacobian matrix corresponding with:
@tex
$$\partial_{\varphi} f(\eta(t),\varphi(t));
$$
@end tex
@ifnottex
f_2(eta,phi);
@end ifnottex

@item aspha
The state advance matrix;
@item dneta
@itemx dphi
the variable increments;
@end vtable
One should be aware of that the linearity test concerns the preceding step. 
We have yet no example of managing the time-step.

@page
@node Controlling the printout and data output
@subsection Controlling the printout and data output

@cindex printing
@vindex zprint
@vindex modzprint

The printout on standard output is performed if the variable @code{zprint} 
of type @code{logical} is true. Therefore it is possible to control this
printout by setting @code{zprint} false or true. For example the following 
code, in sequence @file{zsteer}, triggers printing for every 
@code{modzprint} time step and the two following time steps:

@example
ZPRINT = mod(istep+1,modzprint).eq.0;
Zprint = zprint .or. mod(istep+1,modzprint).eq.1;
Zprint = zprint .or. mod(istep+1,modzprint).eq.2;
@end example

The data output to @file{.data} files described in @ref{Simulation and output,,
Running a simulation and using the output} is performed if the
@code{logical} variable @code{zout} is true. For example the following
code, in @file{zsteer}, triggers output to @file{.data} files every 
@code{modzout} step.

@example
Zout = mod(istep,modzout).eq.0;
@end example

@node Advanced programming
@chapter Advanced @Minik{} programming

@menu
* Selecting features::
* Calling the model code::
* 1D gridded model::
* Double precision::
* Partial Derivatives::
* Rule of programming non continuous models::
* Parameters::
* Observations and data::
* Explicit model size::
* Programming with cmz directives::
@end menu

@node Selecting features
@section Overview of additional features setting

@cindex feature setting
@cindex select flag
@cindex logical flags
@cindex @file{selseq.kumac}

It is possible to enable some features by selecting which code should 
be part of the principal program. Each of these optionnal features are 
associated with a @dfn{select flag}. 
For example 
@c the optimisation with minuit is associated with the select
@c flag @samp{minuik}, 
double precision is used instead of simple precision with 
the @samp{double} select flag,
the model is a subroutine with the select flag @samp{monitor},
the Kalman filter code is set with @samp{kalman} and the 1D gridded
model capabilities are associated with @samp{grid1d}. 
@c Currently it is only possible 
@c to select features in cmz. 
To select a given feature the cmz statement @code{sel @var{select_flag}} should
be written down in the @file{selseq.kumac} found in the model directory.
With make either the corresponding variable should be set to 1 or it
should be added to the @code{SEL} make variable, depending on the feature.

Other features don't need different or additional code to be used. 
Most of the features are enabled by setting specific logical variables to
@samp{.true.}. This is the case for
@code{zback} for the adjoint model, @code{zcommand} if the command is in a file
and @code{zlaw} if it is a function and @code{zkalman} for the Kalman filter.
These select and logical flags are described in the corresponding sections.

In cmz an alternative of writing select flags to @file{selseq.kumac} is to
drive the compilation with @code{smod @var{sel_flag}}. In that case the
@var{sel_flag} is selected and the files and executable goes to a directory
named @file{sel_flag}. 

The select flags are taken into account during cmz directives preprocessing.
Therefore you have the possibility to use these flags to conditionnaly 
include pieces of code. In most cases you don't need to include code conditionally 
yourself though, but if you want to, this is covered in 
@ref{Programming with cmz directives}.

@node Calling the model code
@section Calling the model code

When the model code is a subroutine, it can be called from another fortran
program unit (or another program), and the model will be 
run each time the subroutine is called.
This technique could be used, for example to perform optimization 
(@pxref{Adjoint model and optimisation,,Adjoint model and optimisation 
with @Minik{}}), or to run the model with different parameters.

@menu
* Turning the model into a subroutine::
* The model subroutine::
@end menu

@node Turning the model into a subroutine
@subsection Turning the model into a subroutine

@c This feature is allready enabled with @command{make}, and to 
@c override the default program a file called @file{monitor.f} has to be created 
@c in the working directory. The default program simple calls the model
@c subroutine.

With cmz, one has to do a 
@example
sel monitor
@end example
in the @file{selseq.kumac} file and create the KEEP that call the 
model code. @xref{Selecting features, Overview of additional features 
setting}.

With make @samp{monitor} should be added to the @code{SEL} variable in
the @file{Makefile}, for example:

@example
SEL = monitor
@end example

A file that call the principal subroutine should also be written, using
the prefered language of the user. The additional object files should
then be linked with the @Minik{} objects. To that aim they may be added
to the @code{miniker_user_objects} variable.

@node The model subroutine
@subsection Calling the model subroutine

The model subroutine is called @samp{principal} and is called with the 
following arguments:

@deffn Subroutine principal (Cost, ncall, integer_flag, file_suffix, info, idxerror)
Where @var{Cost} is a real number, @code{real} or @code{double precision}, 
and is set by the @code{principal}
subroutine. It holds the value of the cost function if such function has been
defined (the use and setting of a cost function is covered later, 
see @ref{Cost function coding and adjoint modeling}). 
@var{ncall} is an integer which corresponds with the number of 
call to @code{principal} done so far, it should be initialized to 0 and 
its value should not be changed, as it is changed in the @code{principal}
subroutine.
@var{integer_flag} is an integer that can be set by the user to be accessed 
in the @code{principal} subroutine. For example its value could be used to
set some flags in the @file{zinit} sequence.
@var{file_suffix} is a character string, that is suffixed to the output files
names instead of @samp{.data}. If the first character is the null character 
@samp{char(0)}, the default suffix, @samp{.data} is appended.
@var{info} and @var{idxerror} are integer used for error reporting.
@var{idxerror} value is 0 if there was no error. It is negative for
an alert, positive for a very serious error. The precise value determines
where the error occured.
@var{info} is an integer holding more precise information about the
error. It is usually the information value from lapack.
The precise meaning of these error codes is in @ref{tab:error_codes}.
@end deffn

@float table, tab:error_codes 
@multitable {kalman analysis state matrix advance in phase space, @math{(I-D)} inversion} {inversion} {@code{idxerror}}
@headitem Source of error or warning @tab @code{info} @tab @code{idxerror}
@c @item @code{} @tab @file{.data} @tab  @tab
@item state matrix inversion in ker @tab inversion @tab 1
@item time advance system resolution in ker  @tab system @tab 2
@item transfer propagator, @math{(I-D)} inversion @tab inversion @tab 3
@item kalman analysis state matrix advance in phase space, @math{(I-D)} inversion @tab inversion @tab 21
@item kalman analysis variance covariance matrix non positive @tab Choleski @tab 22
@item kalman analysis error matrix inversion @tab inversion @tab 23
@item kalman error matrix advance @tab system @tab 24
@item transfers determination linearity problem for transfers @tab  @tab -1
@item transerts determination Newton D_loop does not converge @tab  @tab -2
@end multitable
@caption{Meaning of error codes returned by principal.}
@end float

In general more information than the provided arguments has to be passed 
to the @code{principal} subroutine, in that case a @code{common} block,
to be written in the @file{zinit} sequence can be used.

@page
@node 1D gridded model
@section Describing 1D gridded model

Specific macros have been built that allow generic description of 1D gridded models.
Because of the necessity of defining left and right limiting conditions, the models
are partitionned in three groups for cell and transfer components. In the following example,
a chain of masselottes linked by springs and dumps is bounded to a wall on the left,
and open at right. The @acronym{TEF} formulation of the problem is written in the phase space (position-shift, velocity)
for node @math{k}, with bounding conditions:
@ifset latex
@tex
\begin{equation}
\left\{
\begin{array}{cc}
\partial_t \eta _{k} ^{pos}  =  \eta _{k} ^{vel}\qquad& \\
\partial_t \eta _{k} ^{vel}  = ( \varphi_k ^{spr} -\varphi _{k+1} ^{spr}&+\,\,\varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k \nonumber\\
\end{array}
\right.
\end{equation}
\begin{equation}
\left\{
\begin{array}{cc}
\varphi_k ^{spr} = -k_k (\eta _{k} ^{pos}- \eta _{k-1} ^{pos})\\
\varphi_k ^{spr} = -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\end{array}
\right.
\label{mass}
\end{equation}
\begin{equation}
\left\{
\begin{array}{cc}
\eta ^{pos}_{0} =& 0\\
\eta ^{vel}_{0} =& 0\\
\varphi  ^{spr}_{N+1} =& 0\\
\varphi ^{dmp}_{N+1} =& 0
\end{array}
\right.
\end{equation}
@end tex
@end ifset
@ifclear latex
@tex
$$\left\{\eqalign{\partial_t \eta _{k} ^{pos}  &=  \eta _{k} ^{vel} \cr
\partial_t \eta _{k} ^{vel}  &= ( \varphi_k ^{spr} -\varphi _{k+1} ^{spr} + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k  \cr}\right.$$
$$\left\{\eqalign{
\varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1} ^{pos})\cr
\varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\cr}\right.$$
$$\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
\eta ^{vel}_{0} &= 0\cr
\varphi  ^{spr}_{N+1} &= 0\cr
\varphi ^{dmp}_{N+1} &= 0\cr}\right.$$
@end tex
@ifnottex

States:@*
@noindent @math{d position(t,k)/d t = velocity(t,k)@* 
d velocity (t,k)/d t =(spring(t,k) - spring(t,k+1)+ dump(t,k)- dump(t,k+1))/m_k}

Transfers:@*
@noindent @math{spring(t,k) = -k_k (position(t,k)- position(t,k-1))@*
dump(k,t) &= -d_k (velocity(t,k)- velocity(t,k-1))}

Bounding conditions:@*
@noindent @math{position(t,0) = 0@*
velocity(t,0) = 0@*
spring(t,N+1) = 0@*
dump(t,N+1) =0}

@end ifnottex
@end ifclear

@cindex down node
@cindex up node

where @math{m_k} is the mass of node @math{k}, @math{r_k} and @math{d_k} 
the rigidity of springs and dumping coefficients. There are @math{N} nodes
in the grid, from 1 to @math{N}, and two nodes outside of the grid,
a limiting node 0, and a limiting node @math{N+1}. The limiting node
corresponding with node 0 is called the @dfn{down} node, while the limiting 
node corresponding with node @math{N+1} is called the @dfn{up} node.
Other models not part of the 1D grid may be added if any.

To enable 1D gridded models, one should set the select flag @samp{grid1d}.
In cmz it is achieved setting the select flag in
@file{selseq.kumac}, like

@example
sel grid1d
@end example

With make, the @code{SEL} variable should contain @code{grid1d}. For example
to select @code{grid1d} and @code{monitor}, it could be
@example
SEL = grid1d,monitor
@end example


@menu
* 1D gridded Model size::
* 1D gridded model code::
@end menu

@node 1D gridded Model size
@subsection Setting dimensions for 1D gridded model

@c FIXME grid1d sans dimetaphi?
In that case the number of nodes, the number of states and tranferts 
per node, and the number of limiting transfers and states are required.
These dimensions has to be entered in the
@file{DimEtaPhi} sequence. The parameters for cells are
@vtable @code
@item n_node
Number of cell nodes in the 1D grid.
@item n_dwn
Number of limiting cells with index -1, @i{i.e.} number of cells in the
limiting down node.
@item n_up
Number of limiting cells with index +1, @i{i.e.} number of cells in the
limiting up node.
@item n_mult
Number of cells in each node (multiplicity).
@end vtable

@vindex m_node
@vindex m_dwn
@vindex m_up
@vindex m_mult
The parameters for transfers, are similarly 
@code{m_node}, @code{m_dwn}, @code{m_up}, @code{m_mult}.
The layout of their declaration should be respected as 
the precompiler matches the line. Also this procedure is tedious, it
should be selected for debuging processes (use the flag @code{sel dimetaphi}
in ``selsequ.kumac''. Otherwise, the dimensioning sequence will be automaticaly
generated, which is smart but can lead to diffculty in interpreting syntax errors.
Once a model is correctly entred, turn off the sel flag and further modifications
will automatically generate the proper dimensions. The correctness of dimensionning
should nevertheless always be checked in @code{principal.f}, where you can also
check that null valued parameters as @code{lp, mobs, nxp} will suppress parts
of the code - this is signaled as Fortran comment cards.

In our example, there are three grids of cell and
transfer variables (@code{n_node=m_node=3}). 
There are two cells and two transfers in each node 
(@code{n_mult=2} and @code{m_mult=2}). There is no limiting condition 
for the states in the down node therefore @code{n_up=0}. 
There is no transfer for the first limiting node, and 
therefore @code{m_dwn=0}.
There are two states in the limiting node 0, the down node, 
@code{n_dwn=2}, and two transfers in the limiting last node the node up, 
and @code{m_up=2}:

@example 
! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! nodes parameters, and Limiting Conditions (Low and High)
! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++
      parameter (n_node=3,n_dwn=2,n_up=0,n_mult=2);
      parameter (m_node=3,m_dwn=0,m_up=2,m_mult=2);
! ________________________________________________________
@end example

@ignore
@c FIXME enleve par al1
The dimension of the parameter arrays should also be declared in the
@file{dimetaphi} sequence. Here we have 3 parameters, for
@math{m_k}, @math{r_k} and @math{d_k}:

@example
dimension rk(n_node),rd(n_node),rmassm1(n_node);
@end example
@end ignore

@node 1D gridded model code
@subsection 1D gridded Model coding

The model code and parameters go in the @file{zinit} sequence.

@subsubheading Parameters

A value for the @Minik{} parameters and the model parameters should be given in
@file{zinit}, in our example we have

@example
!%%%%%%%%%%%%%%%%%%%%%%
! Parameters
!%%%%%%%%%%%%%%%%%%%%%%
real rk(n_node),rd(n_node),rmassm1(n_node);

data rk/n_node*1./;
data rd/n_node*0.1/;
data rmassm1/n_node*1./;
     dt=.01;
     nstep=5 000;
     modzprint = 1000;
     time=0.;
@end example

@subsubheading Limiting conditions

@cindex limiting conditions

@c The limiting states and transfer variables and the corresponding equations are
@c declared using
@c the symbolic model description 
@c (@pxref{Symbolic model description}).
There are four mortran blocks for @code{node} and @code{up} and @code{down}, both
for states and transfers:

@findex set_dwn_eta
@findex set_dwn_phi
@findex set_up_eta
@findex set_up_phi

@table @code
@item set_dwn_eta
down node cells
@item set_up_eta
up node cells
@item set_dwn_phi
down node transfers
@item set_up_phi
up node transfers
@end table

The following scheme illustrates the example:
@smallexample
!%%%%%%%%%%%%%%%%%%%%%%%%%%================================================
! Maillage convention inode
!%%%%%%%%%%%%%%%%%%%%%%%%%%                                 Open ended
!(2 Down    Phi    Eta                         (n_node)
! Eta)  \|       .-----.       .-----.          .-----.        /
! wall  \|-\/\/\-|     |-\/\/\-|     |  . . .  -|     |-\/\/\- |dummy
!  pos  \|--***--|  1  |--***--|  2  |  . . .  -|  n  |--***-- |Phis
! speed \|   1   |_____|   2   |_____|      n   |_____|  n+1   \(2 Up Phi)
!
@end smallexample

Two states are associated with the down node, they correspond to the position
and speed of the wall. As the wall don't move these states are initialized to
be 0, and the cells are stationnary cells, therefore these values remain 0.

@example
! Down cells (wall)
! -----------------
eta_pos_wall = 0; eta_speed_wall = 0.;

set_dwn_eta
< var: eta_pos_wall,  fun: deta_pos_wall  = 0.;
  var: eta_speed_wall, fun: deta_speed_wall= 0.;
>;
@end example

There are 2 limiting transfers in the up node. They correspond with an open
end and are therefore set to 0.

@example
! limiting Transfers : dummy ones
! -------------------------------
set_Up_Phi
< var:ff_dummy_1, fun: f_dummy_1=0.;
  var:ff_dummy_2, fun: f_dummy_2=0.;
>;
@end example

@subsubheading Starting points

The cell node state values are initialized. They are in an array 
indexed by the @code{inode} variable. In the example the variable 
corresponding with position is @code{eta_move} and the variable corresponding
with speed is @code{eta_speed}. Their initial values are set with the 
following mortran code

@example
!---------------
! Initialisation
!---------------
;
do inode=1,n_node <eta_move(inode)=0.01; eta_speed(inode)=0.0;>;
@end example

If any transfer needs to be given a first-guess value, this is also done 
using @code{inode} as the node index.

@subsubheading Grid node equations

@findex set_node_Phi
@findex set_node_eta
@cindex equations, grid

Each node is associated with an index @code{inode}. It allows to refer to the 
preceding node, with @code{inode-1} and the following node @code{inode+1}.
The node states are declared in @code{set_node_Eta} block and the transfers are
in @code{set_node_Phi} blocks.

In the example, the cells are declared with

@example
! node cells
! ----------
;
set_node_Eta
< var: eta_move(inode),  fun: deta_move(inode) = eta_speed(inode);
  var: eta_speed(inode),
  fun: deta_speed(inode) = rmassm1(inode)
                             *( - ff_spring(inode+1) + ff_spring(inode)
                               - ff_dump(inode+1)  + ff_dump(inode)
                              );
>;
@end example
Note that the @code{inode} is dummy in the @code{var:} definition and can as
well be written as: @code{var: eta_move(.)}.


The transfers are (@code{ff_spring} corresponds with springs and 
@code{ff_dump} with dumps):

@example
!%%%%%%%%%%%%%%%%%%%%%%
! Transfer definition
!%%%%%%%%%%%%%%%%%%%%%%
! node transfers
! --------------
! convention de signe spring : comprime:= +
set_node_Phi
< var: ff_spring(.),
  fun:
   f_spring(inode)= -rk(inode)*(eta_move(inode) - eta_move(inode-1));
  var: ff_dump(.),
  fun:
   f_dump(inode)  = -rd(inode)*(eta_speed(inode) - eta_speed(inode-1));
>;
@end example

The limiting states and transfers are associated with the states or transfers
with index @code{inode+1} or @code{inode-1} appearing in node cell and 
transfer equations (@code{inode-1} for down limiting conditions and 
@code{inode+1} for up limiting conditions) in their order of appearance.
In our example, in the @code{eta_speed} state node equation
@code{ff_spring(inode+1)} appears before @code{ff_dump(inode+1)} and is
therefore associated with @code{ff_dummy_1} while @code{ff_dump(inode+1)} is
associated with the @code{ff_dummy_2} limiting transfer, as @code{ff_dummy_1}
appears before @code{ff_dummy_2} in the limiting up transfers definitions.
Verification of the grid index coherence should be eased with the following
help printed in the listing header:

@example




  --------------- Informing on Dwn Eta definition ---------------
 Var-name,            Function-name, index in eta vector
         eta_pos_wall        deta_pos_wall  1 [
       eta_speed_wall      deta_speed_wall  2 [

  -------------- Informing on Eta Nodes definition --------------
 Var-name,     Function, k2index of (inode: 0 [ 1,...n_node ] n_node+1)
            eta_move           deta_move    1 [   3 ...   7 ]   9
           eta_speed          deta_speed    2 [   4 ...   8 ]  10

  ---------------- Informing on Up  Phi  definition -------------
 Var-name,             Function-name, index in ff vector
          ff_dummy_1           f_dummy_1 ]    7
          ff_dummy_2           f_dummy_2 ]    8
         ff_move_sum          f_move_sum ]    9
        ff_speed_sum         f_speed_sum ]   10
  ----------------------------------------------------

 -------------- Informing on Phi Nodes definition ---------------
 Var-name,     Function, k2index of (inode: 0 [ 1,...m_node ] m_node+1)
           ff_spring            f_spring   -1 [   1 ...   5 ]   7
             ff_dump              f_dump    0 [   2 ...   6 ]   8
  ----------------------------------------------------
@end example

All variable names and functions are free but has to be
different.
Any particular node-attached variable @math{k} is referred to as: @samp{(inode:k)},
where @math{k} has to be a Fortran expression allowed in arguments. The symbol 
@samp{inode} is
reserved. As usual other Fortran instructions can be written within the
Mortran block @samp{< >} of each @code{set_} block.

@node Double precision
@section Double precision

The default for real variables is the @code{real} Fortran type. It is possible to
use double precision instead. In that case all the occurences of @samp{real@ }
in mortran code is substituted with @samp{double precision@ } at 
precompilation stage,
and the Lapack subroutine names are replaced by the double precision names.
Eventual users'declaration of @code{complex@ } Fortran variables is also 
changed to @code{double complex@ }.

This feature is turned on by @code{sel double} in @file{selseq.kumac} with cmz
and @code{double = 1} in the @file{Makefile} with make.

In order for the model to run as well in double as in simple precision, 
some care should be taken to use the generic intrinsic functions, like 
@code{sin} and not @code{dsin}. No numerical constant should be passed directly 
to subroutines or functions, but instead a variable with the right type should
be used to hold the constant value, taking advantage of the implicit casts 
to the variable type.

@node Partial Derivatives
@section  Partial Derivatives

The partial derivative rules are included in a @code{Mortran} macro series
in @file{Derive_mac} of @Minik{} files. When using an anusual function,
one should verify that the corersponding rules are in that file.
It is easy to understand and add new rules in analogy with the already existing ones.

For instance, suppose one wants to use the intrinsic Fortran function @code{ abs()}.
Its derivatives uses the other function @code{sign()} this way:

@example
 &'(ABS(#))(/#)' = '((#1)(/#2)*SIGN(1.,#1))'
@end example

In such cases when one is adding a new rule, it is important to use the generic function names
only (i.e. @code{sin} not @code{dsin}), because when compilating @Minik{} in the double precision
version, or complex version, the generic names will correctly handle the different variable
types - which is not the case when coding with specific function names.

@menu
* Derivating a power function::
@end menu

@node Derivating a power function
@subsection Derivating a power function

Partial derivative of a function in exponent is not secure in its Fortran form
@code{g(x,y)**(f(y))}. It should be replaced by @code{power(g,f)} of 
the @Minik{} @file{mathlib},
or by the explicit form @code{exp(f(y)*log(g(x,y)))}.

Its derivative will have the following form:


@ifset latex                         @c ***PBAl1
@tex
\begin{equation}
\partial_x f^g=g f^{g-1}\partial_x f +  f^g \log f\partial_x g = f^{g-1}(g\partial_x f + f\partial_x g)
\end{equation}
@end tex
@end ifset
@ifclear latex
@tex
$$\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f +  f^g \log f\partial_x g\cr
 &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}$$
@end tex
@end ifclear

and is in the macros list already defined in: @file{DERIVE_MAC}.

@node Rule of programming non continuous  models
@section  Rule of programming non continuous  models

Some models may originally be non continuous, as the ones using a Fortran instruction @code{IF}.
Some may use implicitly a step function on a variable. In such cases, the model has to be
set in a derivable form, and use a ``smooth step'' instead.
 One should be aware of that this apparently mathematical treatment currently
indeed leads to a physical question about the macroscopic form of a physical law.
At a macroscipic level, a step function is usually a nonsense.
@cindex Heaviside function
Taking
the example of phase-change, a fluid volume does not change phase at once, and a ``smooth
change of state'' is a correct macroscopic model.

@Minik{} provides with the smooth step function 
@emph{Heavyside}@footnote{This naming is a joke
for ``Inert'' Heaviside function.} in the @Minik{} @file{mathlib}:

@example
        Delta = -1."K";
        A_Ice =  heavyside("in:" (T_K-Tf), Delta, "out:" dAIce_dT);
@end example

in this example, @code{Tf} is the ice fusion-temperature, @code{A_ice} 
gives the ice-fraction
of the mesh-volume of water at temperature @code{T_k}. 
The smooth-step function is a quasi
hyperbolic tangent function of @math{x/\Delta}, 
normalised from 0 to 1, with a maximum slope
of 2.5, see figure @ref{heavy}.

@float Figure, heavy
@image{heavyside}
@caption{Heaviside function and derivative}
@end float
@ignore
@tex                                  PBAl1
\begin{figure}[h]
\psfig{figure=heavyside.ps,%
bbllx=60pt,bblly=180pt,bburx=526pt,bbury=650pt,width=10cm,clip=}
\caption{La fonction ``domptée'' de Heaviside et sa dérivée pour une variable adimensionnée}
\label{heavy}
\end{figure}
@end tex
@end ignore

For @code{Mortran} to be able to symbolicaly compute the partial derivarives, the rule
is in the table of macros as:

@example
&'(HEAVYSIDE(#,#,#))(/#)' = '((#1)(/#4)*HEAVYDELTA(#1,#2,#3))'
@end example

which uses the Foratn entry point @code{HeavyDelta} in the Fortrsan function @code{heavyside}.

Another type of problem arises when coding a  
@code{var=min(f(x),g(x))} Fortran instruction.
In such a case one does not want a derivative and one will code:

@example
var = HeavySide(f(x)-g(x),Delta,dum)*g(x) + (1.-HeavySide(f(x)-g(x),Delta,dum)*f(x);
@end example

or equivalently:

@example
var = HeavySide(f(x)-g(x),Delta,dum)*g(x) + HeavySide(g(x)-f(x),-Delta,dum)*f(x);
@end example

@strong{Warning}: the value of the argument @var{Delta} is important because 
it will fix the maximum
slope of the function that will appear as a coefficient in the 
Jacbian matrices.

@node Parameters
@section Parameters

It is possible to specify some Fortran variables as specific model parameters.
Model parameters
may be used in sensitivity studies (@pxref{Sensitivity to a parameter})
and in the adjoint model (@pxref{Sensitivity of cost function to parameters}). 
Nothing special is done with parameters with Kalman filtering.


@findex Free_parameter

The parameters are fortran variables that should be initialized somewhere 
in @file{zinit}. For a variable to be considered as a parameter, it should 
be passed as an
argument to the @code{Free_parameters} macro. For example if 
@code{apar} and @code{cpar} (from the predator example) are to be considered
as parameters, @code{Free_parameters} should be called with:

@example
Free_parameter: apar, cpar;
@end example

@c Forward sensitivities are explained later (@pxref{Sensitivity to a parameter}), 
@c the syntax only is described here.


When used with grid1d models (@pxref{1D gridded model,,
Describing 1D gridded model}) the @code{inode} number may appear in 
parenthesis:

@example
Free_parameter: rd(1), rk(2);
@end example

@node Observations and data
@section Observations and data

Some support for observations and interactions with data is available.
The observations are functions of the model variables. They don't have 
any action on the model result, but they may (in theory) be observed 
and measured. The natural use of these observations is to be compared
with data that correspond with the values from real measurements.
They are used in the Kalman filter (@pxref{Kalman filter}).

The (model) observation vector is noted @math{\omega}
@c FIXME is seems untrue?
@c in this section ($\mu$ elsewhere,
and the observation function is noted @math{h}:

@tex
$$
\omega = h ( \eta , \varphi) 
$$
@end tex
@ifnottex

@noindent @math{omega(t) = h(eta(t), phi(t))}

@end ifnottex

@menu
* Observations::
* Data::
@end menu

@node Observations
@subsection Observations

@vindex mobs

The observation functions are set in a @code{set_probe} block in 
the @file{zinit} sequence.

@cindex observation function

@c FIXME doesn't exist anymore
@c @defmac eqn: Obs_tef(@var{i}) = f(eta(.),ff(.))
@c This macro defines the observation equation as usual in a @code{set_block<}.
@c @code{f} is a fortran 
@c expression which may be function of cell state variables, 
@c @samp{eta(1)}@dots{}@samp{eta(np)} and transfers 
@c @samp{ff(1)}@dots{}@samp{ff(mp)}, or of course their symbolic names.
@c @end defmac

For example suppose that, in the predator-prey model, we only 
have access to the total population of preys and predators, we would have:

@example
set_probe
< eqn: pop = eta_pred + eta_pray;
>;
@end example

@c it is always turned on, now
@c The corresponding code is used with @code{sel obs} in @file{selseq.kumac} 
@c with cmz and @code{obs = 1} in @file{Makefile} with make. And the feature
@c is turned on and off at run time with the logical flag @code{zobs} corresponding
@c to an available data from measurement

@c @vindex etaobs(.)
@cindex @file{obs.data}

The number of observations is put in the integer variable @code{mobs}.
The observation vector corresponds with the part of the @code{ff(.)} 
array situated past the regular transferts, @code{ff(mp+.)}, and is output
in the file @file{obs.data}.

@c @vindex obetad(.,.)
@c @vindex obephid(.,.)
@c @vindex obspha(.,.)

@node Data
@subsection Data

@vindex zgetobs
@vindex vobs(.)
@cindex @file{data.data}

Currently this code is only used if the Kalman code is activated. This
may be changed in the future.

The convention for data is that whenever some data are available, the 
logical variable @code{zgetobs} should be set to @samp{.true.}. And the
@code{vobs(.)} vector should be filled with the data values. This 
vector has the same dimension than the observation
vector and each coordinate is meant to correspond with one
coordinate of the observation vector.

This feature is turned on by setting the logical variable @code{zdata}
to @samp{.true.}, and the @code{zgetobs} flag is typically set in the
@file{zsteer} sequence (@pxref{End of time step,,Executing code at
the end of each time step}).
Every instant data are available (@code{zgetobs} is true) the observations
are written to the file @file{data.data}. With the Kalman filter more 
informations are output to the @file{data.data} file, 
see @ref{Kalman filter results}.


@node Explicit model size
@section Entering model size explicitely

It is possible to enter the model dimensions explicitely, instead of 
generating them automatically, as it was done previously.
This feature is turned on by @code{sel dimetaphi} 
in @file{selseq.kumac} with cmz
and @code{dimetaphi} added to the @code{SEL} variable in 
the @file{Makefile} with make.

@menu
* Size sequence::
* Model with explicit size::
@end menu

@node Size sequence
@subsection The explicit size sequence

@cindex dimetaphi
@cindex model size
@vindex np
@vindex mp
@vindex maxstep
@cindex @file{dimetaphi}

The dimension of the model is entered in the sequence @file{dimetaphi},
using the fortran @code{parameter np} for @code{eta(.)} and
@code{mp} for @code{ff(.)}.
For the Lotka-Volterra model, we have two cell components and only one transfer.

@example
parameter (np=2,mp=1);
@end example

You should not change the layout of the parameter statement as the 
mortran preprocessor matches the line.

You also have to provide other parameters even if you don't have any 
use for them. If you don't it will trigger fortran errors.
It includes the @code{maxstep} parameter that can have any value but 0,
@code{lp} and @code{mobs} that should be 0 in the example, and  @code{nxp},
@code{nyp} and @code{nzp} that should also be 0.
The layout is the following:

@example
parameter (np=2,mp=1);
parameter (mobs=0);

parameter (nxp=0,nyp=0,nzp=0);
parameter (lp=0);
parameter (maxstep=1);
@end example

If there are observations, (@pxref{Observations}), the
size of the observation vector is set in the @file{dimetaphi} sequence
by the @code{mobs} parameter. For example if there is one observation:

@example
parameter (mobs=1);
@end example

To specify parameters (@pxref{Parameters}), the number of such parameters 
has to be declared in @file{dimetaphi} with the parameter @code{lp}. 
Then, if there are two parameters, they are first declared with

@example
parameter (lp=2);
@end example

@node Model with explicit size
@subsection Entering the model equations, with explicit sizes

@cindex model equations
@findex Phi_tef(.)
@findex deta_tef(.)
@vindex eta(.), explicit sizes
@vindex ff(.), explicit sizes

When sizes are explicit, another possibility exists for entering
the model equations. The use of symbolic names, as described in
@ref{Model equations} is still possible, and it also becomes possible to
set directly the equations associated with the @code{eta(.)}
and @code{ff(.)} vectors.

In case the symbolic names are not used, 
the model equations for cells and transfers are entered using a mortran macro,
@code{f_set}@footnote{@code{fun_set}, or equivalently @code{f_set}, is a 
general mortran macro associating a symbol with a fortran expression. 
Here, it is the name of the symbol (@code{eta}) that has a particular meaning
for the building of the model.}, setting the @code{eta(.)} evolution with 
@code{deta_tef(.)}
and the transfer definitions @code{ff(.)} with @code{Phi_tef(.)}.

@defmac f_set Phi_tef(@var{i}) = f(eta(.),ff(.))
This macro defines the transfer @var{i} static equation.
@code{f} is a fortran 
expression which may be function of cell state variables, 
@samp{eta(1)}@dots{}@samp{eta(np)} and transfers 
@samp{ff(1)}@dots{}@samp{ff(mp)}.
@end defmac

In the case of the predator-prey model, the transfer definition for 
@math{\varphi_{meet}} is:
@example
f_set Phi_tef(1) = eta(1)*eta(2);  
@end example

@defmac f_set deta_tef(@var{i}) = g(eta(@var{i}),ff(.))
This macro defines the cell state component @var{i} time evolution model. 
@code{g} is a expression which may be function of cell state variables, 
@samp{eta(1)}@dots{}@samp{eta(np)} and transfers 
@samp{ff(1)}@dots{}@samp{ff(mp)}.
@end defmac

The two cell equations of the predator-prey model are, with index 1 for the
prey (@math{\eta_{prey}}) and index 2 for the predator (@math{\eta_{pred}}):

@example
f_set  deta_tef(1) = apar*eta(1)-apar*ff(1);
f_set  deta_tef(2) = - cpar*eta(2) + cpar*ff(1);
@end example

The whole model is:

@example
!%%%%%%%%%%%%%%%%%%%%%%
! Transfer definition
!%%%%%%%%%%%%%%%%%%%%%%
! rencontres (meeting)
    f_set Phi_tef(1) = eta(1)*eta(2); 

!%%%%%%%%%%%%%%%%%%%%%%
! Cell definition
!%%%%%%%%%%%%%%%%%%%%%%
! eta(1) : prey
! eta(2) : predator      

    f_set  deta_tef(1) = apar*eta(1)-apar*ff(1);
    f_set  deta_tef(2) = - cpar*eta(2) + cpar*ff(1);
@end example

The starting points for cells are entered like:
@example
!     initial state
!     -------------
     eta(1) = 1.;
     eta(2) = 1.;
@end example

If there are observations, they are entered as special transferts with
index above @code{mp}, for example:

@example
f_set Phi_tef(mp+1) = ff(1) ;
@end example

@node Programming with cmz directives
@section Programming with cmz directives

@menu
* Cmz directives used with @Minik{}::
* Using cmz directives in @Minik{}::
@end menu

@node Cmz directives used with @Minik{}
@subsection Cmz directives used with @Minik{}

The main feature of cmz directive is to use code conditionnaly for a given
select flag. For example when the double precision is selected
(@pxref{Double precision}) the use of the conditionnal 
@code{double} flag may be required in case there is a different subroutine 
name for different types. If, for example, the user use the subroutine
@code{smysub} for simple precision and @code{dmysub} for double
precision the following code is an example of what could appear in the
user code:

@verbatim
+IF,double
 call dmysub(eta);
+ELSE
 call smysub(eta);
+ENDIF
@end verbatim

For a complete reference on cmz directives see the appendix
@ref{Cmz directives reference}.

@node Using cmz directives in @Minik{}
@subsection Using cmz directives in @Minik{}

In cmz the KEEP and DECK have their cmz directives preprocessed as part
of the source files extraction. And the +KEEP and +DECK 
directives are automatically
set when creating the KEEP or DECK. With make, files with these directives 
has to be created within the files that are to be preprocessed by the
cmz directives preprocessor.

To be processed by make, a file that contains cmz directives 
should have a file suffix corresponding
with the language of the resulting file and with the normal file suffix of
that language. More precisely @samp{cm} should be added before the normal
file suffix and after the @samp{.}. Therefore if the resulting file language
is associated with a suffix @samp{.@var{suf}}, the file with cmz directives
should have a @samp{.cm@var{suf}} suffix. The tradition is to have
a different suffix for main files and include files. 
To add directories searched for @dfn{cmfiles} (files with cmz directives) 
they should be added to the @code{CMFDIRS} makefile variable, separated 
by @samp{:}.

Rules for preprocessing of the files are defined in the file 
@file{Makefile.miniker} for the file types described in 
@ref{tab:cmfile_suffix}:

@float table, tab:cmfile_suffix
@multitable {fortran preprocessed} {include/keep} {cmfile suffix} {suffix} {language}
@headitem language  @tab file type @tab cmfile suffix @tab suffix @tab language
@item fortran @tab main/deck @tab .cmf @tab .f @tab ftn
@item fortran preprocessed @tab main/deck @tab .cmF @tab .F @tab f77
@item fortran preprocessed @tab include/keep @tab .cminc @tab .inc @tab f77
@item mortran @tab main/deck @tab .cmmtn @tab .mtn @tab mtn
@item mortran @tab include/keep @tab .cmmti @tab .mti @tab mtn
@end multitable
@caption{Association between file language, file type, file suffixes and 
language identifier in cmz directives. A main file is called a @dfn{deck}
in cmz and an include file is called a @dfn{keep}.}
@end float

@node Dynamic system analysis
@chapter Dynamic analysis of systems in @Minik{}

@menu
* Sensitivities::
* Adjoint model and optimisation::
* Kalman filter::
* Feedback gain::
* Stability of fastest modes::
* Generalized TLS::
@end menu

@node Sensitivities
@section Automatic sensitivity computation

@cindex sensitivities

An obvious advantage of having acces to the Jacobian matrices along the
system trajectory concerns automatic sensitivity analyses, as either:
@itemize @bullet
@item the sensitivity of all variables to perturbation in the initial condition
      of one state variable;
@item the same sensitivities to an initial pulse (or step) on a transfer;
@item the same sensitivities to a series of pulses (or steps) on a transfer;
@item the same for a change in a parameter, eventually during the run;
@item the sensitivity of the matrix of advance in state space to a change
 in a parameter.
@end itemize

This is declared in Zinit as:

@example
! -------------
! Sensitivities
! -------------
Sensy_to_var
< var: eta_pray, pert: INIT;
  var: eta_pred, pert: INIT;
>;
@end example

Each variable at origin of a perturbation is declared as @code{var:},
and the type of perturbation in @code{pert:}. Here, INIT conditions are
only allowed because the two variables are states variables. For transfers,
@code{pert: pulse} corresponds to an initial pulse, @code{pert: step_resp}
and  @code{pert: step_eff} to initial steps, the difference between 
@code{_resp} (response form)
and @code{_eff} (effect form) concerns the 
diagonal only of the sensitivity matrix
(see Feedback gains in non-linear models).

Non initial perturbation can also be asked for:

@example
  Sensy_to_var
  <
!*     var: eta_courant_L, pert: init at 100;
!*     var: ff_T_czcx,     pert: pulse at 100 every 20;
!*     var: ff_Psi_Tczcx,  pert: step_eff;
!*     var: ff_Psi_Tczcx,  pert: step_Resp at 10 every 100;
! *** premiers tests identiques a lorhcl.ref
    var: ff_courant_L , pert: step_eff;
    var: ff_T_czcx    , pert: step_eff;
    var: ff_Psi_Tczcx , pert: step_eff;
    var: ff_Psi_Tsz   , pert: pulse at 100 every 50;
  >;
@end example

In this example taken from @file{lorhcl}, a sensitivity can increase so as to
trespass the Fortran capacity, so that each  sensitivity vector (matrix column)
can be reset at some time-increment @code{at III every JJJ;}

It is noteworthy that these sensitivity analyses are not based
on difference between two runs with different initial states or
parameter values, but on the formal derivatives of the model. This method 
is not only numerically robust, but is also rigorously funded as based on 
the TLS of the model@footnote{For a short introduction to automatic 
sensitivity analysis, see the document:@*
@url{http://lmd.jussieu.fr/zoom/doc/sensibilite.ps}, in French,
or ask for the more complete research document to a member of the TEF-ZOOM
collaboration}.

If the @code{dimetaphi} sequence is built by the users, he should declare
the number of perturbing variables as @code{nxp=}:

@example
      parameter (nxp=np,nyp=0,nzp=0);
@end example
here, all state variables are considered as perturbing variables.

@cindex sensitivity, output
@cindex output, sensitivity
@cindex @file{sens.data}
@cindex @file{sigma.data}

The sensitivity vectors are output in the result files @file{sens.data} for 
cells and @file{sigma.data} for transfers. In those files the first column
corresponds again with time, and the other columns are relative sensitivities of the cell
states (in @file{sens.data}) and transfers (in @file{sigma.data})
with respect to the initial value of the perturbed state. 

In our predator-prey example, the second column of  @file{sens.data} will contain
the derivative of @math{\eta_1(t)} with respect to @math{\eta_1(t=0)}.
Drawing the
second column of @file{sens.data} against the first one
gives the time evolution of the sensitivity of @code{eta-pred}
to a change in the initial value of @code{eta-pray}. One can check
in that it is set to 1 at @math{t=0}:

@example
#    Sensy_to: eta_pray         3        eta_pred         5       
# time \\  of: eta_pray     eta_pred     eta_pray     eta_pred    
  0.00000E+00  1.00000E+00  0.00000E+00  0.00000E+00  1.00000E+00
  1.00000E-02  9.90868E-01  1.11905E-02 -1.26414E-02  9.98859E-01
@end example
The two last columns are the state sensitivity to a change in initial conditions
of the number of predators.

In the same way, the @var{j+1}th column of @file{sigma.data} will be the
derivative of @math{\phi_{j}(t)} with respect to @math{\eta_i(t=0)}. Here:
@example
#    Sensy_to: eta_pray     eta_pred    
# time \\  of: ff_interact  ff_interact 
  0.00000E+00  1.60683E+00  8.47076E-01
  1.00000E-02  1.59980E+00  8.18164E-01
@end example

the unique transfer variable gives rise to two sensitivity columns. 

Sensitivity studies are usefull to assess the
predictability properties of the corresponding system.

@menu
@c * Initial state sensitivity::
@c * Sensitivity to a pulse or a step on transfer::
@c * Extended Sensitivity studies::
* Sensitivity to a parameter::
* Advance matrix sensitivity::
@end menu



@node Sensitivity to a parameter
@subsection Sensitivity to a parameter

A forward sensitivity to a parameter will be computed when specified as
described in @ref{Parameters}. For example, suppose that
the sensitivity to an initial change in the @code{apar} parameter of
the predator model is of interest. 
@c In that case the number of
@c parameters should be set to 1 in @file{dimetaphi}:
@c 
@c @example
@c parameter (lp=2);
@c @end example

The sensitivity calculs is turned on as a forward
parameter specified on the @code{Free_parameter} list:

@example
Free_parameter: [fwd: apar, cpar];
@end example

The result are in @file{sensp.data} for cells and @file{sigmap.data}
for transfers.

@example
#    Sensy_to: pi_prandtl       3            4        pi_rayleigh_     6
# time \\  of: eta_courant_ eta_T_czcx   eta_T_sz     eta_courant_ eta_T
  0.00000E+00  0.00000E+00  0.00000E+00  0.00000E+00  0.00000E+00  0.000
  2.00000E-03 -4.77172E-03 -3.99170E-05  3.55971E-05 -9.94770E-05 -1.004
@end example
In the above example from @file{lorhcl} sensitivity of the three states with respect
to an initial change in two parameters are independantly given (first line also numbers
the column to easy gnuplot using).

@node Advance matrix sensitivity
@subsection Advance matrix sensitivity


It is possible to look at the sensitivity of the matrix of advance in 
states space (the matrix @code{aspha}) with regard to a parameter. 
The parameter must be accounted for in the parameter number and be in the 
parameter list, flagged as the matrix @code{mx} parameter, like in
@example
Free_parameter: [mx: apar], cpar;
@end example

@vindex d_pi_aspha(.,.)

This feature is associated with a selecting flag, @samp{dPi_aspha}. One gets
the result in the matrix @code{d_pi_aspha(.,.)} of dimension 
(@code{np},@code{np}).

This matrix may be used to compute other quantities, for example
it may be used to compute the sensitivity of the eigenvalues of
the state-advance matrix with regard to the @code{[fwd]} parameter.
These additional computations have to be programmed by the user in 
@file{zsteer} with matrices declared and initialized in 
@file{zinit}. An example is given in the example @file{lorhcl}
provided with the @Minik{} installation files, following a method proposed
by Stephane Blanco.

@node Adjoint model and optimisation
@section Adjoint model and optimisation with @Minik{}

In the following a possible use of @Minik{} for optimisation is discussed.
More precisely the use of adjoint and control laws in @Minik{} are presented.
Optimisation isn't the only application of these tools, but it is the most
common one. In that case the adjoint may be used to determine the gradient of a
functional to perturbations in the control laws, and an optimisation process
can use this
information to search for the optimum.
Another application of the adjoint is to compute the sensitivity of a
cost function to parameters (the ones declared in the @code{free_parameters:}' list.
Note that the cost function can be sensitive to probe's variables, even if these are
uncoupled with standard variables in the forward calculations; this is the case
when minimizing a quadratic distance function between probes (from the  model)
and the corresponding measurements.

The code is close transcription of the mathematical calculus described
in@* @url{http://www.lmd.jussieu.fr/ZOOM/doc/Adjoint.pdf} . It essentialy reverse time and
transpose the four Jacobian matrices: states and transfers are saved in array dimensionned
with @code{maxstep} Fortran parameter.
@menu
* Overview of optimisation with @Minik{}::
* Control laws::
* Cost function coding and adjoint modeling::
* Sensitivity of cost function to parameters::
@end menu

@node Overview of optimisation with @Minik{}
@subsection Overview of optimisation with @Minik{}

@cindex adjoint
@cindex optimisation

In the proposed method, @Minik{} is run twice, one time forward and then
backward to determine the trajectory and the adjoint model. After that the 
control laws are modified by a program external to @Minik{}. The same steps
are repeated until convergence. More pecisely,

@table @strong
@item forward
The command law @math{h(t)} is given (by an explicit law or taken from a file).
The trajectory is computed in a classical way, with the additionnal computation
of the functional to be optimised, @math{J}, prescribed with specific 
@code{f_set} macros. The states, transfers and control laws are stored.  
@item backward
The adjoint variable is computed from the last time @math{T} backward. The
time increment is re-read as it could have changed during the forward 
simulation. The system is solved by using the same technics as in the forward 
simulation, but with a negative time step.
@item external phase
Now the command should be corrected. This step isn't covered here, but, for
example, minuit the optimisation tool from the CERN could be used. 
In order to ease such a use of @Minik{}, the principal program has to be 
compiled as a subroutine to be driven by an external program 
(@pxref{Calling the model code}).
@end table

The functionnal @math{J} to be optimised is defined as

@ifset latex
@tex
\begin{equation}
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
\end{equation}
@end tex
@end ifset
@ifclear latex
@tex
$$
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
$$
@end tex
@ifnottex
@noindent @math{J = psi(eta(T),phi(T),h(T)) + int_0^T l(eta(tau),phi(tau),h(tau)) d tau}
@end ifnottex
@end ifclear

@cindex final cost
@cindex integrand cost

Where @math{\psi} is the final cost function, @math{l} is the integrand
cost function and @math{h} represents the control laws variations.

The general use of the adjoint model of a system is the determination of the 
gradient of this @math{J} functional to be optimised, with respect to perturbations
of the original conditions of the reference trajectory, that is, along its 
GTLS@footnote{General Tangent Linear System, i.e. the TLS circulating along a trajectory.
See the explanation in the document
@url{http://www.lmd.jussieu.fr/Zoom/doc/Adjoint.pdf} (in French).}.

@node Control laws
@subsection Control laws

@vindex zcommand
@cindex command law

Each control law is associated with one cell or transfer equation, meaning that a command
associated with an equation does not appear in any other equation.
It is still possible
to add commands acting anywhere by defining a transfer equal to that command.


The control laws associated with states are in the @code{ux_com(.)} array, 
control laws associated with transfers are in the @code{uy_com(.)} array.
The control laws may be prescribed even when there is no adjoint computed, 
nor any optimisation, and they are used during simulation, in which case they will
act as external sources. To enable
the use of commands, the logical flag @code{Zcommand} should be @code{.true.}.

@cindex @file{uxcom.data}
@cindex @file{uycom.data}

The command can be given either as: 
@enumerate 
@item a table of numerical
values in the files @file{uxcom.data} and @file{uycom.data}.
@item a function
@vindex zlaw
@cindex @file{zcmd_law}
@cindex @file{zcmd_law.inc}
of the problem variables. To turn that feature on the logical flag 
@code{Zlaw} should be set to @code{.true.} in @file{zinit}. The sequence 
@file{zcmd_law} should hold
the code filling the @code{ux_com(.)} and @code{uy_com(.)} arrays, as the code
from that sequence is used whenever the control laws are needed.
In that case the files  @file{uxcom.data} and @file{uycom.data} will 
be filled by the command values generated by the function along the trajectory.
@end enumerate

For example in the Lotka-Volterra model, the parameter @code{apar} could 
be a control variable.
In that case, @code{apar} would be defined as the variable @code{ux_com(1)}, 
and either entered as a law
in the sequence @file{zcmd_law} , either written in the file @file{uxcom.data} 
step by step. In that case, there must be a perfect corresponodence between time
of the commands and time of the run.

@node Cost function coding and adjoint modeling
@subsection Cost function coding and adjoint modeling

@vindex zback
@findex cout_Psi
@findex cout_l

First of all the flag @code{zback} should be set to @code{.true.} in order to
allow adjoint model computation:

@example
Zback=.true.;
@end example

The two functions @code{cout_Psi} corresponding with the final cost and 
@code{cout_l} corresponding with the integrand cost are set up with the
@code{f_set} macros.

@defmac f_set cout_Psi = f(eta(.),ff(.),ux_com(.),uy_com(.))
This macro defines the final cost function.
@code{f} is a fortran 
expression which may be function of cell state variables, 
@samp{eta(1)}@dots{}@samp{eta(np)}, transfers 
@samp{ff(1)}@dots{}@samp{ff(mp)}, 
state control laws
@samp{ux_com(1)}@dots{}@samp{ux_com(np)}, and transfer control laws
@samp{uy_com(1)}@dots{}@samp{uy_com(mp)}.
@end defmac

@defmac f_set cout_l = f(eta(.),ff(.),ux_com(.),uy_com(.))
This macro defines the integrand cost function.
@code{f} is a fortran 
expression which may be function of cell state variables, 
@samp{eta(1)}@dots{}@samp{eta(np)}, transfers 
@samp{ff(1)}@dots{}@samp{ff(mp)},
state control laws
@samp{ux_com(1)}@dots{}@samp{ux_com(np)}, and transfer control laws
@samp{uy_com(1)}@dots{}@samp{uy_com(mp)}.
@end defmac

For example, the following code sets a cost function for the masselottes
model:

@example
! Initialisation 
  F_set cout_Psi = eta_move(inode:1);
!and f_set cout_l integrand in the functionnal    
  F_set cout_l = 0.;
@end example

In that example the functional is reduced to the final value
of the first state component.
Here, the adjoint vector will correspond to the final sensitivity 
(at @math{t=0}) of 
that component (here the first masselotte position) to a perturbation in 
all initial conditions@footnote{For detailed explanation of the adjoint model,
see the document in 
@uref{http://www.lmd.jussieu.fr/@/ZOOM/doc/Adjoint.pdf,pdf}
or @uref{http://www.lmd.jussieu.fr/@/ZOOM/doc/Adjoint.pdf,.ps.gz}}. 

@c In the code, the variables @code{v_adj(.)} and @code{w_adj(.)}
@c are respectively adjoint to @code{eta(.)} and @code{ff(.)}. They are written in the
@c two files: @file{vadj.data} and @file{wadj.data}.
The following variables are set during the backward phase, and output
in the associated files:


@multitable {@code{gradufj(.)}} {@file{hamilton.data}} {time increment, hamiltonian, cost function increment}
@headitem var @tab file @tab explanation
@c @item @code{} @tab @file{.data} @tab  @tab
@item @code{v_adj(.)} @tab @file{vadj.data} @tab adjoint to @code{eta(.)}
@item @code{w_adj(.)} @tab @file{wadj.data} @tab adjoint to @code{ff(.)}
@item @code{wadj(mp+.)} @tab @file{gradmuj.data} @tab adjoint to @code{ff(mp+.)}
@item @code{graduej(.)} @tab @file{gradxj.data} @tab adjoint to @code{ux_com(.)}
@item @code{gradufj(.)} @tab @file{gradyj.data} @tab adjoint to @code{uy_com(.)}
@item @code{hamilton} @tab @file{hamilton.data} @tab time increment, hamiltonian, cost function increment
@end multitable

@node Sensitivity of cost function to parameters
@subsection Sensitivity of cost function to parameters

@cindex @file{gradpj.data}

The sensitivity of the cost function to all the parameters given as
arguments of @code{Free_parameters} is computed. For the
predator model the sensitivity of a cost function consisting in 
the integral of the predator population with respect with
@code{apar} an @code{cpar} is obtained with a number of parameters
set to 2 in @file{dimetaphi}:

@example
parameter (lp=2);
@end example

And the cost function and @code{Free_parameters} list in @file{zinit}:

@example
f_set cout_Psi = eta(2);
f_set cout_l = eta(2);
Free_parameters: apar,cpar;
@end example

@code{apar} and @code{cpar} also have to be given a value.
The result is output in @file{gradpj.data}.

@node Kalman filter
@section Kalman filter

@cindex Kalman filter
@cindex variance-covariance matrices, general
@cindex observations, general

The Kalman filter allows for data assimilation along the model run. In 
that case it is assumed that there is a real-world model with stochastic
perturbations on the states, and that noisy observations are available. 
The situation implemented in @Minik{} corresponds to a continuous 
stochastic perturbation on the state, and discrete noisy observations.
In the @acronym{TEF} this leads to:

@ifset latex
@tex
\begin{eqnarray}
\partial_t \eta (t) &=&  g(\eta(t),\varphi(t)) + W(t) \mu\\
\varphi(t) &=& f(\eta(t),\varphi(t))\\
\omega(t) &=& h ( \eta(t) , \varphi(t)) + \nu
\end{eqnarray}
@end tex
@end ifset
@ifclear latex
@tex
$$\eqalign{
\partial_t \eta (t) &=  g(\eta(t),\varphi(t)) + W(t) \mu\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
\omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
}$$
@end tex
@ifnottex

@noindent @math{d eta(t)/d t = g(eta(t),phi(t)) + W(t) mu@*
phi(i) = f(eta(t),phi(t))@*
omega(t) = h(eta(t), phi(t)) + nu }

@end ifnottex
@end ifclear

@c FIXME partout omega
@c (notice that in this paragraph, $\omega$ stands for the probe vector $\mu$ elsewhere,
@c and $\mu$ is here a noise source.

The observations @math{\omega} are available at discrete time steps @math{t=s_i}. The
stochastic perturbation on state, @math{\mu} is characterized by a 
variance-covariance matrix @math{Q} and the noise on the observation,
@math{\nu} has a variance-covariance matrix @math{R}. @math{W} relates states
with stochastic perturbations. At each time step the Kalman filter recomputes 
an estimation of the state and the variance-covariance matrix of the state.

In the following we use the example of a linear model with perturbation 
on state and observation of state. The model has 3 states and 3 corresponding
transfers (equal to the states), but the error on the state is of dimension 
2. The 3 states are observed. The corresponding equations read:

@ifset latex
@tex
\begin{equation}
\left\{
\begin{array}{cc}
\partial_t \eta_1 =& a_{11} \eta_1 + a_{12} \varphi_2 + a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\\
\partial_t \eta_2 =& a_{21} \varphi_1 + a_{22} \eta_2 + a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\\
\partial_t \eta_3 =& a_{31} \varphi_1 + a_{32} \varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
\end{array}
\right.
\end{equation}
\begin{equation}
\left\{
\begin{array}{cc}
\varphi _1 =& \eta _1\\
\varphi _2 =& \eta _2\\
\varphi _3 =& \eta _3
\end{array}
\right.
\end{equation}
\begin{equation}
\left\{
\begin{array}{cc}
\omega _1 =& \varphi _1 + \nu_1\\
\omega _2 =& \eta _2 + \nu_2 \\
\omega _3 =& \eta _3 + \nu_3
\end{array}
\right.
\end{equation}
@end tex
@end ifset
@ifclear latex
@tex
$$\left\{\eqalign{
\partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 + a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\cr
\partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 + a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\cr
\partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
}\right.$$
$$\left\{\eqalign{
\varphi _1 &= \eta _1\cr
\varphi _2 &= \eta _2\cr
\varphi _3 &= \eta _3
}\right.$$
$$\left\{\eqalign{
\omega _1 &= \varphi _1 + \nu_1\cr
\omega _2 &= \eta _2 + \nu_2 \cr
\omega _3 &= \eta _3 + \nu_3
}\right.$$
@end tex

@ifnottex

Cells:@*
@noindent @math{d eta_1/dt = a_11 eta_1 + a_12 phi_2 + a_13 phi_3 + W_11 mu_1 + W_12 mu_2@*
d eta_2/dt = a_21 phi_1 + a_22 eta_2 + a_23 phi_3 + W_21 mu_1 + W_22 mu_2@*
d eta_3/dt = a_31 phi_1 + a_32 phi_2 + a_33 eta_3 + W_31 mu_1 + W_32 mu_2}

Transfers:@*
@noindent @math{phi_1 = eta_1@*
phi_2 = eta_2@*
phi_3 = eta_3}

Observations:@*
@noindent @math{omega_1 = phi_1 + nu_1@*
omega_2 = eta_2 + nu_2@*
omega_3 = eta_3 + nu_3}

@end ifnottex
@end ifclear

@menu 
* Coding the Kalman filter::
* Kalman filter run and output::
* Executing code after the analysis::
@end menu

@node Coding the Kalman filter
@subsection Coding the Kalman filter

@vindex zkalman

First of all the Kalman filter code should be activated. The observations
code is also required (@pxref{Observations}).
If cmz is used the code
should be selected with the select flag kalman 
in the @file{selseq.kumac}:

@example
sel kalman
@end example

With make the @code{kalman} variable should be set to 1:

@example
kalman = 1
@end example

The kalman code is actually used by setting the flag
@code{zkalman} to @code{.true.}, for example in the @file{zinit}:

@example
zkalman = .True.;
@end example

@c This will set the @code{zobs} and @code{zdata} flags to @code{.true.} 
@c (@pxref{Observations and data}).

With the Kalman filter the dimension of estimated states, of the error 
on the state and of the
observation, the @math{W} matrix, the observation function,
the initial
variance-covariance matrices on the state and the variance-covariance matrices 
of errors have to be given.

@menu 
* Kalman filter vectors dimensions::
* Error and observation matrices::
@end menu

@node Kalman filter vectors dimensions
@subsubsection Kalman filter vectors dimensions

@cindex error vector dimension
@cindex @file{dimetaphi}, Kalman filter

These dimensions should be set in the @file{zinit} sequence.
The size of the estimated states is given by the parameter @code{nkp}. 
You can set this to @code{np} if all the states are estimated, but in case
there are some deterministic state variables, @code{nkp} may be less than
@code{np}. In that case the first @code{nkp} elements of @code{eta(.)}
will be estimated using the Kalman filter.

The error on state dimension is associated with the parameter @code{nerrp}
and the size of the observations vector is @code{mobs} 
(@pxref{Observations}). In our example the dimensions are set with:

@example
parameter (nkp=np);
parameter (mobs=3);
parameter (nerrp=2);
@end example

All the states are estimated,
there are 3 observation functions and the error on the state vector is of
dimension 2.

If the sizes are set explicitely, the parameters should be set in
@file{dimetaphi}.

@node Error and observation matrices
@subsubsection Error and observation matrices

@cindex variance-covariance matrices
@cindex observations
@cindex @file{zinit}, Kalman filter

@subsubheading Initial variance-covariance matrix on the state

@cindex initial variance-covariance on states
@vindex covfor(.,.)

The variance-covariance on the state matrix is @code{covfor(.,.)}. The initial
values have to be given for this matrix, as in our example:

@example
covfor(1,1) = 1000.; covfor(1,2) = 10.; covfor(1,3) = 10.;
covfor(2,1) = 10.; covfor(2,2) = 5000.; covfor(2,3) = 5.;
covfor(3,1) = 10.; covfor(3,2) = 5.; covfor(3,3) = 2000.;
@end example

This matrix is updated by the filter at each time step because the states
are pertubated by some noise, and when assimilation takes place as new
information reduce the error.

@subsubheading Observations and error on state matrix

@cindex variance-covariance matrix on state
@vindex mereta(.,.)

The matrix that relates errors on states vector components to states,
corresponding with @math{W} is @code{mereta(.,.)}. In our example it is 
set by:

@example
mereta(1,1) = 1.;  mereta(1,2) = 0.;
mereta(2,1) = 0.;  mereta(2,2) = 1.;
mereta(3,1) = 0.5;  mereta(3,2) = 0.5;
@end example

The observation functions are set by a @code{f_set} macro with 
@code{Obs_tef(.)} as described in @ref{Observations}.
In our example the observation functions are set by:

@example
f_set Obs_tef(1) = ff(1) ;
f_set Obs_tef(2) = eta(2);
f_set Obs_tef(3) = eta(3);
@end example

@subsubheading Error variance-covariance matrices

@cindex variance-covariance error
@vindex covobs(.,.)

The variance-covariance matrix on observation noise is @code{covobs(.,.)}
set, in our example, by:

@example
covobs(1,1) = 0.3; covobs(1,2) = 0.; covobs(1,3) = 0.;
covobs(2,1) = 0.; covobs(2,2) = 0.1; covobs(2,3) = 0.;
covobs(3,1) = 0.; covobs(3,2) = 0.; covobs(3,3) = 0.2;
@end example

@vindex coveta(.,.)
The variance-covariance matrix on state noise is @code{coveta(.,.)}
set, in our example, by:

@example
coveta(1,1) = 0.2; coveta(1,2) = 0.001;
coveta(2,1) = 0.001; coveta(2,2) = 0.1;
@end example

These matrices are not changed during the run of the model as part
of the filtering process. They may be changed by the user in @file{zsteer}.

@node Kalman filter run and output
@subsection Kalman filter run and output

@menu
* Feeding the observations::
* Kalman filter results::
@end menu

@node Feeding the observations
@subsubsection Feeding the observations to the model

@vindex vobs(.)
@vindex zgetobs
@cindex @file{zsteer}, Kalman filter

The observations must be made available to the model during the run. These
observations are set in the @code{vobs(.)} array, and the assimilation 
(also called the analysis step of the filter) takes
place if the logical variable @code{zgetobs} is @code{.true.} 
(@pxref{Data}). 

These steps are
typically performed in the @file{zsteer} sequence. In this sequence there should
be some code such that when there are data ready to
be assimilated, @code{zgetobs} is set to @code{.true.} and the data is
stored in @code{vobs(.)}, ready for the next step processing.

@node Kalman filter results
@subsubsection Kalman filter results

@cindex results, Kalman filter
@cindex Kalman filter results
@cindex output, Kalman filter
@cindex Kalman filter output
@cindex @file{data.data}

The estimated states and transfers are still in the same @samp{.data} files, 
@file{res.data} and @file{tr.data} and there is the additional file with
observations, called @file{obs.data} (@pxref{Observations}). 
Each time @code{zgetobs} is @code{.true.} the data, and the optimally 
weighted innovations are output 
in the file associated with data, @file{data.data} (@pxref{Data}).

@node Executing code after the analysis
@subsection Executing code after the analysis

The analysis takes place before the time step advance when @code{zgetobs}
is @code{.true.}. It may be usefull to add some code after the analysis
and before the time step advance. For example the analysis may lead to 
absurd values for some states or parameters, it could be usefull to correct
them in that case. The sequence included after the analysis is called
@file{kalsteer}. At this point, in addition to the usual variables 
the following variables could be usefull:

@vtable @code
@item etafor(.)
The state before the analysis.
@item kgain(.)
The Kalman gain.
@item innobs(.)
The innovation vector (observations coherent with the states minus data
values).
@item covana(.,.)
The variance-covariance error matrix after the analysis.
@end vtable

At each time step the derivative of the observation function with respect
to transfer and cells variables are recomputed. The elimination of
transfers is also performed to get the partial derivative of the observation
function of the equivalent model, with states only, with respect to the 
states. In other words, the Kalman filter does not follow the TEF formalism, because
the advance of the var-covar matrix could not yet be set in the TEF form.
@c There is a corresponding additional matrix:

@vtable @code
@c @item obetad(.,.)
@c derivative of observation function with respect to transfers.
@c @item obphid(.,.)
@c derivative of observation function with respect to cell variables.
@item obspha(.,.)
derivative of observation function in state space with respect to
cell variables.
@end vtable


@node Feedback gain
@section Feedback gain


@cindex Borel sweep
@cindex Feedback gain

The feedback dynamic gain associated with a feedback loop
can be expressed as the inverse Borel 
transform of the coefficient of the reduced scalar 
coupling matrix, @math{g(\tau)},
associated with a transfer. 
A Borel sweep provides this @math{g(\tau)}. Therefore it is
an interesting tool for the characterization of the feedback loop@footnote{
More generally, the Borel sweep allows 
the numerical study of the dependency in @math{\tau} of the Borel transform 
of various coefficients in the system coupling  matrix.}. 

As explained in the 
ZOOM web page document 
@url{http://www.lmd.jussieu.fr/@/ZOOM/doc/@/Feedback_Gain.pdf},
this allows for the calculation of the
dynamic gain and factor of any feedback that goes through a unique
transfer variable. An example of the conclusions that can be drawn from such
an analysis is provided in the same document.

@ignore
The Borel sweep allows the numerical study of the dependency in @math{\tau}
of the Borel transform of various
coefficients in the system coupling  matrix. For example, the coefficient
@math{g(\tau)} of the reduced scalar coupling matrix
associated with a transfer defines a feedback gain. 
@end ignore


For linear systems -- whose GTLS are autonomous along the whole trajectory -- 
the @math{\tau} function of the
feedback gain is independent of the position on the system trajectory. 
But in general it is dependant, and one can analyse the function 
@math{g(\tau;t)} defined on a segment @math{t} of the trajectory.

The document introducing the TEF-ZOOM technique explains how a Crank-Nicolson
scheme for the time discretisation
symbolically gives the solution of the Borel transform of the system. One can
identify the @code{dt} variable with the Borel @math{\tau} within a 
factor @math{2}. Hence, to numerically study the @math{\tau} dependency of 
the transform of various coefficients in the system coupling  matrix at one 
point in time, one can calculate the Borel transform of the TLS solutions 
by making a time-step sweep.

The function @math{g(\tau;t)} is simply output for the feedback gain 
attached to a unique @code{ff(k)} transfer variable. 
All the relevant informations should be entered in the @file{zinit} sequence.

@menu
* Specifying the Borel sweep::
* Borel sweep results::
@end menu

@node Specifying the Borel sweep
@subsection Specifying the Borel sweep

@vindex ZBorel

First of all the logical flag @code{ZBorel} should be raised:

@example
ZBorel=.true.;
@end example

@vindex index_ff_gain
The index of the studied transfer is given in the @code{index_ff_gain}
variable
@example
index_ff_gain=7;
@end example

At each time step a Borel sweep may be performed. The time steps of interest 
are
specified with three variables, one for the first step, one for the last step
and one for the number of steps between two Borel sweeps:

@vtable @code
@item istep_B_deb
First time step for the Borel sweep.
@item  istep_B_fin
Last time step for the Borel sweep.
@item istep_B_inc
Number of time steps between Borel sweeps.
@end vtable

In the following examples Borel sweeps are performed from the 
time step 1000 up to the time step 1200, with a sweep at each time step:
@example
istep_B_deb=1000;   
istep_B_fin=1200;  
istep_B_inc=1;      
@end example


For each Borel sweep, the range of the @math{\tau} variable should be 
set. As this is a multiplicative variable the initial value, a multiplicative
factor and the number of values are to be given.

@vtable @code
@item tau_B_ini
Initial value for @math{\tau}.
@item tau_B_mult
Multiplicative factor for sweep in @math{tau}.
@item itau_max
Number of @math{\tau} values.
@end vtable

For example, in the following, at each time step, the Borel
transform will be computed for @math{\tau} values
starting at @math{0.2} and then multiplied a hundred times by @math{\sqrt{\sqrt{2}}}

@example
tau_B_ini=0.2;    
tau_B_mult=sqrt(sqrt(2.)); 
itau_max=100;             
@end example

When the initial value of @math{\tau} is set to a negative value 
(@i{i.e.} @code{tau_B_ini=-0.2;}),
the Borel sweep will first be applied with @code{itau_max} negative values 
for @code{-0.2}, @code{tau_B_mult*(-0.2)},..., then for the zero value, 
and finally for the symetric positive values, resulting in @code{2*itau_max+1} 
values for @math{\tau}.

The whole example reads

@example
! -------------------
! Feedback gain
! Borel
! -------------------
ZBorel=.true.;
if ZBorel           
<  istep_B_deb=1000; 
   istep_B_fin=1200;
   istep_B_inc=1;  
;
   index_ff_gain=7; 
   tau_B_ini=0.2;    
   tau_B_mult=sqrt(sqrt(2.)); 
   itau_max=100;             
   z_pr/Borel/:tau_B_mult,tau_B_ini*(tau_B_mult)**itau_max;
>;
@end example

@findex zborel for

Instead of using the index of the transfer in @code{index_ff_gain} it is 
possible to specify the name of the transfer.@c , whenever 
@c the symbolic model description is used (@pxref{Symbolic model description}). 
In that case the transfer is specified
by the @code{zborel for} macro. For example if the transfer selected for the
feedback gain computation is @var{b_transfer}, it can be selected
with:

@example
zborel for: @var{b_transfer};
@end example

@node Borel sweep results
@subsection Borel sweep results

@cindex Borel sweep results
@cindex results, Borel sweep
@cindex Borel sweep graphics
@cindex graphics, Borel sweep

The file @file{tau_Borel.data} gives the @math{\tau} values of the @var{tau} sweep, 
and the file @file{gains.data} records the feedback gain function values of 
@math{g(\tau)}, with 
one line for each sweep along the trajectory. In the 1.01 version, a new
feature is also provided giving the poles and residuals of the Borel
transform in the file @file{vpgains.data}. Consult the subroutine 
@code{Boreleig}
for (not definitive) output description.

One can easily obtain the surface contours of @math{g(t,\tau)} using
the Fortran program provided as @file{gains.f} and its compilation shell 
@file{gains.xqt},
that builds 2D histograms for PAW, in which one uses the
@file{borels.kumac} provided kumac.

@node Stability of fastest modes
@section Stability analysis of fastest modes

@cindex SVD
@cindex Singular Value Decomposition
@cindex state matrix
@cindex @file{sltc.exe}

The preceding analyses are done along with a simulation. One has also the
possibility of using in a more classical fashion the state advance matrix
@math{A_{st}}, after the end of the simulation. Code to perform the 
@acronym{SVD, Singular Value Decomposition} of the state matrix @math{A_{st}}
and also of @math{A_{st} + A_{st}^\dagger} is provided with @Minik{}.
The singular elements of these two matrices correspond to the most
rapid modes of instability of the perturbed system.

The Singular value decomposition of a matrix is noted

@tex
$$
 U w V^\dagger
$$
@end tex
@ifnottex

@noindent @math{U w V^t}

@end ifnottex

An executable file, @file{sltc.exe} is generated and running this file will
produce the corresponding results.

@menu
* SVD with cmz::
* SVD with make::
* SVD run and output::
@end menu

@node SVD with cmz
@subsection Singular Value Decomposition with cmz

@cindex @command{smod}

The cmz macro @code{smod SLTC} prepares a main program
(@file{circul} of +PATCH SLTC), provided as a base for user's own analysis,
in the directory @file{sltc/}.

@node SVD with make
@subsection Singular Value Decomposition with make

@cindex @file{Makefile.sltc}

To compile the singular value decomposition executable with @command{make} you 
can do
@example
make sltc.exe
@end example

If you want to have a separate directory for the SVD, you should copy 
the sequence @file{dimetaphi.inc} (or make a link to that file) to the
directory. You should also copy the file @file{Makefile.sltc} from the 
@file{template/} directory in this directory, rename it @file{Makefile}
and set the @Minik{} directory path in the 
@code{miniker_dir} variable. For
example, if the @Minik{} directory is in @file{/u/src/mini_ker}:

@example
miniker_dir = /u/src/mini_ker
@end example

@node SVD run and output
@subsection Singular Value Decomposition run and output

@cindex SVD run
@cindex run, SVD
@cindex SVD output
@cindex output, SVD
@cindex @file{sltc.exe}
@cindex @file{title.tex}, SVD
@cindex @file{aspha.data}, SVD

As it is, the @file{sltc.exe} executable generated by the compilation 
determines the SVD. This program requires @file{title.tex} (@pxref{Title file}) to
transmit a title for output and graphics, and @file{aspha.data} 
(@pxref{Simulation and output,,Running a simulation and using the output})
to access the
state matrix. To get access to these files (in case they are not in the current
directory) it is possible to make a link to 
the corresponding files in the model directory. Once it is done 
the program may be run:

@example
./sltc.exe
@end example

The files @file{u.data}, @file{w.data}, and @file{v.data} holds the singular elements 
for @math{A_{st}} (@math{U}, @math{w} and @math{V}), 
and @file{us.data}, @file{ws.data}, and @file{vs.data}
holds the singular elements of @math{A_{st} + A_{st}^\dagger}.
The corresponding macros @samp{.kumac} for PAW@footnote{Explanation in
the research paper about SLTC (Al1 2003) available on request.} 
are also generated.

@node Generalized TLS
@section Generalized linear tangent system analysis

@cindex Generalized linear tangent system
@cindex GTLS
@cindex propagator
@cindex Lyapunov exponents
@cindex @file{sltcirc.exe}

The state matrix @math{A_{st}} may also be used to compute the
GTLS propagator (or state transition matrix applied to perturbation), after the simulation.
The algorithm is a finite product of 
5th order development of
@math{\Phi(t+\delta t,t)=\exp{A_{st} \delta t}}.
Numerous element of analysis are given, in particular the determination
of the Lyapunov exponents of the system.

An executable file, @file{sltcirc.exe} is generated and running this file will
produce the corresponding results.

@menu
* GTLS with cmz::
* GTLS with make::
* GTLS run and output::
@end menu

@node GTLS with cmz
@subsection Generalized tangent linear system with cmz

@cindex @command{smod}

The cmz macro @code{smod SLTCIRC} prepares a main program
(@file{circule} of +PATCH SLTCIRC), in the directory @file{sltcirc/}.

@node GTLS with make
@subsection Generalized tangent linear system with make

@cindex @file{Makefile.sltcirc}

To compile the GTLS analysis executable with @command{make} you 
can do
@example
make sltcirc.exe
@end example

If you want to have a separate directory for the GTLS analysis, you should copy 
the sequence @file{dimetaphi.inc} (or make a link to that file) to the
directory. You should also copy the file @file{Makefile.sltcirc} from the 
@file{template/} directory in this directory and  rename it @file{Makefile}
and set the @Minik{} directory path in the @code{miniker_dir} variable.

@node GTLS run and output
@subsection Generalized tangent linear system analysis run and output

@cindex GTLS run
@cindex run, GTLS
@cindex GTLS output
@cindex output, GTLS
@cindex @file{sltcirc.exe}
@cindex @file{title.tex}, GTLS
@cindex @file{dres.data}, GTLS
@cindex @file{aspha.data}, GTLS

The @file{sltcirc.exe} executable generated by the compilation 
computes the elements of analysis of the system. This program requires 
@file{title.tex} to
transmit a title for output and graphics (@pxref{Title file}), 
@file{aspha.data} to access the
state matrix and @file{dres.data}, because time-step can be changed along the 
simulation 
(@pxref{Simulation and output,,Running a simulation and using the output})
@footnote{cf our research texts about propagator analyses in
SLTC, and ``les Gains sur champs (Al1 2003-2004)''}. To get access to these files 
(in case they are not in the current
directory) it is possible to make a link to 
the corresponding files in the model directory. Once it is done 
the program may be run:

@example
./sltcirc.exe
@end example

The following table gives the correspondence between variable name, 
result file and ntuple number, with a short explanation:

@multitable {@code{lwr(.,.)}} {@file{wlphit.data}} {ntuple} {eigen factors of @math{w} in the SVD of @math{\Phi}}
@headitem var @tab file @tab ntuple @tab explanation
@item @code{p(.,.)} @tab @file{phit.data} @tab 55 @tab propagator from 0 to @math{t}, @math{\Phi(t,0)}
@item @code{up(.,.)} @tab @file{uphit.data} @tab 50 @tab Left singular vectors @math{U} in the SVD of @math{\Phi}
@item @code{wp(.)} @tab @file{wphit.data} @tab 51 @tab singulat values @math{w} in the SVD of @math{\Phi}
@item @code{vp(.,.)} @tab @file{vphit.data} @tab 52 @tab Right Singular Vectors @math{V} in the SVD of @math{\Phi}
@item @code{wr(.)} @tab @file{wr.data} @tab 53  @tab real part of eigen values of @math{\Phi(t,0)}
@item @code{wi(.)} @tab @file{wi.data} @tab 54  @tab imaginary part of eigen values of @math{\Phi(t,0)}
@item @code{lwp(.)} @tab @file{lwphit.data} @tab 67  @tab Lyapunov exponents
@c @item @code{lwr(.,.)} @tab @file{lwr.data} @tab 68  @tab
@c @item @code{lwi(.,.)} @tab @file{lwi.data} @tab 69  @tab 
@c @item @code{} @tab @file{.data} @tab  @tab
@end multitable

@ignore
              ntuple # var name
---------------------------------------------------------------------
uphit.data      50       up       SVD of Phit 
wphit.data      51       wp        = U[w_diag]V'
vphit.data      52       vp
wr.data         53       wr[i]    eigen factors   
wi.data         54       wi       real and imaginary parts   
---------------------------------------------------------------------
lphit.data      65       lp[ij]   1/t Ln Phit     + 1/LnDetPhit,<TrA>
ulphit.data     60       ulp      SVD of 1/t LnPhit
wlphit.data     61       wlp
vlphit.data     62       vlp
wrl.data        63       wrl[i]   VP of 1/t LnPhit
wil.data        64       wil
---------------------------------------------------------------------
lwphit.data     67       lwp[i]   1/t Ln w : Phit=UwV'
lwr.data        68       lwr      1/t VP de UV'
lwi.data        69       lwi
---------------------------------------------------------------------

The bloc 60-65 contains the elements of analysis of the propagator when
it is assimilated to the autonomous system such that @math{\Phi(t,0)=\exp{At}},
where @math{A} is a state matrix of an autonomous system giving the same
sensitivity to perturbation at time @math{t}.
Last bloc  67-69 gives the Lyapunov exponents along with their
corresponding eigen vectors.
@end ignore

@node  Advanced use of @Minik{} with make
@chapter Advanced use of @Minik{} with make

@menu
* Make variables::
* Rules::
* Linking rule::
@end menu

@node Make variables
@section Make variables

@cindex @file{Makefile.miniker}

The @file{Makefile.miniker} Makefile provided in the
distribution should be included as it defines a lot of important 
variables and rules. 

The following make variables can be set by the user:

@table @code
@item miniker_dir
that variable should hold the @Minik{} sources directory. If you installed
@Minik{} that variable should be set to @file{$(includedir)/mini_ker}. 
If you use the sources right from the sources directory it should be set to 
the sources package directory.
@item MTNDIRS
This variable can hold a @samp{:} delimited list of directories that will
be searched for mortran include files.
@item CMFDIRS
This variable can hold a @samp{:} delimited list of directories that will
be searched for cmz directive include files.
@item SEL
This variable holds a @samp{,} delimited list of select flags, for example
@code{monitor}, @code{grid1d}, @code{debug}.
@item LDADD
This variable can be used to add libraries flags and files. It is used in 
the default linking command/rule.
@item miniker_user_objects
This variable should hold a space separated list of additional object files
to be linked with the model and helper object files.
@item CAR2TXTFLAGS
cmz directives preprocessor flag.
@item kalman
This variable should be set to 1 if you want to use the kalman filter
(@pxref{Kalman filter}).
@item double
This variable should be set to 1 if you want to have a double precision
code (@pxref{Double precision}).
@end table

The following variables are allready set and may be used
(some are set by ./configure see @ref{Configuration}):

@table @code
@item miniker_principal_objects
The list of object files needed for the model build, together with some 
helper object files often used but not strictly required for the linking.
@item DEPDIR
The name of a hidden directory containing the dependencies computed
for the main mortran files.
@itemx F77
@itemx FC
@itemx FFLAGS
@item LDFLAGS
Compiler and linker related variables set by ./configure.
@item LIBS
This variable should hold the link flags and files required to build 
@Minik{}, set by ./configure.
@item CAR2TXT
@itemx MORTRAN
@itemx MTNFLAGS
@itemx MTNDEPEND
Preprocessor and preprocessor flags, set by ./configure.
@end table

@node Rules
@section Rules

The following rules are defined in the @file{Makefile.miniker} file.
@table @code
@item miniker-clean
remove the fortran files generated from the mortran files. Remove 
the object files.
@item miniker-mtn-clean
remove the mortran files generated from the files with cmz directives.
@item 
Various rules to preprocess files with cmz directives and mortran files and
to compile fortran files.
@end table

If the user needs a mortran main file, he may take advantage of the rule
used to compute the dependencies of a mortran file. If the file is called,
say, @file{mtnfile.mtn} leading to @file{mtnfile.f}, the following include
should lead to the automatic creation, updating and inclusion of a 
file describing the dependencies of @file{mtnfile.mtn} in the 
@file{Makefile}:

@example
include $(DEPDIR)/mtnfile.Pf
@end example

@node Linking rule
@section Linking rule

The rule used for the linking of the model file is not in the 
@file{Makefile.miniker} file but 
should be provided in the user @file{Makefile} for more flexibility. 
The default rule 
uses the variables @code{miniker_user_objects} for additional object files
and @code{LDADD} for additionnal linking flags and files, those 
variables are there to be changed by the user.

The object files required by the @Minik{} code are in the make variable
@code{miniker_principal_objects}, this variable is also used. 
The value of the variables @code{FC}
for the Fortran compiler, @code{FFLAGS} for the Fortran compiler
flags and @code{LDFLAGS} for the linker flags should be set to right
values; @code{LIBS} should also be right and hold the link flags and link
files required to compile the @Minik{} model. These variables are 
set by by @command{./configure} during configuration (@pxref{Configuration}) 
and used in the default rule: 

@verbatim
$(model_file): $(miniker_user_objects) $(miniker_principal_objects)
	$(FC) $(FFLAGS) $(LDFLAGS) $^ $(LDADD) $(LIBS) -o $@
@end verbatim

In case this isn't right it may be freely changed. You should certainly 
refer to the @ref{Top,,Top,make,GNU Make Manual} manual to understand what
that rule exactly means and make your own.


@node Concepts index
@unnumbered Concepts index

@printindex cp

@node Variables macros and functions index
@unnumbered Variables, macros and functions index

@printindex vr

@node Installation
@appendix Installation

@menu
* Programming environments::
* Common requisites::
* @Minik{} with cmz::
* @Minik{} with make::
@end menu

@node Programming environments
@appendixsec Programming environments
@cindex Programming environments

@Minik{} is not a traditionnal software in that it isn't a library
or an interpreter but rather a set of source and macro file that 
combines with the user model code and enable
to build a binary program corresponding with the model. It
requires a build environment with a preprocessor, a compiler
and facilities that automate these steps.

Two different environment are proposed. One use
@command{cmz} (@url{http://wwwcmz.web.cern.ch/@/wwwcmz/index.html}),
while the other is based on @command{make}. Other libraries
are needed, the CERN Program Library (cernlib) and lapack.

@node Common requisites
@appendixsec Common requisites

@cindex cernlib
@cindex lapack

Whatever method is used a fortran 77 compiler is required. The compilers
that have been used so far are g77, gfortran and the sun solaris compiler.

When usng CMZ, the CERN Program Library, available at
@url{http://wwwasd.web.cern.ch/@/wwwasd/cernlib/}, has to be installed.
With make, internal source files copied from the cernlib may be used instead
but then some examples won't be available, since they rely on some
mathematical functions provided by the CERN library.
On windows, in case you want to use the compiler from the GNU compiler 
collection with cygwin or MINGW/MSYS you can use the binaries provided at
@url{http://zyao.home.cern.ch/@/zyao/cernlib.html}.
On Mac OS X, the cernlib provided by fink (package @code{cernlib-devel})
can be used.

You should also have LAPACK, available at @url{http://www.netlib.org/@/lapack/}.
LAPACK can also be installed as part of the CERN Library or as part of
the @uref{ATLAS,http://math-atlas.sourceforge.net/} implementation.
On most linux distributions a lapack package is available.
On Mac OS X, the ATLAS implementation provided by fink or the frameworks
from Xcode can be used.


@node @Minik{} with cmz
@appendixsec @Minik{} with cmz

@cindex @file{mini_ker.cmz}
@cindex @file{selseq.kumac}

First of all you have to get the cmz file @file{mini_ker.cmz} and put it
in a directory. In that same directory you should create a directory for
each of your models. In the model directory you should copy the file 
@file{selseq.kumac} available with @Minik{}, and create your own cmz 
file for your model, called for example @file{mymodel.cmz}. You should also 
have installed the kumac macro files
handling mortan compilation, the associated shell scripts and the mortran
preprocessor.

@node @Minik{} with make
@appendixsec @Minik{} with make

@menu
* Additional requirements::
* Configuration::
* Installation with make::
@end menu

@node Additional requirements
@appendixsubsec Additional requirements for @Minik{} with make

@cindex @command{mortran}, with make
@cindex requirements, with make

The package has been tested with GNU @command{make} and solaris
@command{make}. 

Suitable preprocessors should also be installed. Two preprocessors are 
required, one that preprocess the cmz directives, and a mortran 
preprocessor. A cmz directives processor written in @command{perl},
is distributed in the @command{car2txt} package available at
@value{myurl}. A @command{mortran}
package with a command able to preprocess a mortran file given on
the command line with a syntax similar with the @command{cpp} command line 
syntax is also required.
Such a mortran is available at @value{myurl}.

@c All the @command{make} commands are not suitable, for example the distribution
@c doesn't work with solaris @command{make}. GNU @command{make} works, however,
@c and should be available on most platforms, often called @command{gmake}.


@node Configuration
@appendixsubsec Configuration

@cindex configuration of source

The package is available at @value{myurl}. It is 
available as a compresssed tar archive. On UNIX, with GNU @command{tar} it 
may be unpacked using 

@example
$ tar xzvf mini_ker-@value{VERSION}.tar.gz
@end example

The detection of the compiler, the preprocessors (car2txt and mortran), 
and the libraries are performed by the configure script. This script
sets the
apropriate variables in makefiles. It can be run with:

@example
$ cd mini_ker-@value{VERSION}
$ ./configure
@end example

If the output of @command{./configure} doesn't show any error it means that
all the components are here. It is possible to give @command{./configure}
switches and also specify environment variables (see also 
@command{./configure --help}):

@table @code
@item --disable-cernlib
Use the internal cernlib source files, even if a cernlib is detected.
@item --with-static-cernlib
This command line switch forces a static linking with the cernlib (or a dynamic linking 
if set to no).
@item --with-cernlib
This command line switch can be used to specify the cernlib location
(if not detected or you want to use a specific cernlib).
@item --with-blas
@itemx --with-lapack
With this command switch, you can specify the location of the blas and lapack
libraries.

For example, on mac OS X this can be used to specify the blas and lapack from 
the Apple frameworks:

@example
./configure \
--with-blas=/System/Library/Frameworks/vecLib.framework/versions/A/vecLib \
--with-lapack=/System/Library/Frameworks/vecLib.framework/versions/A/vecLib
@end example

@item F77
@itemx FC
@itemx FFLAGS
@itemx LDFLAGS 
Classical compiler, compiler flags and linker flags.
@item MORTRAN
This environment variable holds the mortran preprocessor command
(default is @command{mortran}).
@item MTNFLAGS
This environment variable holds command line arguments for the mortran
preprocessor. It is empty in the default case.
@item MTN
This environment variable may be used to specify the mortran executable
name and/or path, it should be used by the @command{mortran} commmand.
(default is empty, which leads to a mortran executable called @command{mtn}).
@item MTNDEPEND
This environment variable may be used to specify the mortran dependencies
checker executable. It should be used by the @command{mortran} commmand.
(default is empty, which leads to a mortran dependencies checker 
called @command{mtndepend}).
@end table

After a proper configuration, if @command{make} is run then the example 
models should be build. You have to perform the configuration only once.

@node Installation with make
@appendixsubsec Installation with make
@cindex installation with make

@Minik{} can be installed by running 
@example
make install
@end example


It should copy the sources 
and the @file{Makefile.miniker} file in 
a @file{mini_ker} directory in the @code{$(includedir)} directory, and
copy the templates in @file{$(datadir)/mini_ker}. The default for
@code{$(includedir)} is @file{/usr/local/include} and the default for
@code{$(datadir)} is @file{/usr/local/share}, these defaults may be 
changed by @command{./configure} switches @samp{--prefix},
@samp{--includedir}  and @samp{--datadir}. See @command{./configure --help}
and the @file{INSTALL} file for more informations. The helper script
@file{start_miniker} should also be installed.


 
The installation is not required to use comfortably @Minik{}. Indeed
the only thing that changes with the sources and the @file{Makefile.miniker}
directory location is the @code{miniker_dir} variable in a 
project @code{Makefile}.


@node Cmz directives reference
@appendix Cmz directives reference

The cmz directives are described together with the other 
features of cmz in the cmz manual at
@url{http://wwwcmz.web.cern.ch/wwwcmz/}, the important ones are 
nevertheless recalled here,
especially for those that use make and don't need the whole
features of cmz.

After the description of the generic features, we turn
to the cmz directive of interest.
There are three kinds of cmz directives that are of use
within @Minik{}: one kind
that introduce files, the other for conditionnal compilation and
the third for sequence inclusion. 


@menu
* Cmz directives general syntax::
* Conditional expressions::
* File introduction directives::
* Conditional directives::
* File inclusion directive::
* The self directive::
@end menu

@node Cmz directives general syntax
@appendixsec Cmz directives general syntax

The cmz directives always begin with a @samp{+} in the first column,
optionnaly followed by any number of @samp{_} that may be used for 
indentation, then the directive label, case insensitive, followed
by the directive arguments separated by @samp{,}. The arguments
are also case insensitive.
Optional spaces may be around directive arguments. 
An optionnal @samp{.} ends the directive
arguments and begin a comment, everything that follows that @samp{.} is 
ignored.

@node Conditional expressions
@appendixsec Conditional expressions

A directive argument common to all the directives is the conditionnal
expression. A conditionnal expression may be true or false, it is a 
combination of select flags. the select flags are combined with 
logical operators. A
select flag itself is true if it was selected. A select flag @var{selflag} 
is selected by using the @code{sel @var{selflag}} instruction in cmz. It is
selected by passing the @code{-D @var{selflag}} command line switch to
the call of the cmz directives preprocessor when using make.


A @samp{-} negates
the expression that follows. Parenthesis @samp{(} and @samp{)} are used 
for the grouping of subexpressions. @samp{|} and @samp{,} are for the 
boolean or: an expression with a or is true
if the expression on the left or the expression on the right of the or
is true. 
@samp{&} is for the boolean and: an expression with an and is true if
the expression on the left and the expression on the right are true.

The grouping is left to right when there is no parenthesis, with or and
@samp{&} having the same precedence. Therefore

@example
a&b|c    @equiv{}    (a&b)|c
a|b&c    @equiv{}    (a|b)&c
a|b&c  is not  a|(b&c)
a&b|c  is not  a&(b|c)
@end example 

@node File introduction directives
@appendixsec File introduction directives

A file (or sequence) introduction directive appears at the beginning
of the file. There are two different directives, one is @code{DECK}
for normal files, the other is @code{KEEP} for include files (sequences).
The first argument is the name of the file. The file name may not be larger
than 32 characters and is converted to lower case in the general case.
The optionnal following arguments may be
of 2 type (and may be mixed, separated by @samp{,}):

@table @asis
@item conditional
A conditionnal is introduced by @code{IF=} followed by a conditionnal 
expression described in 
@ref{Conditional expressions}. The 
file is preprocessed if the conditionnal expression is true.
@item language specification
A language specification is introduced by a @code{T=}. The most
common languages are @samp{mtn} for the mortran, @samp{ftn} for
fortran not preprocessed, @samp{f77} for preprocessed fortran,
@samp{c} for the c language and @samp{txt} for text files.
In general the language of the file determines the name of files 
the preprocessed file is extracted to, the comment style and
the command for inclusions.
@end table

It is a common practice to have wrong language type in @code{KEEP} 
as the language may be determined from the @code{DECK} that include
them with cmz, or from their file name with make. This is not recommended
and considered a bad practice.

Such a directive will always appear in cmz, as it is built-in. It
is recommended to have one when using make too, even though it is not 
required in most cases. Indeed make uses the file name directly
and finds the language and file type by looking at the file extension.
make should then pass the language type with a 
@code{--lang @var{lang}} command
line switch when calling the cmz directives preprocessor.
With make, the convention is to have @samp{cm} added before the normal
file suffix and after the @samp{.}. The table @ref{tab:cmfile_suffix}
shows the matching between suffixes, file type and file language.

For example, a file beginning with

@verbatim
+Deck, subroutine_foo, If=monitor&-simple, T=f77. 
@end verbatim

is a main preprocessed fortran file that will only be generated if
@samp{monitor} is selected and @samp{simple} is not selected. The 
file to be preprocessed by make should have the @samp{.cmF} suffix,
and be called @file{subroutine_foo.cmF}.

A file beginning with

@verbatim
+KEEP,inc_common,If=monitor|interface,T=mtn
@end verbatim

is an mortran include file that should be processed only if @samp{monitor}
or @samp{interface} is selected. The file to be preprocessed by make
should have the @samp{cmmti} suffix and be called @file{inc_common.cmmti}.
The resulting file when make is used will be called @file{inc_common.mti}.

@node Conditional directives
@appendixsec Conditional directives

Conditional directives may be used to conditionnaly skip blocks of
code. There are 4 conditional directives: @code{if}, @code{elseif},
@code{else} and @code{endif}. @code{+if} begins a conditional directives
sequence, with argument a conditional expression. If the expression is
true the block of code following the @code{+if} is output in the
resulting file, up to another conditional directive, if it is false
the code block is skipped. If the 
expression is false and the following conditional directive is 
@code{+elseif}, the same procedure is followed with the argument of 
@code{+elseif} 
which is also a conditionnal expression. More than one @code{+elseif}
may follow a @code{+if}. If a @code{+if} or @code{+elseif} expression 
is true the following
code block is output and all
the following @code{+elseif} code blocks are skipped. If all the @code{+if}
and @code{+elseif} expressions are false and
the following coditionnal 
directive is @code{+else} then the block following the 
@code{+else} is output. If a previous expression was true the 
code block following the @code{+else} is skipped. The last code block
is closed by @code{+endif}.

Conditionnal directives may be nested, a @code{+if} begins a deeper 
conditionnal sequences directives that is ended by the corresponding 
@code{+endif}.

The simplest example is:

@verbatim
     some code;
+IF,monitor
     code output only if monitor is true;
+ENDIF
@end verbatim

If @samp{monitor} is selected, the @code{+if} block is output, it leads to

@verbatim
     some code;
     code output only if monitor is true;
@end verbatim

If @samp{monitor} isn't selected the @code{+if} block is skipped, it leads to

@verbatim
     some code;
@end verbatim

An example with  @code{+else} may be:

@verbatim
+IF,double
 call dmysub(eta);
+ELSE
 call smysub(eta);
+ENDIF
@end verbatim

If @samp{double} is selected the code output is @code{call dmysub(eta);},
if @samp{double} isn't selected the code output is @code{call dmysub(eta);}.

Here is a self explanatory example of use of @code{+elseif}:

@verbatim
+IF,monitor
  code used if monitor is selected;
+ELSEIF,kalman
  code used if kalman is selected and monitor is not;
+ELSE
  code used if kalman and monitor are not selected;
+ENDIF
@end verbatim

And last an example of nested conditional directives:

@verbatim
+IF,monitor
  code used if monitor is selected;
+_IF,kalman. deep if
    code used if monitor and kalman are selected;
+_ELSE. deep else
    code used if monitor is selected and kalman is not;
+_ENDIF. end the deep conditionnals sequence
+ELSE
  code used if monitor is not selected;
+_IF,kalman
    code used if monitor is not selected but kalman is;
+_ELSE
    code used if monitor and kalman are not selected;
+_ENDIF
  other code used if monitor is not selected;
+ENDIF
@end verbatim

@node File inclusion directive
@appendixsec File inclusion directive

The file (sequence) inclusion directive is @code{seq}. The argument of
@code{seq} is an include files @samp{,} separated list. The include 
files are @code{Keep} in cmz. The following optional arguments may be
mixed:

@table @asis
@item conditional
A conditionnal is introduced by @code{IF=} followed by a conditionnal 
expression described in 
@ref{Conditional expressions}. The 
directive is ignored if the conditionnal expression is false.
@item T=noinclude
When this argument is present the text of the sequence will 
always be included in the file where the @code{+seq} appears.
@end table

When there is no @code{T=noinclude} argument, the @code{+seq} 
directive may be replaced with an inclusion command suitable 
for the language of the file being processed, if such 
command has been specified.

For example if we have the following sequence
@verbatim
+KEEP,inc,lang=C
typedef struct incstr {char* msg};
@end verbatim

And the following code in the file being processed:

@verbatim
+DECK,mainf,lang=C
+SEQ,inc
int main (int argc, char* argv) { exit(0); }
@end verbatim

the processing of @file{mainf} should lead to the file 
@file{mainf.c}, containing
an include command for @file{inc}:

@verbatim
#include "inc.h"
int main (int argc, char* argv) { exit(0); }
@end verbatim

In case the @code{+seq} has the @code{T=noinclude}:

@verbatim
+DECK,mainf,lang=C
+SEQ,inc,T=noinclude
int main (int argc, char* argv) { exit(0); }
@end verbatim

The processing of @file{mainf} should lead to the file @file{mainf.c} 
containing the text of @file{inc}:

@verbatim
typedef struct incstr {char* msg};
int main (int argc, char* argv) { exit(0); }
@end verbatim

@node The self directive
@appendixsec The @samp{self} directive

The @code{self} directive is an obsolete directive that may be used for
conditionnal skipping of code. For a better approach see 
@ref{Conditional directives}. 
The optionnal argument of @code{+SELF} is @code{If=} followed by
a conditionnal expression. If the conditionnal expression is true the 
code following the directive is output, if it is false the code 
is skipped up to any directive (including another @code{+SELF})
except @code{+seq}.

@ignore
@node Resolution method
@appendix Overview of resolution method

@node @Minik{} macros
@appendix @Minik{} macros
@end ignore

@node Copying This Manual
@appendix Copying This Manual

@menu
* GNU Free Documentation License::  License for copying this manual.
@end menu

@include fdl.texi

@bye
