# Annealing

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## Annealing

This document describes version 0.4a0 of Annealing, a C language library extension the for GSL, the GNU Scientific Library, attempting a redesign of the simulated annealing module.

http://gna.org/projects/annealing
http://github.com/marcomaggi/annealing/tree/master

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with Invariant Sections being “GNU Free Documentation License” and “GNU General Public License”, no Front–Cover Texts, and no Back–Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.

Appendices

Indexes

--- The Detailed Node Listing ---

Overview of the package

Simulated annealing

Miscellaneous functions

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## 1 Overview of the package

Annealing is a C language extension library for GSL, the GNU Scientific Library, attempting a redesign of the simulated annealing module.

The library is developed and tested under the Linux+GNU system and officially it supports only the GNU infrastructure: requires the GNU C library and compiles fine with the GNU C compiler (-std=c99 -pedantic switches). The building infrastructure requires GNU Make, GNU Bash and the other common tools (rm, mkdir, ...).

Notice that Annealing's API violates the GSL's Design Document; specifically the requirement of section 3.18: “Algorithm decomposition”. Annealing uses callbacks everywhere, while the Document requires for the algorithm to be split into an initialise, iterate, test form.

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### 1.1 Using the library

In the following we assume that the library is installed under the /usr/local hierarchy.

A shell script is installed on the system to provide informations about version numbers and installation directories; it should be /usr/local/bin/annealing-config. Run annealing-config without arguments for the usage screen. An example of output for the important options is:

     $annealing-config --package-version 0.2.0$ annealing-config --library-interface-version
1.0

$annealing-config --pkgincludedir /usr/local/include/annealing/0.2a1$ annealing-config --libdir
/usr/local/lib

$annealing-config --library-id annealing1.0  There is a single header file that must be included in C sources: annealing.h; in our source code we can put:  #include <annealing.h>  and use the output of annealing-config --pkgincludedir as argument to the -I option of the C preprocessor. The libraries (shared and static) are installed under the directory obtained by running annealing-config --libdir; we can use its output as argument of the -L option to the linker. If we want to link with the shared or static library we take the output of annealing-config --library-id and use it as value for the -l option to the linker. Next: , Previous: overview using, Up: overview ### 1.2 Using GNU Autoconf If we use GNU Autoconf to configure our application we can embed in our project the file annealing.m4, installed under$prefix/share/aclocal, and load it in our configure.ac by putting:

     m4_include(annealing.m4)


in aclocal.m4. Notice that GSL distributes its own Autoconf plug–in, a file named gsl.m4. The annealing.m4 macro file defines a macro that can be invoked with:

     ANNEALING_LIBRARY(1,0)


it finds and invokes annealing-config to acquire installation directories and library names. The macro accepts two arguments: the major and minor interface version numbers that are needed (library version); both of the arguments are optional, but it is recommended to specify them.

ANNEALING_LIBRARY defines the following symbols:

@ANNEALING_INCLUDEDIR@
The directory under which header files are installed, example:
          /usr/local/include/annealing/0.2a1


@ANNEALING_CPPFLAGS@
The preprocessor option -I with the value of @ANNEALING_INCLUDEDIR@ attached, example:
          -I/usr/local/include/annealing/0.2a1


@ANNEALING_LIBDIR@
The directory under which shared and static libraries are installed, example: /usr/local/lib.
@ANNEALING_LDFLAGS@
The linker option -L with the value of @ANNEALING_LIBDIR@ appended, example: -L/usr/local/lib.
@ANNEALING_LIBRARY_ID@
The identifier of the shared or static library, example: annealing1.0.
@ANNEALING_LIBS@
The linker option -l with the value of @ANNEALING_LIBRARY_ID@ appended, example: -lannealing1.0.

Previous: overview autoconf, Up: overview

### 1.3 Using pkgconfig

pkg-config is a program (a compiled executable) that inspects a database of package meta informations and prints to its standard output channel informations about installed packages and libraries. Annealing installs a meta data file for use with pkg-config, it should be:

     ${libdir}/pkgconfig/annealing.pc  notice that the GSL distributes its own pkg-config meta file, a file named gsl.pc. For the full list of available informations look in the file itself, and remember that the value of all the variables set in the meta file can be printed with: $ pkg-config annealing --variable=<VARNAME>


an example of output for the important variables is:

     $pkg-config annealing --variable=PACKAGE_VERSION 0.2.0$ pkg-config annealing --variable=library_interface_version
1.0

$pkg-config annealing --variable=pkgincludedir /usr/local/include/annealing/0.2a1$ pkg-config annealing --variable=libdir
/usr/local/lib

$pkg-config annealing --variable=library_id annealing1.0  The --libs option outputs something like: $ pkg-config annealing --libs
-L/usr/local/lib -lannealing1.0 -lgsl -lgslcblas -lm


while the list of preprocessor options is:

     $pkg-config annealing --cflags -I/usr/local/include/annealing/0.2a1 -I/usr/local/include  Notice that: $ pkg-config annealing --modversion
1.0


because what matters is the version of the API, not the package distribution version.

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## 2 Simulated annealing

Stochastic search techniques are used when the structure of a space is not well understood or is not smooth, so that techniques like Newton's method (which requires calculating Jacobian derivative matrices) cannot be used. In particular, these techniques are frequently used to solve combinatorial optimization problems, such as the traveling salesman problem.

The goal is to find a point in the space at which a real valued “energy function” (or “cost function”) is minimized. Simulated annealing is a minimization technique which has given good results in avoiding local minima; it is based on the idea of taking a random walk through the space at successively lower temperatures, where the probability of taking a step is given by a Boltzmann distribution.

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### 2.1 The basic algorithm

Here is an explanation of the basic algorithm. The idea is to generate at random new configurations in the solution's space, and apply two criteria to decide if the a new configuration must be accepted or not. The criteria are: if lower energy accept, if upper energy accept with a probability from a Boltzmann distribution.

It goes like this:

1. interpret the start configuration as current and also register it as best so far;
2. start a number of iterations at fixed temperature:
1. compute a new configuration by taking a random step from the current configuration;
2. evaluate new configuration's energy:
• if new_E <= best_E: register the new configuration as both best so far and current;
• else if new_E <= current_E: register the new configuration as current;
• else compute:
                    R = exp(-(new_E - best_E)/(k * T))


where T is the current temperature and k the Boltzmann constant; if R is greater than a random number in the range [0, 1): register the new configuration as current;

• else: discard the new configuration;
3. if not done all the iterations: go back to a;
3. cool the temperature, the new value is:
          T_new = T / mu


where mu is the damping factor: it must be a number greater than 1;

4. if the new temperature is less than the selected minimum value: stop and report the better configuration as result;
5. go to 2.

Additionally some algorithms allow restarting, between steps 4 and 5: if the new temperature is less than the selected restart value, discard the current configuration replacing it with the best so far.

Notes:

• there are two nested loops: the outer does as many iterations as required to cool the temperature from the initial value to the minimum; the inner does a fixed number of iterations at the same temperature;
• at least one outer iteration is performed, so the number of outer iterations can be computed like this:
          initial = 100;
minimum = 1;
damping = 1.005;
count   = 1;

for (l = initial; l >= miminum; l /= damping, ++count)
;
/* count = 925 */

• the cooling waveform of the temperature is somewhat like an exponential; for the example above the application of nonlinear multifitting with the exponential model returns the following:
          T(x) = A exp(-lam x)    A=99.9999999173883  lam=4.60350080940739


which fits very well the curve;

• the value:
          R = exp(-(new_E - best_E)/(k * T))


is computed only if new_E > best_E, so it is always:

          new_E - best_E > 0  => R \in [0, 1)


with R near 1 when k * T is high, and near 0 when k * T is low;

• during the first iterations: the temperature drops rapidly and its value is high; this means that a worse configuration is more likely to be accepted;
• during the last iterations: the temperature drops slowly and its value is low; this means that a worse configuration is less likely to be accepted;
• with the basic algorithm only three configuration objects are in existence at any instant: current, best so far, new; so the memory space required to hold them can be allocated at the beginning of the algorithm and released at the end by the user code;
1. we avoid memory allocation in the code that builds new configurations;
2. building a new configuration from an existent one cannot fail.

Next: , Previous: annealing algorithm, Up: annealing

### 2.2 Data types

#### 2.2.1 Data structures

— Struct Typedef: annealing_configuration_t

A container for the configuration object. Public fields:

void * data
pointer to a block of memory holding the configuration data; it can be any type of data: the annealing code does not access it directly, it is handled only by the user supplied functions;
double energy
the energy level associated to the configuration; it is computed by a user supplied function.

#### 2.2.2 User supplied functions

In the following function types:

• when the function is invoked by annealing_simple_solve() the argument S references a structure of type annealing_simple_workspace_t.
• when the function is invoked by annealing_multibest_solve() the argument S references a structure of type annealing_multibest_workspace_t;
• when the function is invoked by annealing_manytries_solve() the argument S references a structure of type annealing_manytries_workspace_t.

The configuration parameters are the values in the data field of annealing_configuration_t structures.

— Function: double annealing_energy_fun_t (void * S, void * configuration)

Must compute and return the energy value of configuration.

— Function: void annealing_step_fun_t (void * S, void * configuration)

Must use the data in S to transform configuration to a new value.

— Function: double annealing_metric_fun_t (void * S, void * configuration_a, void * configuration_b)

Must compute and return the distance between the two configurations. The metric should be such that: the order in which the configurations are handed to this function is not important.

— Function: void annealing_cooling_fun_t (void * S)

Invoked to cool the temperature. Must update the temperature field of the workspace structure.

— Function: void annealing_log_fun_t (void * S)

Invoked once before the first iteration and then at the end of each internal loop.

— Function: void annealing_copy_fun_t (void * S, void * dst_configuration, void * src_configuration)

Invoked to clone a configuration.

It is responsibility of this function to free resources in dst_configuration before overwriting them with values from src_configuration.

Next: , Previous: annealing types, Up: annealing

### 2.3 Interface to the basic algorithm

— Struct Typedef: annealing_simple_workspace_t

Holds all the data required to run a basic simulated annealing algorithm. It must be allocated and freed by the user code. Public fields:

size_t number_of_iterations_at_fixed_temperature
self explaining;
void * max_step_value
pointer to the value used to limit the “distance” between new configurations and the current configuration; it can reference any type of value, it is accessed only by the user supplied functions;
double boltzmann_constant
the constant used in the energy computation; it must be positive; when tuning the parameters for a search a good start value is 1.0;
double temperature
the initial value for the temperature;
double minimum_temperature
the minimum value for the temperature: when the value in the field temperature drops below this level the outer loop stops;
double restart_temperature
the temperature level that causes the current configuration to be reset to the best so far;

restarting is useful to prevent the search to get lost at low temperature in a region with no interesting minima;

restarting is optional: if we select DBL_MIN as value, it is never temperature < restart_temperature and the configuration is never reset (DBL_MIN is defined in float.h, but any negative value will do);

the value must be less than temperature, or the algorithm will restart at each iteration;

int restart_flag
initially set to 0 by annealing_simple_solve(), it is set to 1 if/when the algorithm is restarted from the best configuration;
double damping_factor
the coefficient used to cool the temperature; it must be a number slightly above 1.0;
annealing_configuration_t current_configuration
holds the current configuration value; its data field must be initialised with a pointer to the start configuration;
annealing_configuration_t best_configuration
holds the best so far configuration value; its data field must be initialised with a pointer to a memory block large enough to hold a configuration value; the memory block must be initialised with an invalid value that can be recognised by copy_function;
annealing_configuration_t new_configuration
holds the new configuration value; its data field must be initialised with a pointer to a memory block large enough to hold a configuration value; the memory block must be initialised with an invalid value that can be recognised by copy_function;
gsl_rng * numbers_generator
the random number generator used to apply the Boltzmann distribution acceptance criterion; it can be used also to generate the random step;
annealing_energy_fun_t * energy_function
the user supplied function used to compute the energy of a configuration;
annealing_step_fun_t * step_function
the user supplied function used to compute the new configuration from the current configuration;
annealing_cooling_fun_t * cooling_function
the user supplied function used to cool the temperature; it can be NULL to select the default cooling algorithm;
annealing_log_fun_t * log_function
the user supplied function used to log the search path; it can be NULL if no logging is required;
annealing_copy_fun_t * copy_function
the user supplied function used to copy a configuration from an allocation space to another;
void * params
this is a free field that can be used to hand data to the user supplied functions.

— Function: void annealing_simple_solve (annealing_simple_workspace_t * S)

Does the search.

The structure referenced by S must be allocated and initialised by the user before invoking this function, and freed by the user after this function has returned.

Space for the data referenced by the current_configuration, best_configuration and new_configuration fields in S must be allocated by the user before invoking this function, and freed by the user after this function has returned.

This function does no resource allocation, so it is perfectly all right if the user supplied functions use a dynamic wind mechanism, like setjmp() and longjmp(), to report errors.

When this function returns: the best configuration found is in the best_configuration field.

#### 2.3.1 Examples

Find the minimum of f(t) = -sin(t)/t.

     /** ------------------------------------------------------------
** ----------------------------------------------------------*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <float.h>
#include <gsl/gsl_rng.h>
#include "annealing.h"

static  annealing_energy_fun_t      energy_function;
static  annealing_step_fun_t        step_function;
static  annealing_log_fun_t         log_function;
static  annealing_copy_fun_t        copy_function;

/** ------------------------------------------------------------
** Main.
** ----------------------------------------------------------*/

int
main (void)
{
annealing_simple_workspace_t      S;
double        configurations[3];
double        max_step = 10.0;

printf("sinc minimisation with simulated annealing\n");

S.number_of_iterations_at_fixed_temperature = 10;
S.max_step_value              = &max_step;

S.temperature                 = 10.0;
S.minimum_temperature         = 1.0e-6;
S.restart_temperature         = DBL_MIN; /* do not restart */
S.boltzmann_constant          = 1.0;
S.damping_factor              = 1.005;

S.energy_function             = energy_function;
S.step_function               = step_function;
S.copy_function               = copy_function;
S.log_function                = log_function;
S.cooling_function            = NULL;

S.numbers_generator           = gsl_rng_alloc(gsl_rng_rand);
gsl_rng_set(S.numbers_generator, 15);

S.current_configuration.data  = &(configurations[0]);
S.best_configuration.data     = &(configurations[1]);
S.new_configuration.data      = &(configurations[2]);

configurations[0] = 100.0;

annealing_simple_solve(&S);

printf("final best solution: %f, expected 0.0\n",
configurations[1]);

gsl_rng_free(S.numbers_generator);
exit(EXIT_SUCCESS);
}

/** ------------------------------------------------------------
** Iteration functions.
** ----------------------------------------------------------*/

static double
alea (annealing_simple_workspace_t * S)
{
double        max_step = *((double *)S->max_step_value);

return (2.0 * gsl_rng_uniform(S->numbers_generator) - 1.0) * max_step;
}

/* ------------------------------------------------------------ */

double
energy_function (void * dummy, void * configuration)
{
double        C = *((double *)configuration);

return -sin(C)/C;
}
void
step_function (void * W, void * configuration)
{
annealing_simple_workspace_t * S = W;
double *      C = (double *)configuration;
double        c;

do c = *C + alea(S); while (fabs(c) > 120.0);
*C = c;
}
void
log_function (void * W)
{
annealing_simple_workspace_t * S = W;
double        current = *((double *)S->current_configuration.data);
double        best    = *((double *)S->best_configuration.data);

printf("current %f (energy %f), best %f (energy %f)\n",
current, S->current_configuration.energy,
best,    S->best_configuration.energy);
}

/** ------------------------------------------------------------
** Configuration handling functions.
** ----------------------------------------------------------*/

void
copy_function (void * dummy,
void * dst_configuration, void * src_configuration)
{
double *      dst = dst_configuration;
double *      src = src_configuration;

*dst = *src;
}

/* end of file */


Next: , Previous: annealing basic, Up: annealing

### 2.4 Interface to the multi–best algorithm

To understand this section you have to read annealing algorithm first.

The multi–best algorithm works like the simple one, but keeps as result a number of best configurations. It allows a two step search:

1. find N promising configurations using annealing_multibest_solve();
2. explore the neighbourhood of each of the best configurations using annealing_simple_solve();

When using the multi–best algorithm there is no restart option.

— Struct Typedef: annealing_multibest_workspace_t

Holds all the data required to run a multi–best simulated annealing algorithm. It must be allocated and freed by the user code. Public fields:

size_t number_of_iterations_at_fixed_temperature
self explaining;
size_t max_number_of_best_configurations
maximum number of best configurations to register;
size_t best_configurations_count
number of best configurations currently registered; at the beginning of the search this field is set to 1, to account that the start configuration is registered as best so far;
void * max_step_value
pointer to the value used to limit the “distance” between new configurations and the current configuration; it can reference any type of value, it is accessed only by the user supplied functions;
double minimum_acceptance_distance
when evaluating a new promising configuration: the vector of better configurations is split in two: better than the promising, worse than the promising; the promising configuration is accepted if its distance from the better configurations is greater than this value;
double boltzmann_constant
the constant used in the energy computation; it must be positive; when tuning the parameters for a search a good start value is 1.0;
double temperature
the initial value for the temperature;
double minimum_temperature
the minimum value for the temperature: when the value in the field temperature drops below this level the outer loop stops;
double damping_factor
the coefficient used to cool the temperature; it must be a number slightly above 1.0;
annealing_configuration_t current_configuration
holds the current configuration value; its data field must be initialised with a pointer to the start configuration;
annealing_configuration_t new_configuration
holds the new configuration value; its data field must be initialised with a pointer to a memory block large enough to hold a configuration value; the memory block must be initialised with an invalid value that can be recognised by copy_function;
annealing_configuration_t * best_configurations
pointer to an array of max_number_of_best_configurations structures, allocated by the user, that will reference the best so far configurations; the data fields of each structure must be initialised with a pointer to a memory block large enough to hold a configuration value; the memory block must be initialised with an invalid value that can be recognised by copy_function;
gsl_rng * numbers_generator
the random number generator used to apply the Boltmann distribution acceptance criterion; it can be used also to generate the random step;
annealing_energy_fun_t * energy_function
the user supplied function used to compute the energy of a configuration;
annealing_step_fun_t * step_function
the user supplied function used to compute the new configuration from the current configuration;
annealing_cooling_fun_t * cooling_function
the user supplied function used to cool the temperature; it can be NULL to select the default cooling algorithm;
annealing_log_fun_t * log_function
the user supplied function used to log the search path; it can be NULL if no logging is required;
annealing_copy_fun_t * copy_function
the user supplied function used to copy a configuration from an allocation space to another;
annealing_metric_fun_t * metric_function
the user supplied function used to compute the distance between two configurations;
void * params
this is a free field that can be used to hand data to the user supplied functions.

— Function: void annealing_multibest_solve (annealing_multibest_workspace_t * S)

Does the search.

The structure referenced by S must be allocated and initialised by the user before invoking this function, and freed by the user after this function has returned.

Space for current_configuration, new_configuration and the best_configurations array in S must be allocated by the user before invoking this function, and freed by the user after this function has returned.

This function does no resource allocation, so it is perfectly all right if the user supplied functions use a dynamic wind mechanism, like setjmp() and longjmp(), to report errors.

When this function returns: the best configurations found are in the best_configurations array.

#### 2.4.1 Examples

Find 4 local minima of f(t) = -sin(t)/t.

     /** ------------------------------------------------------------
** ----------------------------------------------------------*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <float.h>
#include <gsl/gsl_rng.h>
#include "annealing.h"

static  annealing_energy_fun_t      energy_function;
static  annealing_step_fun_t        step_function;
static  annealing_metric_fun_t      metric_function;
static  annealing_log_fun_t         log_function;
static  annealing_copy_fun_t        copy_function;

/** ------------------------------------------------------------
** Main.
** ----------------------------------------------------------*/

int
main (void)
{
annealing_multibest_workspace_t   S;
annealing_configuration_t         array[4];
double        configurations[2+4]; /* new, current and 4 best */
double        max_step = 10.0;

printf("multi-best sinc minimisation with simulated annealing\n");

S.number_of_iterations_at_fixed_temperature = 10;
S.max_step_value              = &max_step;
S.minimum_acceptance_distance = 2.0;

S.temperature                 = 10.0;
S.minimum_temperature         = 1.0e-6;
S.boltzmann_constant          = 1.0;
S.damping_factor              = 1.005;

S.energy_function             = energy_function;
S.step_function               = step_function;
S.copy_function               = copy_function;
S.log_function                = log_function;
S.metric_function             = metric_function;
S.cooling_function            = NULL;

S.numbers_generator           = gsl_rng_alloc(gsl_rng_rand);
gsl_rng_set(S.numbers_generator, 15);

S.max_number_of_best_configurations = 4;
S.current_configuration.data  = &(configurations[0]);
S.new_configuration.data      = &(configurations[1]);
S.best_configurations         = array;

array[0].data         = &(configurations[2]);
array[1].data         = &(configurations[3]);
array[2].data         = &(configurations[4]);
array[3].data         = &(configurations[5]);
configurations[0] = 100.0;

annealing_multibest_solve(&S);

printf("found %u best solutions:", S.best_configurations_count);
for (size_t i=0; i<S.best_configurations_count; ++i)
{
double *	value = array[i].data;
printf(" %.5f", *value);
}

gsl_rng_free(S.numbers_generator);
exit(EXIT_SUCCESS);
}

/** ------------------------------------------------------------
** Iteration functions.
** ----------------------------------------------------------*/

static double
alea (annealing_multibest_workspace_t * S)
{
double        max_step = *((double *)S->max_step_value);

return (2.0 * gsl_rng_uniform(S->numbers_generator) - 1.0) * max_step;
}

/* ------------------------------------------------------------ */

double
energy_function (void * dummy, void * configuration)
{
double        C = *((double *)configuration);

return -sin(C)/C;
}
void
step_function (void * W, void * configuration)
{
annealing_multibest_workspace_t * S = W;
double *      C = (double *)configuration;
double        c;

do c = *C + alea(S); while (fabs(c) > 120.0);
*C = c;
}
double
metric_function (void * dummy, void * configuration_a, void * configuration_b)
{
double        A = *((double *)configuration_a);
double        B = *((double *)configuration_b);

return fabs(A - B);
}
void
log_function (void * W)
{
annealing_multibest_workspace_t * S = W;
double        current = *((double *)S->current_configuration.data);

printf("current %5.5g (energy %.4f), worst best energy %.4f, best (%u):",
current, S->current_configuration.energy,
S->best_configurations[S->best_configurations_count-1].energy,
S->best_configurations_count);
for (size_t i=0; i<S->best_configurations_count; ++i)
{
double *	value = S->best_configurations[i].data;
printf(" %.5f", *value);
}
printf("\n");
}

/** ------------------------------------------------------------
** Configuration handling functions.
** ----------------------------------------------------------*/

void
copy_function (void * dummy, void * dst_configuration, void * src_configuration)
{
double *      dst = dst_configuration;
double *      src = src_configuration;

*dst = *src;
}

/* end of file */


Previous: annealing multi, Up: annealing

### 2.5 Interface to the many–tries algorithm

To understand this section you have to read annealing algorithm first.

The many–tries algorithm makes use of the Monte Carlo method to select the new configuration. The algorithm itself is very simple, but its meaning is not immediate to understand if one does not know the Boltzmann distribution.

#### 2.5.1 The Boltzmann distribution

Let's consider a volume of gas composed by N particles at a temperature T. We can partition the particles in I sets, indexed by i \in [1, I], such that the number of particles in the i–th set is N_i and:

         I
N = S N_i
i=1


We do the partition so that: in each set all the particles share the same energy level E_i.

The Boltzmann distribution equation is:

              g_i exp(-E_i / (k T))
N_i/N = ------------------------
I
S g_j exp(-E_j / (k T))
j=1


where k is the Boltzmann constant; if we know the temperature T and all the possible energy states E_i, we can compute the fraction of particles in each state.

Altenatively: the fraction N_i/N is the probability that a particle randomly selected in the gas is in the state E_i.

Given two numbers N_i and N_j, and assuming g_i = g_j, we can write:

               g_i exp(-E_i / (k T))
N_i/N_j = --------------------- = exp(-(E_i - E_j) / (k T))
g_j exp(-E_j / (k T))


which gives us the probability centred around the state E_j.

#### 2.5.2 Monte Carlo method

Let's say that we have a gas with I possible states, for which we can compute the energies E_i, i \in [1, I]. Using the Boltzmann distribution centred around a known level E having N particles:

     N_i/N = exp(-(E_i - E) / (k T))            i \in [1, I]


we can compute the fractions N_i/N.

The Monte Carlo method simulates the random extraction of a particle and the determination of its energy state:

1. compute samples of a coordinate X:
          X_0 = 0      X_i = N_i/N + X_{i-1}       i \in [1, I]

2. generate a random number x from a uniform distribution in the range [0, X_I];
3. select the interval for which x \in [X_{i-1}, X_i];

the result of the simulation is that the particle has energy state E_i, so it is in the state i.

#### 2.5.3 Application to simulated annealing

To apply the method we substitute the number of iterations at fixed temperature with the following process:

1. create a number I-1 of new configurations; add the current configuration, obtaining a set of I configurations;
2. for each configuration: compute the energy level E_i centred around the energy of the current configuration;
3. apply the Monte Carlo method to select a state;

the configuration associated to the selected state is the new configuration.

We notice that:

• the fraction associated to the current configuration is:
          N_cur/N = exp(-(E - E) / (k T)) = 1

• the fraction associated to a configuration with energy higher than E is:
          E_hi > E     =>      (E_hi - E) > 0
=>      N_hi/N = exp(-(E_hi - E) / (k T)) < 1


when the temperature is low the exponential becomes infinitesimal and the library forces the fraction to the value of 0.0;

• the fraction associated to a configuration with energy lower than E is:
          E_lo < E     =>      (E_lo - E) < 0
=>      N_lo/N = exp(-(E_lo - E) / (k T)) > 1


when the temperature is low the exponential becomes huge and the library forces the fraction to the value of 10.0;

so the lower energy configurations have more probability to be selected than the current and higher energy ones, but still there is the possibility to select a higher energy configuration.

#### 2.5.4 Interface

— Struct Typedef: annealing_manytries_workspace_t

Holds all the data required to run a many-tries simulated annealing algorithm. It must be allocated and freed by the user code. Public fields:

size_t number_of_tries
the number of new configurations to generate at fixed temperature; this is the I-1 number;
void * max_step_value
pointer to the value used to limit the “distance” between new configurations and the current configuration; it can reference any type of value, it is accessed only by the user supplied functions;
double boltzmann_constant
the constant used in the energy computation; it must be positive; when tuning the parameters for a search a good start value is 1.0;
double temperature
the initial value for the temperature;
double minimum_temperature
the minimum value for the temperature: when the value in the field temperature drops below this level the loop stops;
double restart_temperature
the temperature level that causes the current configuration to be reset to the best so far;

restarting is useful to prevent the search to get lost at low temperature in a region with no interesting minima;

restarting is optional: if we select DBL_MIN as value, it is never temperature < restart_temperature and the configuration is never reset (DBL_MIN is defined in float.h, but any negative value will do);

the value must be less than temperature, or the algorithm will restart at each iteration;

int restart_flag
initially set to 0 by annealing_manytries_solve(), it is set to 1 if/when the algorithm is restarted from the best configuration;
double damping_factor
the coefficient used to cool the temperature; it must be a number slightly above 1.0;
annealing_configuration_t current_configuration
holds the current configuration value; its data field must be initialised with a pointer to the start configuration;
annealing_configuration_t best_configuration
holds the best so far configuration value; its data field must be initialised with a pointer to a memory block large enough to hold a configuration value; the memory block must be initialised with an invalid value that can be recognised by copy_function;
annealing_configuration_t * new_configurations
pointer to an array of number_of_tries structures used to hold references to the new configurations; the data field of the structures must be initialised with a pointer to a memory block large enough to hold a configuration value; the memory block must be initialised with an invalid value that can be recognised by copy_function;
double * monte_carlo_coordinates
pointer to an array of number_of_tries values used to store the Monte Carlo method coordinates for the new configurations (the coordinate for the current configuration is always 1.0, so it does not need a variable);
gsl_rng * numbers_generator
the random number generator used to apply the Boltzmann distribution acceptance criterion; it can be used also to generate the random step;
annealing_energy_fun_t * energy_function
the user supplied function used to compute the energy of a configuration;
annealing_step_fun_t * step_function
the user supplied function used to compute the new configuration from the current configuration;
annealing_cooling_fun_t * cooling_function
the user supplied function used to cool the temperature; it can be NULL to select the default cooling algorithm;
annealing_log_fun_t * log_function
the user supplied function used to log the search path; it can be NULL if no logging is required;
annealing_copy_fun_t * copy_function
the user supplied function used to copy a configuration from an allocation space to another;
void * params
this is a free field that can be used to hand data to the user supplied functions.

— Function: void annealing_manytries_solve (annealing_manytries_workspace_t * S)

Does the search.

The structure referenced by S must be allocated and initialised by the user before invoking this function, and freed by the user after this function has returned.

Space for the data referenced by the current_configuration, best_configuration and new_configurations fields in S must be allocated by the user before invoking this function, and freed by the user after this function has returned.

This function does no resource allocation, so it is perfectly all right if the user supplied functions use a dynamic wind mechanism, like setjmp() and longjmp(), to report errors.

When this function returns: the best configuration found is in the best_configuration field.

#### 2.5.5 Examples

Find minimum of f(t) = -sin(t)/t.

     /** ------------------------------------------------------------
** ----------------------------------------------------------*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <float.h>
#include <gsl/gsl_rng.h>
#include "annealing.h"

static	annealing_energy_fun_t	energy_function;
static	annealing_step_fun_t	step_function;
static	annealing_log_fun_t		log_function;
static	annealing_copy_fun_t	copy_function;

/** ------------------------------------------------------------
** Main.
** ----------------------------------------------------------*/

#define TRIES   10

int
main (int argc, char ** argv)
{
annealing_manytries_workspace_t   S;
annealing_configuration_t         array[TRIES];
double        configurations[2 + TRIES]; /* best, current and new tries */
double        monte_carlo_coordinates[TRIES];
double        max_step = 10.0;

printf("many-tries sinc minimisation with simulated annealing\n");

S.max_step_value              = &max_step;

S.temperature                 = 10.0;
S.minimum_temperature         = 1.0e-6;
S.restart_temperature         = DBL_MIN; /* do not restart */
S.boltzmann_constant          = 1.0;
S.damping_factor              = 1.005;

S.energy_function             = energy_function;
S.step_function               = step_function;
S.copy_function               = copy_function;
S.log_function                = log_function;
S.cooling_function            = NULL;

S.numbers_generator           = gsl_rng_alloc(gsl_rng_rand);
gsl_rng_set(S.numbers_generator, 15);

S.current_configuration.data  = &(configurations[0]);
S.best_configuration.data     = &(configurations[1]);
S.new_configurations          = array;
S.monte_carlo_coordinates     = monte_carlo_coordinates;
S.number_of_tries             = TRIES;

for (size_t i=0; i<S.number_of_tries; ++i)
array[i].data = &(configurations[i+2]);

configurations[0] = 100.0;

annealing_manytries_solve(&S);

printf("final best solution: %f, global 0.0\n", configurations[1]);

gsl_rng_free(S.numbers_generator);
exit(EXIT_SUCCESS);
}

/** ------------------------------------------------------------
** Iteration functions.
** ----------------------------------------------------------*/

static double
alea (annealing_manytries_workspace_t * S)
{
double        max_step = *((double *)S->max_step_value);

return (2.0 * gsl_rng_uniform(S->numbers_generator) - 1.0) * max_step;
}

/* ------------------------------------------------------------ */

double
energy_function (void * dummy, void * configuration)
{
double        C = *((double *)configuration);

return -sin(C)/C;
}
void
step_function (void * W, void * configuration)
{
annealing_manytries_workspace_t * S = W;
double *      C = (double *)configuration;
double        c;

do c = *C + alea(S); while (fabs(c) > 120.0);
*C = c;
}
void
log_function (void * W)
{
annealing_manytries_workspace_t * S = W;
double *      current = (double *)S->current_configuration.data;
double *      best    = (double *)S->best_configuration.data;

printf("current %f (energy %f), best %f (energy %f)\n",
*current, S->current_configuration.energy,
*best,    S->best_configuration.energy);
}

/** ------------------------------------------------------------
** Configuration handling functions.
** ----------------------------------------------------------*/

void
copy_function (void * dummy, void * dst_configuration, void * src_configuration)
{
double *      dst = dst_configuration;
double *      src = src_configuration;

*dst = *src;
}

/* end of file */


Next: , Previous: annealing, Up: Top

## 3 Miscellaneous functions

Up: misc

### 3.1 Version functions

library version for details on version numbers.

— Function: const char * annealing_version (void)

Return a statically allocated string representing the package version.

— Function: int annealing_library_major_version (void)

Return an integer representing the interface major number version.

— Function: int annealing_library_minor_version (void)

Return an integer representing the interface minor number version.

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## Appendix A Bibliography and references

“Boltzmann distribution.” Wikipedia, The Free Encyclopedia. 24 Jan 2007, 14:52 UTC. Wikimedia Foundation, Inc. 26 Feb 2007 http://en.wikipedia.org/w/index.php?title=Boltzmann_distribution&oldid=102908405.

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## Appendix B GNU Free Documentation License

Version 1.3, 3 November 2008
     Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
http://fsf.org/

Everyone is permitted to copy and distribute verbatim copies
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## Appendix C GNU General Public License

Version 3, 29 June 2007
     Copyright © 2007 Free Software Foundation, Inc. http://fsf.org/

Everyone is permitted to copy and distribute verbatim copies of this
license document, but changing it is not allowed.


### Preamble

The GNU General Public License is a free, copyleft license for software and other kinds of works.

The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program—to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.

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Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.

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Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users' freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.

Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.

The precise terms and conditions for copying, distribution and modification follow.

### TERMS AND CONDITIONS

1. Definitions.

“This License” refers to version 3 of the GNU General Public License.

To “modify” a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a “modified version” of the earlier work or a work “based on” the earlier work.

A “covered work” means either the unmodified Program or a work based on the Program.

To “propagate” a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.

To “convey” a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.

An interactive user interface displays “Appropriate Legal Notices” to the extent that it includes a convenient and prominently visible feature that (1) displays an appropriate copyright notice, and (2) tells the user that there is no warranty for the work (except to the extent that warranties are provided), that licensees may convey the work under this License, and how to view a copy of this License. If the interface presents a list of user commands or options, such as a menu, a prominent item in the list meets this criterion.

2. Source Code.

The “source code” for a work means the preferred form of the work for making modifications to it. “Object code” means any non-source form of a work.

A “Standard Interface” means an interface that either is an official standard defined by a recognized standards body, or, in the case of interfaces specified for a particular programming language, one that is widely used among developers working in that language.

The “System Libraries” of an executable work include anything, other than the work as a whole, that (a) is included in the normal form of packaging a Major Component, but which is not part of that Major Component, and (b) serves only to enable use of the work with that Major Component, or to implement a Standard Interface for which an implementation is available to the public in source code form. A “Major Component”, in this context, means a major essential component (kernel, window system, and so on) of the specific operating system (if any) on which the executable work runs, or a compiler used to produce the work, or an object code interpreter used to run it.

The “Corresponding Source” for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include the work's System Libraries, or general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work. For example, Corresponding Source includes interface definition files associated with source files for the work, and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require, such as by intimate data communication or control flow between those subprograms and other parts of the work.

The Corresponding Source need not include anything that users can regenerate automatically from other parts of the Corresponding Source.

The Corresponding Source for a work in source code form is that same work.

3. Basic Permissions.

All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law.

You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you.

Conveying under any other circumstances is permitted solely under the conditions stated below. Sublicensing is not allowed; section 10 makes it unnecessary.

4. Protecting Users' Legal Rights From Anti-Circumvention Law.

No covered work shall be deemed part of an effective technological measure under any applicable law fulfilling obligations under article 11 of the WIPO copyright treaty adopted on 20 December 1996, or similar laws prohibiting or restricting circumvention of such measures.

When you convey a covered work, you waive any legal power to forbid circumvention of technological measures to the extent such circumvention is effected by exercising rights under this License with respect to the covered work, and you disclaim any intention to limit operation or modification of the work as a means of enforcing, against the work's users, your or third parties' legal rights to forbid circumvention of technological measures.

5. Conveying Verbatim Copies.

You may convey verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice; keep intact all notices stating that this License and any non-permissive terms added in accord with section 7 apply to the code; keep intact all notices of the absence of any warranty; and give all recipients a copy of this License along with the Program.

You may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee.

6. Conveying Modified Source Versions.

You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:

1. The work must carry prominent notices stating that you modified it, and giving a relevant date.
2. The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to “keep intact all notices”.
3. You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.
4. If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so.

A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an “aggregate” if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation's users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.

7. Conveying Non-Source Forms.

You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways:

1. Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange.
2. Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge.
3. Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b.
4. Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements.
5. Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d.

A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work.

A “User Product” is either (1) a “consumer product”, which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, “normally used” refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product.

“Installation Information” for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made.

If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM).

The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network.

Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying.

“Additional permissions” are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions.

When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission.

Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms:

1. Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License; or
2. Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it; or
3. Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or
4. Limiting the use for publicity purposes of names of licensors or authors of the material; or
5. Declining to grant rights under trademark law for use of some trade names, trademarks, or service marks; or
6. Requiring indemnification of licensors and authors of that material by anyone who conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors.

All other non-permissive additional terms are considered “further restrictions” within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying.

If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms.

Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way.

9. Termination.

You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11).

However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10.

10. Acceptance Not Required for Having Copies.

You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.

11. Automatic Licensing of Downstream Recipients.

Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License.

An “entity transaction” is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party's predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts.

You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it.

12. Patents.

A “contributor” is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor's “contributor version”.

A contributor's “essential patent claims” are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, “control” includes the right to grant patent sublicenses in a manner consistent with the requirements of this License.

Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor's essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version.

In the following three paragraphs, a “patent license” is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To “grant” such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party.

If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. “Knowingly relying” means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient's use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.

If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.

A patent license is “discriminatory” if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.

Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.

13. No Surrender of Others' Freedom.

If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.

14. Use with the GNU Affero General Public License.

Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.

15. Revised Versions of this License.

The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License “or any later version” applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.

If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program.

16. Disclaimer of Warranty.

THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

17. Limitation of Liability.

IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

18. Interpretation of Sections 15 and 16.

If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.

### How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

     one line to give the program's name and a brief idea of what it does.
Copyright (C) year name of author

This program is free software: you can redistribute it and/or modify
the Free Software Foundation, either version 3 of the License, or (at

This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see http://www.gnu.org/licenses/.


Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:

     program Copyright (C) year name of author
This program comes with ABSOLUTELY NO WARRANTY; for details type ‘show w’.
This is free software, and you are welcome to redistribute it
under certain conditions; type ‘show c’ for details.


The hypothetical commands ‘show w’ and ‘show c’ should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an “about box”.

You should also get your employer (if you work as a programmer) or school, if any, to sign a “copyright disclaimer” for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see http://www.gnu.org/licenses/.

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read http://www.gnu.org/philosophy/why-not-lgpl.html.

Next: , Previous: Package License, Up: Top

## Appendix D How to interpret the library version

NOTE This section is not specific to this package; it is an explanation of library interface version numbering that the author uses for all its C libraries.

This package is distributing one or more libraries; this package has a version number and each library has an interface version number, package and interface versions are independent.

### D.1 Package version

The package version tracks the development of the source code: every time the source code is modified the package version is incremented. There are three numbers: major, minor, patch level.

major
Incremented before big rewritings or additions.
minor
Incremented to track incremental development of the code.
patch level
For stable versions: incremented for bug fixes and very small additions that were forgotten in the last minor update.

The package has also a letter that indicates the status of the release: a for experimental or “alpha”, b for testing or “beta”, . for stable.

A full version number looks like this:

1.2a3
major 1, minor 2, patch level 3, status alpha;
10.23b34
major 10, minor 23, patch level 34, status beta;
1.2.3
major 1, minor 2, patch level 3, status stable.

The reason for the letter to be between the minor number and the patch level is that it is more important:

• when starting to develop version 1.2 (major 1, minor 2), the last stable version of the 1.1 branch is taken and the project starts with alpha versions; each alpha has a patch level: 1.2a0, 1.2a1, ...;
• when experimental development, is finished the package is moved to testing and the status to beta: 1.2b0, 1.2b1, ... so the first beta version is newer than the last alpha version;
• when testing is finished, the package is moved to stable status: 1.2.0, 1.2.1, ... so the first stable version is newer than the last beta version.

summary:

     1.1.x < 1.2a0 < 1.2a1 < ... < 1.2b0 < 1.2b1 < ... < 1.2.0 < 1.2.1


### D.2 Interface version

The interface version changes only when the public interface of a library (public function prototypes and data types) is modified. There are two numbers: major and minor.

major
Establishes a set of: functions, data types, code behaviour that will not change until the next major number update. Code written for the first release of the library with a major interface number (1.0), will work unchanged with all the libraries with the same major interface number (1.1, 1.2, ...); if it does not: it is a library bug, please signal this to the author using the bug tracking system.

The only exception to this rule is the wrong behaviour caused by bugs; code that relies on buggy behaviour of a package version may not work with subsequent package versions with the same major interface number.

minor
Establishes a set of: functions, data types, code behaviour that will not change until the next major number update. This set is added to the set of the previous interface version.

The library file installed on our system has name composed with the interface number; example: if the package has version 1.2.3 and the interface has version 1.1, the library file is called libmy1.1.so.

Even if the interface number has not changed, and the library file has the same name: when an installed shared library is updated to a new package version, programs and libraries depending on it will have to be recompiled, because the internals of the library file will have changed.

Next: , Previous: library version, Up: Top

## Appendix E An entry for each concept

Next: , Previous: Concept Index, Up: Top

## Appendix F An entry for each function.

Next: , Previous: Function Index, Up: Top

## Appendix G An entry for each variable.

Previous: Variable Index, Up: Top