     [file version.texi not included]

This file documents GNU libmatheval library.

Copyright © 2002, 2003, 2004, 2005 Aleksandar B. Samardžić

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 the Invariant Sections being ``Rationale and history'', with no Front-Cover Texts, and with no Back-Cover Texts.  A copy of the license is included in the section entitled ``Copying''.

GNU libmatheval manual
**********************

GNU libmatheval is small library of procedures for evaluating mathematical functions.  This manual documents how to use the library; this is manual edition EDITION, last updated UPDATED, corresponding to library version VERSION.

License
*******

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

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, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

Introduction
************

GNU libmatheval is library comprising several procedures that makes possible to create in-memory tree representation of mathematical functions over single or multiple variables and later use this representation to evaluate function for specified variable values, to create corresponding tree for function derivative over specified variable or to get back textual representation of in-memory tree.

This section discuss use of programming interface exposed by library from C programs. Readers interested in Fortran interface should switch immediately to Fortran interface section.

In order to use GNU libmatheval library from C code, it is necessary first to include header file matheval.h from all files calling GNU libmatheval procedures and then refer to libmatheval library among other linker option.  Thus, command to compile C program using library and stored in file example.c using GNU C compiler would look like (supposing that library is installed using default prefix /usr/local/lib):

     gcc example.c -I/usr/local/include -L/usr/local/lib -lmatheval -o example

First step in actually utilizing library after including appropriate header file would be to declare variable of void * type to point to evaluator object that will represent given mathematical function:

     void *f;

Then, given that textual representation of function is stored into string buffer, evaluator object corresponding to given mathematical function could be created using evaluator_create() procedure (see evaluator_create) as follows (see documentation for this procedure also for description of notation that should be used to describe mathematical functions):

     f = evaluator_create (buffer); assert (f);

Return value should be always checked, because above procedure will return null pointer if there exist syntax errors in notation.  After that, one could utilize evaluator_get_variables() (see evaluator_get_variables) procedure to obtain a list of variable names appearing in function:

     {
       char **names;
       int count;
       int i;

       evaluator_get_variables (f, &names, &count);
       for (i = 0; i < count; i++)
         printf ("%s ", names[i]);
       printf ("\n");
     }

Procedure evaluator_evaluate() (see evaluator_evaluate) could be used to evaluate function for specific variable values.  Say that above function is over variable ``x'' only, then following code will evaluate and print function value for x = 0.1:

     {
       char *names[] = { "x" };
       double values[] = { 0.1 };

       printf ("f(0.1) = %g\n", evaluator_evaluate (f, 1, names,
                                                    values));
     }

Or alternatively, since function is over variable with standard name ``x'', convenience procedure evaluator_evaluate_x() (evaluator_evaluate_x) could be used to accomplish same by following:

     printf ("f(0.1) = %g\n", evaluator_evaluate_x (f, 0.1));

Evaluator object for function derivative over some variable could be created from evaluator object for given function.  In order to accomplish this, a declaration for derivative evaluator object should be added to variable declarations section:

     void *f_prim;

After that (supposing that ``x'' is used as derivation variable), derivative evaluator object could be created using evaluator_derivative() procedure (see evaluator_derivative):

     f_prim = evaluator_derivative (f, "x");

or alternatively using evaluator_derivative_x() convenience procedure (see evaluator_derivative_x):

     f_prim = evaluator_derivative_x (f);

Derivative evaluator object could be used to evaluate derivative values or say textual representation of derivative could be written to standard output through utilizing evaluator_get_string() procedure (see evaluator_get_string) to get string representing given evaluator.  Following code would accomplish this:

     printf ("  f'(x) = %s\n", evaluator_get_string (f_prim));

All evaluator objects must be destroyed after finished with using them and evaluator_destroy() procedure (see evaluator_destroy) is intended for this:

     evaluator_destroy (f);
     evaluator_destroy (f_prim);

Here follows complete program connecting above fragments.  Program read from standard input string representing function over variable ``x'', create evaluators for function and its first derivative, print textual representation of function derivative to standard output, then read value of variable ``x'' and finally print to standard output values of function and its first derivative for given value of variable ``x''.

     #include <stdio.h>
     #include <stdlib.h>
     #include <string.h>
     #include <assert.h>
     #include <matheval.h>

     /* Size of input buffer.  */
     #define BUFFER_SIZE 256

     /* Program is demonstrating use of GNU libmatheval library of procedures
        for evaluating mathematical functions.  */
     int
     main (int argc, char **argv)
     {
       char buffer[BUFFER_SIZE];	/* Input buffer.  */
       int length;			/* Length of above buffer. */
       void *f, *f_prim;		/* Evaluators for function and function derivative.  */
       char **names;			/* Function variables names. */
       int count;			/* Number of function variables. */
       double x;			/* Variable x value.  */
       int i;			/* Loop counter. */

       /* Read function.  Function has to be over variable x, or result may
          be undetermined.  Size of textual represenatation of function is
          bounded here to 256 characters, in real conditions one should
          probably use GNU readline() instead of fgets() to overcome this
          limit.  */
       printf ("f(x) = ");
       fgets (buffer, BUFFER_SIZE, stdin);
       length = strlen (buffer);
       if (length > 0 && buffer[length - 1] == '\n')
         buffer[length - 1] = '\0';

       /* Create evaluator for function.  */
       f = evaluator_create (buffer);
       assert (f);

       /* Print variable names appearing in function. */
       evaluator_get_variables (f, &names, &count);
       printf ("  ");
       for (i = 0; i < count; i++)
         printf ("%s ", names[i]);
       printf ("\n");

       /* Create evaluator for function derivative and print textual
          representation of derivative.  */
       f_prim = evaluator_derivative_x (f);
       printf ("  f'(x) = %s\n", evaluator_get_string (f_prim));

       /* Read variable x value.  */
       printf ("x = ");
       scanf ("%lf", &x);

       /* Calculate and print values of function and its derivative for given
          value of x.  */
       printf ("  f(%g) = %g\n", x, evaluator_evaluate_x (f, x));
       printf ("  f'(%g) = %g\n", x, evaluator_evaluate_x (f_prim, x));

       /* Destroy evaluators.  */
       evaluator_destroy (f);
       evaluator_destroy (f_prim);

       exit (EXIT_SUCCESS);
     }

Above example exercise most of library main procedures (see Main entry points), as well as some of convenience procedures (see Convenience procedures).  For full documentation, see Reference.

Reference
*********

This section documents procedures constituting GNU libmatheval library.  The convention is that all procedures have evaluator_ prefix.

Main entry points
=================

evaluator_create()
------------------

Synopsis
........

     #include <matheval.h>

     void *evaluator_create (char *string);

Description
...........

Create evaluator object from string containing mathematical representation of function.  Evaluator object could be used later to evaluate function for specific variable values or to calculate function derivative over some variable.

String representation of function is allowed to consist of decimal numbers, constants, variables, elementary functions, unary and binary operations.

Supported constants are (names that should be used are given in parenthesis): e (e), log2(e) (log2e), log10(e) (log10e), ln(2) (ln2), ln(10) (ln10), pi (pi), pi / 2 (pi_2), pi / 4 (pi_4), 1 / pi (1_pi), 2 / pi (2_pi), 2 / sqrt(pi) (2_sqrtpi), sqrt(2) (sqrt) and sqrt(1 / 2) (sqrt1_2).

Variable name is any combination of alphanumericals and _ characters beginning with a non-digit that is not elementary function name.

Supported elementary functions are (names that should be used are given in parenthesis): exponential (exp), logarithmic (log), square root (sqrt), sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), cosecant (csc), inverse sine (asin), inverse cosine (acos), inverse tangent (atan), inverse cotangent (acot), inverse secant (asec), inverse cosecant (acsc), hyperbolic sine (sinh), cosine (cosh), hyperbolic tangent (tanh), hyperbolic cotangent (coth), hyperbolic secant (sech), hyperbolic cosecant (csch), hyperbolic inverse sine (asinh), hyperbolic inverse cosine (acosh), hyperbolic inverse tangent (atanh), hyperbolic inverse cotangent (acoth), hyperbolic inverse secant (asech), hyperbolic inverse cosecant (acsch), absolute value (abs), Heaviside step function (step) with value 1 defined for x = 0 and Dirac delta function with infinity (delta) and not-a-number (nandelta) values defined for x = 0.

Supported unary operation is unary minus ('-').

Supported binary operations are addition ('+'), subtraction ('+'), multiplication ('*'), division multiplication ('/') and exponentiation ('^').

Usual mathematical rules regarding operation precedence apply. Parenthesis ('(' and ')') could be used to change priority order.

Blanks and tab characters are allowed in string representing function; newline characters must not appear in this string.

Return value
............

Pointer to evaluator object if operation successful, null pointer otherwise.  Evaluator object is opaque, one should only use return pointer to pass it to other functions from library.

See also
........

evaluator_destroy, evaluator_evaluate, evaluator_get_string, evaluator_get_variables, evaluator_derivative

evaluator_destroy()
-------------------

Synopsis
........

     #include <matheval.h>

     void evaluator_destroy (void *evaluator);

Description
...........

Destroy evaluator object pointer by evaluator pointer.  After returning from this call evaluator pointer must not be dereferenced because evaluator object gets invalidated.

Return value
............

None.

See also
........

evaluator_create

evaluator_evaluate()
--------------------

Synopsis
........

     #include <matheval.h>

     double evaluator_evaluate (void *evaluator, int count, char **names,
     				    double *values);

Description
...........

Calculate value of function represented by evaluator object for given variable values.  Evaluator object is pointed by evaluator pointer.  Variable names and corresponding values are given by names and values array respectively.  Length of arrays is given by count argument.

Return value
............

Function value for given variable values.  If some variable that appears in function is not mentioned in arguments, result is indeterminate.  If all variables that appear in function are given, presence of variable or variables that doesn't appear in function in arguments has no effect, i.e. result is still exact.

See also
........

evaluator_create, evaluator_destroy, evaluator_evaluate_x, evaluator_evaluate_x_y, evaluator_evaluate_x_y_z

evaluator_get_string()
----------------------

Synopsis
........

     #include <matheval.h>

     char *evaluator_get_string (void *evaluator);

Description
...........

Return textual representation (i.e. mathematical function) of evaluator object pointed by evaluator.  For notation used, see evaluator_create documentation.

Return value
............

String with textual representation of evaluator object.  This string is stored in evaluator object and caller must not free pointer returned by this function.  Returned string is valid until evaluator object destroyed.

See also
........

evaluator_create, evaluator_destroy, evaluator_get_variables

evaluator_get_variables()
-------------------------

Synopsis
........

     #include <matheval.h>

     void evaluator_get_variables (void *evaluator, char ***names, int *count);

Description
...........

Return array of strings with names of variables appearing in function represented by evaluator.  Address of array first element is stored by function in location pointed by second argument and number of array elements is stored in location pointed by third argument.  Array with function variable names is stored in evaluator object and caller must not free any of strings returned by this function nor array itself. Returned values are valid until evaluator object destroyed.

Return value
............

None.

See also
........

evaluator_create, evaluator_destroy, evaluator_get_string

evaluator_derivative()
----------------------

Synopsis
........

     #include <matheval.h>

     void *evaluator_derivative (void *evaluator, char *name);

Description
...........

Create evaluator for derivative of function represented by given evaluator object.  Evaluator object is pointed to by evaluator pointer and derivation variable is determined by name argument. Calculated derivative is in mathematical sense correct no matters of fact that derivation variable appears or not in function represented by evaluator.

Return value
............

Pointer to evaluator object representing derivative of given function.

See also
........

evaluator_create, evaluator_destroy, evaluator_derivative_x, evaluator_derivative_y, evaluator_derivative_z

Convenience procedures
======================

evaluator_evaluate_x()
----------------------

Synopsis
........

     #include <matheval.h>

     double evaluator_evaluate_x (void *evaluator, double x);

Description
...........

Convenience function to evaluate function for given variable ``x'' value.  Function is equivalent to following:

     char *names[] = { "x" };
     double values[] = { x };

     evaluator_evaluate (evaluator, sizeof (names) / sizeof(names[0]),
     			     names, values);

See evaluator_evaluate for further information.

Return value
............

Value of function for given value of variable ``x''.

See also
........

evaluator_create, evaluator_destroy, evaluator_evaluate

evaluator_evaluate_x_y()
------------------------

Synopsis
........

     #include <matheval.h>

     double evaluator_evaluate_x_y (void *evaluator, double x, double y);

Description
...........

Convenience function to evaluate function for given variables ``x'' and ``y'' values.  Function is equivalent to following:

     char *names[] = { "x", "y" };
     double values[] = { x, y };

     evaluator_evaluate (evaluator, sizeof (names) / sizeof(names[0]),
     			     names, values);

See evaluator_evaluate for further information.

Return value
............

Value of function for given values of variables ``x'' and ``y''.

See also
........

evaluator_create, evaluator_destroy, evaluator_evaluate

evaluator_evaluate_x_y_z()
--------------------------

Synopsis
........

     #include <matheval.h>

     double evaluator_evaluate_x_y_z (void *evaluator, double x, double y,
     					  double z);

Description
...........

Convenience function to evaluate function for given variables ``x'', ``y'' and ``z'' values.  Function is equivalent to following:

     char *names[] = { "x", "y", "z" };
     double values[] = { x, y, z };

     evaluator_evaluate (evaluator, sizeof (names) / sizeof(names[0]),
     			     names, values);

See evaluator_evaluate for further information.

Return value
............

Value of function for given values of variables ``x'', ``y'' and ``z''.

See also
........

evaluator_create, evaluator_destroy, evaluator_evaluate

evaluator_derivative_x()
------------------------

Synopsis
........

     #include <matheval.h>

     void *evaluator_derivative_x (void *evaluator);

Description
...........

Convenience function to differentiate function using ``x'' as derivation variable.  Function is equivalent to:

     evaluator_derivative (evaluator, "x");

See evaluator_derivative for further information.

Return value
............

Evaluator object representing derivative of function over variable ``x''.

See also
........

evaluator_create, evaluator_destroy, evaluator_derivative

evaluator_derivative_y()
------------------------

Synopsis
........

     #include <matheval.h>

     void *evaluator_derivative_y (void *evaluator);

Description
...........

Convenience function to differentiate function using ``y'' as derivation variable.  Function is equivalent to:

     evaluator_derivative (evaluator, "y");

See evaluator_derivative for further information.

Return value
............

Evaluator object representing derivative of function over variable ``y''.

See also
........

evaluator_create, evaluator_destroy, evaluator_derivative

evaluator_derivative_z()
------------------------

Synopsis
........

     #include <matheval.h>

     void *evaluator_derivative_z (void *evaluator);

Description
...........

Convenience function to differentiate function using ``z'' as derivation variable.  Function is equivalent to:

     evaluator_derivative (evaluator, "z");

See evaluator_derivative for further information.

Return value
............

Evaluator object representing derivative of function over variable ``z''.

See also
........

evaluator_create, evaluator_destroy, evaluator_derivative

Fortran interface
*****************

Fortran interface to GNU libmatheval library is very similar to C interface; still, complete documentation from Reference is reproduced here using Fortran terms in order to have Fortran programmer not to mess with C terms that he may not understand. Besides documentation for all library exported procedures, an example Fortran program of structure similar to sequence of code fragments presented for C programmers in Introduction section as well as notes on how to link library with Fortran programs are presented here.

Since passing arguments between C and Fortran is not (yet) standardized, Fortran interface of library applies only to GNU Fortran 77 compiler; but note that same interface is working fine for GNU Fortran 95 compiler.  Requests to adapt interface to other Fortran compilers are welcome (see section Bugs for contact information), under condition that access to corresponding compiler is provided.

Fortran main entry points
=========================

evaluator_create()
------------------

Synopsis
........

     integer*8 function evaluator_create (string)
       character(len=*) :: string
     end function evaluator_create

Description
...........

Create evaluator object from string containing mathematical representation of function.  Evaluator object could be used later to evaluate function for specific variable values or to calculate function derivative over some variable.

String representation of function is allowed to consist of decimal numbers, constants, variables, elementary functions, unary and binary operations.

Supported constants are (names that should be used are given in parenthesis): e (e), log2(e) (log2e), log10(e) (log10e), ln(2) (ln2), ln(10) (ln10), pi (pi), pi / 2 (pi_2), pi / 4 (pi_4), 1 / pi (1_pi), 2 / pi (2_pi), 2 / sqrt(pi) (2_sqrtpi), sqrt(2) (sqrt) and sqrt(1 / 2) (sqrt1_2).

Variable name is any combination of alphanumericals and _ characters beginning with a non-digit that is not elementary function name.

Supported elementary functions are (names that should be used are given in parenthesis): exponential (exp), logarithmic (log), square root (sqrt), sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), cosecant (csc), inverse sine (asin), inverse cosine (acos), inverse tangent (atan), inverse cotangent (acot), inverse secant (asec), inverse cosecant (acsc), hyperbolic sine (sinh), cosine (cosh), hyperbolic tangent (tanh), hyperbolic cotangent (coth), hyperbolic secant (sech), hyperbolic cosecant (csch), hyperbolic inverse sine (asinh), hyperbolic inverse cosine (acosh), hyperbolic inverse tangent (atanh), hyperbolic inverse cotangent (acoth), hyperbolic inverse secant (asech), hyperbolic inverse cosecant (acsch), absolute value (abs), Heaviside step function (step) with value 1 defined for x = 0 and Dirac delta function with infinity (delta) and not-a-number (nandelta) values defined for x = 0.

Supported unary operation is unary minus ('-').

Supported binary operations are addition ('+'), subtraction ('+'), multiplication ('*'), division multiplication ('/') and exponentiation ('^').

Usual mathematical rules regarding operation precedence apply. Parenthesis ('(' and ')') could be used to change priority order.

Blanks and tab characters are allowed in string representing function; newline characters must not appear in this string.

Return value
............

Positive 64-bit integer representing evaluator object unique handle if operation successful, 0 otherwise.  Return value should be used only to pass it to other functions from library.

See also
........

Fortran evaluator_destroy, Fortran evaluator_evaluate, Fortran evaluator_get_string_length, Fortran evaluator_get_string_chars, Fortran evaluator_get_variables_length, Fortran evaluator_get_variables_chars, Fortran evaluator_derivative

evaluator_destroy()
-------------------

Synopsis
........

     subroutine evaluator_destroy (evaluator)
       integer*8 :: evaluator
     end subroutine evaluator_destroy

Description
...........

Destroy evaluator object denoted by evaluator handle.  After returning from this call evaluator object gets invalidated, so value of evaluator handle should not be used any more.

Return value
............

None.

See also
........

Fortran evaluator_create

evaluator_evaluate()
--------------------

Synopsis
........

     double precision function evaluator_evaluate (evaluator, count, names, values)
       integer*8 :: evaluator
       integer :: count
       character(len=*) :: names
       double precision :: values
       dimension values(*)
     end function evaluator_evaluate

Description
...........

Calculate value of function represented by evaluator object for given variable values.  Evaluator object is identified by evaluator handle.  Variable names are given by names string and corresponding values are given by values array respectively. Number of variables is given by count argument.  Variable names in names string should be delimited by one or more blank characters.

Return value
............

Function value for given variable values.  If some variable that appears in function is not mentioned in arguments, result is indeterminate.  If all variables that appear in function are given, presence of variable or variables that doesn't appear in function in arguments has no effect, i.e. result is still exact.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_evaluate_x, Fortran evaluator_evaluate_x_y, Fortran evaluator_evaluate_x_y_z

evaluator_get_string_length()
-----------------------------

Synopsis
........

     integer function evaluator_get_string_length (evaluator)
       integer*8 :: evaluator
     end function evaluator_get_string_length

Description
...........

Return length of textual representation (i.e. mathematical function) of evaluator object pointed by evaluator.

Return value
............

Evaluator textual representation string length.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_get_string_chars

evaluator_get_string_chars()
----------------------------

Synopsis
........

     subroutine evaluator_get_string_chars (evaluator)
       integer*8 :: evaluator
       character(len=*) :: string
     end subroutine evaluator_get_string_chars

Description
...........

Write textual representation (i.e. mathematical function) of evaluator object pointed by evaluator to string specified.  For notation used, see Fortran evaluator_create documentation.  In order to declare string of appropriate length to be passed to this function, Fortran evaluator_get_string_length function should be utilized.

Return value
............

None.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_get_string_length

evaluator_get_variables_length()
--------------------------------

Synopsis
........

     integer function evaluator_get_variables_length (evaluator)
       integer*8 :: evaluator
     end function evaluator_get_variables_length

Description
...........

Return length of string with names of all variables (separated by a blank character) appearing in evaluator object pointed by evaluator.

Return value
............

Variable names string length.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_get_variables_chars

evaluator_get_variables_chars()
-------------------------------

Synopsis
........

     subroutine evaluator_get_variables_chars (evaluator)
       integer*8 :: evaluator
       character(len=*) :: string
     end subroutine evaluator_get_variables_chars

Description
...........

Write names of all variables appearing in evaluator object pointed by evaluator into given string (separated by a blank character). In order to declare string of appropriate length to be passed to this function, Fortran evaluator_get_variables_length function should be utilized.

Return value
............

None.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_get_variables_length

evaluator_derivative()
----------------------

Synopsis
........

     integer*8 function evaluator_derivative (evaluator, name)
       integer*8 :: evaluator
       character(len=*) ::  name
     end function evaluator_derivative

Description
...........

Create evaluator for derivative of function represented by given evaluator object.  Evaluator object is identified by evaluator handle and derivation variable is determined by name argument. Calculated derivative is in mathematical sense correct no matters of fact that derivation variable appears or not in function represented by evaluator.

Return value
............

64-bit integer uniquely identifying evaluator object representing derivative of given function.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_derivative_x, Fortran evaluator_derivative_y, Fortran evaluator_derivative_z

Fortran convenience procedures
==============================

evaluator_evaluate_x()
----------------------

Synopsis
........

     double precision function evaluator_evaluate_x (evaluator, x)
       integer*8 :: evaluator
       double precision ::  x
     end function evaluator_evaluate_x

Description
...........

Convenience function to evaluate function for given variable ``x'' value.  Function is equivalent to following:

     evaluator_evaluate (evaluator, 1, 'x', (/ x /))

See Fortran evaluator_evaluate for further information.

Return value
............

Value of function for given value of variable ``x''.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_evaluate

evaluator_evaluate_x_y()
------------------------

Synopsis
........

     double precision function evaluator_evaluate_x_y (evaluator, x, y)
        integer*8 :: evaluator
        double precision ::  x, y
     end function evaluator_evaluate_x_y

Description
...........

Convenience function to evaluate function for given variables ``x'' and ``y'' values.  Function is equivalent to following:

     evaluator_evaluate (evaluator, 2, 'x y', (/ x, y /))

See Fortran evaluator_evaluate for further information.

Return value
............

Value of function for given values of variables ``x'' and ``y''.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_evaluate

evaluator_evaluate_x_y_z()
--------------------------

Synopsis
........

     double precision function evaluator_evaluate_x_y_z (evaluator, x, y, z)
       integer*8 :: evaluator
       double precision :: x, y, z
     end function evaluator_evaluate_x_y_z

Description
...........

Convenience function to evaluate function for given variables ``x'', ``y'' and ``z'' values.  Function is equivalent to following:

     evaluator_evaluate (evaluator, 2, 'x y z', (/ x, y, z /))

See Fortran evaluator_evaluate for further information.

Return value
............

Value of function for given values of variables ``x'', ``y'' and ``z''.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_evaluate

evaluator_derivative_x()
------------------------

Synopsis
........

     integer*8 function evaluator_derivative_x (evaluator)
       integer*8 :: evaluator
     end function evaluator_derivative_x

Description
...........

Convenience function to differentiate function using ``x'' as derivation variable.  Function is equivalent to:

     evaluator_derivative (evaluator, 'x');

See Fortran evaluator_derivative for further information.

Return value
............

Evaluator object representing derivative of function over variable ``x''.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_derivative

evaluator_derivative_y()
------------------------

Synopsis
........

     integer*8 function evaluator_derivative_y (evaluator)
       integer*8 :: evaluator
     end function evaluator_derivative_y

Description
...........

Convenience function to differentiate function using ``y'' as derivation variable.  Function is equivalent to:

     evaluator_derivative (evaluator, 'y');

See Fortran evaluator_derivative for further information.

Return value
............

Evaluator object representing derivative of function over variable ``y''.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_derivative

evaluator_derivative_z()
------------------------

Synopsis
........

     integer*8 function evaluator_derivative_z (evaluator)
       integer*8 :: evaluator
     end function evaluator_derivative_z

Description
...........

Convenience function to differentiate function using ``z'' as derivation variable.  Function is equivalent to:

     evaluator_derivative (evaluator, 'z');

See Fortran evaluator_derivative for further information.

Return value
............

Evaluator object representing derivative of function over variable ``z''.

See also
........

Fortran evaluator_create, Fortran evaluator_destroy, Fortran evaluator_derivative

Fortran sample program
======================

Here follows sample program demonstrating use of library Fortran interface.  Hopefully, comments throughout code will be enough for Fortran programmer to get acquainted with library usage.  Basic functioning of program is equivalent to code presented for C programmer in Introduction sequence, except that textual representation of function derivative is not printed to standard output and this is avoided simply because of Fortran 77 ugly string handling.  Following code is written in Fortran 77 with GNU Fortran 77 compiler extensions (most notable of these certainly is free form of source code).

     ! Program is demonstrating use of GNU libmatheval library of procedures
     ! for evaluating mathematical functions.
     program evaluator
       implicit none

       ! Declarations of GNU libmatheval procedures used.
       integer*8 evaluator_create
       integer*8 evaluator_derivative_x
       double precision evaluator_evaluate_x
       external evaluator_destroy

       ! Size of input buffer.
       integer :: BUFFER_SIZE
       parameter(BUFFER_SIZE = 256)

       character(len = BUFFER_SIZE) :: buffer ! Input buffer.
       integer*8 :: f, f_prim ! Evaluators for function and function derivative.
       double precision :: x ! Variable x value.

       ! Read function.  Function has to be over variable x, or result may
       ! be undetermined.  Size of textual represenatation will be truncated
       ! here to BUFFER_SIZE characters, in real conditions one should
       ! probably come with something smarter to avoid this limit.
       write (*, '(A)') 'f(x) = '
       read (*, '(A)') buffer

       ! Create evaluator for function.
       f = evaluator_create (buffer);
       if (f == 0) stop

       ! Create evaluator for function derivative.
       f_prim = evaluator_derivative_x (f);
       if (f_prim == 0) stop

       ! Read variable x value.
       write (*, '(A)') 'x = '
       read (*, *) x

       ! Calculate and print values of function and its derivative for given
       ! value of x.
       write (*,*) '  f (', x, ') = ', evaluator_evaluate_x (f, x)
       write (*,*) '  f'' (', x, ') = ', evaluator_evaluate_x (f_prim, x)

       ! Destroy evaluators.
       call evaluator_destroy (f)
       call evaluator_destroy (f_prim)
     end program evaluator

Fortran build process
=====================

In order to be able to reference GNU libmatheval procedures from Fortran code, declarations of procedures that will be used should be repeated, like demonstrated by Fortran sample program (once when interface upgraded to Fortran 90, modules and use statement will be employed here).  Command for compilation Fortran program using library and stored in file example.f using GNU Fortran 77 compiler would look like (again supposing that library is installed using default prefix /usr/local/lib):

     f77 example.f -ff90 -ffree-form -L/usr/local/lib -lmatheval -o example

Hacking
*******

Design notes
============

As usual with an open source project, ultimate reference for anyone willing to hack on it is its source code.  Every effort is put to have source code properly commented; having in mind that GNU libmatheval is rather simple project, it is reasonable to expect that this would be enough for anyone interested in project internals to get acquainted with it.  Still, this section will briefly explain project design. See Project structure section for description of where each functionality is located in source code.

Mathematical functions are represented as trees in computer memory. There are five different nodes in such a tree: number, constants, variables, functions, unary operations and binary operations.  Single data structure is employed for tree nodes, while union is used over what is different among them.  Numbers have unique value, unary and binary operations have unique pointer(s) to their operand(s) node(s). To represent constants, variables and functions, a symbol table is employed; thus constants, variables and functions have unique pointers to corresponding symbol table records (functions also have unique pointer to their argument node).  All operations related to functions (e.g. evaluation or derivative calculation) are implemented as recursive operations on tree nodes.  There exist a node operation that is not visible as external procedure and this is node simplification; this operation is very important regarding overall efficiency of other operations and is employed each time when new tree created.

Symbol table is implemented as hash table, where each bucket has linked list of records stored in it.  Records store information of symbol name and type (variable or function), as well as some unique information related to evaluation: variable records store temporary variable value and function records store pointer to procedure used to actually calculate function value during evaluation.  Hashing function described in A.V. Aho, R. Sethi, J.D. Ullman, ``Compilers - Principle, Techniques, and Tools'', Addison-Wesley, 1986, pp 435-437 is used.  Symbol tables are reference counted objects, i.e. could be shared.

Evaluator objects actually consists of function tree and reference to symbol table.  Most of operations on evaluator objects are simply delegated to function tree root node.

For parsing strings representing mathematical functions, Lex and Yacc are employed.  Scanner is creating symbol table records for variables, while for constants and functions it is only looking up existing symbol table records (before starting scanning, symbol table should be populated with records for constants and functions recognized by scanner).  Parser is responsible for building function tree representation.

Couple error reporting procedures, as well as replacements for standard memory allocation routines are also present.  These are rather standard for all GNU projects, so deserve no further discussion.  Further present in project are couple procedures for mathematical functions not implemented by C standard library, like cotangent, inverse cotangent and some hyperbolic and inverse hyperbolic functions.

Also present in project are stubs for Fortran code calling library. These stubs uses knowledge of GNU Fortran 77 compiler calling conventions, take parameters from Fortran 77 calls, eventually mangle them to satisfy primary C library interface and call library procedures to actually do the work, finally eventually mangling return values to satisfy Fortran 77 calling conventions again.

Most important thing to know before criticizing library design is that it is intentionally left as simple as it could be.  Decision is now that eventual library usage should direct its improvements.  Some obvious and intended improvements if enough interest for library arise are enumerated in Intended improvements section.  If having further suggestions, pleas see Bugs sections for contact information.

Project structure
=================

Interesting source files are mostly concentrated in lib subdirectory of distribution.  Basic arrangement is rather standard for GNU projects, thus scanner is in scanner.l file, parser in parser.y, error handling routines are in error.c and error.h files, replacements for standard memory allocation routines are in xmalloc.c and xmalloc.h, additional mathematical functions are in xmath.c and xmath.c. Project specific files are: node.h and node.c files for tree representing mathematical function data structures and procedures, symbol_table.c and symbol_table.h for symbol table data structures and procedures and finally evaluator.c and matheval.h for evaluator object data structures and procedures (evaluator object data structure is moved to .c file because matheval.h is public header file and this data structure should be opaque).  Fortran interface is implemented in f77_interface.c file.

File libmatheval.texi under doc subdirectory of distribution contains Texinfo source of project documentation (i.e. what you are reading now).

Subdirectory tests contains library test suite.  Kind of mixed design is employed here - GNU autotest is used for test framework in order to achieve more portability, while number of small Guile scripts are performing tests.  File matheval.c in tests subdirectory contains program extending Guile interpreter with GNU libmatheval procedures.  Files with .at extension in same subdirectory in turn consist of fragments of Guile code that this extended Guile interpreter executes in order to conduct tests.  File matheval.sh is shell wrapper for program contained in matheval.c file; this wrapper is used by autotest during testing instead of original program.  Most interesting aspect of code from tests subdirectory is certainly Guile interface for library that is implemented in matheval.c file; anyone intending to write more tests must before approaching this task become familiar with this interface.

Intended improvements
=====================

As stated in Design notes section, GNU libmatheval is designed with intention to be simple and understandable and to eventually have its usage to govern improvements.  Thus, further work will be primarily directed by user requests and of course, as usual with open source projects, with amount of spare time of primary developer (see Bugs for contact information).  However, there exist several obvious improvements that I'm willing to work on immediately if any interest of library arise and these are (in random order) listed below:

  * Extend scanner to recognize more mathematical functions, to recognize alternative names for existing functions (e.g. to recognize both tg and tan as names for tangent function) and to recognize more constants.

  * Implement variable hash table length for symbol table.  As for now, hash table length is fixed to 211 that is reasonable for most cases, but it would certainly be more robust to have hash table to be constructed of length proportional say to length of string representing function.

  * Add more simplifications to function tree representation.  Only basic simplifications, mostly related to numbers subtrees consolidation and binary operations neutral elements are employed now.  More ambitious optimization, using commutative, associative and distributive rules for binary operations would be desirable.

  * Improve output when evaluator object is printed.  Presently, parenthesis are always used around operations, while using them when necessary to establish proper evaluation priority order only would give prettier output

  * Add more tests.  Basic functionality of library is exercised through existing test suite, but present number of tests is certainly far from enough.

  * Extend and improve error handling.  There are couple assert()s left in code that may be replaced with some other mechanism, also probably error handling of more error conditions should be added to library.

  * Add command line interface to library, i.e. write a program that will make possible to evaluate expression for given variable values where both specified in command line, as program arguments (for expressions without variables this program could be useful as a calculator).

There exists also an improvement that is obvious and necessary but because I'm not native speaker I'm unfortunately not able to accomplish it anything more than I already tried:

  * Clean up English used in documentation.

Bugs
****

If you encounter something that you think is a bug, please report it immediately.  Try to include a clear description of the undesired behavior.  A test case that exhibits the bug or maybe even patch fixing it, would too be of course very useful.

Suggestions on improving library would be also more than welcome.  Please see Hacking, for further information.

Please direct bug reports and eventual patches to bug-libmatheval@gnu.org mailing list.  For suggestions regarding improvements and other libmatheval related conversation use author e-mail address asamardzic@matf.bg.ac.yu.

Rationale and history
*********************

The library is developed as a back-end for ``Numerical Analysis'' course taught during 1999/2000, 2000/2001 and 2001/2002 school years at Faculty of Mathematics, University of Belgrade.  Most numerical libraries (library accompanying ``Numerical Recipes'' book most notably example) are asking programmer to write corresponding C code when it comes to evaluate mathematical functions.  It seemed to me that it would be more appropriate (well, at least for above mentioned course) to have library that will make possible to specify functions as strings and then have them evaluated for given variable values, so I wrote first version of library during November 1999.  Fortran interface is added to the library later; during January 2001 interface for Pacific Sierra VAST Fortran 90 translator was implemented and during September 2001 it was replaced by interface for Intel Fortran 90 compiler  (That was in turn replaced by interface for GNU Fortran 77 compiler in order to meet requirement that no GNU project should require use of non-free software).  This library eventually went into rather stable state and was tested by number of other programs implementing various numerical methods and developed for the same course.

After completing engagement with this course, I thought it may be interesting for someone else to use this code and decided to make it publicly available.  So, having some spare time during June 2002, I re-wrote whole library in preparation for public release, now employing simpler overall design and also using GNU auto-tools and what else was necessary according to GNU guidelines.  The benefit is that final product looks much better now (well, at least to me and at least at the very moment of this writing), the drawback is that code is not thoroughly tested again.  But certainly author would be more than happy to further improve and maintain it.  Please see Bugs, for contact information.

The library source code was hosted on Savannah (http://savannah.gnu.org/) since Septembar 2002.  In September 2003, library officially became part of GNU project.

GNU Free Documentation License
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