@c -----------------------------------------------------------------------------
@c File     : simplifications.de.texi
@c License  : GNU General Public License (GPL)
@c Language : German
@c Original : simplifications.texi revision 03.07.2011
@c Date     : 08.11.2010
@c Revision : 03.07.2011
@c 
@c This file is part of Maxima -- GPL CAS based on DOE-MACSYMA
@c -----------------------------------------------------------------------------

@menu
* Introduction to simplification::
* Package absimp::
* Package facexp::
* Package functs::
* Package ineq::
* Package rducon::
* Package scifac::
* Package sqdnst::
@end menu

@c -----------------------------------------------------------------------------
@node Introduction to simplification, Package absimp, simplification, simplification
@section Introduction to simplification
@c -----------------------------------------------------------------------------

The directory @code{maxima/share/simplification} contains several scripts
which implement simplification rules and functions,
and also some functions not related to simplification.

@c Adapted from absimp.usg ----------------------

@c -----------------------------------------------------------------------------
@node Package absimp, Package facexp, Introduction to simplification, simplification
@section Package absimp
@c -----------------------------------------------------------------------------

The @code{absimp} package contains pattern-matching rules that
extend the built-in simplification rules for the @code{abs} and @code{signum}
functions.
@code{absimp} respects relations
established with the built-in @code{assume} function and by declarations such
as @code{modedeclare (m, even, n, odd)} for even or odd integers.

@code{absimp} defines @code{unitramp} and @code{unitstep} functions
in terms of @code{abs} and @code{signum}.

@code{load(absimp)} loads this package.
@code{demo(absimp)} shows a demonstration of this package.

Examples:

@example
(%i1) load (absimp)$
(%i2) (abs (x))^2;
                                       2
(%o2)                                 x
(%i3) diff (abs (x), x);
                                      x
(%o3)                               ------
                                    abs(x)
(%i4) cosh (abs (x));
(%o4)                               cosh(x)
@end example

@c disol.usg: "disolate" already in doc/info/Expressions.texi
@c elim.usg: "eliminate" already in doc/info/Polynomials.texi

@c Adapted from facexp.usg ----------------------
@c ALL OF THE TEXT IN FACEXP.USG IS VERY VAGUE.
@c I HAVE NO IDEA WHAT THESE FUNCTIONS DO.
@c ALL OF THESE ITEMS NEED TO BE HEAVILY REVISED
@c (ASSUMING THIS PACKAGE IS SOMETHING WE WANT TO INVEST TIME IN)

@c -----------------------------------------------------------------------------
@node Package facexp, Package functs, Package absimp, simplification
@section Package facexp
@c -----------------------------------------------------------------------------

@c THIS IS VERY VAGUE. JUST WHAT DOES THIS DO?
The @code{facexp} package contains several related functions that
provide the user with the ability to structure expressions by controlled
expansion.  This capability is especially useful when the expression
contains variables that have physical meaning, because it is often true
that the most economical form of such an expression can be obtained by
fully expanding the expression with respect to those variables, and then
factoring their coefficients.  While it is true that this procedure is
not difficult to carry out using standard Maxima functions, additional
fine-tuning may also be desirable, and these finishing touches can be
more difficult to apply.

The function @code{facsum} and its related forms provide a convenient means
for controlling the structure of expressions in this way.  Another function,
@code{collectterms}, can be used to add two or more expressions that have
already been simplified to this form, without resimplifying the whole expression
again.  This function may be useful when the expressions are very large.

@c CAN'T FIND ANY SUCH FILE "DIAGEVAL".
@c THERE ARE COMMENTED-OUT DEFNS OF FACTENEXPAND, FACEXPTEN, AND FACTORFACEXPTEN
@c IN FACEXP (AND NOWHERE ELSE).
@c COMMENTING OUT THIS TEXT FOR NOW.
@c Note: @code{factenexpand}, @code{facexpten}, and @code{factorfacexpten} 
@c are available only
@c after loading @code{diageval}. They are special functions used for tensor
@c manipulation.

@code{load(facexp)} loads this package.
@code{demo(facexp)} shows a demonstration of this package.

@c THIS IS VERY VAGUE. JUST WHAT DOES THIS DO?
@c SOME EXAMPLES WOULD HELP HERE

@c -----------------------------------------------------------------------------
@anchor{facsum}
@deffn {Function} facsum (@var{expr}, @var{arg_1}, @dots{}, @var{arg_n})

Returns a form of @var{expr} which depends on the arguments @var{arg_1},
@dots{}, @var{arg_n}.  The arguments can be any form suitable for 
@code{ratvars}, or they can be lists of such forms.  If the arguments are not
lists, then the form returned is fully expanded with respect to the arguments,
and the coefficients of the arguments are factored.  These coefficients are
free of the arguments, except perhaps in a non-rational sense.

If any of the arguments are lists, then all such lists are combined
into a single list, and instead of calling @code{factor} on the coefficients of
the arguments, @code{facsum} calls itself on these coefficients, using this
newly constructed single list as the new argument list for this recursive call.
This process can be repeated to arbitrary depth by nesting the desired elements
in lists.

It is possible that one may wish to @code{facsum} with respect to more
complicated subexpressions, such as @code{log(x + y)}.  Such arguments are also
permissible.

@c CUTTING THIS OUT, BECAUSE IT IS OBVIOUSLY NOT CORRECT
@c SEE EXAMPLE IN BUG REPORT ID: 2834336 - ratsimp vs facsum
@c With no variable specification, for example @code{facsum (@var{expr})}, the 
@c result returned is the same as that returned by @code{ratsimp (@var{expr})}.

Occasionally the user may wish to obtain any of the above forms
for expressions which are specified only by their leading operators.
For example, one may wish to @code{facsum} with respect to all @code{log}'s.
In this situation, one may include among the arguments either the specific
@code{log}'s which are to be treated in this way, or alternatively, either
the expression @code{operator (log)} or @code{'operator (log)}.  If one
wished to @code{facsum} the expression @var{expr} with respect to the operators
@var{op_1}, @dots{}, @var{op_n}, one would evaluate @code{facsum (@var{expr},
operator (@var{op_1}, ..., @var{op_n}))}.  The @code{operator} form may also
appear inside list arguments.

In addition, the setting of the switches @code{facsum_combine} and
@code{nextlayerfactor} may affect the result of @code{facsum}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{nextlayerfactor}
@defvr {Global variable} nextlayerfactor
Default value: @code{false}

When @code{nextlayerfactor} is @code{true}, recursive calls of @code{facsum}
are applied to the factors of the factored form of the
coefficients of the arguments.

When @code{false}, @code{facsum} is applied to
each coefficient as a whole whenever recusive calls to @code{facsum} occur.

Inclusion of the atom @code{nextlayerfactor} in the argument list of
@code{facsum} has the effect of @code{nextlayerfactor: true}, but for the next
level of the expression @i{only}.  Since @code{nextlayerfactor} is always bound
to either @code{true} or @code{false}, it must be presented single-quoted
whenever it appears in the argument list of @code{facsum}.
@end defvr

@c -----------------------------------------------------------------------------
@anchor{facsum_combine}
@defvr {Global variable} facsum_combine
Default value: @code{true}

@code{facsum_combine} controls the form of the final result returned by
@code{facsum} when its argument is a quotient of polynomials.  If
@code{facsum_combine} is @code{false} then the form will be returned as a fully
expanded sum as described above, but if @code{true}, then the expression
returned is a ratio of polynomials, with each polynomial in the form
described above.

The @code{true} setting of this switch is useful when one
wants to @code{facsum} both the numerator and denominator of a rational
expression, but does not want the denominator to be multiplied
through the terms of the numerator.
@end defvr

@c -----------------------------------------------------------------------------
@anchor{factorfacsum}
@deffn {Function} factorfacsum (@var{expr}, @var{arg_1}, @dots{} @var{arg_n})

Returns a form of @var{expr} which is obtained by calling @code{facsum} on the
factors of @var{expr} with @var{arg_1}, @dots{} @var{arg_n} as arguments.  If
any of the factors of @var{expr} is raised to a power, both the factor and the
exponent will be processed in this way.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{collectterms}
@deffn {Function} collectterms (@var{expr}, @var{arg_1}, @dots{}, @var{arg_n})

If several expressions have been simplified with the following functions: 
@code{facsum}, @code{factorfacsum}, @code{factenexpand}, @code{facexpten} or
@code{factorfacexpten}, and they are to be added together, it may be
desirable to combine them using the function @code{collecterms}.
@code{collecterms} can take as arguments all of the arguments that can be
given to these other associated functions with the exception of
@code{nextlayerfactor}, which has no effect on @code{collectterms}.  The 
advantage of @code{collectterms} is that it returns a form similar to 
@code{facsum}, but since it is adding forms that have already been processed by 
@code{facsum}, it does not need to repeat that effort.  This capability is
especially useful when the expressions to be summed are very large.
@end deffn

@c Adapted from functs.usg ----------------------

@c -----------------------------------------------------------------------------
@node Package functs, Package ineq, Package facexp, simplification
@section Package functs
@c -----------------------------------------------------------------------------

@c -----------------------------------------------------------------------------
@anchor{rempart}
@deffn {Function} rempart (@var{expr}, @var{n})

Removes part @var{n} from the expression @var{expr}.

If @var{n} is a list of the form @code{[@var{l}, @var{m}]}
then parts @var{l} thru @var{m} are removed.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{wronskian}
@deffn {Function} wronskian ([@var{f_1}, @dots{}, @var{f_n}], @var{x})

Returns the Wronskian matrix of the list of expressions [@var{f_1}, @dots{}, 
@var{f_n}] in the variable @var{x}.  The determinant of the Wronskian matrix is
the Wronskian determinant of the list of expressions.

To use @code{wronskian}, first @code{load(functs)}.  Example:

@c ===beg===
@c load (functs)$
@c wronskian([f(x), g(x)],x);
@c ===end===
@example
(%i1) load(functs)$
(%i2) wronskian([f(x), g(x)],x);
(%o2) matrix([f(x),g(x)],['diff(f(x),x,1),'diff(g(x),x,1)])
@end example
@end deffn

@c adjoint already described in doc/info/Matrices.texi

@c -----------------------------------------------------------------------------
@anchor{tracematrix}
@deffn {Function} tracematrix (@var{M})

Returns the trace (sum of the diagonal elements) of matrix @var{M}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{function_rational}
@deffn {Function} rational (@code{z})

Multiplies numerator and denominator of @var{z} by the complex conjugate of
denominator, thus rationalizing the denominator.  Returns canonical rational
expression (CRE) form if given one, else returns general form.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{nonzeroandfreeof}
@deffn {Function} nonzeroandfreeof (@var{x}, @var{expr})

Returns @code{true} if @var{expr} is nonzero and @code{freeof (@var{x}, 
@var{expr})} returns @code{true}.  Returns @code{false} otherwise.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{function_linear}
@deffn {Function} linear (@var{expr}, @var{x})

When @var{expr} is an expression linear in variable @var{x}, @code{linear}
returns @code{@var{a}*@var{x} + @var{b}} where @var{a} is nonzero,
and @var{a} and @var{b} are free of @var{x}.
Otherwise, @code{linear} returns @var{expr}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{gcdivide}
@deffn {Function} gcdivide (@var{p}, @var{q})

When the option variable @code{takegcd} is @code{true} which is the default,
@code{gcdivide} divides the polynomials @var{p} and @var{q} by their greatest
common divisor and returns the ratio of the results.  @code{gcdivde} calls the
function @mref{ezgcd} to divide the polynomials by the greatest common divisor.

When @code{takegcd} is @code{false}, @code{gcdivide} returns the ratio
@code{@var{p}/@var{q}}.

To use this function write first @code{load(functs)}.

See also @mrefcomma{ezgcd} @mrefcomma{gcd} @mrefcomma{gcdex} and
@mrefdot{poly_gcd}

Example:

@example
(%i1) load(functs)$

(%i2) p1:6*x^3+19*x^2+19*x+6; 
                        3       2
(%o2)                6 x  + 19 x  + 19 x + 6
(%i3) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                  5       4       3       2
(%o3)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
(%i4) gcdivide(p1, p2);
                             x + 1
(%o4)                        ------
                              3
                             x  + x
(%i5) takegcd:false;
(%o5)                         false
(%i6) gcdivide(p1, p2);
                       3       2
                    6 x  + 19 x  + 19 x + 6
(%o6)          ----------------------------------
                  5       4       3       2
               6 x  + 13 x  + 12 x  + 13 x  + 6 x
(%i7) ratsimp(%);
                             x + 1
(%o7)                        ------
                              3
                             x  + x
@end example
@end deffn

@c -----------------------------------------------------------------------------
@anchor{arithmetic}
@deffn {Function} arithmetic (@var{a}, @var{d}, @var{n})

Returns the @var{n}-th term of the arithmetic series @code{@var{a}, @var{a} +
@var{d}, @var{a} + 2*@var{d}, ..., @var{a} + (@var{n} - 1)*@var{d}}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{geometric}
@deffn {Function} geometric (@var{a}, @var{r}, @var{n})

Returns the @var{n}-th term of the geometric series
@code{@var{a}, @var{a}*@var{r}, @var{a}*@var{r}^2, ...,
@var{a}*@var{r}^(@var{n} - 1)}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{harmonic}
@deffn {Function} harmonic (@var{a}, @var{b}, @var{c}, @var{n})

Returns the @var{n}-th term of the harmonic series
@code{@var{a}/@var{b}, @var{a}/(@var{b} + @var{c}), @var{a}/(@var{b} +
2*@var{c}), ..., @var{a}/(@var{b} + (@var{n} - 1)*@var{c})}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{arithsum}
@deffn {Function} arithsum (@var{a}, @var{d}, @var{n})

Returns the sum of the arithmetic series from 1 to @var{n}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{geosum}
@deffn {Function} geosum (@var{a}, @var{r}, @var{n})

Returns the sum of the geometric series from 1 to @var{n}.  If @var{n} is
infinity (@code{inf}) then a sum is finite only if the absolute value
of @var{r} is less than 1.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{gaussprob}
@deffn {Function} gaussprob (@var{x})

Returns the Gaussian probability function
@code{%e^(-@var{x}^2/2) / sqrt(2*%pi)}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{gd}
@deffn {Function} gd (@var{x})

Returns the Gudermannian function @code{2*atan(%e^x)-%pi/2}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{agd}
@deffn {Function} agd (@var{x})

Returns the inverse Gudermannian function @code{log (tan (%pi/4 + x/2)))}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{vers}
@deffn {Function} vers (@var{x})

Returns the versed sine @code{1 - cos (x)}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{covers}
@deffn {Function} covers (@var{x})

Returns the coversed sine @code{1 - sin (@var{x})}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{exsec}
@deffn {Function} exsec (@var{x})

Returns the exsecant @code{sec (@var{x}) - 1}.

To use this function write first @code{load(functs)}.
@end deffn

@c -----------------------------------------------------------------------------
@anchor{hav}
@deffn {Function} hav (@var{x})

Returns the haversine @code{(1 - cos(x))/2}.

To use this function write first @code{load(functs)}.
@end deffn

@c REDUNDANT WITH BINOMIAL COEFFICIENT; CUT IT ??

@c -----------------------------------------------------------------------------
@anchor{combination}
@deffn {Function} combination (@var{n}, @var{r})

Returns the number of combinations of @var{n} objects taken @var{r} at a time.

To use this function write first @code{load(functs)}.
@end deffn

@c REDUNDANT WITH PERMUTATIONS FUNCTION IN NSET; CUT IT ??

@c -----------------------------------------------------------------------------
@anchor{permutation}
@deffn {Function} permutation (@var{n}, @var{r})

Returns the number of permutations of @var{r} objects selected from a set of
@var{n} objects.

To use this function write first @code{load(functs)}.
@end deffn

@c Adapted from ineq.usg ----------------------
@c THIS PACKAGE IS INTERESTING BUT THIS TEXT NEEDS WORK AND EXAMPLES

@c -----------------------------------------------------------------------------
@node Package ineq, Package rducon, Package functs, simplification
@section Package ineq
@c -----------------------------------------------------------------------------

The @code{ineq} package contains simplification rules for inequalities.

Example session:

@c ===beg===
@c load(ineq)$
@c a>=4;  /* a sample inequality */
@c (b>c)+%; /* add a second, strict inequality */
@c 7*(x<y); /* multiply by a positive number */
@c -2*(x>=3*z); /* multiply by a negative number */
@c (1+a^2)*(1/(1+a^2)<=1); /* Maxima knows that 1+a^2 > 0 */
@c assume(x>0)$ x*(2<3); /* assuming x>0 */
@c a>=b; /* another inequality */
@c 3+%; /* add something */
@c %-3; /* subtract it out */
@c a>=c-b; /* yet another inequality */
@c b+%; /* add b to both sides */
@c %-c; /* subtract c from both sides */
@c -%;  /* multiply by -1 */
@c (z-1)^2>-2*z; /* determining truth of assertion */
@c expand(%)+2*z; /* expand this and add 2*z to both sides */
@c %,pred;
@c ===end===
@example
(%i1) load(ineq)$
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
(%i2) a>=4;  /* a sample inequality */
(%o2)                               a >= 4
(%i3) (b>c)+%; /* add a second, strict inequality */
(%o3)                            b + a > c + 4
(%i4) 7*(x<y); /* multiply by a positive number */
(%o4)                              7 x < 7 y
(%i5) -2*(x>=3*z); /* multiply by a negative number */
(%o5)                           - 2 x <= - 6 z
(%i6) (1+a^2)*(1/(1+a^2)<=1); /* Maxima knows that 1+a^2 > 0 */
                                        2
(%o6)                             1 <= a  + 1
(%i7) assume(x>0)$ x*(2<3); /* assuming x>0 */
(%o7)                              2 x < 3 x
(%i8) a>=b; /* another inequality */
(%o8)                               a >= b
(%i9) 3+%; /* add something */
(%o9)                           a + 3 >= b + 3
(%i10) %-3; /* subtract it out */
(%o10)                              a >= b
(%i11) a>=c-b; /* yet another inequality */
(%o11)                            a >= c - b
(%i12) b+%; /* add b to both sides */
(%o12)                            b + a >= c
(%i13) %-c; /* subtract c from both sides */
(%o13)                         - c + b + a >= 0
(%i14) -%;  /* multiply by -1 */
(%o14)                          c - b - a <= 0
(%i15) (z-1)^2>-2*z; /* determining truth of assertion */
                                      2
(%o15)                         (z - 1)  > - 2 z
(%i16) expand(%)+2*z; /* expand this and add 2*z to both sides */
                                   2
(%o16)                            z  + 1 > 0
(%i17) %,pred;
(%o17)                               true
@end example

Be careful about using parentheses
around the inequalities: when the user types in @code{(A > B) + (C = 5)} the
result is @code{A + C > B + 5}, but @code{A > B + C = 5} is a syntax error,
and @code{(A > B + C) = 5} is something else entirely.

Do @code{disprule (all)} to see a complete listing
of the rule definitions.

The user will be queried if Maxima is
unable to decide the sign of a quantity multiplying an inequality.

The most common mis-feature is illustrated by:

@c ===beg===
@c eq: a > b;
@c 2*eq;
@c % - eq;
@c ===end===
@example
(%i1) eq: a > b;
(%o1)                              a > b
(%i2) 2*eq;
(%o2)                            2 (a > b)
(%i3) % - eq;
(%o3)                              a > b
@end example

Another problem is 0 times an inequality; the default to have this
turn into 0 has been left alone. However, if you type 
@code{X*@var{some_inequality}} and Maxima asks about the sign of @code{X} and
you respond @code{zero} (or @code{z}), the program returns
@code{X*@var{some_inequality}} and not use the information that @code{X} is 0.
You should do @code{ev (%, x: 0)} in such a case, as the database will only be
used for comparison purposes in decisions, and not for the purpose of evaluating
@code{X}.

The user may note a slower response when this package is loaded, as
the simplifier is forced to examine more rules than without the
package, so you might wish to remove the rules after making use of
them.  Do @code{kill (rules)} to eliminate all of the rules (including any
that you might have defined); or you may be more selective by
killing only some of them; or use @code{remrule} on a specific rule.

Note that if you load this package after defining your own rules you will
clobber your rules that have the same name.  The rules in this package are:
@code{*rule1}, @dots{}, @code{*rule8}, @code{+rule1}, @dots{}, @code{+rule18},
and you must enclose the rulename in quotes to refer to it, as in
@code{remrule ("+", "+rule1")} to specifically remove the first rule on
@code{"+"} or @code{disprule ("*rule2")} to display the definition of the
second multiplicative rule.

@c lrats.usg:"lratsubst" and "fullratsubst" already in doc/info/Polynomials.texi

@c Adapted from rducon.usg ----------------------
@c THIS IS AN INTERESTING FUNCTION BUT THIS TEXT NEEDS WORK AND EXAMPLES

@c -----------------------------------------------------------------------------
@node Package rducon, Package scifac, Package ineq, simplification
@section Package rducon
@c -----------------------------------------------------------------------------

@c -----------------------------------------------------------------------------
@anchor{reduce_consts}
@deffn {Function} reduce_consts (@var{expr})

Replaces constant subexpressions of @var{expr} with
constructed constant atoms, saving the definition of all these
constructed constants in the list of equations @code{const_eqns}, and
returning the modified @var{expr}.  Those parts of @var{expr} are constant which
return @code{true} when operated on by the function @code{constantp}.  Hence,
before invoking @code{reduce_consts}, one should do

@example
declare ([@var{objects to be given the constant property}], constant)$
@end example

to set up a database of the constant quantities occurring in your
expressions.

If you are planning to generate Fortran output after these symbolic
calculations, one of the first code sections should be the calculation
of all constants.  To generate this code segment, do

@example
map ('fortran, const_eqns)$
@end example

Variables besides @code{const_eqns} which affect @code{reduce_consts} are:

@code{const_prefix} (default value: @code{xx}) is the string of characters used to prefix all
symbols generated by @code{reduce_consts} to represent constant subexpressions.

@code{const_counter} (default value: 1) is the integer index used to generate unique
symbols to represent each constant subexpression found by @code{reduce_consts}.

@code{load(rducon)} loads this function.
@code{demo(rducon)} shows a demonstration of this function.
@end deffn

@c rncomb.usg: "rncombine" already in doc/info/Miscellaneous.texi

@c Adapted from scifac.usg ----------------------

@c -----------------------------------------------------------------------------
@node Package scifac, Package sqdnst, Package rducon, simplification
@section Package scifac
@c -----------------------------------------------------------------------------

@c -----------------------------------------------------------------------------
@anchor{gcfac}
@deffn {Function} gcfac (@var{expr})

@code{gcfac} is a factoring function that attempts to apply the same heuristics
which scientists apply in trying to make expressions simpler.  @code{gcfac} is
limited to monomial-type factoring.  For a sum, @code{gcfac} does the following:

@enumerate
@item
Factors over the integers.
@item
Factors out the largest powers of terms occurring as
coefficients, regardless of the complexity of the terms.
@item
Uses (1) and (2) in factoring adjacent pairs of terms.
@item
Repeatedly and recursively applies these techniques until
the expression no longer changes.
@end enumerate

Item (3) does not necessarily do an optimal job of pairwise
factoring because of the combinatorially-difficult nature of finding
which of all possible rearrangements of the pairs yields the most
compact pair-factored result.

@code{load(scifac)} loads this function.
@code{demo(scifac)} shows a demonstration of this function.
@end deffn

@c Adapted from sqdnst.usg ----------------------
@c THIS FUNCTION IS INTERESTING BUT THIS TEXT NEEDS WORK. HOW DEEPLY CAN SQRT
@c BE NESTED ??

@c -----------------------------------------------------------------------------
@node Package sqdnst,  , Package scifac, simplification
@section Package sqdnst
@c -----------------------------------------------------------------------------

@c -----------------------------------------------------------------------------
@anchor{sqrtdenest}
@deffn {Function} sqrtdenest (@var{expr})

Denests @code{sqrt} of simple, numerical, binomial surds, where possible.  E.g.

@c ===beg===
@c load (sqdnst)$
@c sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12);
@c sqrtdenest(%);
@c ===end===
@example
(%i1) load (sqdnst)$
(%i2) sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12);
                                    sqrt(3)
                               sqrt(------- + 1)
                                       2
(%o2)                        ---------------------
                             sqrt(11 sqrt(2) - 12)
(%i3) sqrtdenest(%);
                                  sqrt(3)   1
                                  ------- + -
                                     2      2
(%o3)                            -------------
                                    1/4    3/4
                                 3 2    - 2
@end example

Sometimes it helps to apply @code{sqrtdenest} more than once, on such as
@code{(19601-13860 sqrt(2))^(7/4)}.

@code{load(sqdnst)} loads this function.
@end deffn

@c stopex.usg: "expandwrt", "expandwrt_denom", and "expandwrt_factored" already
@c in doc/info/Simplification.texi

@c --- End of file simplifications.de.texi -------------------------------------

