From: Robert D. <rob...@us...> - 2005-12-06 02:45:08
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Update of /cvsroot/maxima/maxima/share/simplification In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv10502 Added Files: simplification.texi Removed Files: absimp.usg elim.usg functs.usg lrats.usg rncomb.usg sqdnst.usg disol.usg facexp.usg ineq.usg rducon.usg scifac.usg stopex.usg Log Message: Concatenate the *.usg files in share/simplification/ and convert to texinfo markup. Created simplification.texi and removed the *.usg files. --- NEW FILE: simplification.texi --- @c Adapted from absimp.usg ---------------------- @defvr {Add-on package} absimp 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); (%o1) /usr/share/maxima/5.9.2/share/simplification/absimp.mac (%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 @end defvr @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) @defvr {Add-on package} facexp @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} 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. @end defvr @c THIS IS VERY VAGUE. JUST WHAT DOES THIS DO? @c SOME EXAMPLES WOULD HELP HERE @deffn {Function} facsum (@var{expr}, @var{arg_1}, ..., @var{arg_n}) Returns a form of @var{expr} which depends on the arguments @var{arg_1}, ..., @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. With no variable specification, for example @code{facsum (@var{expr})}, the 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}, ..., @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 @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 @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 @deffn {Function} factorfacsum (@var{expr}, @var{arg_1}, ... @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}, ... @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 @deffn {Function} collectterms (@var{arg_1}, ..., @var{arg_n}) If several expressions have been simplified with @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 conjugate already described in doc/info/Matrices.texi @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. @end deffn @deffn {Function} wronskian ([@var{f_1}, ..., @var{f_n}], @var{x}) Returns the Wronskian matrix of the functions @var{f_1}, ..., @var{f_n} in the variable @var{x}. @var{f_1}, ..., @var{f_n} may be the names of user-defined functions, or expressions in the variable @var{x}. The determinant of the Wronskian matrix is the Wronskian determinant of the set of functions. The functions are linearly dependent if this determinant is zero. @end deffn @c adjoint already described in doc/info/Matrices.texi @deffn {Function} tracematrix (@var{M}) Returns the trace (sum of the diagonal elements) of matrix @var{M}. @end deffn @deffn {Function} rational (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. @end deffn @deffn {Function} logand (x,y) Returns logical (bit-wise) "and" of arguments x and y. @end deffn @deffn {Function} logor (x,y) Returns logical (bit-wise) "or" of arguments x and y. @end deffn @deffn {Function} logxor (x,y) Returns logical (bit-wise) exclusive-or of arguments x and y. @end deffn @c uprobe calls ?uprobe and assumes file is a list => obsolete, not common lisp @c kronecker superseded by kron_delta in src/nset.lisp @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. @end deffn @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}. @end deffn @deffn {Function} quadratic (@var{expr}, @var{x}) When @var{expr} is an expression quadratic in variable @var{x}, @code{quadratic} returns @code{@var{a}*@var{x}^2 + @var{b}*x + @var{c}} where @var{a} is nonzero, and @var{a}, @var{b}, and @var{c} are free of @var{x}. Otherwise, @code{quadratic} returns @var{expr}. @end deffn @deffn {Function} gcdivide (@var{p}, @var{q}) When @code{takegcd} is @code{true}, @code{gcdivide} divides the polynomials @var{p} and @var{q} by their greatest common divisor and returns the ratio of the results. When @code{takegcd} is @code{false}, @code{gcdivide} returns the ratio @code{@var{p}/@var{q}}. @end deffn @c lcm already described in doc/info/Number.texi @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}}. @end deffn @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)}. @end deffn @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})}. @end deffn @deffn {Function} arithsum (@var{a}, @var{d}, @var{n}) Returns the sum of the arithmetic series from 1 to @var{n}. @end deffn @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 value of @var{r} is not equal to 1. @end deffn @deffn {Function} gaussprob (@var{x}) Returns the Gaussian probability function @code{%e^(-@var{x}^2/2) / sqrt(2*%pi)}. @end deffn @deffn {Function} gd (@var{x}) Returns the Gudermannian function @code{2 * atan(%e^@var{x} - %pi/2)}. @end deffn @deffn {Function} agd (@var{x}) Returns the inverse Gudermannian function @code{log (tan (%pi/4 + x/2)))}. @end deffn @deffn {Function} vers (@var{x}) Returns the versed sine @code{1 - cos (x)}. @end deffn @deffn {Function} covers (@var{x}) Returns the coversed sine @code{1 - sin (@var{x})}. @end deffn @deffn {Function} exsec (@var{x}) Returns the exsecant @code{sec (@var{x}) - 1}. @end deffn @deffn {Function} hav (@var{x}) Returns the haversine @code{(1 - cos(x))/2}. @end deffn @c REDUNDANT WITH BINOMIAL COEFFICIENT; CUT IT ?? @deffn {Function} combination (@var{n}, @var{r}) Returns the number of combinations of @var{n} objects taken @var{r} at a time. @end deffn @c REDUNDANT WITH PERMUTATIONS FUNCTION IN NSET; CUT IT ?? @deffn {Function} permutation (@var{n}, @var{r}) Returns the number of permutations of @var{r} objects selected from a set of @var{n} objects. @end deffn @c Adapted from ineq.usg ---------------------- @c THIS PACKAGE IS INTERESTING BUT THIS TEXT NEEDS WORK AND EXAMPLES @defvr {Add-on package} ineq The @code{ineq} package contains simplification rules for inequalities. 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: @example eq: a > b; 2*eq; % - eq; @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}, ..., @code{*rule8}, @code{+rule1}, ..., @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. @end defvr @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 @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 ---------------------- @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 BE NESTED ?? @deffn {Function} sqrtdenest (@var{expr}) Denests @code{sqrt} of simple, numerical, binomial surds, where possible. E.g. @example (c1) sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12); sqrt(3) sqrt(------- + 1) 2 (d1) --------------------- sqrt(11 sqrt(2) - 12) (c2) sqrtdenest(%); sqrt(3) 1 ------- + - 2 2 (d2) ------------- 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 in doc/info/Simplification.texi --- absimp.usg DELETED --- --- elim.usg DELETED --- --- functs.usg DELETED --- --- lrats.usg DELETED --- --- rncomb.usg DELETED --- --- sqdnst.usg DELETED --- --- disol.usg DELETED --- --- facexp.usg DELETED --- --- ineq.usg DELETED --- --- rducon.usg DELETED --- --- scifac.usg DELETED --- --- stopex.usg DELETED --- |