69.3 Package facexp

The 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 facsum and its related forms provide a convenient means for controlling the structure of expressions in this way. Another function, 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.

load("facexp") loads this package. demo(facexp) shows a demonstration of this package.

Function: facsum (expr, arg_1, …, arg_n) ¶

Returns a form of expr which depends on the arguments arg_1, …, arg_n. The arguments can be any form suitable for 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 factor on the coefficients of the arguments, 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 facsum with respect to more complicated subexpressions, such as log(x + y). Such arguments are also permissible.

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 facsum with respect to all log’s. In this situation, one may include among the arguments either the specific log’s which are to be treated in this way, or alternatively, either the expression operator (log) or 'operator (log). If one wished to facsum the expression expr with respect to the operators op_1, …, op_n, one would evaluate facsum (expr, operator (op_1, ..., op_n)). The operator form may also appear inside list arguments.

In addition, the setting of the switches facsum_combine and nextlayerfactor may affect the result of facsum.

Global variable: nextlayerfactor ¶

Default value: false

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

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

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

Global variable: facsum_combine ¶

Default value: true

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

The true setting of this switch is useful when one wants to 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.

Function: factorfacsum (expr, arg_1, … arg_n) ¶

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

Function: collectterms (expr, arg_1, …, arg_n) ¶

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