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| 69.1 Introduction to simplification | ||
| 69.2 Package absimp | ||
| 69.3 Package facexp | ||
| 69.4 Package functs | ||
| 69.5 Package ineq | ||
| 69.6 Package rducon | ||
| 69.7 Package scifac | ||
| 69.8 Package sqdnst |
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The directory maxima/share/simplification contains several scripts
which implement simplification rules and functions,
and also some functions not related to simplification.
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The absimp package contains pattern-matching rules that
extend the built-in simplification rules for the abs and signum
functions.
absimp respects relations
established with the built-in assume function and by declarations such
as modedeclare (m, even, n, odd) for even or odd integers.
absimp defines unitramp and unitstep functions
in terms of abs and signum.
load(absimp) loads this package.
demo(absimp) shows a demonstration of this package.
Examples:
(%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)
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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.
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.
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.
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.
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.
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.
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Removes part n from the expression expr.
If n is a list of the form [l, m]
then parts l thru m are removed.
To use this function write first load(functs).
Returns the Wronskian matrix of the list of expressions [f_1, …, f_n] in the variable x. The determinant of the Wronskian matrix is the Wronskian determinant of the list of expressions.
To use wronskian, first load(functs). 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)])
Returns the trace (sum of the diagonal elements) of matrix M.
To use this function write first load(functs).
z)
Multiplies numerator and denominator of 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 load(functs).
x,y)
Returns logical (bit-wise) "and" of arguments x and y.
To use this function write first load(functs).
x,y)
Returns logical (bit-wise) "or" of arguments x and y.
To use this function write first load(functs).
x,y)
Returns logical (bit-wise) exclusive-or of arguments x and y.
To use this function write first load(functs).
Returns true if expr is nonzero and freeof (x,
expr) returns true. Returns false otherwise.
To use this function write first load(functs).
When expr is an expression linear in variable x, linear
returns a*x + b where a is nonzero,
and a and b are free of x.
Otherwise, linear returns expr.
To use this function write first load(functs).
When the option variable takegcd is true which is the default,
gcdivide divides the polynomials p and q by their greatest
common divisor and returns the ratio of the results. gcdivde calls the
function ezgcd
to divide the polynomials by the greatest common divisor.
When takegcd is false, gcdivide returns the ratio
p/q.
To use this function write first load(functs).
See also ezgcd,
gcd,
gcdex,
and
poly_gcd.
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
Returns the n-th term of the arithmetic series a, a +
d, a + 2*d, ..., a + (n - 1)*d.
To use this function write first load(functs).
Returns the n-th term of the geometric series
a, a*r, a*r^2, ...,
a*r^(n - 1).
To use this function write first load(functs).
Returns the n-th term of the harmonic series
a/b, a/(b + c), a/(b +
2*c), ..., a/(b + (n - 1)*c).
To use this function write first load(functs).
Returns the sum of the arithmetic series from 1 to n.
To use this function write first load(functs).
Returns the sum of the geometric series from 1 to n. If n is
infinity (inf) then a sum is finite only if the absolute value
of r is less than 1.
To use this function write first load(functs).
Returns the Gaussian probability function
%e^(-x^2/2) / sqrt(2*%pi).
To use this function write first load(functs).
Returns the Gudermannian function 2*atan(%e^x)-%pi/2.
To use this function write first load(functs).
Returns the inverse Gudermannian function log (tan (%pi/4 + x/2))).
To use this function write first load(functs).
Returns the versed sine 1 - cos (x).
To use this function write first load(functs).
Returns the coversed sine 1 - sin (x).
To use this function write first load(functs).
Returns the exsecant sec (x) - 1.
To use this function write first load(functs).
Returns the haversine (1 - cos(x))/2.
To use this function write first load(functs).
Returns the number of combinations of n objects taken r at a time.
To use this function write first load(functs).
Returns the number of permutations of r objects selected from a set of n objects.
To use this function write first load(functs).
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The ineq package contains simplification rules for inequalities.
Example session:
(%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
Be careful about using parentheses
around the inequalities: when the user types in (A > B) + (C = 5) the
result is A + C > B + 5, but A > B + C = 5 is a syntax error,
and (A > B + C) = 5 is something else entirely.
Do 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:
(%i1) eq: a > b; (%o1) a > b (%i2) 2*eq; (%o2) 2 (a > b) (%i3) % - eq; (%o3) a > b
Another problem is 0 times an inequality; the default to have this
turn into 0 has been left alone. However, if you type
X*some_inequality and Maxima asks about the sign of X and
you respond zero (or z), the program returns
X*some_inequality and not use the information that X is 0.
You should do 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
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 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 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:
*rule1, …, *rule8, +rule1, …, +rule18,
and you must enclose the rulename in quotes to refer to it, as in
remrule ("+", "+rule1") to specifically remove the first rule on
"+" or disprule ("*rule2") to display the definition of the
second multiplicative rule.
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Replaces constant subexpressions of expr with
constructed constant atoms, saving the definition of all these
constructed constants in the list of equations const_eqns, and
returning the modified expr. Those parts of expr are constant which
return true when operated on by the function constantp. Hence,
before invoking reduce_consts, one should do
declare ([objects to be given the constant property], constant)$
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
map ('fortran, const_eqns)$
Variables besides const_eqns which affect reduce_consts are:
const_prefix (default value: xx) is the string of characters used to prefix all
symbols generated by reduce_consts to represent constant subexpressions.
const_counter (default value: 1) is the integer index used to generate unique
symbols to represent each constant subexpression found by reduce_consts.
load(rducon) loads this function.
demo(rducon) shows a demonstration of this function.
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gcfac is a factoring function that attempts to apply the same heuristics
which scientists apply in trying to make expressions simpler. gcfac is
limited to monomial-type factoring. For a sum, gcfac does the following:
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.
load(scifac) loads this function.
demo(scifac) shows a demonstration of this function.
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Denests sqrt of simple, numerical, binomial surds, where possible. E.g.
(%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
Sometimes it helps to apply sqrtdenest more than once, on such as
(19601-13860 sqrt(2))^(7/4).
load(sqdnst) loads this function.
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