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| 9.1 Functions and Variables for Simplification |
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If declare(f,additive) has been executed, then:
(1) If f is univariate, whenever the simplifier encounters f
applied to a sum, f will be distributed over that sum. I.e.
f(y+x) will simplify to f(y)+f(x).
(2) If f is a function of 2 or more arguments, additivity is defined as
additivity in the first argument to f, as in the case of sum or
integrate, i.e. f(h(x)+g(x),x) will simplify to
f(h(x),x)+f(g(x),x). This simplification does not occur when f is
applied to expressions of the form sum(x[i],i,lower-limit,upper-limit).
Example:
(%i1) F3 (a + b + c); (%o1) F3(c + b + a) (%i2) declare (F3, additive); (%o2) done (%i3) F3 (a + b + c); (%o3) F3(c) + F3(b) + F3(a)
Categories: Operators · Declarations and inferences
If declare(h,antisymmetric) is done, this tells the simplifier that
h is antisymmetric. E.g. h(x,z,y) will simplify to
- h(x, y, z). That is, it will give (-1)^n times the result given by
symmetric or commutative, where n is the number of interchanges
of two arguments necessary to convert it to that form.
Examples:
(%i1) S (b, a); (%o1) S(b, a) (%i2) declare (S, symmetric); (%o2) done (%i3) S (b, a); (%o3) S(a, b) (%i4) S (a, c, e, d, b); (%o4) S(a, b, c, d, e) (%i5) T (b, a); (%o5) T(b, a) (%i6) declare (T, antisymmetric); (%o6) done (%i7) T (b, a); (%o7) - T(a, b) (%i8) T (a, c, e, d, b); (%o8) T(a, b, c, d, e)
Categories: Operators · Declarations and inferences
Simplifies the sum expr by combining terms with the same denominator into a single term.
Example:
(%i1) 1*f/2*b + 2*c/3*a + 3*f/4*b +c/5*b*a;
5 b f a b c 2 a c
(%o1) ----- + ----- + -----
4 5 3
(%i2) combine (%);
75 b f + 4 (3 a b c + 10 a c)
(%o2) -----------------------------
60
Categories: Expressions
If declare(h, commutative) is done, this tells the simplifier that
h is a commutative function. E.g. h(x, z, y) will simplify to
h(x, y, z). This is the same as symmetric.
Exemplo:
(%i1) S (b, a); (%o1) S(b, a) (%i2) S (a, b) + S (b, a); (%o2) S(b, a) + S(a, b) (%i3) declare (S, commutative); (%o3) done (%i4) S (b, a); (%o4) S(a, b) (%i5) S (a, b) + S (b, a); (%o5) 2 S(a, b) (%i6) S (a, c, e, d, b); (%o6) S(a, b, c, d, e)
Categories: Operators · Declarations and inferences
The function demoivre (expr) converts one expression
without setting the global variable demoivre.
When the variable demoivre is true, complex exponentials are
converted into equivalent expressions in terms of circular functions:
exp (a + b*%i) simplifies to %e^a * (cos(b) + %i*sin(b))
if b is free of %i. a and b are not expanded.
The default value of demoivre is false.
exponentialize converts circular and hyperbolic functions to exponential
form. demoivre and exponentialize cannot both be true at the same
time.
Categories: Complex variables · Trigonometric functions · Hyperbolic functions
Distributes sums over products. It differs from expand in that it works
at only the top level of an expression, i.e., it doesn't recurse and it is
faster than expand. It differs from multthru in that it expands
all sums at that level.
Examples:
(%i1) distrib ((a+b) * (c+d));
(%o1) b d + a d + b c + a c
(%i2) multthru ((a+b) * (c+d));
(%o2) (b + a) d + (b + a) c
(%i3) distrib (1/((a+b) * (c+d)));
1
(%o3) ---------------
(b + a) (d + c)
(%i4) expand (1/((a+b) * (c+d)), 1, 0);
1
(%o4) ---------------------
b d + a d + b c + a c
Categories: Expressions
Default value: true
distribute_over controls the mapping of functions over bags like lists,
matrices, and equations. At this time not all Maxima functions have this
property. It is possible to look up this property with the command
properties.
The mapping of functions is switched off, when setting distribute_over
to the value false.
Examples:
The sin function maps over a list:
(%i1) sin([x,1,1.0]); (%o1) [sin(x), sin(1), .8414709848078965]
mod is a function with two arguments which maps over lists. Mapping over
nested lists is possible too:
(%i2) mod([x,11,2*a],10); (%o2) [mod(x, 10), 1, 2 mod(a, 5)] (%i3) mod([[x,y,z],11,2*a],10); (%o3) [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]
Mapping of the floor function over a matrix and an equation:
(%i4) floor(matrix([a,b],[c,d]));
[ floor(a) floor(b) ]
(%o4) [ ]
[ floor(c) floor(d) ]
(%i5) floor(a=b);
(%o5) floor(a) = floor(b)
Functions with more than one argument map over any of the arguments or all arguments:
(%i6) expintegral_e([1,2],[x,y]);
(%o6) [[expintegral_e(1, x), expintegral_e(1, y)],
[expintegral_e(2, x), expintegral_e(2, y)]]
Check if a function has the property distribute_over:
(%i7) properties(abs); (%o7) [integral, distributes over bags, noun, rule, gradef]
The mapping of functions is switched off, when setting distribute_over
to the value false.
(%i1) distribute_over; (%o1) true (%i2) sin([x,1,1.0]); (%o2) [sin(x), sin(1), 0.8414709848078965] (%i3) distribute_over : not distribute_over; (%o3) false (%i4) sin([x,1,1.0]); (%o4) sin([x, 1, 1.0]) (%i5)
Categories: Simplification flags and variables
Default value: real
When domain is set to complex, sqrt (x^2) will remain
sqrt (x^2) instead of returning abs(x).
Categories: Simplification flags and variables
declare(f, evenfun) or declare(f, oddfun) tells Maxima to recognize
the function f as an even or odd function.
Examples:
(%i1) o (- x) + o (x); (%o1) o(x) + o(- x) (%i2) declare (o, oddfun); (%o2) done (%i3) o (- x) + o (x); (%o3) 0 (%i4) e (- x) - e (x); (%o4) e(- x) - e(x) (%i5) declare (e, evenfun); (%o5) done (%i6) e (- x) - e (x); (%o6) 0
Expand expression expr. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplication (commutative and non-commutative) are distributed over addition at all levels of expr.
For polynomials one should usually use ratexpand which uses a
more efficient algorithm.
maxnegex and maxposex control the maximum negative and
positive exponents, respectively, which will expand.
expand (expr, p, n) expands expr,
using p for maxposex and n for maxnegex.
This is useful in order to expand part but not all of an expression.
expon - the exponent of the largest negative power which is
automatically expanded (independent of calls to expand). For example
if expon is 4 then (x+1)^(-5) will not be automatically expanded.
expop - the highest positive exponent which is automatically expanded.
Thus (x+1)^3, when typed, will be automatically expanded only if
expop is greater than or equal to 3. If it is desired to have
(x+1)^n expanded where n is greater than expop then
executing expand ((x+1)^n) will work only if maxposex is not
less than n.
expand(expr, 0, 0) causes a resimplification of expr. expr
is not reevaluated. In distinction from ev(expr, noeval) a special
representation (e. g. a CRE form) is removed. See also ev.
The expand flag used with ev causes expansion.
The file `share/simplification/facexp.mac'
contains several related functions (in particular facsum,
factorfacsum and collectterms, which are autoloaded) and variables
(nextlayerfactor and facsum_combine) that provide the user with
the ability to structure expressions by controlled expansion.
Brief function descriptions are available in `simplification/facexp.usg'.
A demo is available by doing demo("facexp").
Examples:
(%i1) expr:(x+1)^2*(y+1)^3;
2 3
(%o1) (x + 1) (y + 1)
(%i2) expand(expr);
2 3 3 3 2 2 2 2 2
(%o2) x y + 2 x y + y + 3 x y + 6 x y + 3 y + 3 x y
2
+ 6 x y + 3 y + x + 2 x + 1
(%i3) expand(expr,2);
2 3 3 3
(%o3) x (y + 1) + 2 x (y + 1) + (y + 1)
(%i4) expr:(x+1)^-2*(y+1)^3;
3
(y + 1)
(%o4) --------
2
(x + 1)
(%i5) expand(expr);
3 2
y 3 y 3 y 1
(%o5) ------------ + ------------ + ------------ + ------------
2 2 2 2
x + 2 x + 1 x + 2 x + 1 x + 2 x + 1 x + 2 x + 1
(%i6) expand(expr,2,2);
3
(y + 1)
(%o6) ------------
2
x + 2 x + 1
Resimplify an expression without expansion:
(%i7) expr:(1+x)^2*sin(x);
2
(%o7) (x + 1) sin(x)
(%i8) exponentialize:true;
(%o8) true
(%i9) expand(expr,0,0);
2 %i x - %i x
%i (x + 1) (%e - %e )
(%o9) - -------------------------------
2
Categories: Expressions
Expands expression expr with respect to the
variables x_1, …, x_n.
All products involving the variables appear explicitly. The form returned
will be free of products of sums of expressions that are not free of
the variables. x_1, …, x_n
may be variables, operators, or expressions.
By default, denominators are not expanded, but this can be controlled by
means of the switch expandwrt_denom.
This function is autoloaded from `simplification/stopex.mac'.
Categories: Expressions
Default value: false
expandwrt_denom controls the treatment of rational
expressions by expandwrt. If true, then both the numerator and
denominator of the expression will be expanded according to the
arguments of expandwrt, but if expandwrt_denom is false,
then only the numerator will be expanded in that way.
Categories: Expressions
is similar to expandwrt, but treats expressions that are products
somewhat differently. expandwrt_factored expands only on those factors
of expr that contain the variables x_1, …, x_n.
This function is autoloaded from `simplification/stopex.mac'.
Categories: Expressions
Default value: 0
expon is the exponent of the largest negative power which
is automatically expanded (independent of calls to expand). For
example, if expon is 4 then (x+1)^(-5) will not be automatically
expanded.
Categories: Expressions
The function exponentialize (expr) converts
circular and hyperbolic functions in expr to exponentials,
without setting the global variable exponentialize.
When the variable exponentialize is true,
all circular and hyperbolic functions are converted to exponential form.
The default value is false.
demoivre converts complex exponentials into circular functions.
exponentialize and demoivre cannot
both be true at the same time.
Categories: Complex variables · Trigonometric functions · Hyperbolic functions
Default value: 0
expop is the highest positive exponent which is automatically expanded.
Thus (x + 1)^3, when typed, will be automatically expanded only if
expop is greater than or equal to 3. If it is desired to have
(x + 1)^n expanded where n is greater than expop then
executing expand ((x + 1)^n) will work only if maxposex is not
less than n.
Categories: Expressions
declare (g, lassociative) tells the Maxima simplifier that g is
left-associative. E.g., g (g (a, b), g (c, d)) will simplify to
g (g (g (a, b), c), d).
Categories: Declarations and inferences · Operators · Simplification
One of Maxima's operator properties. For univariate f so
declared, "expansion" f(x + y) yields f(x) + f(y),
f(a*x) yields a*f(x) takes
place where a is a "constant". For functions of two or more arguments,
"linearity" is defined to be as in the case of sum or integrate,
i.e., f (a*x + b, x) yields a*f(x,x) + b*f(1,x)
for a and b free of x.
Example:
(%i1) declare (f, linear); (%o1) done (%i2) f(x+y); (%o2) f(y) + f(x) (%i3) declare (a, constant); (%o3) done (%i4) f(a*x); (%o4) a f(x)
linear is equivalent to additive and outative.
See also opproperties.
Example:
(%i1) 'sum (F(k) + G(k), k, 1, inf);
inf
====
\
(%o1) > (G(k) + F(k))
/
====
k = 1
(%i2) declare (nounify (sum), linear);
(%o2) done
(%i3) 'sum (F(k) + G(k), k, 1, inf);
inf inf
==== ====
\ \
(%o3) > G(k) + > F(k)
/ /
==== ====
k = 1 k = 1
Categories: Declarations and inferences · Operators · Simplification
Default value: 1000
maxnegex is the largest negative exponent which will
be expanded by the expand command, see also maxposex.
Categories: Expressions
Default value: 1000
maxposex is the largest exponent which will be
expanded with the expand command, see also maxnegex.
Categories: Expressions
declare(f, multiplicative) tells the Maxima simplifier that f
is multiplicative.
f is univariate, whenever the simplifier encounters f applied
to a product, f distributes over that product. E.g., f(x*y)
simplifies to f(x)*f(y).
This simplification is not applied to expressions of the form f('product(...)).
f is a function of 2 or more arguments, multiplicativity is
defined as multiplicativity in the first argument to f, e.g.,
f (g(x) * h(x), x) simplifies to f (g(x) ,x) * f (h(x), x).
declare(nounify(product), multiplicative) tells Maxima to simplify symbolic products.
Example:
(%i1) F2 (a * b * c); (%o1) F2(a b c) (%i2) declare (F2, multiplicative); (%o2) done (%i3) F2 (a * b * c); (%o3) F2(a) F2(b) F2(c)
declare(nounify(product), multiplicative) tells Maxima to simplify symbolic products.
(%i1) product (a[i] * b[i], i, 1, n);
n
/===\
! !
(%o1) ! ! a b
! ! i i
i = 1
(%i2) declare (nounify (product), multiplicative);
(%o2) done
(%i3) product (a[i] * b[i], i, 1, n);
n n
/===\ /===\
! ! ! !
(%o3) ( ! ! a ) ! ! b
! ! i ! ! i
i = 1 i = 1
Categories: Declarations and inferences · Expressions · Simplification
Multiplies a factor (which should be a sum) of expr by the other factors
of expr. That is, expr is f_1 f_2 ... f_n
where at least one factor, say f_i, is a sum of terms. Each term in that
sum is multiplied by the other factors in the product. (Namely all the factors
except f_i). multthru does not expand exponentiated sums.
This function is the fastest way to distribute products (commutative or
noncommutative) over sums. Since quotients are represented as products
multthru can be used to divide sums by products as well.
multthru (expr_1, expr_2) multiplies each term in
expr_2 (which should be a sum or an equation) by expr_1. If
expr_1 is not itself a sum then this form is equivalent to
multthru (expr_1*expr_2).
(%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
1 x f(x)
(%o1) - ----- + -------- - --------
x - y 2 3
(x - y) (x - y)
(%i2) multthru ((x-y)^3, %);
2
(%o2) - (x - y) + x (x - y) - f(x)
(%i3) ratexpand (%);
2
(%o3) - y + x y - f(x)
(%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
10 2 2 2
(b + a) s + 2 a b s + a b
(%o4) ------------------------------
2
a b s
(%i5) multthru (%); /* note that this does not expand (b+a)^10 */
10
2 a b (b + a)
(%o5) - + --- + ---------
s 2 a b
s
(%i6) multthru (a.(b+c.(d+e)+f));
(%o6) a . f + a . c . (e + d) + a . b
(%i7) expand (a.(b+c.(d+e)+f));
(%o7) a . f + a . c . e + a . c . d + a . b
Categories: Expressions
declare(f, nary) tells Maxima to recognize the function f as an
n-ary function.
The nary declaration is not the same as calling the
nary function. The sole effect of
declare(f, nary) is to instruct the Maxima simplifier to flatten nested
expressions, for example, to simplify foo(x, foo(y, z)) to
foo(x, y, z). See also declare.
Example:
(%i1) H (H (a, b), H (c, H (d, e))); (%o1) H(H(a, b), H(c, H(d, e))) (%i2) declare (H, nary); (%o2) done (%i3) H (H (a, b), H (c, H (d, e))); (%o3) H(a, b, c, d, e)
Default value: true
When negdistrib is true, -1 distributes over an expression.
E.g., -(x + y) becomes - y - x. Setting it to false
will allow - (x + y) to be displayed like that. This is sometimes useful
but be very careful: like the simp flag, this is one flag you do not
want to set to false as a matter of course or necessarily for other
than local use in your Maxima.
Example:
(%i1) negdistrib; (%o1) true (%i2) -(x+y); (%o2) - y - x (%i3) negdistrib : not negdistrib ; (%o3) false (%i4) -(x+y); (%o4) - (y + x)
Categories: Simplification flags and variables
opproperties is the list of the special operator properties recognized
by the Maxima simplifier:
Example:
(%i1) opproperties;
(%o1) [linear, additive, multiplicative, outative, evenfun, oddfun,.
commutative, symmetric, antisymmetric, nary, lassociative, rassociative]
Categories: Global variables · Operators
declare(f, outative) tells the Maxima simplifier that constant factors
in the argument of f can be pulled out.
f is univariate, whenever the simplifier encounters f applied
to a product, that product will be partitioned into factors that are constant
and factors that are not and the constant factors will be pulled out. E.g.,
f(a*x) will simplify to a*f(x) where a is a constant.
Non-atomic constant factors will not be pulled out.
f is a function of 2 or more arguments, outativity is defined as in
the case of sum or integrate, i.e., f (a*g(x), x) will
simplify to a * f(g(x), x) for a free of x.
sum, integrate, and limit are all outative.
Example:
(%i1) F1 (100 * x); (%o1) F1(100 x) (%i2) declare (F1, outative); (%o2) done (%i3) F1 (100 * x); (%o3) 100 F1(x) (%i4) declare (zz, constant); (%o4) done (%i5) F1 (zz * y); (%o5) zz F1(y)
Categories: Declarations and inferences · Operators
Simplifies expr, which can contain logs, exponentials, and radicals, by
converting it into a form which is canonical over a large class of expressions
and a given ordering of variables; that is, all functionally equivalent forms
are mapped into a unique form. For a somewhat larger class of expressions,
radcan produces a regular form. Two equivalent expressions in this class
do not necessarily have the same appearance, but their difference can be
simplified by radcan to zero.
For some expressions radcan is quite time consuming. This is the cost
of exploring certain relationships among the components of the expression for
simplifications based on factoring and partial-fraction expansions of exponents.
Examples:
(%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
a/2
(%o1) log(x + 1)
(%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
(%o2) 2
(%i3) radcan((%e^x-1)/(1+%e^(x/2)));
x/2
(%o3) %e - 1
Categories: Simplification functions
Default value: true
radexpand controls some simplifications of radicals.
When radexpand is all, causes nth roots of factors of a product
which are powers of n to be pulled outside of the radical. E.g. if
radexpand is all, sqrt (16*x^2) simplifies to 4*x.
More particularly, consider sqrt (x^2).
radexpand is all or assume (x > 0) has been executed,
sqrt(x^2) simplifies to x.
radexpand is true and domain is real
(its default), sqrt(x^2) simplifies to abs(x).
radexpand is false, or radexpand is true and
domain is complex, sqrt(x^2) is not simplified.
Note that domain only matters when radexpand is true.
Categories: Simplification flags and variables
declare (g, rassociative) tells the Maxima
simplifier that g is right-associative. E.g.,
g(g(a, b), g(c, d)) simplifies to g(a, g(b, g(c, d))).
Categories: Declarations and inferences · Operators
Sequential Comparative Simplification (method due to Stoute).
scsimp attempts to simplify expr
according to the rules rule_1, …, rule_n.
If a smaller expression is obtained, the process repeats. Otherwise after all
simplifications are tried, it returns the original answer.
example (scsimp) displays some examples.
Categories: Simplification functions
Default value: true
simp enables simplification. This is the default. simp is also
an evflag, which is recognized by the function ev. See ev.
When simp is used as an evflag with a value false, the
simplification is suppressed only during the evaluation phase of an expression.
The flag does not suppress the simplification which follows the evaluation
phase.
Many Maxima functions and operations require simplification to be enabled to work normally. When simplification is disabled, many results will be incomplete, and in addition there may be incorrect results or program errors.
Examples:
The simplification is switched off globally. The expression sin(1.0) is
not simplified to its numerical value. The simp-flag switches the
simplification on.
(%i1) simp:false; (%o1) false (%i2) sin(1.0); (%o2) sin(1.0) (%i3) sin(1.0),simp; (%o3) .8414709848078965
The simplification is switched on again. The simp-flag cannot suppress
the simplification completely. The output shows a simplified expression, but
the variable x has an unsimplified expression as a value, because the
assignment has occurred during the evaluation phase of the expression.
(%i4) simp:true; (%o4) true (%i5) x:sin(1.0),simp:false; (%o5) .8414709848078965 (%i6) :lisp $X ((%SIN) 1.0)
Categories: Evaluation flags
declare (h, symmetric) tells the Maxima
simplifier that h is a symmetric function. E.g., h (x, z, y)
simplifies to h (x, y, z).
commutative is synonymous with symmetric.
Categories: Declarations and inferences · Operators
Combines all terms of expr (which should be a sum) over a common
denominator without expanding products and exponentiated sums as ratsimp
does. xthru cancels common factors in the numerator and denominator of
rational expressions but only if the factors are explicit.
Sometimes it is better to use xthru before ratsimping an
expression in order to cause explicit factors of the gcd of the numerator and
denominator to be canceled thus simplifying the expression to be
ratsimped.
Examples:
(%i1) ((x+2)^20 - 2*y)/(x+y)^20 + (x+y)^(-19) - x/(x+y)^20;
20
1 (x + 2) - 2 y x
(%o1) --------- + --------------- - ---------
19 20 20
(y + x) (y + x) (y + x)
(%i2) xthru (%);
20
(x + 2) - y
(%o2) -------------
20
(y + x)
Categories: Expressions
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