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The abs
function represents the mathematical absolute value function and
works for both numerical and symbolic values. If the argument, z, is a
real or complex number, abs
returns the absolute value of z. If
possible, symbolic expressions using the absolute value function are
also simplified.
Maxima can differentiate, integrate and calculate limits for expressions
containing abs
. The abs_integrate
package further extends
Maxima’s ability to calculate integrals involving the abs function. See
(%i12) in the examples below.
When applied to a list or matrix, abs
automatically distributes over
the terms. Similarly, it distributes over both sides of an
equation. To alter this behaviour, see the variable distribute_over
.
See also cabs
.
Examples:
Calculation of abs
for real and complex numbers, including numerical
constants and various infinities. The first example shows how abs
distributes over the elements of a list.
(%i1) abs([-4, 0, 1, 1+%i]); (%o1) [4, 0, 1, sqrt(2)] (%i2) abs((1+%i)*(1-%i)); (%o2) 2 (%i3) abs(%e+%i); 2 (%o3) sqrt(%e + 1) (%i4) abs([inf, infinity, minf]); (%o4) [inf, inf, inf]
Simplification of expressions containing abs
:
(%i5) abs(x^2); 2 (%o5) x (%i6) abs(x^3); 2 (%o6) x abs(x) (%i7) abs(abs(x)); (%o7) abs(x) (%i8) abs(conjugate(x)); (%o8) abs(x)
Integrating and differentiating with the abs
function. Note that more
integrals involving the abs
function can be performed, if the
abs_integrate
package is loaded. The last example shows the Laplace
transform of abs
: see laplace
.
(%i9) diff(x*abs(x),x),expand; (%o9) 2 abs(x) (%i10) integrate(abs(x),x); x abs(x) (%o10) -------- 2 (%i11) integrate(x*abs(x),x); / [ (%o11) I x abs(x) dx ] / (%i12) load("abs_integrate")$ (%i13) integrate(x*abs(x),x); 2 3 x abs(x) x signum(x) (%o13) --------- - ------------ 2 6 (%i14) integrate(abs(x),x,-2,%pi); 2 %pi (%o14) ---- + 2 2 (%i15) laplace(abs(x),x,s); 1 (%o15) -- 2 s
When x is a real number, return the least integer that is greater than or equal to x.
If x is a constant expression (10 * %pi
, for example),
ceiling
evaluates x using big floating point numbers, and
applies ceiling
to the resulting big float. Because ceiling
uses
floating point evaluation, it’s possible, although unlikely, that ceiling
could return an erroneous value for constant inputs. To guard against errors,
the floating point evaluation is done using three values for fpprec
.
For non-constant inputs, ceiling
tries to return a simplified value.
Here are examples of the simplifications that ceiling
knows about:
(%i1) ceiling (ceiling (x)); (%o1) ceiling(x)
(%i2) ceiling (floor (x)); (%o2) floor(x)
(%i3) declare (n, integer)$
(%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))]; (%o4) [n, abs(n), max(6, n)]
(%i5) assume (x > 0, x < 1)$
(%i6) ceiling (x); (%o6) 1
(%i7) tex (ceiling (a)); $$\left \lceil a \right \rceil$$ (%o7) false
The ceiling
function distributes over lists, matrices and equations.
See distribute_over
.
Finally, for all inputs that are manifestly complex, ceiling
returns
a noun form.
If the range of a function is a subset of the integers, it can be declared to
be integervalued
. Both the ceiling
and floor
functions
can use this information; for example:
(%i1) declare (f, integervalued)$
(%i2) floor (f(x)); (%o2) f(x)
(%i3) ceiling (f(x) - 1); (%o3) f(x) - 1
Example use:
(%i1) unitfrac(r) := block([uf : [], q], if not(ratnump(r)) then error("unitfrac: argument must be a rational number"), while r # 0 do ( uf : cons(q : 1/ceiling(1/r), uf), r : r - q), reverse(uf)); (%o1) unitfrac(r) := block([uf : [], q], if not ratnump(r) then error("unitfrac: argument must be a rational number"), 1 while r # 0 do (uf : cons(q : ----------, uf), r : r - q), 1 ceiling(-) r reverse(uf))
(%i2) unitfrac (9/10); 1 1 1 (%o2) [-, -, --] 2 3 15
(%i3) apply ("+", %); 9 (%o3) -- 10
(%i4) unitfrac (-9/10); 1 (%o4) [- 1, --] 10
(%i5) apply ("+", %); 9 (%o5) - -- 10
(%i6) unitfrac (36/37); 1 1 1 1 1 (%o6) [-, -, -, --, ----] 2 3 8 69 6808
(%i7) apply ("+", %); 36 (%o7) -- 37
Returns the largest integer less than or equal to x where x is
numeric. fix
(as in fixnum
) is a synonym for this, so
fix(x)
is precisely the same.
When x is a real number, return the largest integer that is less than or equal to x.
If x is a constant expression (10 * %pi
, for example), floor
evaluates x using big floating point numbers, and applies floor
to
the resulting big float. Because floor
uses floating point evaluation,
it’s possible, although unlikely, that floor
could return an erroneous
value for constant inputs. To guard against errors, the floating point
evaluation is done using three values for fpprec
.
For non-constant inputs, floor
tries to return a simplified value. Here
are examples of the simplifications that floor
knows about:
(%i1) floor (ceiling (x)); (%o1) ceiling(x)
(%i2) floor (floor (x)); (%o2) floor(x)
(%i3) declare (n, integer)$
(%i4) [floor (n), floor (abs (n)), floor (min (n, 6))]; (%o4) [n, abs(n), min(6, n)]
(%i5) assume (x > 0, x < 1)$
(%i6) floor (x); (%o6) 0
(%i7) tex (floor (a)); $$\left \lfloor a \right \rfloor$$ (%o7) false
The floor
function distributes over lists, matrices and equations.
See distribute_over
.
Finally, for all inputs that are manifestly complex, floor
returns
a noun form.
If the range of a function is a subset of the integers, it can be declared to
be integervalued
. Both the ceiling
and floor
functions
can use this information; for example:
(%i1) declare (f, integervalued)$
(%i2) floor (f(x)); (%o2) f(x)
(%i3) ceiling (f(x) - 1); (%o3) f(x) - 1
A synonym for entier (x)
.
The Heaviside unit step function, equal to 0 if x is negative, equal to 1 if x is positive and equal to 1/2 if x is equal to zero.
If you want a unit step function that takes on the value of 0 at x
equal to zero, use unit_step
.
When L is a list or a set, return apply ('max, args (L))
.
When L is not a list or a set, signal an error.
See also lmin
and max
.
When L is a list or a set, return apply ('min, args (L))
.
When L is not a list or a set, signal an error.
See also lmax
and min
.
Return a simplified value for the numerical maximum of the expressions x_1
through x_n. For an empty argument list, max
yields minf
.
The option variable maxmin_effort
controls which simplification methods are
applied. Using the default value of twelve for maxmin_effort
,
max
uses all available simplification methods. To to inhibit all
simplifications, set maxmin_effort
to zero.
When maxmin_effort
is one or more, for an explicit list of real numbers,
max
returns a number.
Unless max
needs to simplify a lengthy list of expressions, we suggest using
the default value of maxmin_effort
. Setting maxmin_effort
to zero
(no simplifications), will cause problems for some Maxima functions; accordingly,
generally maxmin_effort
should be nonzero.
See also min
, lmax
., and lmin
..
Examples:
In the first example, setting maxmin_effort
to zero suppresses simplifications.
(%i1) block([maxmin_effort : 0], max(1,2,x,x, max(a,b))); (%o1) max(1,2,max(a,b),x,x) (%i2) block([maxmin_effort : 1], max(1,2,x,x, max(a,b))); (%o2) max(2,a,b,x)
When maxmin_effort
is two or more, max
compares pairs of members:
(%i1) block([maxmin_effort : 1], max(x,x+1,x+3)); (%o1) max(x,x+1,x+3) (%i2) block([maxmin_effort : 2], max(x,x+1,x+3)); (%o2) x+3
Finally, when maxmin_effort
is three or more, max
compares triples
members and excludes those that are in between; for example
(%i1) block([maxmin_effort : 4], max(x, 2*x, 3*x, 4*x)); (%o1) max(x,4*x)
Return a simplified value for the numerical minimum of the expressions x_1
through x_n. For an empty argument list, minf
yields inf
.
The option variable maxmin_effort
controls which simplification methods are
applied. Using the default value of twelve for maxmin_effort
,
max
uses all available simplification methods. To to inhibit all
simplifications, set maxmin_effort
to zero.
When maxmin_effort
is one or more, for an explicit list of real numbers,
min
returns a number.
Unless min
needs to simplify a lengthy list of expressions, we suggest using
the default value of maxmin_effort
. Setting maxmin_effort
to zero
(no simplifications), will cause problems for some Maxima functions; accordingly,
generally maxmin_effort
should be nonzero.
See also max
, lmax
., and lmin
..
Examples:
In the first example, setting maxmin_effort
to zero suppresses simplifications.
(%i1) block([maxmin_effort : 0], min(1,2,x,x, min(a,b))); (%o1) min(1,2,a,b,x,x) (%i2) block([maxmin_effort : 1], min(1,2,x,x, min(a,b))); (%o2) min(1,a,b,x)
When maxmin_effort
is two or more, min
compares pairs of members:
(%i1) block([maxmin_effort : 1], min(x,x+1,x+3)); (%o1) min(x,x+1,x+3) (%i2) block([maxmin_effort : 2], min(x,x+1,x+3)); (%o2) x
Finally, when maxmin_effort
is three or more, min
compares triples
members and excludes those that are in between; for example
(%i1) block([maxmin_effort : 4], min(x, 2*x, 3*x, 4*x)); (%o1) max(x,4*x)
When x is a real number, returns the closest integer to x.
Multiples of 1/2 are rounded to the nearest even integer. Evaluation of
x is similar to floor
and ceiling
.
The round
function distributes over lists, matrices and equations.
See distribute_over
.
For either real or complex numbers x, the signum function returns
0 if x is zero; for a nonzero numeric input x, the signum function
returns x/abs(x)
.
For non-numeric inputs, Maxima attempts to determine the sign of the input.
When the sign is negative, zero, or positive, signum
returns -1,0, 1,
respectively. For all other values for the sign, signum
a simplified but
equivalent form. The simplifications include reflection (signum(-x)
gives -signum(x)
) and multiplicative identity (signum(x*y)
gives
signum(x) * signum(y)
).
The signum
function distributes over a list, a matrix, or an
equation. See sign
and distribute_over
.
When x is a real number, return the closest integer to x not
greater in absolute value than x. Evaluation of x is similar
to floor
and ceiling
.
The truncate
function distributes over lists, matrices and equations.
See distribute_over
.
Next: Functions for Complex Numbers, Previous: Mathematical Functions, Up: Mathematical Functions [Contents][Index]