Next: Functions for Complex Numbers, Up: Elementary Functions [Contents][Index]
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, Up: Elementary Functions [Contents][Index]