10.1 Functions for Numbers
Function: abs (z)
このabs関数は数学的な絶対値関数を表し、数値と記号値の両方で機能します。引数zが実数または複素数の場合、zabsの絶対値を返します。可能な場合は、絶対値関数を使用する記号式も簡略化されます。
Maxima は、 を含む式の微分、積分、および極限の計算を行うことができますabs。このabs_integrateパッケージは、abs 関数を含む積分を計算する Maxima の機能をさらに拡張します。以下の例の (%i12) を参照してください。
リストまたは行列に適用すると、abs項に自動的に分配されます。同様に、方程式の両辺に分配されます。この動作を変更するには、変数 を参照してくださいdistribute_over。
も参照してくださいcabs。
例:
abs数値定数やさまざまな無限大を含む実数と複素数の の計算。最初の例はabs 、 がリストの要素にどのように分配されるかを示しています。
(%i1) 絶対値([-4, 0, 1, 1+%i]);
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
Categories: Mathematical functions ·
Function: ceiling (x)
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
Categories: Mathematical functions ·
Function: entier (x)
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.
Categories: Mathematical functions ·
Function: floor (x)
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
Categories: Mathematical functions ·
Function: fix (x)
A synonym for entier (x).
Categories: Mathematical functions ·
Function: hstep (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.
Categories: Laplace transform · Mathematical functions ·
Function: lmax (L)
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.
Categories: Mathematical functions · Lists · Sets ·
Function: lmin (L)
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.
Categories: Mathematical functions · Lists · Sets ·
Function: max (x_1, …, x_n)
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)
Categories: Mathematical functions ·
Function: min (x_1, …, x_n)
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)
Categories: Mathematical functions ·
Function: round (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.
Categories: Mathematical functions ·
Function: signum (x)
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.
Categories: Mathematical functions ·
Function: truncate (x)
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.
Categories: Mathematical functions ·