Trac #14801: Piecewise functions done right

Rewrite piecewise functions as symbolic functions.

For a (late) discussion about interface issues see
https://groups.google.com/forum/?hl=en#!topic/sage-devel/dgwUMsdiHfM

URL: http://trac.sagemath.org/14801
Reported by: vbraun
Ticket author(s): Volker Braun, Ralf Stephan
Reviewer(s): Volker Braun, Ralf Stephan

src/doc/de/thematische_anleitungen/sage_gymnasium.rst

@@ -572,7 +572,7 @@ Tupel besteht, welches das Interval des Definitionsbereichs angibt, also z.B. ``
 `[-\infty, 0]` und einer für das Interval geltende Funktionsgleichung. Als letztes Argument muss angegeben werden,
 welche Variable durch die Funktion gebunden werden soll::
 
-    sage: f = Piecewise([[(-oo,0), -x^2],[(0,oo), x^2]], x)
+    sage: f = piecewise([[(-oo,0), -x^2],[(0,oo), x^2]], var=x)
     sage: f(3)
     9
     sage: f(-3)

src/doc/en/constructions/calculus.rst

@@ -81,7 +81,7 @@ You can find critical points of a piecewise defined function:
     sage: f2 = 1-x
     sage: f3 = 2*x
     sage: f4 = 10*x-x^2
-    sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
+    sage: f = piecewise([((0,1),f1), ((1,2),f2), ((2,3),f3), ((3,10),f4)])
     sage: f.critical_points()
     [5.0]
 
@@ -177,9 +177,10 @@ where :math:`f(x)=1`, :math:`0<x<1`:
 ::
 
     sage: x = PolynomialRing(QQ, 'x').gen()
-    sage: f = Piecewise([[(0,1),1*x^0]])
+    sage: f = piecewise([((0,1),1*x^0)])
     sage: g = f.convolution(f)
     sage: h = f.convolution(g)
+    sage: set_verbose(-1)
     sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1))
 
 To view this, type ``show(P+Q+R)``.
@@ -212,18 +213,13 @@ where :math:`f` is a piecewise defined function, can
 
     sage: f1(x) = x^2
     sage: f2(x) = 5-x^2
-    sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
-    sage: f.trapezoid(4)
-    Piecewise defined function with 4 parts, [[(0, 1/2), 1/2*x],
-    [(1/2, 1), 9/2*x - 2], [(1, 3/2), 1/2*x + 2],
-    [(3/2, 2), -7/2*x + 8]]
-    sage: f.riemann_sum_integral_approximation(6,mode="right")
-    19/6
-    sage: f.integral()
-    Piecewise defined function with 2 parts,
-    [[(0, 1), x |--> 1/3*x^3], [(1, 2), x |--> -1/3*x^3 + 5*x - 13/3]]
-    sage: f.integral(definite=True)
-    3
+    sage: f = piecewise([[[0,1], f1], [RealSet.open_closed(1,2), f2]])
+    sage: t = f.trapezoid(2); t
+    piecewise(x|-->1/2*x on (0, 1/2), x|-->3/2*x - 1/2 on (1/2, 1), x|-->7/2*x - 5/2 on (1, 3/2), x|-->-7/2*x + 8 on (3/2, 2); x)
+    sage: t.integral()
+    piecewise(x|-->1/4*x^2 on (0, 1/2), x|-->3/4*x^2 - 1/2*x + 1/8 on (1/2, 1), x|-->7/4*x^2 - 5/2*x + 9/8 on (1, 3/2), x|-->-7/4*x^2 + 8*x - 27/4 on (3/2, 2); x)
+    sage: t.integral(definite=True)
+    9/4
 
 .. index: Laplace transform
 
@@ -242,7 +238,7 @@ computation.
     (x, s)
     sage: f1(x) = 1
     sage: f2(x) = 1-x
-    sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
+    sage: f = piecewise([((0,1),f1), ((1,2),f2)])
     sage: f.laplace(x, s)
     -e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2
 
@@ -382,7 +378,7 @@ illustrating how the Gibbs phenomenon is mollified).
 
     sage: f1 = lambda x: -1
     sage: f2 = lambda x: 2
-    sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
+    sage: f = piecewise([((0,pi/2),f1), ((pi/2,pi),f2)])
     sage: f.fourier_series_cosine_coefficient(5,pi)
     -3/5/pi
     sage: f.fourier_series_sine_coefficient(2,pi)

src/doc/en/constructions/plotting.rst

@@ -34,13 +34,13 @@ You can plot piecewise-defined functions:
 
 ::
 
-    sage: f1 = lambda x:1
-    sage: f2 = lambda x:1-x
-    sage: f3 = lambda x:exp(x)
-    sage: f4 = lambda x:sin(2*x)
-    sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
-    sage: f.plot()
-    Graphics object consisting of 4 graphics primitives
+    sage: f1 = 1
+    sage: f2 = 1-x
+    sage: f3 = exp(x)
+    sage: f4 = sin(2*x)
+    sage: f = piecewise([((0,1),f1), ((1,2),f2), ((2,3),f3), ((3,10),f4)])
+    sage: f.plot(x,0,10)
+    Graphics object consisting of 1 graphics primitive
 
 Other function plots can be produced as well:
 

src/sage/calculus/wester.py

@@ -526,7 +526,7 @@
     sage: #      Verify(Simplify(Integrate(x)
     sage: #        if(x<0) (-x) else x),
     sage: #        Simplify(if(x<0) (-x^2/2) else x^2/2));
-    sage: f = piecewise([ [[-10,0], -x], [[0,10], x]])
+    sage: f = piecewise([ ((-10,0), -x), ((0,10), x)])
     sage: f.integral(definite=True)
     100
 

src/sage/functions/all.py

@@ -1,4 +1,7 @@
-from piecewise import piecewise, Piecewise
+from sage.misc.lazy_import import lazy_import
+
+lazy_import('sage.functions.piecewise_old', 'Piecewise')   # deprecated
+lazy_import('sage.functions.piecewise', 'piecewise')
 
 from trig import ( sin, cos, sec, csc, cot, tan,
                    asin, acos, atan,

src/sage/functions/other.py

@@ -1554,7 +1554,7 @@ def __init__(self):
             sage: loads(dumps(binomial(n,k)))
             binomial(n, k)
         """
-        GinacFunction.__init__(self, "binomial", nargs=2,
+        GinacFunction.__init__(self, "binomial", nargs=2, preserved_arg=1,
                 conversions=dict(maxima='binomial',
                                  mathematica='Binomial',
                                  sympy='binomial'))