Changes between Version 3 and Version 4 of Piecewise

Timestamp:

Jun 13, 2014, 1:59:09 PM (11 years ago)

Author:

Pedro Gea

Comment:

--

Piecewise

@@ -91 +91 @@
 Podemos reagrupar los términos en función de los parámetros {{{b_k}}} del siguiente modo:
 {{{
-        |         / (x_k-x_{k+1})  ,,  x<x_k
-        |  b_k * |  (x-x_{k+1})    ,,  x<:[x_k, x_{k+1}]  ,,  k<q
-        |         \ 0              ,,  x>x_{k+1}
-        |         / x_q      ,,  x<x_q
-E(x) = <|  b_q * |  x        ,,  x<:[x_q, x_{q+1}]
-        |         \ x_{q+1}  ,,  x>x_{q+1}
-        |         / 0              ,,  x<x_m
-        |  b_m * |  (x-x_m)        ,,  x<:[x_m, x_{m+1}]  ,,  m>q
-        |         \ (x_{m+1}-x_m)  ,,  x>x_{m+1}
+        |         / x_k  -x_{k+1}   ,,  x<x_k
+        |  b_k * |  x - x_{k+1}     ,,  x<:[x_k, x_{k+1}]  ,,  k<q
+        |         \ 0               ,,  x>x_{k+1}
+        |         / x_k             ,,  x<x_k
+E(x) = <|  b_k * |  x               ,,  x<:[x_k, x_{k+1}]  ,, k==q
+        |         \ x_{k+1}         ,,  x>x_{k+1}
+        |         / 0               ,,  x<x_k
+        |  b_k * |  x - x_k         ,,  x<:[x_k, x_{k+1}]  ,,  k>q
+        |         \ x_{k+1} - x_k   ,,  x>x_{k+1}
 }}} 
 A estos nuevos inputs los denominamos ''inputs-piecewise''.
+
+Podemos observar que el particionamiento del input, puede escribirse en los tres diferentes casos siguientes como:
+{{{
+si x <: [x_a, x_{a+1}]   ,,  a>q  ,,  0 <: [x_q, x_{q+1}]
+
+         | 0             ,, k < q
+P_k(x) = | x_{k+1}       ,, k == q
+         | x_{k+1} - x_k ,, q < k < a
+         | x - x_k       ,, k == a
+         | 0             ,, k > a
+}}}
+{{{
+si x <: [x_a, x_{a+1}]   ,,  a==q  ,,  0 <: [x_q, x_{q+1}]
+
+         | 0             ,, k < q
+P_k(x) = | x_{k+1}       ,, k == q
+         | x_{k+1} - x_k ,, q < k < a
+         | x - x_k       ,, k == a
+         | 0             ,, k > a
+}}}
+{{{
+si x <: [x_a, x_{a+1}]   ,,  a>q  ,,  0 <: [x_q, x_{q+1}]
+
+         | 0           ,, k < q
+P_k(x) = | x           ,, k == q
+         | 0           ,, k > q
+}}}
+y comprobar que:
+{{{Sum_k(P_k(x)) == x}}}
+y de ahí el término particionamiento.