Dual numbers for automatic differentiation
Example : (5 + 3 * epsilon) * (8 + epsilon) = 40 + 29 * epsilon
+-----+
|40  0|
|29 40|
+-----+
Vector a = 3+epsilon 5+epsilon :
+---+---+
|3 0|5 0|
|1 3|1 5|
+---+---+
Scalar product of a by itself = (3+epsilon)^2 + (5+epsilon)^2 = 34+16*epsilon :
+-----+
|34  0|
|16 34|
+-----+
f(6+epsilon) = 59+15*epsilon
+-----+
|59  0|
|15 59|
+-----+
Example : 5 + 3*epsilon + zeta
+-----+
|5 0 0|
|3 5 0|
|1 0 5|
+-----+
g(10+epsilon,20+zeta) = 330+23*epsilon+15*zeta
+-----------+
|330   0   0|
| 23 330   0|
| 15   0 330|
+-----------+
+-----------+
|330   0   0|
| 23 330   0|
| 15   0 330|
+-----------+
m(2+epsilon,3+zeta)
+-----------------------+
|5.85888       0       0|
|      3 5.85888       0|
|2.98999       0 5.85888|
+-----------------------+
h(10+epsilon) = 1350 + 365*epsilon + 1/2 66 * epsilon^2
+--------------+
|1350    0    0|
| 365 1350    0|
|  33  365 1350|
+--------------+
epsilon :
+-----------+
|0 0 0 0 0 0|
|1 0 0 0 0 0|
|0 0 0 0 0 0|
|0 1 0 0 0 0|
|0 0 1 0 0 0|
|0 0 0 0 0 0|
+-----------+
zeta :
+-----------+
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|1 0 0 0 0 0|
|0 0 0 0 0 0|
|0 1 0 0 0 0|
|0 0 1 0 0 0|
+-----------+
epsilon^2 :
+-----------+
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|1 0 0 0 0 0|
|0 0 0 0 0 0|
|0 0 0 0 0 0|
+-----------+
zeta^2 :
+-----------+
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|1 0 0 0 0 0|
+-----------+
epsilon*zeta :
+-----------+
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|0 0 0 0 0 0|
|1 0 0 0 0 0|
|0 0 0 0 0 0|
+-----------+
k(10+epsilon,1+zeta) = 454 + 74*epsilon + 79*zeta + 3*epsilon^2 + 5*zeta^2 + 6*epsilon*zeta
+-----------------------+
|454   0   0   0   0   0|
| 74 454   0   0   0   0|
| 79   0 454   0   0   0|
|  3  74   0 454   0   0|
|  6  79  74   0 454   0|
|  5   0  79   0   0 454|
+-----------------------+