error: métodes numérics

f(x_{i}) = | x+(-1)·x_{i} |

f(x_{i}+y_{i}) [< f(x_{i})+f(y_{i})

| ( x+y )+(-1)·( x_{i}+y_{i} ) | [< | x+(-1)·x_{i} |+| y+(-1)·y_{i} |

1+1 = 2

0.75+1.25 = 2

0 = f(0.75+1.25) [< f(0.75)+f(1.25) = 0.50

f(x_{i}·y_{i}) [< |y_{i}|·f(x_{i})+f(x_{i})·f(y_{i})+|x_{i}|·f(y_{i})

| ( x·y )+(-1)·( x_{i}·y_{i} ) | [< |y|| x+(-1)·x_{i} |+|x_{i}|| y+(-1)·y_{i} |

| ( x·y )+(-1)·( x_{i}·y_{i} ) | [< |y_{i}|| x+(-1)·x_{i} |+|x|| y+(-1)·y_{i} |

si ( x [< (-1) || 1 [< x ) ==>

f(1/x_{i}) [< (1/|x_{i}|)·f(x_{i})

si (-1) < x < 1 ==>

f(1/x_{i}) >] (1/|x_{i}|)·f(x_{i})

| (1/x)+(-1)·(1/x_{i}) | = |x_{i}+(-x)|/( |x|·|x_{i}| ) [< ...

... |x_{i}+(-x)|/( |x_{i}+(-x)|·|x_{i}|+(-1)·|x_{i}|^{2} )

f(1/x_{i}) [< (1/|x_{i}|)·f(x_{i})/( f(x_{i})+(-1)·|x_{i}| )

f(y_{i}/x_{i}) [< (1/|x_{i}|)·( ( |x_{i}|·f(y_{i})+|y_{i}|·f(x_{n}) )/( f(x_{i})+(-1)·|x_{i}| ) )

| (y/x)+(-1)·(y_{i}/x_{i}) | [< ( |x_{i}||y+(-1)·y_{i}|+|y_{i}||x_{i}+(-x)| )/( |x|·|x_{i}| )

si x [< (-1) || 1 [< x ==>

f(y_{i}/x_{i}) [< (1/|x_{i}|)·( |x_{i}|·f(y_{i})+|y_{i}|·f(x_{n}) )

si (-1) < x || x < 1 ==>

f(y_{i}/x_{i}) >] (1/|x_{i}|)·( |x_{i}|·f(y_{i})+|y_{i}|·f(x_{n}) )

f( (x_{i})^{2} ) [< ( f(x_{i}) )^{2}+2·|x_{i}|·f(x_{i})

| x^{2}+(-1)·(x_{i})^{2} | = |x+(-1)·x_{i}|·|x+x_{i}|

Si x >] x_{i} ==>

f( ln(x_{i}) ) [< ln( f(x_{i})/|x_{i}|+1 )

| ln(x)+(-1)·ln(x_{i}) | = |ln(x/x_{i})|

f( e^{x_{i}} ) [< ( e^{f(x_{i})}+1 )·e^{|x_{i}|}

| e^{x}+(-1)·e^{x_{i}} | [< |e^{x+(-1)·x_{i}+x_{i}}|+|e^{x_{i}}|