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http://en.wikipedia.org/wiki/Numerical_differentiation#Practical_considerations mentions the formula

$h=\sqrt \epsilon * x$

where $\epsilon$ is the machine epsilon (approx. 2.2e-16 for 64-bit IEEE 754 doubles), to calculate the optimum "small number" h to be use used in differentiation, such as

$\frac{f(x+h)-f(x)}{h}$

But what if x is zero? Then h will be zero too, and division by zero is certainly not a way to do numerical differentiation. Is the article wrong? Is it otherwise correct, except that near zero (how near?) some small enough constant (how small?) should be used?

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# Optimum small number for numerical differentiation

http://en.wikipedia.org/wiki/Numerical_differentiation#Practical_considerations mentions the formula

$h=\sqrt \epsilon * x$

where $\epsilon$ is the machine epsilon (approx. 2.2e-16 for 64-bit IEEE 754 doubles), to calculate the optimum "small number" h to be use in differentiation, such as

$\frac{f(x+h)-f(x)}{h}$

But what if x is zero? Then h will be zero too, and division by zero is certainly not a way to do numerical differentiation. Is the article wrong? Is it otherwise correct, except that near zero (how near?) some small enough constant (how small?) should be used?