Numerically stable forms

[Dimitri Terryn]

Hi.

I have an assignment lying around, in which I have to find numerically stable forms of some expressions. A few still elude me, so I was wondering if someone might have a suggestion.


e^{x}-e


This has large rounding errors if x is close to 1


sinh (x) - tanh (x)


Large errors for x close to 0


log(x+\sqrt{x^2+1})


No idea...

[pervect]

[QUOTE=Dimitri Terryn]Hi.

I have an assignment lying around, in which I have to find numerically stable forms of some expressions. A few still elude me, so I was wondering if someone might have a suggestion.


e^{x}-e


This has large rounding errors if x is close to 1

did you try a Taylor series expansion, or is that not what's being asked for?

[Dimitri Terryn]

Yes, a Taylor expansion does seem obvious; but alas, the'yre asking analytical forms...

[arildno]

2.:
sinh(x)-tanh(x)=sinh(x)(\frac{cosh(x)-1}{cosh(x)})=2tanh(x)sinh^{2}(\frac{x}{2})

Use a similar trick for 1, by noting sinh(y)=\frac{e^{y}-e^{-y}}{2}

[Dimitri Terryn]

Thanks! This is just what I needed!