Multiplication of large number with small number

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Filip Trönnberg 2012-5-9

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I'm trying to compute a rather ugly integral using MATLAB. What I'm having problem with though is a part where I multiply a very big number (>10^300) with a very small number (<10^-300). MATLAB returns 'inf' for this even though it should be in the range of 0-0.0005. This is what I have

    besselFunction = @(u)besseli(qb,2*sqrt(lambda*(theta + mu)).*u);
    exponentFuncion = @(u)exp(-u.*(lambda + theta + mu));

where qb = 5, lambda = 12, theta = 10, mu = 3. And what I want to find is

besselFunction(u)*exponentFunction(u)

for all real values of u. The problem is that whenever u>28 it will be evaluated as 'inf'. This is because the besselFunction(29) is so large even though exponentFunction(29) is extremely small. I've heared of, and tried, to use MATLAB function 'vpa' but it doesn't seem to work well when I want to use functions...

Any tips will be appreciated at this point!

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laurie 2012-5-9

sorry i feel stupid for even suggesting this, but can't you divide your very large number bessel(u>28) by, say, 10^150, do your multiplication, and then multiply 10^150 again after that ? or is bessel (u>28) evaluated as 'inf' ?

Filip Trönnberg 2012-5-10

Was my first thought too but unfortunately bessel(u>28) is evaluated as inf.

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Answer 1

Walter Roberson 2012-5-9

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在 MATLAB Online 中打开

vpa() is for use with symbolic expressions only.

If you have the symbolic engine,

    besselFunction = @(u)besseli(qb,2*sqrt(lambda*(theta + mu)).*sym(u));
    exponentFuncion = @(u)exp(-sym(u).*(lambda + theta + mu));

Then besselFunction(u)*exponentFunction(u) will return a symbolic value. You can double() it if you want the double-precision representation.

Note: going symbolic will only postpone the problem: the symbolic engine is limited to about a billion decimal places.

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Walter Roberson 2012-5-9

The dynamic range for symbolic numbers only goes up to somewhere around 10 to 1 billion.

Filip Trönnberg 2012-5-10

Great suggestion, thanks! It is correct that the symbolic numbers could pose a limitation but for my needs it was more than enough. What John wrote do also work but requires some extra thinking!

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Answer 2

John D'Errico 2012-5-9

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The simple answer is always to use logs. The log of these numbers will be well behaved. Add the logs, then exponentiate at the end.

You could also use my HPF toolbox. It is now on the FEX, although I've not yet implemented bessel functions.

And, of course, the symbolic toolbox is an option with vpa.

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Sargondjani 2012-5-9

he means: take logs, add them, take exponential. this is equivalent to multipying 2 numbers:

a x b = exp (log (a x b)) = exp(log a + log b)

John D'Errico 2012-5-10

Exactly as Sargondjani says. A problem may be you need the log of this bessel function, if it is itself too large for the dynamic range of a double. For that you might need to use some approximations. Perhaps a series approximation would be adequate for the log of the bessel function. Or, one day I'll get bessel functions written for HPF, or maybe someone else will do that.

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