(A+B+C ) ^ (X + Y + Z ) of arbitrary complexity

[jebusv20]

This question relates mostly to computer sciences (my personal field of expertise),
I am trying to find the method for breaking down large floating point (non integer) numbers into a series of integers (so that they can be stored easily), and completing a power operation over them.

for example

1.23 ^ 4.56 = ((1E0)+(2E-1)+(2E-2))^((4E0)+(5E-1)+(6E-2))

I am aware that a program would be able to store quite large floating point numbers (generally up to 16 places of complexity) but, if I were to try and calculate something to 1 million places of accuracy, I will need the above break down)

Im sure it has something to do with pascals triangle which I am fairly sure is good for (a+b)^c, but I don't know how to do (a+b)^(c+d). I know this formula exists because I'm sure calculators use it every day... just not sure how to do it myself. To be clear, this formula does not need to be 'easy' to use, a computer can do billions of integer calculations a second, just need to write the code.

 

[jebusv20]

No replies? If its a matter of me not explaining myself please just ask what I mean and what confuses you and I'll try to further explain. If its something that would be really hard to explain, just give it a shot... and if you don't know... well ... feel free to tell your friends who might.

Should we just start with the simple stuff? Purely algebraically how could the following be expressed without the power operation, preferably with a serious of multiplication and addition operations:

(a+b)^(c+d)

 

[drag12]

If you're starting with something like $(a+b)^{c+d}$ just break it into $(a+b)^{c}(a+b)^{d}$ and use the usual http://en.wikipedia.org/wiki/Binomial_theorem for each part of the product, then multiply them together.

 

[willem2]

If you're starting with something like $(a+b)^{c+d}$ just break it into $(a+b)^{c}(a+b)^{d}$ and use the usual http://en.wikipedia.org/wiki/Binomial_theorem for each part of the product, then multiply them together.

This doesn't help for multiprecision arithmetic with more than 2 digits.
If you want to compute (a+b+c)^n, with n an integer repeated squaring and multiplication
is the best way.

a^(2n) = (a^n)^2
a^(2n+1) = a (a^(2n))

A number of algorithms exist to do fast multiplication of multiprecision numbers. (karatsuba multiplication, fast Fourier transfrom multiplication)

The latter will be much faster for large numbers, but also quite hard to implement. Using a multiprecision library is always an option.

 

[CellCoree]

As a computer scientist, I can offer some insights on this problem. Breaking down large floating point numbers into a series of integers can be achieved using a technique called "floating point representation". In this method, a floating point number is represented as a combination of a sign, a mantissa, and an exponent. The mantissa is the main number, while the exponent indicates the position of the decimal point. This allows for efficient storage and manipulation of floating point numbers.

To perform a power operation on floating point numbers, you can use the "power series" method. This involves expanding the power operation into a series of simpler operations, such as multiplication and addition. For example, to calculate (a+b)^(c+d), you can use the following steps:

1. Convert a and b into their floating point representations.
2. Use the power series method to calculate a^(c+d) and b^(c+d).
3. Add the two results from step 2 to get (a+b)^(c+d).

To achieve high accuracy, you can increase the number of terms in the power series. This means performing more calculations, but with the power of modern computers, this should not be a problem.

As for the use of Pascal's triangle, it can be helpful in some cases, but it is not necessary for this problem. Pascal's triangle is mainly used for calculating the coefficients in binomial expansions, which are not directly related to power operations.

In summary, the key to breaking down large floating point numbers and performing power operations on them is to use floating point representation and the power series method. With these techniques, you can achieve high accuracy and handle numbers of arbitrary complexity.