Why doesn't Matlab's answer to factorial(100) equal Wolfram Alpha's 100!?

When computing:

>> a = sprintf('%f',factorial(100))

a =

      '93326215443944102000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.000000'

However the number above does not equal 100!

why is this and how do I fix it?

[Stephen23]

"why is this..."

Because you are doing a numeric calculation using a variable of class double, which is limited to approximately 16 decimal digits precision.

"...and how do I fix it?"

Do a symbolic calculation.

[Alok Virkar]

Thanks!!

Answer 1

[Andrei Bobrov]

factorial(sym(100))

[Alok Virkar]

Thanks! this resolved my problem but Im now wondering

why doesnt factorial(sym(100)) == sym(factorial(100))

[Walter Roberson]

sym(factorial(100)) calculates factorial(100) in double precision and passes the result to sym(), which then converts that rather wrong double precision number to an extended precision number.

Remember, in MATLAB, arguments are always evaluated before being passed to a function.

The only exception to this that I can think of is that the various int*() and uint*() and single() and double() calls with a literal number (and spacing is important!) is treated as syntax rather than a function call: those restricted cases of using those functions create the values at parse time rather than at execution time. This was needed to be able to write a accurately write integers between 2^53 and 2^64-1.

[Alok Virkar]

got it, Thanks!

Answer 2

[Walter Roberson]

Two reasons:

1) You are using MS Windows, and the native number formatting routines cannot handle more than 16 digits. On OS-X / MacOS, which does the conversion properly,

>> a = sprintf('%f',factorial(100))
a =
    '93326215443944102188325606108575267240944254854960571509166910400407995064242937148632694030450512898042989296944474898258737204311236641477561877016501813248.000000'

If I recall correctly, Linux does better than MS Windows but not as good as Mac.

2)

>> eps(factorial(100))
ans =
     1.21943302746718e+142

By that large of a number, the distance between representable numbers is large enough integers as to badly distort the value.

Work around:

>> factorial(sym(100))
ans =
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

[Alok Virkar]

why is factorial(sym(100)) not equal to sym(factorial(100))?

[Walter Roberson]

I explained that https://www.mathworks.com/matlabcentral/answers/362904-why-doesn-t-matlab-s-answer-to-factorial-100-equal-wolfram-alpha-s-100#comment_496719

Answer 3

[WEEKIAN SOH]

Indeed this is a good question. I had also noticed the same problem while I compare Mathematica's factorial(100) with Matlab's.

Good thing we don't have to work with so large of a numbers most of the time...

Now, if I want high precisions to see the digits, I'll use sym(Number), and pass this sym(Number) into the function to derive the long precisions.

Thank you for sharing the answer