Value of a discontinuous function within the discontinuity point

I have a jump discontinuous scalar function f:R->R

with "a" some real (say positive) number and f1,2 scalar continuous functions.

I wonder what is the smallest value close to the right of "a" which Matlab recognizes, is it given by using eps(a)?

Also, could it be possible to exist some lacunar/missing/"invisible" value for Matlab, in the right-neighborhood of "a" for which f cannot be calculated (even f is defined on R) ?

Thanks!

Answer 1

[Walter Roberson]

Supposing that A is exactly a representable number then the smallest value greater than A that MATLAB can represent is A+eps(A). Keep in mind however that A+eps(A)/2 is probably going to round to A+eps(A)

Any value A+delta will round to something, either to A or to at least A+eps(A), and the function can be calculated on the rounded value.

Notice that I said above "Assuming that A is exactly a representable number". Most decimal numbers are not exactly representable. For example, 0.1 is not exactly representable.

Suppose that you had a function that was described mathematically as having a boundary at 1/10 and the right hand side involves log(A-1/10). Mathematically any value A strictly greater than 1/10 will have a positive subtraction and the log would be of a positive value. But with 1/10 exactly not being exactly representable you have to worry about whether there is any value that is mathematically less than 1/10 whose representation rounds upward and so mathematically A-1/10 would be negative but the representation of A rounds up so the comparison of representation_of_A to 1/10 might succeed. Would the representation minus 1/10 potentially give negative even though the comparison succeeded? The answer depends on exactly how the subtraction is coded. For example (10*A - 1)/10 is algebraically A-1/10 but due to rounding the calculated result could be different.

[Marius-F. Danca]

Thank you for the answer!

And the answer to second part of my question if there could be "untouchable" values within a (small) right neighborhood of a which Matlab cannot access (theoretically)?

[Walter Roberson]

Consider any numeric representation that is limited to finite precision. It could, for example, operate in decimal at 100000000 digits, able to distinguish between values that differ in their hundred millionth decimal place. There would still be an infinite number of values that it would not be able exactly represent.

The difficulty is not in MATLAB itself: the difficulty is in finite representation of a transfinite amount of numbers.

If you have a system able to distinguish values delta apart then there are an infinite number of values delta/integer apart that the system would not be able to represent.

[Marius-F. Danca]

Perfect, thank you!

I intend to use these reasonings in a paper, could we talk (e.g. via email)?

[Walter Roberson]

Note that it is not possible to enter those values into MATLAB, even by calculation. For example, double precision is limited to 53 bits, so

original = 1;
hex1 = num2hex(original)    %ieee 754 representation of double precision 1, convert to hex

hex1 = '3ff0000000000000'

hex1next = hex1;
hex1next(end) = '1';     %next representable number, least change possible
next = hex2num(hex1next)

next = 1.0000

difference = next - original

difference = 2.2204e-16

difference / eps(original)    %eps() can calculate directly

ans = 1

log2(difference)

ans = -52

For any given number (except the largest!) there will be a next representable number. The difference between representable numbers is commonly called eps and in floating point formats it is not a constant in absolute value, but it is a constant in relative value. If you were using an 80 bit floating point representation, the eps would potentially be

difference / 2^(80-64)

ans = 3.3881e-21

but it would still exist. You would not be able to construct a binary floating point number that was approximately (1.0 + 1e-25) because MATLAB would not be able to distinguish it from 1.0

D = @(x) x - 1.0

D = function_handle with value:
    @(x)x-1.0

D(original)

ans = 0

D(next)

ans = 2.2204e-16

D(original + 1e-25)    %notice the result is not 1e-25

ans = 0

If you were using a decimal floating point representation scheme (such as on IBM zOS systems), the exact value of the difference between representable numbers would not be the same, but the same kind of issues would arise in such a system.

You ask about values that MATLAB would be unable to calculate the function for. Well, in the case of the above function D, there is [for example] the algebraic number (10^25 + 1)/10^25 = 1 + 10^(-25) and

D((10^25  + 1)/10^25)

ans = 0

D(1 + 10^(-25))

ans = 0

If MATLAB were working with "algebaric" numbers then it would be able to return the 10^-25 as the difference compared to 1. But that does not mean that the function is not able to calculate the value: the issue is that you cannot construct floating point 1 + 10^(-25) as the input value to be received by the function to calculate with.

Can you work with algebraic numbers in MATLAB? Not exactly, but you can get closer

value = (sym(10)^25  + 1)/10^25

value = 

D(value)

ans = 

sym(10) constructed a software symbolic integer representing 10, and raising it ^25 performed a software symbolic calculation getting an exact software symbolic integer 10000000000000000000000000 that you can add 1 to and get an exact software symbolic integer 10000000000000000000000001 and then create the rational ratio of the two.

Does the available symbolic number system have limits? Yes it does. They can be difficult to find. For example,

str2sym("1"+repmat('0', 1, 1e5))

ans = 

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0

On my system using 1e6 works fine, and using 1e7 does work but is slow. Using 1e8 would probably run out of memory. But even without running out of memory, there are limits, but I do not recall what they are at the moment. Especially if you express a polynomial to a fractional power and ask to expand it, you might encounter messages about "Integer too large in context".

[Marius-F. Danca]

Impressive answer, thank you!

Let me sumarize my problem: suppose the smallest neighborhood at the right of a is (a,a') where a' is determined via eps(a). Then, is it possible to obtain numerically x, by some formula, such that x\in(a,a')?

Thank you

[Walter Roberson]

Only if you cheat on the definitions.

eps(a) is defined for single() and double() as being the least representable number such that a+eps(a) is different from a. If there were a different representable number in the interval (a, a+eps(a)) then eps(a) would have been smaller, the difference between a and what was found. So you cannot construct a double precision number in that interval: if you could then the interval would be smaller

However you might be able to create a symbolic number in the interval.

[Marius-F. Danca]

OK, now is clear: both for mathematically and numerically approach f is defined (and can be calculated) for "all" x \in R, (mathematically "all"=infinity of real numebrs, numerically "all"=a lot of real numbers), right?

Again: I need some of these reasonings into a scientific paper and I want to acknowledge it, but beside your name I need your affiliation

Thanks!

[Walter Roberson]

Another way of looking at it is that by the Pigeon Hole Principle, if you have an N-bit representation of numbers, then that can represent at most 2^N different numbers, and other numbers cannot exactly exist within that system. This should be no more astonishing than if you have an integer data type and notice that the integer data type cannot represent fractions.