Precision and realmax
I was on a recent email thread at the office regarding realmax, eps, and overflow.
What is realmax?
From the help for realmax we learn that
h = help('realmax'); disp(h(1:48))
REALMAX Largest positive floating point number.
Here's the double precision value for realmax shown with about 15 digits.
format long rmax = realmax
rmax = 1.797693134862316e+308
Overflow with Respect to realmax
Let's see what it takes to add to realmax to get to the next number.
epsrmax = eps(rmax) big = rmax+epsrmax
epsrmax = 1.995840309534720e+292 big = Inf
Are you surprised that the next largest number is Inf?
How about Less than eps?
What if we try a number a tenth of eps?
biggish = rmax + epsrmax/10 isequal(biggish, rmax)
biggish = 1.797693134862316e+308 ans = 1
It's not far enough away on the floating point scale to produce a value different from rmax. In fact, the help for eps has a list of values for eps at some interesting double and single values.
EPS Spacing of floating point numbers. D = EPS(X), is the positive distance from ABS(X) to the next larger in magnitude floating point number of the same precision as X. X may be either double precision or single precision. For all X, EPS(X) is equal to EPS(ABS(X)). EPS, with no arguments, is the distance from 1.0 to the next larger double precision number, that is EPS with no arguments returns 2^(-52). EPS('double') is the same as EPS, or EPS(1.0). EPS('single') is the same as EPS(single(1.0)), or single(2^-23). Except for numbers whose absolute value is smaller than REALMIN, if 2^E <= ABS(X) < 2^(E+1), then EPS(X) returns 2^(E-23) if ISA(X,'single') EPS(X) returns 2^(E-52) if ISA(X,'double') For all X of class double such that ABS(X) <= REALMIN, EPS(X) returns 2^(-1074). Similarly, for all X of class single such that ABS(X) <= REALMIN('single'), EPS(X) returns 2^(-149). Replace expressions of the form if Y < EPS * ABS(X) with if Y < EPS(X) Example return values from calling EPS with various inputs are presented in the table below: Expression Return Value =========================================== eps(1/2) 2^(-53) eps(1) 2^(-52) eps(2) 2^(-51) eps(realmax) 2^971 eps(0) 2^(-1074) eps(realmin/2) 2^(-1074) eps(realmin/16) 2^(-1074) eps(Inf) NaN eps(NaN) NaN ------------------------------------------- eps(single(1/2)) 2^(-24) eps(single(1)) 2^(-23) eps(single(2)) 2^(-22) eps(realmax('single')) 2^104 eps(single(0)) 2^(-149) eps(realmin('single')/2) 2^(-149) eps(realmin('single')/16) 2^(-149) eps(single(Inf)) single(NaN) eps(single(NaN)) single(NaN) See also REALMAX, REALMIN. Overloaded methods: codistributed/eps quantizer/eps qfft/eps Reference page in Help browser doc eps
Here are some references to help you learn more about floating point issues.
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