To handle rational number without losing accuracy of computation in Matlab? - matlab

Question

I want to use this rational number in computations without losing the accuracy of the picture in Matlab:
f = 359.0 + 16241/16250.0
I think storing, for instance by f = uint64(359.0 + 16241/16250.0) loses accuracy, seen as 360 in Matlab.
I think the best way to handle the thing is never to store the value but to store its factors like
% f = a + b/c
a = 359
b = 16241
c = 16250
and then doing computation by the variables a, b and c, and giving the result as a picture.
Is this a good way to maintain the accuracy?

Answer 1

As you suggest, if you absolutely don't want to lose accuracy when storing a rational number, the best solution probably is to store the number in terms of its integer components.
Instead of your three components (f = a + b/c) you can reduce the reprentation to two components: f = n/d. Thus each rational number would be defined (and stored) as the two-component integer vector [n d]. For example, the number f in your example corresponds to n=5849991 and d=16250.
To simplify handling rational numbers stored this way, you could define a helper function which converts from the [n d] representation to n/d before applyling the desired operation:
useInteger = #(x, nd, fun) fun(x,double(nd(1))/double(nd(2)));
Then
>> x = sqrt(pi);
>> nd = int64([5849991 16250]);
>> useInteger(x, nd, #plus)
ans =
361.7719
>> useInteger(x, nd, #times)
ans =
638.0824
If you want to achieve arbitrarily high precision in computations, you should consider using variable-precision arithmetic (vpa) with string arguments. With that approach you get to specify how many digits you want:
>> vpa('sqrt(pi)*5849991/16250', 50)
ans =
638.08240465923757600307902117159072301901656248436

Answer 2

Perhaps create a Rational class and define the needed operations (plus,minus,times,etc.). Start with something like this:
Rational.m
classdef Rational
properties
n;
d;
end
methods
function obj = Rational(n,d)
GCD = gcd(n,d);
obj.n = n./GCD;
obj.d = d./GCD;
end
function d = dec(obj)
d = double(obj.n)/double(obj.d);
end
% X .* Y
function R = times(X,Y)
chkxy(X,Y);
if isnumeric(X),
N = X .* Y.n; D = Y.d;
elseif isnumeric(Y),
N = X.n .* Y; D = X.d;
else
N = X.n .* Y.n; D = X.d .* Y.d;
end
R = Rational(N,D);
end
% X * Y
function R = mtimes(X,Y)
R = times(X,Y);
end
% X ./ Y
function R = rdivide(X,Y)
if isnumeric(Y),
y = Rational(1,Y);
else
y = Rational(Y.d,Y.n);
end
R = times(X,y);
end
% X / Y
function R = mrdivide(X,Y)
R = rdivide(X,Y);
end
% X + Y
function R = plus(X,Y)
chkxy(X,Y);
if isnumeric(X),
N = X.*Y.d + Y.n; D = Y.d;
elseif isnumeric(Y),
N = Y.*X.d + X.n; D = X.d;
else
D = lcm(X.d,Y.d);
N = sum([X.n Y.n].*(D./[X.d Y.d]));
end
R = Rational(N,D);
end
% X - Y
function R = minus(X,Y)
R = plus(X,-Y);
end
% -X
function R = uminus(X)
R = Rational(-X.n,X.d);
end
function chkxy(X,Y)
if (~isa(X, 'Rational') && ~isnumeric(X)) || ...
(~isa(Y, 'Rational') && ~isnumeric(Y)),
error('X and Y must be Rational or numeric.');
end
end
end
end
Examples
Construct objects:
>> clear all % reset class definition
>> r1 = Rational(int64(1),int64(2))
r1 =
Rational with properties:
n: 1
d: 2
>> r2 = Rational(int64(3),int64(4))
r2 =
Rational with properties:
n: 3
d: 4
Add and subtract:
>> r1+r2
ans =
Rational with properties:
n: 5
d: 4
>> r1-r2
ans =
Rational with properties:
n: -1
d: 4
Multiply and divide:
>> r1*r2
ans =
Rational with properties:
n: 3
d: 8
>> r1/r2
ans =
Rational with properties:
n: 2
d: 3
Get decimal value:
>> r12 = r1/r2; % 2/3 ((1/2)/(3/4))
>> f = r12.dec
f =
0.6667

Answer 3

Extension to LuisMendo's answer
I got this as the error for your suggestion by py
>>> a = 638.08240465923757600307902117159072301901656248436059
>>> a
638.0824046592376 % do not know if Python is computing here with exact number
>>> b = 638.0824
>>> ave = abs(b+a)/2
>>> diff = abs(b-a)
>>> ave = abs(b+a)/2
>>> diff/ave
7.30193709165014e-09
which is more than the proposed error storing error above.
I run in WolframAlpha
x = sqrt(pi)
x*5849991/16250
and get
509.11609919757198016211937362635174599076143654820109
I am not sure if this is what you meant in your comment of your answer.

Answer 4

Extension to chappjc's answer.
I have now
[B,T,F] = tfrwv(data1, 1:length(data1), length(data1)); % here F double
fs = Rational(uint64(5849991), uint64(16250));
t = 1/fs;
imagesc(T*t, F*fs, B);
I run it
Error using .*
Integers can only be combined with integers of
the same class, or scalar doubles.
Error in .* (line 23)
N = X .* Y.n; D = Y.d;
Error in * (line 34)
R = times(X,Y);
How can you multiply in this class the double with Rational?