Handling a very big difference between numbers(ratio)

Hello,

I was working on a code that demands operating with statistics of long numbers (such as 2000000) vs the errors(such as 0.0003) so different equations that demands both of the numbers are equal 0 or Inf, how can I improve my situation?

I tried to use vpa, it helps but it's not a magic pill and the problems appers later in the code.

Is there anyway to operate with such numbers in terms of division without getting results as Inf or 0?

Thank you and a blessed week,

Vadim

[Chunru]

Have you considered the log scale?

[Mathieu NOE]

hi

why not working with log of your numbers / ratio ? this way the "difference" is much reduced.

[Vadim Patrick Nave]

Hey, and thank you for your answer!

Mainly it is problematic because I need to preform several matrix operations and than the numbers are super important.

[Steven Lord]

I was working on a code that demands operating with statistics of long numbers (such as 2000000) vs the errors(such as 0.0003) so different equations that demands both of the numbers are equal 0 or Inf, how can I improve my situation?

It's likely to be difficult if not impossible to offer any specific suggestions without seeing the specific equations you're using.

[Vadim Patrick Nave]

Thank you Steven, you are right. Sorry.

While dy is a very small number

x is very large number

% Fit line analytically:
matrix = [ sum(x.^2./dy.^2)       sum(x./dy.^2);
    sum(x./dy.^2)          sum(1./dy.^2)];
y_sigma = y./dy.^2;
free_vector = [sum(x.*y_sigma);
    sum(1.*y_sigma)];
solution=matrix\free_vector;           
a=solution(1);b=solution(2);
err=sqrt(diag(inv(matrix)));
da=err(1);db=err(2);
% if there are errors in x, co-add them and re-fit
if nargin>10
    dy = sqrt((a*dx).^2+dy.^2);
    matrix = [ sum(x.^2./dy.^2)       sum(x./dy.^2);
        sum(x./dy.^2)          sum(1./dy.^2)];
    y_sigma = y./dy.^2;
    free_vector = [sum(x.*y_sigma);
        sum(1.*y_sigma)];
    solution=matrix\free_vector;           
    a=solution(1);b=solution(2);
    err=sqrt(diag(inv(matrix)));
    da=err(1);db=err(2);
end

Answer 1

[Infinite_king]

Hi Vadim Patrick Nave,

I understand that you want to work with very large and very small numbers and perform arithmetic operations on them without running into ‘Inf’ or ‘nan’ values.

I suggest you to use ‘sym’ function which was available in ‘Symbolic Math Toolbox’. First convert the number to a symbolic number or matrix to symbolic matrix. Then you can perform simple arithmetic operations and finally you can use ‘double’ function to convert the answer to double.

Refer below code snippet,

% let x be a matrix of numbers
x = rand(5);
disp(x);

    0.6448    0.1331    0.2477    0.0948    0.9077
    0.0157    0.2160    0.5678    0.6035    0.4680
    0.1102    0.4070    0.8393    0.2991    0.9093
    0.2817    0.2833    0.9535    0.0625    0.5194
    0.2113    0.5092    0.7433    0.0416    0.2276

% now convert the matrix to symbolic matrix
x_sym = sym(x);

% now perform simple arithmetic operations
% op 1
% op 2
% for example, addition
x_sym = x_sym + 5;

% now convert the values to double
% make sure the numbers are within range of double
res = double(x_sym);
disp(res);

    5.6448    5.1331    5.2477    5.0948    5.9077
    5.0157    5.2160    5.5678    5.6035    5.4680
    5.1102    5.4070    5.8393    5.2991    5.9093
    5.2817    5.2833    5.9535    5.0625    5.5194
    5.2113    5.5092    5.7433    5.0416    5.2276

For more information on how to use ‘sym’ function and ‘Symbolic Math Toolbox’, please refer the following MATLAB documentations,

• https://www.mathworks.com/help/symbolic/sym.html
• https://www.mathworks.com/products/symbolic.html
• https://www.mathworks.com/help/symbolic/sym.double.html

Hope this is helpful.

[Walter Roberson]

and final you can use ‘eval’ function to convert the answer to double.

No you cannot. There is no documented meaning for eval() of a symbolic expression.

syms x
f = piecewise(1 <= x & x <= 5, x, x^2)

f = 

char(f)

ans = 'piecewise(x in Dom::Interval([1], [5]), x, symtrue, x^2)'

eval(f)

Error using sym/eval
Invalid expression. Check for missing multiplication operator, missing or unbalanced delimiters, or other syntax error. To construct matrices, use brackets instead of parentheses.

Symbolic expressions displayed to the user are in a user-friendly-ish language, not in a programming language. They are not expressions in MATLAB, and by the time they reach the user, they are not expressions in the internal MuPAD programming language. They can use constructs that are not accepted by MATLAB, such as the above in-fix notation use of the "in" operator, and such as the :: construction . The order of parameters they show might not be the same as the order that a numeric function of the same name expects, and might not be the same as the order that a symbolic function of the same name expects, and might not be the same as the order that the internal MuPAD programming language expects.

[Infinite_king]

Hi Walter Roberson,

Here we are not using symbolic varialbes. we are using symbolic numbers.

[Walter Roberson]

@Infinite_king

Still no defined meaning for eval() of a symbolic number.

If you have a symbolic expression, you can vpa() it and hope that the result is a numeric approximation of the result (will not always be, especially if the value involves a very bumpy integral)

If you have a symbolic expression that has no "free variables" in it, you can double() it and hope a numeric approximation could be created (will not always work for values that theoretically exist but are tough to compute.)

[Walter Roberson]

So let us take a closer look to the documentation of eval() that you linked to https://www.mathworks.com/help/matlab/ref/eval.html

Syntax

eval(expression)

[output1,...,outputN] = eval(expression)

And what is an expression?

https://www.mathworks.com/help/matlab/ref/eval.html#mw_da6cc3d7-b0f6-46b1-9e81-f9c96e554611

expression — Expression to evaluate

character vector | string scalar

Notice those are not symbolic expressions or symbolic numbers.

You will not find eval() documented anywhere in the Symbolic Toolbox. https://www.mathworks.com/help/symbolic/index.html

[Walter Roberson]

x = 5;
x_sym = sym(x);
x_sym = x_sym * 5;
x_sym = x_sym + 5;
x_sym = x_sym / 5;
% how to evaluate x_sym, assuming the resulting value is within ranage of
% double.
double(x_sym)

ans = 6

No eval() needed.

See also vpa such as

vpa(cos(sym(pi)^2), 50)

ans = 

[Infinite_king]

Thanks for the information Walter.