Problem adding the large number with small number

Question

Erkan 2022-1-8

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1625170-problem-adding-the-large-number-with-small-number

评论： Star Strider 2022-1-9

采纳的回答： Star Strider

Hello everyone, I have a problem with adding numbers.

I have two numbers;

a=6.3276e+16 and b=17.3304, when adding these numbers, the result is a, that is, no addition is made.

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Erkan 2022-1-8

How can i solve this issue, thanks in advance

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Answer 1

Star Strider 2022-1-8

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1625170-problem-adding-the-large-number-with-small-number#answer_871045

在 MATLAB Online 中打开

The addition is made. It is necessary to display the appropriate precision to see it —

a=6.3276e+16
a = 6.3276e+16
b=17.3304
b = 17.3304
a_plus_b = a + b
a_plus_b = 6.3276e+16
fprintf('%33.25E',a)
  6.3276000000000000000000000E+16
fprintf('%33.25E',a_plus_b)
  6.3276000000000016000000000E+16
fprintf('%33.25E',a_plus_b - a)
  1.6000000000000000000000000E+01

Floating-point numbers aren’t as accurate as necessary in that respect, however thye do their best!

.

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Image Analyst 2022-1-8

在 MATLAB Online 中打开

There will come a point when the one number is so big that when the smaller number is converted to the same exponent (the larger one, which is required so that they can be added together), there won't be enough digits in the computer to hold the smaller number and the result is the smaller number will be zero and when adding it you'll end up with exactly the bigger number no matter how many decimal places you print out:

a=6.3276e+36
a = 6.3276e+36
b=17.3304
b = 17.3304
a_plus_b = a + b
a_plus_b = 6.3276e+36
fprintf('a            = %.75E\n',a)
a            = 6.327600000000000206026557892920868864000000000000000000000000000000000000000E+36
fprintf('a_plus_b     = %.75E\n',a_plus_b)
a_plus_b     = 6.327600000000000206026557892920868864000000000000000000000000000000000000000E+36
fprintf('a_plus_b - a = %.75E\n',a_plus_b - a)
a_plus_b - a = 0.000000000000000000000000000000000000000000000000000000000000000000000000000E+00

This is called "truncation error" (if you'd have taken a linear algebra or numerical analysis course, you should have learned about that.)

https://en.wikipedia.org/wiki/Truncation_error_(numerical_integration)

Erkan 2022-1-9

Thanks for answer

Star Strider 2022-1-9

在 MATLAB Online 中打开

As always, my pleasure!

Espanding on this with the Symbolic Math Toolbox —

a = vpa(sym(6.3276e+16))
a = 63276000000000000.0
b = vpa(sym(17.3304))
b = 17.3304
a_plus_b = vpa(a + b, 30)
a_plus_b = 63276000000000017.3304

So to retain full precision in interim calculations, use symbolic variables.

format long
a_plus_b = double(a_plus_b)
a_plus_b = 
     6.327600000000002e+16
fprintf('%33.25E', a_plus_b)
  6.3276000000000016000000000E+16

The double conversion reverts to double-precision representation. Note that the representation is not the exact value, as computed using symbolic variables.

.

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