Change variable precision used - MATLAB digits (2024)

Syntax

Increase Precision of Results

By default, MATLAB^® uses 16 digits of precision.For higher precision, use vpa.The default precision for vpa is 32 digits. Increaseprecision beyond 32 digits by using digits.

Find pi using vpa, whichuses the default 32 digits of precision. Confirm that the currentprecision is 32 by using digits.

pi32 = vpa(pi)

pi32 =3.1415926535897932384626433832795

currentPrecision = digits

currentPrecision = 32

Save the current value of digits in digitsOld andset the new precision to 100 digits. Find pi using vpa.The result has 100 digits.

digitsOld = digits(100);pi100 = vpa(pi)

pi100 =3.1415926535897932384626433832795028841971693993751058209...74944592307816406286208998628034825342117068

Lastly, restore the old value of digits forfurther calculations.

digits(digitsOld)

For more information, see Increase Precision of Numeric Calculations.

Increase Speed by Decreasing Precision

Increase the speed of MATLAB calculationsby using vpa with a lower precision. Set thelower precision by using digits.

First, find the time taken to perform an operation on a largeinput.

input = 1:0.01:500;ticzeta(input);toc

Elapsed time is 48.968983 seconds.

Now, repeat the operation with a lower precision by using vpa.Lower the precision to 10 digits by using digits.Then, use vpa to reduce the precision of input andperform the same operation. The time taken decreases significantly.

digitsOld = digits(10);vpaInput = vpa(input);ticzeta(vpaInput);toc

Elapsed time is 31.450342 seconds.

Lastly, restore the old value of digits forfurther calculations.

digits(digitsOld)

For more information, see Increase Speed by Reducing Precision.

Guard Digits

The number of digits that you specify using the vpa functionor the digits function is the guaranteed numberof digits. Internally, the toolbox can use a few more digits thanyou specify. These additional digits are called guard digits.For example, set the number of digits to 4, and then display the floating-pointapproximation of 1/3 using four digits:

old = digits(4);a = vpa(1/3)

a =0.3333

Now, display a using 20 digits. The resultshows that the toolbox internally used more than four digits whencomputing a. The last digits in the following resultare incorrect because of the round-off error:

digits(20)vpa(a)digits(old)

ans =0.33333333333303016843

Hidden Round-Off Errors

Hidden round-off errors can cause unexpected results. For example,compute the number 1/10 with the default 32-digit accuracy and with10-digit accuracy:

a = vpa(1/10)old = digits(10);b = vpa(1/10)digits(old)

a =0.1 b =0.1

Now, compute the difference a - b. The resultis not 0:

a - b

ans =0.000000000000000000086736173798840354720600815844403

The difference a - b is not equal to zerobecause the toolbox internally boosts the 10-digit number b= 0.1 to 32-digit accuracy. This process implies round-offerrors. The toolbox actually computes the difference a -b as follows:

b = vpa(b)a - b

b =0.09999999999999999991326382620116 ans =0.000000000000000000086736173798840354720600815844403

Techniques Used to Convert Floating-Point Numbers to Symbolic Objects

Suppose you convert a double number to a symbolic object, andthen perform VPA operations on that object. The results can dependon the conversion technique that you used to convert a floating-pointnumber to a symbolic object. The sym function letsyou choose the conversion technique by specifying the optional secondargument, which can be 'r', 'f', 'd',or 'e'. The default is 'r'.For example, convert the constant π=3.141592653589793... toa symbolic object:

r = sym(pi)f = sym(pi,'f')d = sym(pi,'d')e = sym(pi,'e')

r =pi f =884279719003555/281474976710656 d =3.1415926535897931159979634685442 e =pi - (198*eps)/359

Although the toolbox displays these numbers differently on thescreen, they are rational approximations of pi.Use vpa to convert these rational approximationsof pi back to floating-point values.

Set the number of digits to 4. Three of the four approximationsgive the same result.

digits(4)vpa(r)vpa(f)vpa(d)vpa(e)

ans =3.142 ans =3.142 ans =3.142 ans =3.142 - 0.5515*eps

Now, set the number of digits to 40. The differences betweenthe symbolic approximations of pi become more visible.

digits(40)vpa(r)vpa(f)vpa(d)vpa(e)

ans =3.141592653589793238462643383279502884197 ans =3.141592653589793115997963468544185161591 ans =3.1415926535897931159979634685442 ans =3.141592653589793238462643383279502884197 -...0.5515320334261838440111420612813370473538*eps

Input Arguments

Output Arguments

Version History

Introduced before R2006a

See Also

double | vpa