# Global precision setting

Coming from Maple I do not understand how the precision for numerical computations in Mathematica is specified. I understand that there are various options to commands such as WorkingPrecision and PrecisionGoal. But I would like to use the same precision (above machine precision) for a number of computations including matrix operations and the FindRoot command outside and inside of routines. Also I would like to specify the precision of the output.

• You can look at the documentation for the functions listed when you evaluate ?$*Precision. You can do fixed precision calculations with Block[{$MaxPrecision=..., $MinPrecision=...}, ...] or set these globally to affect all functions that rely on it – rm -rf Commented Jun 4, 2012 at 19:50 • I tried setting$MinPrecision=20 already. Strangely I still get results with ScientificForm[%, 20] with just 16 digits. Do matrix computations and FindRoot depend on it? Commented Jun 4, 2012 at 20:02
• @highsciguy yes, but you have to be careful not to introduce machine-precision numbers at any point, which "poison" the result. That is, all numbers specified as decimals should have a precision annotation, e.g. 1.020. Also, you should be aware that some matrix decompositions are done in machine precision using LAPACK. Commented Jun 4, 2012 at 20:12
• I see. How do I tell mathematica that all numbers e.g. 1.5 are actually 20 Digits precision? SetPrecision on all numbers or add the 20 everywhere? Commented Jun 4, 2012 at 20:16
• You can use either approach. SetPrecision will take the machine-precision value and extend it with base-2 zeros up to the required precision, which may not be what you want (since zeros in base 2 are not necessarily so in base 10; e.g. SetPrecision[1.9, 20] gives a result slightly less than 1.9). If you use the annotation, the zeros are taken to be in base 10 instead. Another possible approach is to use Rationalize. Commented Jun 4, 2012 at 21:53

How do I tell mathematica that all numbers e.g. 1.5 are actually 20 Digits precision? SetPrecision on all numbers or add the 20 everywhere?

You could force this with $PreRead. This naive definition is likely inefficient and probably breaks a number of corner cases I have not considered, but here is a rough demonstration: $PreRead = (# /.
s_String /;
StringMatchQ[s, NumberString] &&
Precision@ToExpression@s == MachinePrecision :> s <> "20." &);

3/1.5 + Pi/7

Precision[%]

2.4487989505128276055

20.0879


As Alexey notes this breaks if the machine number string already has a "NumberMark" after it e.g. 1.23. One could use a more precise string replacement to avoid this.

A different approach is to process at the expression rather than box level, though this simple first attempt probably fails in some cases as well:

$Pre = Function[Null, Unevaluated[#] /. r_Real?MachineNumberQ :> RuleCondition@SetPrecision[r, 25], HoldAllComplete]  Now: MachineNumberQ[2.2] ToString[3.14]  False "3.140000000000000124344979"  • Your solution fails when the number entered has the NumberMark after digits, for example 1.5 . Commented Dec 1, 2013 at 19:24 • @Alexey I never thought it would be robust as written. Let me see if I can improve it some. Commented Dec 1, 2013 at 20:42 • Your second solution works in Mathematica 5.2 and 7.0.1 but not in v.8.0.4. Looks like a bug in $Pre in v.8.0.4. Commented Dec 1, 2013 at 21:07
• @Mr.Wizard: Your first solution also doesn't appear to work with numbers with 18 or more significant figures. For example: if you set the precision in your function equal to say, 50, then evaluate z = 1.2345678901234567 and then Precision[z], you get a precision of 50. By contrast, with z = 1.23456789012345678, you get a precision of 17.0915, which is equal to its native precision. Commented Dec 29, 2016 at 23:19
• @theorist This code was designed to work with machine precision numbers only. Numbers entered with additional digits are automatically interpreted as arbitrary precision; I did not wish to override the precision those. If you leave out && Precision@ToExpression@s == MachinePrecision it should force those too. Commented Dec 31, 2016 at 18:41

There is a quick-n-dirty solution. Set

$MinPrecision = 100  And then enter numbers something like x = 1.012;  You will be getting warnings as Precision::precsm: Requested precision 2. is smaller than$MinPrecision.
Using $MinPrecision instead.  but in this way you if you want to change precision you just change $MinPrecision value.

In[21]:= x

Out[21]= 1.\
0100000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000

• I don't even get the Precision::precsm warning message. However it does not work on machine numbers; the entry form as you rightly noted is critical. +1 for a simple method that may work in a number of applications. Commented Feb 10, 2016 at 18:52
• This is 10x easier than the accepted answer. Commented May 11, 2021 at 12:45

First time posting, but for any googlers of this issue, the "global" precision of machine precision numbers can be set by:

Unprotect[$$MachinePrecision];$$MachinePrecision = 100;
Protect[\$MachinePrecision];

• Does this setting affect any evaluations? Commented Jun 1, 2017 at 9:27
• This does not affect MachinePrecision (no dollar sign) for example. I cannot imagine that it will do anything to change internals when a number like 3.3 is encountered (Try it: In[20]:= Precision[3.3] Out[20]= MachinePrecision). On top of which, it could have unpredictable effects which I would not view as a benefit. Commented Jun 1, 2017 at 16:04