# User Guide¶

## Basic setup¶

Basic mathematical operations involving numbers with uncertainties only require a simple import:

```
>>> from uncertainties import ufloat
```

The `ufloat()`

function creates numbers with uncertainties. Existing
calculation code can usually run with no or little modification and
automatically produce results with uncertainties.

The `uncertainties`

module contains other features, which can be
made accessible through

```
>>> import uncertainties
```

The `uncertainties`

package also contains sub-modules for
advanced mathematical functions, and
arrays and matrices.

## Creating numbers with uncertainties¶

Numbers with uncertainties can be input either numerically, or through one of many string representations, so that files containing numbers with uncertainties can easily be parsed. Thus, x = 0.20±0.01 can be expressed in many convenient ways, including:

```
>>> x = ufloat(0.20, 0.01) # x = 0.20+/-0.01
```

```
>>> from uncertainties import ufloat_fromstr
>>> x = ufloat_fromstr("0.20+/-0.01")
>>> x = ufloat_fromstr("(2+/-0.1)e-01") # Factored exponent
>>> x = ufloat_fromstr("0.20(1)") # Short-hand notation
>>> x = ufloat_fromstr("20(1)e-2") # Exponent notation
>>> x = ufloat_fromstr(u"0.20±0.01") # Pretty-print form
>>> x = ufloat_fromstr("0.20") # Automatic uncertainty of +/-1 on last digit
```

Each number created this way is an **independent (random) variable**
(for details, see the Technical Guide).

More information can be obtained with `pydoc uncertainties.ufloat`

and `pydoc uncertainties.ufloat_fromstr`

(“20(1)×10^{-2}” is
also recognized, etc.).

## Basic math¶

Calculations can be performed directly, as with regular real numbers:

```
>>> square = x**2
>>> print square
0.040+/-0.004
```

## Mathematical operations¶

Besides being able to apply basic mathematical operations to numbers
with uncertainty, this package provides generalizations of **most of
the functions from the standard** `math`

**module**. These
mathematical functions are found in the `uncertainties.umath`

module:

```
>>> from uncertainties.umath import * # Imports sin(), etc.
>>> sin(x**2)
0.03998933418663417+/-0.003996800426643912
```

The list of available mathematical functions can be obtained with the
`pydoc uncertainties.umath`

command.

### NaN testing¶

NaN values can appear in a number with uncertainty. Care must be
taken with such values, as values like NaN±1, 1±NaN and NaN±NaN are by
definition *not* NaN, which is a float.

Testing whether a number with uncertainty has a **NaN nominal value** can
be done with the provided function `uncertainties.umath.isnan()`

,
which generalizes the standard `math.isnan()`

.

Checking whether the *uncertainty* of `x`

is NaN can be done directly
with the standard function: `math.isnan(x.std_dev)`

(or equivalently
`math.isnan(x.s)`

).

## Arrays of numbers with uncertainties¶

It is possible to put numbers with uncertainties in NumPy arrays and matrices:

```
>>> arr = numpy.array([ufloat(1, 0.01), ufloat(2, 0.1)])
>>> 2*arr
[2.0+/-0.02 4.0+/-0.2]
>>> print arr.sum()
3.00+/-0.10
```

Thus, usual operations on NumPy arrays can be performed transparently even when these arrays contain numbers with uncertainties.

More complex operations on NumPy arrays and matrices can be
performed through the dedicated `uncertainties.unumpy`

module.

## Automatic correlations¶

Correlations between variables are **automatically handled** whatever
the number of variables involved, and whatever the complexity of the
calculation. For example, when `x`

is the number with
uncertainty defined above,

```
>>> square = x**2
>>> print square
0.040+/-0.004
>>> square - x*x
0.0+/-0
>>> y = x*x + 1
>>> y - square
1.0+/-0
```

The last two printed results above have a zero uncertainty despite the
fact that `x`

, `y`

and `square`

have a non-zero uncertainty: the
calculated functions give the same value for all samples of the random
variable `x`

.

Thanks to the automatic correlation handling, calculations can be
performed in as many steps as necessary, exactly as with simple
floats. When various quantities are combined through mathematical
operations, the result is calculated by taking into account all the
correlations between the quantities involved. All of this is done
completely **transparently**.

## Access to the uncertainty and to the nominal value¶

The nominal value and the uncertainty (standard deviation) can also be accessed independently:

```
>>> print square
0.040+/-0.004
>>> print square.nominal_value
0.04
>>> print square.n # Abbreviation
0.04
>>> print square.std_dev
0.004
>>> print square.s # Abbreviation
0.004
```

## Access to the individual sources of uncertainty¶

The various contributions to an uncertainty can be obtained through the
`error_components()`

method, which maps the **independent variables
a quantity depends on** to their **contribution to the total
uncertainty**. According to linear error propagation theory (which is the method followed by `uncertainties`

),
the sum of the squares of these contributions is the squared
uncertainty.

The individual contributions to the uncertainty are more easily usable
when the variables are **tagged**:

```
>>> u = ufloat(1, 0.1, "u variable") # Tag
>>> v = ufloat(10, 0.1, "v variable")
>>> sum_value = u+2*v
>>> sum_value
21.0+/-0.223606797749979
>>> for (var, error) in sum_value.error_components().items():
... print "{}: {}".format(var.tag, error)
...
u variable: 0.1
v variable: 0.2
```

The variance (i.e. squared uncertainty) of the result
(`sum_value`

) is the quadratic sum of these independent
uncertainties, as it should be (`0.1**2 + 0.2**2`

).

The tags *do not have to be distinct*. For instance, *multiple* random
variables can be tagged as `"systematic"`

, and their contribution to
the total uncertainty of `result`

can simply be obtained as:

```
>>> syst_error = math.sqrt(sum( # Error from *all* systematic errors
... error**2
... for (var, error) in result.error_components().items()
... if var.tag == "systematic"))
```

The remaining contribution to the uncertainty is:

```
>>> other_error = math.sqrt(result.std_dev**2 - syst_error**2)
```

The variance of `result`

is in fact simply the quadratic sum of
these two errors, since the variables from
`result.error_components()`

are independent.

## Comparison operators¶

Comparison operators behave in a natural way:

```
>>> print x
0.200+/-0.010
>>> y = x + 0.0001
>>> y
0.2001+/-0.01
>>> y > x
True
>>> y > 0
True
```

One important concept to keep in mind is that `ufloat()`

creates a
random variable, so that two numbers with the same nominal value and
standard deviation are generally different:

```
>>> y = ufloat(1, 0.1)
>>> z = ufloat(1, 0.1)
>>> print y
1.00+/-0.10
>>> print z
1.00+/-0.10
>>> y == y
True
>>> y == z
False
```

In physical terms, two rods of the same nominal length and uncertainty
on their length are generally of different sizes: `y`

is different
from `z`

.

More detailed information on the semantics of comparison operators for numbers with uncertainties can be found in the Technical Guide.

## Covariance and correlation matrices¶

### Covariance matrix¶

The covariance matrix between various variables or calculated quantities can be simply obtained:

```
>>> sum_value = u+2*v
>>> cov_matrix = uncertainties.covariance_matrix([u, v, sum_value])
```

has value

```
[[0.01, 0.0, 0.01],
[0.0, 0.01, 0.02],
[0.01, 0.02, 0.05]]
```

In this matrix, the zero covariances indicate that `u`

and `v`

are
independent from each other; the last column shows that `sum_value`

does depend on these variables. The `uncertainties`

package
keeps track at all times of all correlations between quantities
(variables and functions):

```
>>> sum_value - (u+2*v)
0.0+/-0
```

### Correlation matrix¶

If the NumPy package is available, the correlation matrix can be obtained as well:

```
>>> corr_matrix = uncertainties.correlation_matrix([u, v, sum_value])
>>> corr_matrix
array([[ 1. , 0. , 0.4472136 ],
[ 0. , 1. , 0.89442719],
[ 0.4472136 , 0.89442719, 1. ]])
```

## Printing¶

Numbers with uncertainties can be printed conveniently:

```
>>> print x
0.200+/-0.010
```

The resulting form can generally be parsed back with
`ufloat_fromstr()`

(except for the LaTeX form).

The nominal value and the uncertainty always have the **same
precision**: this makes it easier to compare them.

### Standard formats¶

More **control over the format** can be obtained (in Python 2.6+)
through the usual `format()`

method of strings:

```
>>> print 'Result = {:10.2f}'.format(x)
Result = 0.20+/- 0.01
```

(Python 2.6 requires `'{0:10.2f}'`

instead, with the usual explicit
index. In Python 2.5 and earlier versions, `str.format()`

is not
available, but one can use the `format()`

method of numbers with
uncertainties instead: `'Result = %s' % x.format('10.2f')`

.)

**All the float format specifications** are accepted, except those
with the `n`

format type. In particular, a fill character, an
alignment option, a sign or zero option, a width, or the `%`

format
type are all supported.

The usual **float formats with a precision** retain their original
meaning (e.g. `.2e`

uses two digits after the decimal point): code
that works with floats produces similar results when running with
numbers with uncertainties.

### Precision control¶

It is possible to **control the number of significant digits of the
uncertainty** by adding the precision modifier `u`

after the
precision (and before any valid float format type like `f`

, `e`

,
the empty format type, etc.):

```
>>> print '1 significant digit on the uncertainty: {:.1u}'.format(x)
1 significant digit on the uncertainty: 0.20+/-0.01
>>> print '3 significant digits on the uncertainty: {:.3u}'.format(x)
3 significant digits on the uncertainty: 0.2000+/-0.0100
>>> print '1 significant digit, exponent notation: {:.1ue}'.format(x)
1 significant digit, exponent notation: (2.0+/-0.1)e-01
>>> print '1 significant digit, percentage: {:.1u%}'.format(x)
1 significant digit, percentage: (20+/-1)%
```

When `uncertainties`

must **choose the number of significant
digits on the uncertainty**, it uses the Particle
Data Group rounding
rules (these rules keep the number of digits small, which is
convenient for reading numbers with uncertainties, and at the same
time prevent the uncertainty from being displayed with too few
digits):

```
>>> print 'Automatic number of digits on the uncertainty: {}'.format(x)
Automatic number of digits on the uncertainty: 0.200+/-0.010
>>> print x
0.200+/-0.010
```

### Custom options¶

`uncertainties`

provides even more flexibility through custom
formatting options. They can be added at the end of the format string:

`P`

for**pretty-printing**:>>> print '{:.2e}'.format(x) (2.00+/-0.10)e-01 >>> print u'{:.2eP}'.format(x) (2.00±0.10)×10⁻¹

The pretty-printing mode thus uses “±”, “×” and superscript exponents. Note that the pretty-printing mode implies using

**Unicode format strings**(`u'…'`

in Python 2, but simply`'…'`

in Python 3).`S`

for the**shorthand notation**:>>> print '{:+.1uS}'.format(x) # Sign, 1 digit for the uncertainty, shorthand +0.20(1)

In this notation, the digits in parentheses represent the uncertainty on the last digits of the nominal value.

`L`

for a**LaTeX**output:>>> print x*1e7 (2.00+/-0.10)e+06 >>> print '{:L}'.format(x*1e7) # Automatic exponent form, LaTeX \left(2.00 \pm 0.10\right) \times 10^{6}

`p`

is for requiring that parentheses be always printed around the …±… part (without enclosing any exponent or trailing “%”, etc.). This can for instance be useful so as to explicitly factor physical units:>>> print '{:p} kg'.format(x) # Adds parentheses (0.200+/-0.010) kg >>> print("{:p} kg".format(x*1e7)) # No parentheses added (exponent) (2.00+/-0.10)e+06 kg

These custom formatting options **can be combined** (when meaningful).

### Details¶

A **common exponent** is automatically calculated if an exponent is
needed for the larger of the nominal value (in absolute value) and the
uncertainty (the rule is the same as for floats). The exponent is
generally **factored**, for increased legibility:

```
>>> print x*1e7
(2.00+/-0.10)e+06
```

When a *format width* is used, the common exponent is not factored:

```
>>> print 'Result = {:10.1e}'.format(x*1e-10)
Result = 2.0e-11+/- 0.1e-11
```

(Using a (minimal) width of 1 is thus a way of forcing exponents to not be factored.) Thanks to this feature, each part (nominal value and standard deviation) is correctly aligned across multiple lines, while the relative magnitude of the error can still be readily estimated thanks to the common exponent.

An uncertainty which is *exactly* **zero** is always formatted as an
integer:

```
>>> print ufloat(3.1415, 0)
3.1415+/-0
>>> print ufloat(3.1415e10, 0)
(3.1415+/-0)e+10
>>> print ufloat(3.1415, 0.0005)
3.1415+/-0.0005
>>> print '{:.2f}'.format(ufloat(3.14, 0.001))
3.14+/-0.00
>>> print '{:.2f}'.format(ufloat(3.14, 0.00))
3.14+/-0
```

**All the digits** of a number with uncertainty are given in its
representation:

```
>>> y = ufloat(1.23456789012345, 0.123456789)
>>> print y
1.23+/-0.12
>>> print repr(y)
1.23456789012345+/-0.123456789
>>> y
1.23456789012345+/-0.123456789
```

**More information** on formatting can be obtained with ```
pydoc
uncertainties.UFloat.__format__
```

(customization of the LaTeX output,
etc.).

### Global formatting¶

It is sometimes useful to have a **consistent formatting** across
multiple parts of a program. Python’s string.Formatter class
allows one to do just that. Here is how it can be used to consistently
use the shorthand notation for numbers with uncertainties:

```
class ShorthandFormatter(string.Formatter):
def format_field(self, value, format_spec):
if isinstance(value, uncertainties.UFloat):
return value.format(format_spec+'S') # Shorthand option added
# Special formatting for other types can be added here (floats, etc.)
else:
# Usual formatting:
return super(ShorthandFormatter, self).format_field(
value, format_spec)
frmtr = ShorthandFormatter()
print frmtr.format("Result = {0:.1u}", x) # 1-digit uncertainty
```

prints with the shorthand notation: `Result = 0.20(1)`

.

### Customizing the pretty-print and LaTeX outputs¶

The pretty print and LaTeX outputs themselves can be customized.

For example, the pretty-print representation of numbers with uncertainty can
display multiplication with a centered dot (⋅) instead of the default symbol
(×), like in `(2.00±0.10)⋅10⁻¹`

; this is easily done through the global
setting `uncertainties.core.MULT_SYMBOLS["pretty-print"] = "⋅"`

.

Beyond this multiplication symbol, the “±” symbol, the parentheses and the
exponent representations can also be customized globally. The details can be
found in the documentation of `uncertainties.core.format_num()`

.

## Making custom functions accept numbers with uncertainties¶

This package allows **code which is not meant to be used with numbers
with uncertainties to handle them anyway**. This is for instance
useful when calling external functions (which are out of the user’s
control), including functions written in C or Fortran. Similarly,
**functions that do not have a simple analytical form** can be
automatically wrapped so as to also work with arguments that contain
uncertainties.

It is thus possible to take a function `f()`

*that returns a
single float*, and to automatically generalize it so that it also
works with numbers with uncertainties:

```
>>> wrapped_f = uncertainties.wrap(f)
```

The new function `wrapped_f()`

(optionally) *accepts a number
with uncertainty* in place of any float *argument* of `f()`

(note
that floats contained instead *inside* arguments of `f()`

, like
in a list or a NumPy array, *cannot* be replaced by numbers with
uncertainties).
`wrapped_f()`

returns the same values as `f()`

, but with
uncertainties.

With a simple wrapping call like above, uncertainties in the function
result are automatically calculated numerically. **Analytical
uncertainty calculations can be performed** if derivatives are
provided to `wrap()`

.

More details are available in the documentation string of `wrap()`

(accessible through the `pydoc`

command, or Python’s `help()`

shell function).

## Miscellaneous utilities¶

It is sometimes useful to modify the error on certain parameters so as
to study its impact on a final result. With this package, the
**uncertainty of a variable can be changed** on the fly:

```
>>> sum_value = u+2*v
>>> sum_value
21.0+/-0.223606797749979
>>> prev_uncert = u.std_dev
>>> u.std_dev = 10
>>> sum_value
21.0+/-10.00199980003999
>>> u.std_dev = prev_uncert
```

The relevant concept is that `sum_value`

does depend on the
variables `u`

and `v`

: the `uncertainties`

package keeps
track of this fact, as detailed in the Technical Guide, and uncertainties can thus be updated at any time.

When manipulating ensembles of numbers, *some* of which contain
uncertainties while others are simple floats, it can be useful to
access the **nominal value and uncertainty of all numbers in a uniform
manner**. This is what the `nominal_value()`

and
`std_dev()`

functions do:

```
>>> print uncertainties.nominal_value(x)
0.2
>>> print uncertainties.std_dev(x)
0.01
>>> uncertainties.nominal_value(3)
3
>>> uncertainties.std_dev(3)
0.0
```

Finally, a utility method is provided that directly yields the
standard score
(number of standard deviations) between a number and a result with
uncertainty: with `x`

equal to 0.20±0.01,

```
>>> x.std_score(0.17)
-3.0
```

## Derivatives¶

Since the application of linear error propagation theory involves the calculation of **derivatives**, this
package automatically performs such calculations; users can thus
easily get the derivative of an expression with respect to any of its
variables:

```
>>> u = ufloat(1, 0.1)
>>> v = ufloat(10, 0.1)
>>> sum_value = u+2*v
>>> sum_value.derivatives[u]
1.0
>>> sum_value.derivatives[v]
2.0
```

These values are obtained with a fast differentiation algorithm.

## Additional information¶

The capabilities of the `uncertainties`

package in terms of array
handling are detailed in Uncertainties in arrays.

Details about the theory behind this package and implementation information are given in the Technical Guide.