anaflow.tools.laplace¶

Anaflow subpackage providing functions concerning the laplace transformation.

The following functions are provided

`get_lap`(func[, arg_dict])	Callable Laplace transform.
`get_lap_inv`(func[, method, method_dict, …])	Callable Laplace inversion.
`lap_trans`(func, phase[, arg_dict])	The laplace transform.
`stehfest`(func, time[, bound, arg_dict])	The stehfest-algorithm for numerical laplace inversion.

get_lap(func, arg_dict=None, **kwargs)[source]¶

Callable Laplace transform.

Get the Laplace transform of a given function as a callable function.

Parameters

func (callable) –
function that shall be transformed. The first argument needs to be the time-variable: func(t, **kwargs)

func should be capable of taking numpy arrays as input for s and the first shape component of the output of func should match the shape of s.
arg_dict (dict or None, optional) – Keyword-arguments given as a dictionary that are forwarded to the function given in func. Will be merged with **kwargs This is designed for overlapping keywords. Default: None
**kwargs – Keyword-arguments that are forwarded to the function given in func. Will be merged with arg_dict.

Returns

The Laplace transformed of the given function.

Return type

callable

Raises

ValueError – If func is not callable.

get_lap_inv(func, method='stehfest', method_dict=None, arg_dict=None, **kwargs)[source]¶

Callable Laplace inversion.

Get the Laplace inversion of a given function as a callable function.

Parameters

func (callable) –
function in laplace-space that shall be inverted. The first argument needs to be the laplace-variable: func(s, **kwargs)

func should be capable of taking numpy arrays as input for s and the first shape component of the output of func should match the shape of s.
method (str) –
Method that should be used to calculate the inverse. One can choose between
- "stehfest": for the stehfest algorithm
Default: "stehfest"
method_dict (dict or None, optional) – Keyword arguments for the used method.
arg_dict (dict or None, optional) – Keyword-arguments given as a dictionary that are forwarded to the function given in func. Will be merged with **kwargs This is designed for overlapping keywords. Default: None
**kwargs – Keyword-arguments that are forwarded to the function given in func. Will be merged with arg_dict.

Returns

The Laplace inverse of the given function.

Return type

callable

Raises

ValueError – If func is not callable.
ValueError – If method is unknown.

lap_trans(func, phase, arg_dict=None, **kwargs)[source]¶

The laplace transform.

Parameters

func (callable) –
function that shall be transformed. The first argument needs to be the time-variable: func(s, **kwargs)

func should be capable of taking numpy arrays as input for s and the first shape component of the output of func should match the shape of s.
phase (float or numpy.ndarray) – phase-points to evaluate the transformed function at
arg_dict (dict or None, optional) – Keyword-arguments given as a dictionary that are forwarded to the function given in func. Will be merged with **kwargs This is designed for overlapping keywords in stehfest and func.Default: None
**kwargs – Keyword-arguments that are forwarded to the function given in func. Will be merged with arg_dict

Returns

Array with all evaluations in phase-space.

Return type

numpy.ndarray

Raises

ValueError – If func is not callable.

stehfest(func, time, bound=12, arg_dict=None, **kwargs)[source]¶

The stehfest-algorithm for numerical laplace inversion.

The Inversion was derivide in ‘’Stehfest 1970’’[R1] and is given by the formula

$f\left(t\right) &=\frac{\ln2}{t}\sum_{n=1}^{N}c_{n}\cdot\tilde{f} \left(n\cdot\frac{\ln2}{t}\right)\\ c_{n} &=\left(-1\right)^{n+\frac{N}{2}}\cdot \sum_{k=\left\lfloor \frac{n+1}{2}\right\rfloor } ^{\min\left\{ n,\frac{N}{2}\right\} } \frac{k^{\frac{N}{2}+1}\cdot\binom{2k}{k}} {\left(\frac{N}{2}-k\right)!\cdot\left(n-k\right)! \cdot\left(2k-n\right)!}$

In the algorithm $N$ corresponds to bound, $\tilde{f}$ to func and $t$ to time.

Parameters

func (callable) –
function in laplace-space that shall be inverted. The first argument needs to be the laplace-variable: func(s, **kwargs)

func should be capable of taking numpy arrays as input for s and the first shape component of the output of func should match the shape of s.
time (float or numpy.ndarray) – time-points to evaluate the function at
bound (int, optional) – Here you can specify the number of interations within this algorithm. Default: 12
arg_dict (dict or None, optional) – Keyword-arguments given as a dictionary that are forwarded to the function given in func. Will be merged with **kwargs This is designed for overlapping keywords in stehfest and func.Default: None
**kwargs – Keyword-arguments that are forwarded to the function given in func. Will be merged with arg_dict

Returns

Array with all evaluations in Time-space.

Return type

numpy.ndarray

Raises

ValueError – If func is not callable.
ValueError – If time is not positive.
ValueError – If bound is not positive.
ValueError – If bound is not even.

References

R1: Stehfest, H., ‘’Algorithm 368: Numerical inversion of laplace transforms [d5].’’ Communications of the ACM, 13(1):47-49, 1970

Notes

The parameter time needs to be strictly positiv.

The algorithm gets unstable for bound values above 20.

Examples

>>> f = lambda x: x**-1
>>> stehfest(f, [1,10,100])
array([ 1.,  1.,  1.])