WallGo.helpers

Helper functions that are used by WallGo. This includes the derivative functions, a function to compute the step size for solving detonations and other small functions.

Functions

boostVelocity(xi, v)

Lorentz-transformed velocity

derivative(f, x[, n, order, bounds, ...])

Computes numerical derivatives of a callable function.

gammaSq(v)

Lorentz factor \(\gamma^2\) corresponding to velocity \(v\)

gradient(f, x[, order, epsilon, scale, dx, ...])

Computes the gradient of a callable function.

hessian(f, x[, order, epsilon, scale, dx, ...])

Computes the hessian of a callable function.

nextStepDeton(pos1, pos2, pressure1, ...[, ...])

Function used in EOM to find detonation solutions.
boostVelocity(xi, v)[source]

Lorentz-transformed velocity

Parameters:

  • xi (float)

  • v (float)

Return type:

float

derivative(f, x, n=1, order=4, bounds=None, epsilon=1e-16, scale=1.0, dx=None, args=None)[source]

Computes numerical derivatives of a callable function. Uses the epsilon and scale parameters to estimate the optimal value of dx, if the latter is not provided.

Parameters:

  • f (function) – Function to differentiate. Should take a float or an array as argument and return a float or array (the returned array can have a different shape as the input, but the first axis must match).

  • x (float or array-like) – The position at which to evaluate the derivative.

  • n (int, optional) – The number of derivatives to take. Can be 0, 1, 2. The default is 1.

  • order (int, optional) – The accuracy order of the scheme. Errors are of order \(\mathcal{O}({\rm d}x^{\text{order}/(\text{order+n})})\). Can be 2 or 4. Note that the order at the endpoints is reduced by 1 as it would require more function evaluations to keep the same order. The default is 4.

  • bounds (tuple or None, optional) – Interval in which f can be called. If None, can be evaluated anywhere. The default is None.

  • epsilon (float, optional) – Fractional accuracy at which f can be evaluated. If f is a simple function, should be close to the machine precision. Default is 1e-16.

  • scale (float, optional) – Typical scale at which f(x) change by order 1. Default is 1.

  • dx (float or None, optional) – The magnitude of finite differences. If None, use epsilon and scale to estimate the optimal dx. Default is None.

  • args (list, optional) – List of other fixed arguments passed to the function \(f\).

Returns:

res – The value of the derivative of f evaluated at x.

Return type:

float

gammaSq(v)[source]

Lorentz factor \(\gamma^2\) corresponding to velocity \(v\)

Parameters:

v (float)

Return type:

float

gradient(f, x, order=4, epsilon=1e-16, scale=1.0, dx=None, axis=None, args=None)[source]

Computes the gradient of a callable function. Uses the epsilon and scale parameters to estimate the optimal value of dx, if the latter is not provided.

Parameters:

  • f (function) – Function to differentiate. Should take an array as argument and return an array.

  • x (array-like) – The position at which to evaluate the derivative. The size of the last axis must correspond to the number of variables on which f depends.

  • order (int, optional) – The accuracy order of the scheme. Errors are of order \(\mathcal{O}({\rm d}x^{\text{order}/(\text{order}+1)})\). Can be 2 or 4. The default is 4.

  • epsilon (float, optional) – Fractional accuracy at which f can be evaluated. If f is a simple function, should be close to the machine precision. Default is 1e-16.

  • scale (float or array-like, optional) – Typical scale at which f(x) change by order 1. Can be an array, in which case each element corresponds to the scale of a different variable. Default is 1.

  • dx (float or np.ndarray or None, optional) – The magnitude of finite differences. Can be an array, in which case each element corresponds to the dx of a different variable.If None, use epsilon and scale to estimate the optimal dx. Default is None.

  • axis (list, int or None, optional) – Element of the gradient to return. If None, returns the whole gradient. Default is None.

  • args (list, optional) – List of other fixed arguments passed to the function \(f\).

Returns:

res – The value of the gradient of f evaluated at x.

Return type:

float

hessian(f, x, order=4, epsilon=1e-16, scale=1.0, dx=None, xAxis=None, yAxis=None, args=None)[source]

Computes the hessian of a callable function. Uses the epsilon and scale parameters to estimate the optimal value of dx, if the latter is not provided.

Parameters:

  • f (function) – Function to differentiate. Should take an array as argument and return an array.

  • x (array-like) – The position at which to evaluate the derivative. The size of the last axis must correspond to the number of variables on which f depends.

  • order (int, optional) – The accuracy order of the scheme. Errors are of order \(\mathcal{O}({\rm d}x^{\text{order}/(\text{order}+2)})\). Can be 2 or 4. The default is 4.

  • epsilon (float, optional) – Fractional accuracy at which f can be evaluated. If f is a simple function, should be close to the machine precision. Default is 1e-16.

  • scale (float, optional) – Typical scale at which f(x) change by order 1. Default is 1.

  • dx (float or None, optional) – The magnitude of finite differences. If None, use epsilon and scale to estimate the optimal dx. Default is None.

  • xAxis (list, int or None, optional) – Lines of the hessian matrix to return. If None, returns all the lines. Default is None.

  • yAxis (list, int or None, optional) – Columns of the hessian matrix to return. If None, returns all the columns. Default is None.

  • args (list, optional) – List of other fixed arguments passed to the function \(f\).

Returns:

res – The value of the hessian of f evaluated at x.

Return type:

float

nextStepDeton(pos1, pos2, pressure1, pressure2, mean2ndDeriv, std2ndDeriv, pressureTol, posMax, overshootProb=0.05)[source]

Function used in EOM to find detonation solutions. It finds the next point to be sampled to try to bracket a root in such a way that the probability of overshooting a root is roughly equal to overshootProb.

To estimate the overshoot probability, it fits the pressure to a quadratic which is equal to pressure2 at :py:data`pos2`, but with uncertain 1st and 2nd derivatives which are assumed to be normally distributed. The mean of the 1st derivative is computed by finite differences from the last 2 points.

Parameters:

  • pos1 (float) – Position of the first sampled point.

  • pos2 (float) – position of the second sampled point.

  • pressure1 (float) – Pressure at pos1.

  • pressure2 (float) – Pressure at pos2.

  • mean2ndDeriv (float) – Estimate of the 2nd derivative at pos2.

  • std2ndDeriv (float) – Uncertainty on the 2nd derivative at pos2.

  • pressureTol (float) – Relative accuracy at which pressure1 and pressure2 are computed.

  • posMax (float) – Maximal position that the next step can have.

  • overshootProb (float, optional) – Desired overshoot probability. A smaller value will lead to smaller step sizes which will take longer to evaluate, but with less chances of missing a root. The default is 0.05.

Returns:

Position where the overshoot probability is overshootProb (or posMax if there is no solution).

Return type:

float