nitrogen.math

General mathematical tools and functions

nitrogen.math.clenshawcurtis(n, bounds=(- 1, 1))

Calculate the quadrature grid points and weights for an n-point Clenshaw-Curtis quadrature.

Parameters

  • n (integer) – The number of quadrature points.

  • bounds ((2,) tuple, optional) – The integration end-points. The default is (-1,1). If the bounds are in reverse order, the weights will be negative.

Returns

  • x (ndarray) – The quadrature points

  • w (ndarray) – The quadrature weights

Notes

The quadrature rule approximates the integral \(\int_{-1}^{1} f(x) dx\).

See ``Fast Construction of the Fejer and Clenshaw–Curtis Quadrature Rules’’, by Jörg Waldvogel, BIT Num. Math. 46, 195 (2006). doi: 10.1007/s10543-006-0045-4.

nitrogen.math.constrainedFourier(x, df, period=None)

Calculate a single-variable Fourier series that exactly satisfies the value and derivatives at one or more points

Parameters

  • x ((n,) array_like) – The matching positions

  • df ((deriv+1, n) array_like) – The scaled derivative arrays of \(f(x)\) at the matching positions

  • period (float, optional) – The period of the coordinate x. If None, \(2\pi\) is assumed.

Returns

c – The Fourier series coefficients.

Return type

(nc,) ndarray

Notes

The df derivative array contains the scaled derivatives, df[n] = \(f^{(n)} = \partial_x^n f / n!\).

The expansions coefficients are defined as

\[f(x) = c_0 + c_1 \sin \sigma x + c_2 \cos \sigma x + c_3 \sin 2 \sigma x + c_4 \cos 2 \sigma x + \cdots\]

where \(\sigma = 2\pi/\)period.

nitrogen.math.constrainedPolynomial(x, df, x0=None)

Calculate a single-variable polynomial that exactly satisfies the value and derivatives at one or more points

Parameters

  • x ((n,) array_like) – The matching positions

  • df ((deriv+1, n) array_like) – The scaled derivative arrays of \(f(x)\) at the matching positions

  • x0 (scalar, optional) – The expansion point of the polynomial. If None, x0 is zero.

Returns

c – The power series coefficients for the matching polynomial, c[0] + c[1]*(x-x0) + c[2]*(x-x0)**2 + ..., where nc = n * (deriv + 1).

Return type

(nc,) ndarray

Notes

The df derivative array contains the scaled derivatives, df[n] = \(f^{(n)} = \partial_x^n f / n!\).

nitrogen.math.cumsimp(y, x, axis=0)

Calculate the cumulative integral by Simpson’s Rule

Parameters

  • y (array_like) – The uniformly spaced samples

  • x (array_like or scalar) – The sample points along the integration axis or, if scalar, the sample spacing.

  • axis (int, optional) – The integration axis. The default is 0.

Returns

res – The result of cumulative integration.

Return type

ndarray

Notes

The first element of the cumulative integral is fixed to zero. The second element is handled in one of two ways. If the length of y is only 2, then the trapezoid rule is used. Otherwise, the next element in y is used for quadratic interpolation. The remaining elements are evaluated via Simpson’s 1/3 rule.

nitrogen.math.gaussianFWHM(x, fwhm, norm='area')

Calculate a Gaussian function.

Parameters

  • x (array_like) – Input array

  • fwhm (scalar) – The full-width at half-maximum value.

  • norm ({'area', 'amplitude'}) – The normalization convention. The default is ‘area’. See Notes for definitions.

Returns

y – The result.

Return type

ndarray

Notes

For norm = \`area\`, the integrated area is equal to unity. For norm = \`amplitude\`, the peak amplitude is equal to unity.

nitrogen.math.levi3()

Return the 3-index Levi-Civita \(\epsilon_{ijk}\) tensor.

Returns

eps – The Levi-Civita tensor

Return type

(3,3,3) ndarray

nitrogen.math.mpolyfit(x, y, deg, rcond=None)

Multivariable polynomial least-squares fitting.

Parameters

  • x ((N,nx) or (N,) array_like) – The input coordinates.

  • y ((N,) array_like) – The output value

  • deg (int) – The degree of the polynomial

  • rcond (float, optional) – rcond parameter passed to lstsq.

Returns

  • p ((nt,) ndarray) – The polynomial coefficients

  • res ((N,) ndarray) – The residuals.

Notes

The polynomial is ordered using the standard lexical ordering defined by the autodiff and DFun modules.

nitrogen.math.mpolyfit_grad(x, y, yg, deg, scale=None)

Multivariable polynomial least-squares fitting including gradient constraints.

Parameters

  • x ((N,nx) or (N,) array_like) – The input coordinates.

  • y ((N,) array_like) – The output value

  • yg ((N,nx) array_like) – The output gradient

  • deg (int) – The degree of the polynomial

  • scale ((nx,) array_like, optional) – The coordinate scale. The default is 1 for each coordinate.

Returns

  • p ((nt,) ndarray) – The polynomial coefficients

  • res ((N,nx+1) ndarray) – The residuals. The first column is the y residual. The second is the gradient residual including scale.

Notes

The polynomial is ordered using the standard lexical ordering defined by the autodiff and DFun modules.

The gradient data is premultiplied by scale for each coordinate before least-squares optimization.

nitrogen.math.mpolyval(p, x)

Evaluate a polynomial in standard lexical order.

Parameters

  • p (array_like) – The polynomial coefficients.

  • x ((N,nx) or (N,) array_like) – The input values.

Returns

y – The polynomial value.

Return type

(N,) ndarray

nitrogen.math.spech_fft(C, dt, sample_factor=1, damping=0.0)

Calculate the intensity spectrum from a hermitian autocorrelation function C.

Parameters

  • C (ndarray) – The autocorrelation function.

  • dt (float) – The time step.

  • sample_factor (int) – The over-sampling factor. The default is 1.

  • damping (float) – The Gaussian damping factor. The default is 0.

Returns

  • g (ndarray) – The (real) intensity spectrum

  • freq (ndarray) – The angular frequency axis

Notes

The autocorrelation function is provided for \(t \geq 0\), i.e. C[0] is the \(t = 0\) value. It is assumed that \(C(-t) = C(t)^*\).

The returned spectrum approximates the Fourier transformation

\[g(\omega) = \frac{1}{2\pi}\, \int_{-\infty}^{\infty} dt e^{i \omega t} C(t)\]

This normalization means that the angular frequency integral of \(g(\omega)\) equals \(C(0)\), which is typically unity.

The calculated spectrum may be over-sampled by zero-padding the autocorrelation function. This sampling factor is controlled by sample_factor.

A Gaussian window function is also applied to the autocorrelation function, of the form \(C(t) \exp(-a (t/T)^2)\), where \(T\) is length of the correlation function (before any padding). The damping factor \(a\) is controlled by the damping keyword. At sufficiently large \(a\), the line shape becomes Gaussian with an angular frequency full-width at half-maximum equal to \(\omega_\text{FWHM} = 4 T^{-1} \sqrt{a \ln 2}\).