maths pack

33 nodes, in noodlelab[maths] and above.

Math/Arrays

Add Noise

maths.add_noise

Add Gaussian noise. A fixed seed keeps runs reproducible (and cacheable).

Inputs

Name	Type	Description
x	NDArray[floating]
sigma	float	Default 0.1.
seed	int	Default 0.

Outputs

Name	Type	Description
result	NDArray[floating]

Apply Function

maths.apply_function

Apply a common function element-wise.

Inputs

Name	Type	Description
x	NDArray[floating]
function	Literal['sin', 'cos', 'exp', 'log', 'log10', 'sqrt', 'abs', 'gaussian']	Default 'sin'.

Outputs

Name	Type	Description
result	NDArray[floating]

Array Math

maths.array_math

Element-wise math between an array and a scalar or another array.

Inputs

Name	Type	Description
a	NDArray[floating]
b	float | NDArray[floating]	Default 1.0.
operation	Literal['add', 'subtract', 'multiply', 'divide', 'power']	Default 'multiply'.

Outputs

Name	Type	Description
result	NDArray[floating]

Clip Values

maths.clip_values

Limit values to [minimum, maximum].

Inputs

Name	Type	Description
x	NDArray[number]
minimum	float	Default 0.0.
maximum	float	Default 1.0.

Outputs

Name	Type	Description
result	NDArray[floating]

Cumulative Sum

maths.cumulative_sum

Running total of an array (NaNs count as zero).

Inputs

Name	Type	Description
x	NDArray[number]

Outputs

Name	Type	Description
result	NDArray[floating]

Linspace

maths.linspace

Evenly spaced numbers over an interval.

Inputs

Name	Type	Description
start	float	Default 0.0.
stop	float	Default 10.0.
num	int	Default 200.

Outputs

Name	Type	Description
result	NDArray[float64]

Math/Calculus

Derivative

maths.derivative

dy/dx by central differences (numpy.gradient), for uneven spacing too.

Inputs

Name	Type	Description
x	NDArray[number]
y	NDArray[number]

Outputs

Name	Type	Description
result	NDArray[floating]

Integrate

maths.integrate

The area under y(x) by the trapezoidal rule, and its running total (e.g. rainfall rate to accumulated rain, velocity to distance).

Inputs

Name	Type	Description
x	NDArray[number]
y	NDArray[number]

Outputs

Name	Type	Description
total	float
cumulative	NDArray[float64]

Interpolate

maths.interpolate

Values of y at new positions: resample onto another grid, or fill gaps. Positions outside the data give NaN.

Inputs

Name	Type	Description
x	NDArray[number]
y	NDArray[number]
new_x	NDArray[number]
method	Literal['linear', 'cubic', 'nearest']	Default 'linear'.

Outputs

Name	Type	Description
result	NDArray[floating]

Math/Complex

Complex Parts

maths.complex_parts

The parts of a complex number or array: real and imag, magnitude |z| and phase arg z (in radians, from −π to π), and the conjugate. Each keeps the value’s unit, except the phase. A real value gives itself, an imaginary part of 0 and a phase of 0 or π.

Inputs

Name	Type	Description
value	Quantity | NDArray[number] | complex	A complex number or array, with a unit or without: 3+4j ohm Default '(3+4j) Ω'.

Outputs

Name	Type	Description
real	Any
imag	Any
magnitude	Any
phase	Quantity[rad]
conjugate	Any

Math/Fitting

Curve Fit

maths.curve_fit

Non-linear least squares fit of a standard model (scipy.optimize.curve_fit).

Inputs

Name	Type	Description
x	NDArray[number]
y	NDArray[number]
model	Literal['linear', 'quadratic', 'exponential decay', 'exponential growth', 'gaussian', 'logistic', 'power law', 'michaelis-menten', 'damped sine']	Default 'exponential decay'.
initial	str	Starting values, comma-separated; empty: estimated from the data Default ''.
sigma	NDArray[number] | None	Optional.
confidence	float	Default 0.95.
curve_points	int	Default 300.

Outputs

Name	Type	Description
parameters	DataFrame
values	dict[str, float]
fitted	NDArray[float64]
residuals	NDArray[float64]
r_squared	float
rmse	float
curve_x	NDArray[float64]
curve_y	NDArray[float64]
equation	str
summary	dict[str, Any]
estimates	dict[str, Uncertain]

Inverse Prediction

maths.inverse_prediction

Calibration: fit the standards (x = known amount, y = signal) with a straight line, then estimate the amount in each sample from its signal.

Inputs

Name	Type	Description
x	NDArray[number]
y	NDArray[number]
samples	DataFrame
response	str	The measured signal column Default ''.
replicates	int	Readings per sample Default 1.
confidence	float	Default 0.95.

Outputs

Name	Type	Description
results	DataFrame
lod	float
loq	float
summary	dict[str, Any]

Linear Regression

maths.linear_regression

Ordinary least squares y = slope·x + intercept, with standard errors, R², the p-value of the slope and the residual standard deviation. Pairs with a NaN are left out. through_origin fixes the intercept at zero.

Inputs

Name	Type	Description
x	Quantity | NDArray[number]	Numbers, or a quantity holding them (a column with a unit)
y	Quantity | NDArray[number]	Numbers, or a quantity holding them (a column with a unit)
through_origin	bool	Default False.
confidence	float	Default 0.95.

Outputs

Name	Type	Description
slope	float
intercept	float
slope_se	float
intercept_se	float
r_squared	float
p_value	float
residual_std	float
n	int
fitted	NDArray[float64]
residuals	NDArray[float64]
summary	dict[str, Any]
slope_q	Uncertain
intercept_q	Uncertain

Polynomial Fit

maths.polynomial_fit

Least-squares polynomial fit, highest power first. Pairs with a NaN are left out; fitted has a value for every x.

Inputs

Name	Type	Description
x	NDArray[number]
y	NDArray[number]
degree	int	Default 1.

Outputs

Name	Type	Description
coefficients	NDArray[float64]
fitted	NDArray[float64]
r_squared	float

Statistics

maths.statistics

Summary statistics of an array, ignoring NaNs: mean, standard deviation, minimum, maximum, sum (total) and the number of values.

Inputs

Name	Type	Description
x	NDArray[number]
ddof	int	Default 1.

Outputs

Name	Type	Description
mean	float
std	float
minimum	float
maximum	float
total	float
count	int
summary	dict[str, float]

Math/Fourier

Fourier Transform

maths.fourier_transform

The frequencies in a signal, with the discrete Fourier transform (NumPy’s FFT). amplitude is in the signal’s unit, scaled so that a sine of amplitude a reads a at its frequency (with a window, exactly so only at a bin; flattop reads amplitudes best, hann separates peaks best). spectrum is the complex DFT itself, for Inverse Fourier Transform.

Inputs

Name	Type	Description
signal	Quantity | NDArray[number]	Evenly sampled values: an array, or a quantity holding one
time	Quantity | NDArray[floating] | None	When each sample was taken, evenly spaced; else use sample_spacing Optional.
sample_spacing	Quantity	Time (or distance) between samples, when time is not linked Default '1.0 s'.
window	Literal['hann', 'hamming', 'blackman', 'flattop', 'none']	Tapers the ends against leakage; none for an exact inverse Default 'hann'.
detrend	Literal['mean', 'linear', 'none']	Taken off before transforming Default 'mean'.
padding	int	Pad with zeros to this many times the length Default 1.
sides	Literal['one-sided', 'two-sided']	One-sided: frequencies from 0 (real signals); two-sided: ± Default 'one-sided'.
scaling	Literal['amplitude', 'rms', 'raw']	amplitude: a sine of amplitude a reads a; rms: a/√2; raw: |DFT| Default 'amplitude'.
peaks	int	How many peaks to find Default 3.

Outputs

Name	Type	Description
frequency	Quantity
amplitude	Quantity
spectrum	NDArray[complex128]
phase	NDArray[float64]
plot	Figure
peak_frequencies	Quantity
dominant_frequency	Quantity
dominant_amplitude	Quantity
resolution	Quantity
summary	dict[str, float]

Inverse Fourier Transform

maths.inverse_fourier_transform

The signal back from its complex spectrum (the spectrum of Fourier Transform, perhaps filtered on the way). One-sided or two-sided is told from the frequencies. The round trip is exact with window none and detrend none; otherwise the result is the windowed, detrended signal. With a one-sided spectrum of an odd-length signal, give samples.

Inputs

Name	Type	Description
spectrum	NDArray[complexfloating]
frequency	Quantity
samples	int	Length of the signal; 0: from the spectrum Default 0.

Outputs

Name	Type	Description
time	Quantity
signal	NDArray[float64]
imaginary	NDArray[float64]

Power Spectrum

maths.power_spectrum

How the power of a signal is spread over frequency: its power spectral density (unit² per Hz), or with spectrum scaling the power of each tone (unit²). rms is the signal’s root mean square about its mean, from the whole spectrum (Parseval).

Inputs

Name	Type	Description
signal	Quantity | NDArray[number]	Evenly sampled values: an array, or a quantity holding one
time	Quantity | NDArray[floating] | None	When each sample was taken, evenly spaced; else use sample_spacing Optional.
sample_spacing	Quantity	Time (or distance) between samples, when time is not linked Default '1.0 s'.
method	Literal['welch', 'periodogram']	welch: averaged over overlapping segments, less noisy Default 'welch'.
window	Literal['hann', 'hamming', 'blackman', 'flattop', 'none']	Default 'hann'.
segment	int	Welch: samples per segment (finer frequency: longer) Default 1024.
scaling	Literal['density', 'spectrum']	density: power per Hz (noise); spectrum: power per peak (tones) Default 'density'.

Outputs

Name	Type	Description
frequency	Quantity
psd	Quantity
plot	Figure
rms	Quantity
dominant_frequency	Quantity
summary	dict[str, float]

Math/Matrices

Determinant

maths.determinant

The determinant, in the matrix’s unit to the power of its size. Zero means the matrix is singular: its equations are not independent.

Inputs

Name	Type	Description
matrix	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers

Outputs

Name	Type	Description
result	Quantity

Diagonal Matrix

maths.diagonal_matrix

A matrix with diagonal on its diagonal and zeros elsewhere, such as the mass matrix of masses on springs. A linked vector replaces the text.

Inputs

Name	Type	Description
diagonal	str	The diagonal, comma separated: 1, 1 Default '1, 1'.
unit	str	One unit for every entry; empty: plain numbers Default 'kg'.
values	Quantity | NDArray[floating] | None	A vector for the diagonal, instead of the text (and its unit) Optional.

Outputs

Name	Type	Description
result	Quantity

Eigenvalues

maths.eigenvalues

The eigenvalues λ and eigenvectors v of A (A v = λ v), or of A against B (A v = λ B v). λ is in A’s unit over B’s. Column j of vectors belongs to value j, scaled to length 1.

Inputs

Name	Type	Description
a	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers
b	Quantity | NDArray[floating] | None	For the generalized problem A v = λ B v, such as K v = ω² M v Optional.
sort	Literal['ascending', 'descending', 'as computed']	Default 'ascending'.

Outputs

Name	Type	Description
values	Quantity
vectors	NDArray[number]
real	bool
symmetric	bool
summary	dict[str, float]

Element

maths.element

One entry of a matrix or vector, with its unit. Rows and columns are numbered from 1, so row 2 of a solution vector is x2. An entry of a complex matrix is complex.

Inputs

Name	Type	Description
matrix	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers
row	int	Numbered from 1 Default 1.
column	int	Numbered from 1; 1 for a vector Default 1.

Outputs

Name	Type	Description
result	Quantity

Identity Matrix

maths.identity_matrix

The identity matrix: ones on the diagonal, zeros elsewhere.

Inputs

Name	Type	Description
size	int	Default 2.
unit	str	One unit for every entry; empty: plain numbers Default ''.

Outputs

Name	Type	Description
result	Quantity

Inverse

maths.inverse

The inverse A⁻¹, in the reciprocal unit: the inverse of a stiffness matrix (N/m) is a flexibility matrix (m/N). To solve A x = b, Solve Linear System is more accurate than multiplying by the inverse.

Inputs

Name	Type	Description
matrix	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers

Outputs

Name	Type	Description
result	Quantity

Matrix

maths.matrix

A matrix typed as text, MATLAB style: 2, -1; -1, 2 is a 2×2 matrix, 10; 0 a column. Every entry has the one unit.

Inputs

Name	Type	Description
text	str	Rows end at ; or a new line, entries are separated by commas or spaces: 2, -1; -1, 2 Default '2, -1; -1, 2'.
unit	str	One unit for every entry; empty: plain numbers Default ''.

Outputs

Name	Type	Description
result	Quantity

Matrix Multiply

maths.matrix_multiply

The matrix product A B, with the units multiplied too: a stiffness matrix times a displacement vector is a force vector. A vector counts as a column. For element-by-element products use Quantity Math.

Inputs

Name	Type	Description
a	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers
b	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers

Outputs

Name	Type	Description
result	Quantity

Natural Frequencies

maths.natural_frequencies

The natural frequencies and mode shapes of an undamped system of masses and springs, from its stiffness matrix K and mass matrix M: the solutions of K φ = ω² M φ. Frequencies are in ascending order, and column j of shapes is mode j, the way the masses move at that frequency.

Inputs

Name	Type	Description
stiffness	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers
mass	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers
normalise	Literal['largest = 1', 'mass']	Scale each mode shape so its largest entry is 1, or so φᵀ M φ = 1 Default 'largest = 1'.

Outputs

Name	Type	Description
frequency	Quantity[Hz]
angular_frequency	Quantity[rad/s]
shapes	NDArray[float64]
first	Quantity[Hz]
plot	Figure
summary	dict[str, float]

Solve Linear System

maths.solve_linear

Solve A x = b for x, such as K x = F for the displacements of a structure. x is in b’s unit over A’s (N over N/m is m), and has b’s shape.

Inputs

Name	Type	Description
a	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers
b	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers
unit	str	Unit of x; empty: b’s unit over A’s Default ''.

Outputs

Name	Type	Description
x	Quantity
residual	Quantity
condition	float
rank	int
method	str
summary	dict[str, float]

Transpose

maths.transpose

Rows become columns. A vector (a column) becomes a row, 1×n.

Inputs

Name	Type	Description
matrix	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array (one unit for every entry), or plain numbers

Outputs

Name	Type	Description
result	Quantity

Math/Plot

Heatmap

maths.heatmap

A matrix as coloured cells, such as a correlation matrix. A first text column names the rows; the other columns must be numeric.

Inputs

Name	Type	Description
table	DataFrame
colormap	Literal['viridis', 'plasma', 'cividis', 'magma', 'coolwarm', 'RdBu_r', 'YlOrRd', 'Blues']	Default 'RdBu_r'.
annotate	bool	Default True.
symmetric	bool	Centre the colours on zero (for correlations, anomalies) Default True.
title	str	Default ''.
colorbar_label	str	Default ''.

Outputs

Name	Type	Description
result	Figure

Histogram Plot

maths.histogram_plot

Distribution of values, optionally with the normal curve of the same mean and standard deviation for comparison.

Inputs

Name	Type	Description
x	NDArray[number]
bins	int	Default 30.
normal_curve	bool	Overlay a normal distribution Default False.
title	str	Default ''.
x_label	str	Default 'value'.
log_y	bool	Default False.

Outputs

Name	Type	Description
result	Figure

XY Plot

maths.xy_plot

Plot y (and optionally y2, e.g. a fitted curve or marked peaks) against x. A y with uncertainties gets error bars of ±u unless error is linked.

Inputs

Name	Type	Description
x	NDArray[number]
y	Uncertain[NDArray[number]]
y2	NDArray[number] | None	Optional.
x2	NDArray[number] | None	x for y2, if it differs from x Optional.
error	NDArray[number] | None	Error bars for y Optional.
style	Literal['line', 'scatter', 'scatter + line']	Default 'line'.
title	str	Default ''.
x_label	str	Default 'x'.
y_label	str	Default 'y'.
label	str	Default 'data'.
label2	str	Default 'fit'.
log_x	bool	Default False.
log_y	bool	Default False.
style2	Literal['line', 'markers']	markers: e.g. to mark peaks on y Default 'line'.

Outputs

Name	Type	Description
result	Figure

Output

Save Figure

maths.save_figure

Write a figure into this run’s output folder. The extension picks the format: .png, .svg or .pdf.

Inputs

Name	Type	Description
figure	Figure
filename	str	Default 'figure.png'.
dpi	int	Default 200.

Outputs

Name	Type	Description
result	Path