Polynomials and Interpolation (polyfun)

Polynomial curve fitting, roots, derivatives, integration, 1D/2D/3D interpolation, and piecewise cubic splines. More...

Files
file	polyfun.hpp

Classes
struct	nk::builtin::PolyDiv
	Result of polynomial division and deconvolution ([q, r] = polydiv(b, a)). More...
struct	nk::builtin::PadeCoef
	Padé approximation numerator and denominator polynomials. More...
struct	nk::builtin::ResidueResult
	Result of partial fraction expansion (residues, poles, direct term). More...
struct	nk::builtin::MatchpairsResult
	Result of linear assignment problem (matchpairs). More...
struct	nk::builtin::InpolygonResult
	Result of 2-D polygon inclusion test (inpolygon). More...
struct	nk::builtin::PolyxpolyResult
	Result of polygon intersection (polyxpoly). More...

Functions
Value	nk::builtin::roots (const Value &p, std::pmr::memory_resource *mr=nullptr)
	Computes polynomial roots (roots(p)).
Value	nk::builtin::poly (const Value &r, std::pmr::memory_resource *mr=nullptr)
	Polynomial with specified roots or characteristic polynomial of a matrix.
Value	nk::builtin::polyval (const Value &p, const Value &x, std::pmr::memory_resource *mr=nullptr)
	Evaluates polynomial p at points x via Horner's scheme.
Value	nk::builtin::polyder (const Value &p, std::pmr::memory_resource *mr=nullptr)
	Computes polynomial derivative (polyder(p)).
std::tuple< Value, Value >	nk::builtin::polyder (const Value &b, const Value &a, std::pmr::memory_resource *mr=nullptr)
	Computes derivative of polynomial quotient b / a ([q, d] = polyder(b, a)).
Value	nk::builtin::polyint (const Value &p, double k=0.0, std::pmr::memory_resource *mr=nullptr)
	Computes polynomial integral (polyint(p, k)).
Value	nk::builtin::polyfit (const Value &x, const Value &y, int n, std::pmr::memory_resource *mr=nullptr)
	Polynomial least-squares curve fitting (polyfit(x, y, n)).
Value	nk::builtin::polyvalm (const Value &p, const Value &A, std::pmr::memory_resource *mr=nullptr)
	Matrix polynomial evaluation (polyvalm(p, A)).
Value	nk::builtin::poly_of_matrix (const Value &A, std::pmr::memory_resource *mr=nullptr)
	Characteristic polynomial of square matrix (poly_of_matrix(A)).
PadeCoef	nk::builtin::padecoef (double T, int N, std::pmr::memory_resource *mr=nullptr)
	Padé approximation of time delay ([num, den] = padecoef(T, N)).
PolyDiv	nk::builtin::polydiv (const Value &b, const Value &a, std::pmr::memory_resource *mr=nullptr)
	Polynomial deconvolution and division ([q, r] = polydiv(b, a)).
std::tuple< Value, Value, Value >	nk::builtin::tf2zp (const Value &b, const Value &a, std::pmr::memory_resource *mr=nullptr)
	Transfer function to zero-pole-gain conversion ([z, p, k] = tf2zp(b, a)).
std::tuple< Value, Value >	nk::builtin::zp2tf (const Value &z, const Value &p, double k, std::pmr::memory_resource *mr=nullptr)
	Zero-pole-gain to transfer function conversion ([b, a] = zp2tf(z, p, k)).
ResidueResult	nk::builtin::residue (const Value &b, const Value &a, std::pmr::memory_resource *mr=nullptr)
	Partial fraction expansion (residues) in Laplace s-domain ([r, p, k] = residue(b, a)).
ResidueResult	nk::builtin::residuez (const Value &b, const Value &a, std::pmr::memory_resource *mr=nullptr)
	Partial fraction expansion in Z-domain ([r, p, k] = residuez(b, a)).
Value	nk::builtin::interp1 (const Value &x, const Value &v, const Value &xq, const std::string &method="linear", std::pmr::memory_resource *mr=nullptr)
	1-D table lookup and data interpolation (interp1(x, v, xq, method)).
Value	nk::builtin::interp2 (const Value &V, const Value &Xq, const Value &Yq, const std::string &method="linear", std::pmr::memory_resource *mr=nullptr)
	2-D data interpolation on a grid (interp2(x, y, v, xq, yq, method)).
Value	nk::builtin::interp2 (const Value &x, const Value &y, const Value &v, const Value &xq, const Value &yq, const std::string &method="linear", std::pmr::memory_resource *mr=nullptr)
	2-D data interpolation on a grid (interp2(x, y, v, xq, yq, method)).
Value	nk::builtin::interp3 (const Value &V, const Value &Xq, const Value &Yq, const Value &Zq, const std::string &method="linear", std::pmr::memory_resource *mr=nullptr)
	3-D data interpolation on a uniform grid (interp3(V, Xq, Yq, Zq, method)).
Value	nk::builtin::interp3 (const Value &X, const Value &Y, const Value &Z, const Value &V, const Value &Xq, const Value &Yq, const Value &Zq, const std::string &method="linear", std::pmr::memory_resource *mr=nullptr)
	3-D data interpolation on a grid (interp3(X, Y, Z, V, Xq, Yq, Zq, method)).
Value	nk::builtin::spline (const Value &x, const Value &y, const Value &xq=Value(), std::pmr::memory_resource *mr=nullptr)
	Cubic spline data interpolation (spline(x, y, xq)).
Value	nk::builtin::pchip (const Value &x, const Value &y, const Value &xq=Value(), std::pmr::memory_resource *mr=nullptr)
	Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) (pchip(x, y, xq)).
Value	nk::builtin::makima (const Value &x, const Value &y, const Value &xq, std::pmr::memory_resource *mr=nullptr)
	Modified Akima cubic Hermite interpolation (makima(x, y, xq)).
Value	nk::builtin::mkpp (const Value &breaks, const Value &coefs, std::pmr::memory_resource *mr=nullptr)
	Constructs a piecewise polynomial structure (ppform).
Value	nk::builtin::unmkpp (const Value &pp, std::pmr::memory_resource *mr=nullptr)
	Extracts piecewise polynomial structure fields ([breaks, coefs, l, k, d] = unmkpp(pp)).
Value	nk::builtin::ppval (const Value &pp, const Value &xq, std::pmr::memory_resource *mr=nullptr)
	Evaluates a piecewise polynomial structure at query points xq.
Value	nk::builtin::gradient (const Value &f, double h=1.0, std::pmr::memory_resource *mr=nullptr)
	Numerical gradient of 1-D array or uniform multi-dimensional array (gradient(f, h)).
std::tuple< Value, Value >	nk::builtin::gradient2 (const Value &f, double hx=1.0, double hy=1.0, std::pmr::memory_resource *mr=nullptr)
	Numerical 2-D gradient along rows and columns ([fx, fy] = gradient(f, hx, hy)).
Value	nk::builtin::del2 (const Value &u, double h=1.0, std::pmr::memory_resource *mr=nullptr)
	Five-point discrete Laplacian / Laplace operator (del2(u, h)).
Value	nk::builtin::trapz (const Value &y, std::pmr::memory_resource *mr=nullptr)
	Trapezoidal numerical integration with unit spacing (trapz(y)).
Value	nk::builtin::trapz (const Value &x, const Value &y, std::pmr::memory_resource *mr=nullptr)
	Trapezoidal numerical integration over coordinates x (trapz(x, y)).
Value	nk::builtin::trapz (const Value &x, const Value &y, int dim, std::pmr::memory_resource *mr=nullptr)
	Trapezoidal numerical integration along specified dimension (trapz(x, y, dim)).
Value	nk::builtin::cumtrapz (const Value &y, std::pmr::memory_resource *mr=nullptr)
	Cumulative trapezoidal numerical integration with unit spacing (cumtrapz(y)).
Value	nk::builtin::cumtrapz (const Value &x, const Value &y, std::pmr::memory_resource *mr=nullptr)
	Cumulative trapezoidal numerical integration over coordinates x (cumtrapz(x, y)).
Value	nk::builtin::cumtrapz (const Value &x, const Value &y, int dim, std::pmr::memory_resource *mr=nullptr)
	Cumulative trapezoidal numerical integration along dimension (cumtrapz(x, y, dim)).
Value	nk::builtin::cumtrapzDim (const Value &y, int dim, std::pmr::memory_resource *mr=nullptr)
	Cumulative trapezoidal numerical integration with unit spacing along dimension (cumtrapz(y, dim)).
Value	nk::builtin::integral (FnHandle fn, double a, double b, double absTol=1e-10, std::pmr::memory_resource *mr=nullptr)
	Numerical 1-D adaptive quadrature integration (integral(fn, a, b)).
Value	nk::builtin::integral2 (FnHandle fn, double a, double b, double c, double d, double absTol=1e-10, std::pmr::memory_resource *mr=nullptr)
	Numerical 2-D double integration over rectangular region (integral2(fn, a, b, c, d)).
Value	nk::builtin::integral3 (FnHandle fn, double a, double b, double c, double d, double e, double f, double absTol=1e-10, std::pmr::memory_resource *mr=nullptr)
	Numerical 3-D triple integration over cuboid region (integral3(fn, a, b, c, d, e, f)).
Value	nk::builtin::polyarea (const Value &x, const Value &y, std::pmr::memory_resource *mr=nullptr)
	Area of 2-D polygonal contour (polyarea(x, y)).
Value	nk::builtin::inpolygon (const Value &xq, const Value &yq, const Value &xv, const Value &yv, std::pmr::memory_resource *mr=nullptr)
	Tests if 2-D query points are inside polygonal region (inpolygon(xq, yq, xv, yv)).
InpolygonResult	nk::builtin::inpolygon2 (const Value &xq, const Value &yq, const Value &xv, const Value &yv, std::pmr::memory_resource *mr=nullptr)
	Tests points inside and on edge of 2-D polygonal region ([in, on] = inpolygon(xq, yq, xv, yv)).
Value	nk::builtin::convhull (const Value &x, const Value &y, std::pmr::memory_resource *mr=nullptr)
	Computes 2-D convex hull of planar point set (convhull(x, y)).
PolyxpolyResult	nk::builtin::polyxpoly (const Value &x1, const Value &y1, const Value &x2, const Value &y2, std::pmr::memory_resource *mr=nullptr)
	Intersection points of polygonal lines or boundaries ([xi, yi] = polyxpoly(x1, y1, x2, y2)).
MatchpairsResult	nk::builtin::matchpairs (const Value &C, double cU, const std::string &mode="min", std::pmr::memory_resource *mr=nullptr)
	Solves linear sum assignment problem with cost threshold (matchpairs(C, cU, mode)).
Value	nk::builtin::boundary (const Value &x, const Value &y, double shrink=0.5, std::pmr::memory_resource *mr=nullptr)
	Computes boundary / alpha-shape around 2-D point cloud (boundary(x, y, shrink)).
Value	nk::builtin::delaunay (const Value &x, const Value &y, std::pmr::memory_resource *mr=nullptr)
	2-D Delaunay triangulation of planar point set (delaunay(x, y)).
Value	nk::builtin::griddata (const Value &x, const Value &y, const Value &v, const Value &xq, const Value &yq, std::pmr::memory_resource *mr=nullptr)
	Interpolates scattered 2-D data onto query points (griddata(x, y, v, xq, yq, method)).
Value	nk::builtin::griddata (const Value &x, const Value &y, const Value &v, const Value &xq, const Value &yq, const std::string &method, std::pmr::memory_resource *mr=nullptr)
	Interpolates scattered 2-D data onto query points with specified method (griddata(x, y, v, xq, yq, method)).
Value	nk::builtin::griddatan (const Value &Xv, const Value &vv, const Value &xi, std::pmr::memory_resource *mr=nullptr)
	N-D scattered data interpolation (griddatan(Xv, vv, xi, method)).
Value	nk::builtin::griddatan (const Value &Xv, const Value &vv, const Value &xi, const std::string &method, std::pmr::memory_resource *mr=nullptr)
	N-D scattered data interpolation with method (griddatan(Xv, vv, xi, method)).
Value	nk::builtin::histcounts2 (const Value &x, const Value &y, const Value &xedgesV, const Value &yedgesV, std::pmr::memory_resource *mr=nullptr)
	Computes 2-D bivariate histogram bin counts (histcounts2(x, y, xedges, yedges)).

Detailed Description

Polynomial curve fitting, roots, derivatives, integration, 1D/2D/3D interpolation, and piecewise cubic splines.

Function Documentation

boundary()

Value nk::builtin::boundary	(	const Value &	x,
		const Value &	y,
		double	shrink = 0.5,
		std::pmr::memory_resource *	mr = nullptr )

Computes boundary / alpha-shape around 2-D point cloud (boundary(x, y, shrink)).

Parameters

x	Point x-coordinates.
y	Point y-coordinates.
shrink	Shrink factor between 0.0 (convex hull) and 1.0 (compact boundary).
mr	Memory resource.

Returns

Indices of boundary points.

See also

convhull

convhull()

Value nk::builtin::convhull	(	const Value &	x,
		const Value &	y,
		std::pmr::memory_resource *	mr = nullptr )

Computes 2-D convex hull of planar point set (convhull(x, y)).

Parameters

x	Point x-coordinates.
y	Point y-coordinates.
mr	Memory resource.

Returns

Indices of vertices forming the convex hull in counterclockwise order.

See also

delaunay, boundary

cumtrapz() [1/3]

Value nk::builtin::cumtrapz	(	const Value &	x,
		const Value &	y,
		int	dim,
		std::pmr::memory_resource *	mr = nullptr )

Cumulative trapezoidal numerical integration along dimension (cumtrapz(x, y, dim)).

Parameters

x	Coordinate points.
y	Function values.
dim	Operating dimension.
mr	Memory resource.

Returns

Cumulative integral array.

cumtrapz() [2/3]

Value nk::builtin::cumtrapz	(	const Value &	x,
		const Value &	y,
		std::pmr::memory_resource *	mr = nullptr )

Cumulative trapezoidal numerical integration over coordinates x (cumtrapz(x, y)).

Parameters

x	Coordinate points.
y	Function values.
mr	Memory resource.

Returns

Cumulative integral array.

cumtrapz() [3/3]

Value nk::builtin::cumtrapz	(	const Value &	y,
		std::pmr::memory_resource *	mr = nullptr )

Cumulative trapezoidal numerical integration with unit spacing (cumtrapz(y)).

Parameters

y	Function values.
mr	Memory resource.

Returns

Cumulative integral array.

See also

trapz

cumtrapzDim()

Value nk::builtin::cumtrapzDim	(	const Value &	y,
		int	dim,
		std::pmr::memory_resource *	mr = nullptr )

Cumulative trapezoidal numerical integration with unit spacing along dimension (cumtrapz(y, dim)).

Parameters

y	Function values.
dim	Operating dimension.
mr	Memory resource.

Returns

Cumulative integral array.

del2()

Value nk::builtin::del2	(	const Value &	u,
		double	h = 1.0,
		std::pmr::memory_resource *	mr = nullptr )

Five-point discrete Laplacian / Laplace operator (del2(u, h)).

Parameters

u	Input array.
h	Spacing between grid points (default 1.0).
mr	Memory resource.

Returns

Discrete Laplacian array.

See also

gradient

delaunay()

Value nk::builtin::delaunay	(	const Value &	x,
		const Value &	y,
		std::pmr::memory_resource *	mr = nullptr )

2-D Delaunay triangulation of planar point set (delaunay(x, y)).

Parameters

x	Point x-coordinates.
y	Point y-coordinates.
mr	Memory resource.

Returns

M x 3 matrix where each row defines a triangle by point indices.

See also

convhull

gradient()

Value nk::builtin::gradient	(	const Value &	f,
		double	h = 1.0,
		std::pmr::memory_resource *	mr = nullptr )

Numerical gradient of 1-D array or uniform multi-dimensional array (gradient(f, h)).

Parameters

f	Input array.
h	Grid step size (default 1.0).
mr	Memory resource.

Returns

Approximate derivative along the operating dimension.

See also

del2, diff

gradient2()

std::tuple< Value, Value > nk::builtin::gradient2	(	const Value &	f,
		double	hx = 1.0,
		double	hy = 1.0,
		std::pmr::memory_resource *	mr = nullptr )

Numerical 2-D gradient along rows and columns ([fx, fy] = gradient(f, hx, hy)).

Parameters

f	2-D input matrix.
hx	Grid spacing along X.
hy	Grid spacing along Y.
mr	Memory resource.

Returns

Tuple containing {fx, fy} partial derivatives.

griddata() [1/2]

Value nk::builtin::griddata	(	const Value &	x,
		const Value &	y,
		const Value &	v,
		const Value &	xq,
		const Value &	yq,
		const std::string &	method,
		std::pmr::memory_resource *	mr = nullptr )

Interpolates scattered 2-D data onto query points with specified method (griddata(x, y, v, xq, yq, method)).

Parameters

x	Scattered point x-coordinates.
y	Scattered point y-coordinates.
v	Sample values at scattered points.
xq	Query point x-coordinates.
yq	Query point y-coordinates.
method	Interpolation method ("linear", "nearest", "cubic", "natural").
mr	Memory resource.

Returns

Interpolated surface values.

griddata() [2/2]

Value nk::builtin::griddata	(	const Value &	x,
		const Value &	y,
		const Value &	v,
		const Value &	xq,
		const Value &	yq,
		std::pmr::memory_resource *	mr = nullptr )

Interpolates scattered 2-D data onto query points (griddata(x, y, v, xq, yq, method)).

Parameters

x	Scattered point x-coordinates.
y	Scattered point y-coordinates.
v	Sample values at scattered points.
xq	Query point x-coordinates.
yq	Query point y-coordinates.
mr	Memory resource.

Returns

Interpolated surface values.

See also

griddatan, interp2

griddatan() [1/2]

Value nk::builtin::griddatan	(	const Value &	Xv,
		const Value &	vv,
		const Value &	xi,
		const std::string &	method,
		std::pmr::memory_resource *	mr = nullptr )

N-D scattered data interpolation with method (griddatan(Xv, vv, xi, method)).

Parameters

Xv	M x N sample point coordinates.
vv	Sample values.
xi	Query point coordinates.
method	Interpolation method ("linear", "nearest").
mr	Memory resource.

Returns

Interpolated values.

griddatan() [2/2]

Value nk::builtin::griddatan	(	const Value &	Xv,
		const Value &	vv,
		const Value &	xi,
		std::pmr::memory_resource *	mr = nullptr )

N-D scattered data interpolation (griddatan(Xv, vv, xi, method)).

Parameters

Xv	M x N matrix of scattered sample point coordinates.
vv	M x 1 sample values.
xi	P x N matrix of query point coordinates.
mr	Memory resource.

Returns

Interpolated values vector P x 1.

See also

griddata

histcounts2()

Value nk::builtin::histcounts2	(	const Value &	x,
		const Value &	y,
		const Value &	xedgesV,
		const Value &	yedgesV,
		std::pmr::memory_resource *	mr = nullptr )

Computes 2-D bivariate histogram bin counts (histcounts2(x, y, xedges, yedges)).

Parameters

x	Data points x-coordinates.
y	Data points y-coordinates.
xedgesV	Bin edges vector along X.
yedgesV	Bin edges vector along Y.
mr	Memory resource.

Returns

2-D matrix of bivariate count values.

inpolygon()

Value nk::builtin::inpolygon	(	const Value &	xq,
		const Value &	yq,
		const Value &	xv,
		const Value &	yv,
		std::pmr::memory_resource *	mr = nullptr )

Tests if 2-D query points are inside polygonal region (inpolygon(xq, yq, xv, yv)).

Parameters

xq	Query points x-coordinates.
yq	Query points y-coordinates.
xv	Polygon vertices x-coordinates.
yv	Polygon vertices y-coordinates.
mr	Memory resource.

Returns

Logical array indicating whether query points lie inside or on edge.

See also

inpolygon2

inpolygon2()

InpolygonResult nk::builtin::inpolygon2	(	const Value &	xq,
		const Value &	yq,
		const Value &	xv,
		const Value &	yv,
		std::pmr::memory_resource *	mr = nullptr )

Tests points inside and on edge of 2-D polygonal region ([in, on] = inpolygon(xq, yq, xv, yv)).

Parameters

xq	Query points x-coordinates.
yq	Query points y-coordinates.
xv	Polygon vertices x-coordinates.
yv	Polygon vertices y-coordinates.
mr	Memory resource.

Returns

Struct containing {in, on} logical arrays.

integral()

Value nk::builtin::integral	(	FnHandle	fn,
		double	a,
		double	b,
		double	absTol = 1e-10,
		std::pmr::memory_resource *	mr = nullptr )

Numerical 1-D adaptive quadrature integration (integral(fn, a, b)).

Parameters

fn	Function handle taking numeric vector argument.
a	Lower integration limit.
b	Upper integration limit.
absTol	Absolute error tolerance.
mr	Memory resource.

Returns

Integral value.

See also

integral2, integral3, trapz

integral2()

Value nk::builtin::integral2	(	FnHandle	fn,
		double	a,
		double	b,
		double	c,
		double	d,
		double	absTol = 1e-10,
		std::pmr::memory_resource *	mr = nullptr )

Numerical 2-D double integration over rectangular region (integral2(fn, a, b, c, d)).

Parameters

fn	Function handle taking (x, y).
a	Lower limit in x.
b	Upper limit in x.
c	Lower limit in y.
d	Upper limit in y.
absTol	Absolute error tolerance.
mr	Memory resource.

Returns

Double integral value.

See also

integral, integral3

integral3()

Value nk::builtin::integral3	(	FnHandle	fn,
		double	a,
		double	b,
		double	c,
		double	d,
		double	e,
		double	f,
		double	absTol = 1e-10,
		std::pmr::memory_resource *	mr = nullptr )

Numerical 3-D triple integration over cuboid region (integral3(fn, a, b, c, d, e, f)).

Parameters

fn	Function handle taking (x, y, z).
a	Lower limit in x.
b	Upper limit in x.
c	Lower limit in y.
d	Upper limit in y.
e	Lower limit in z.
f	Upper limit in z.
absTol	Absolute error tolerance.
mr	Memory resource.

Returns

Triple integral value.

See also

integral, integral2

interp1()

Value nk::builtin::interp1	(	const Value &	x,
		const Value &	v,
		const Value &	xq,
		const std::string &	method = "linear",
		std::pmr::memory_resource *	mr = nullptr )

1-D table lookup and data interpolation (interp1(x, v, xq, method)).

Parameters

x	Sample grid points (strictly monotonic).
v	Sample values.
xq	Query points.
method	Interpolation method: "linear", "nearest", "next", "previous", "spline", "pchip".
mr	Memory resource.

Returns

Interpolated values at query points xq.

See also

interp2, spline, pchip

interp2() [1/2]

Value nk::builtin::interp2	(	const Value &	V,
		const Value &	Xq,
		const Value &	Yq,
		const std::string &	method = "linear",
		std::pmr::memory_resource *	mr = nullptr )

2-D data interpolation on a grid (interp2(x, y, v, xq, yq, method)).

Parameters

x	Grid coordinates along columns.
y	Grid coordinates along rows.
v	Matrix of values on grid.
xq	Query points x-coordinates.
yq	Query points y-coordinates.
method	Interpolation method: "linear", "nearest", "cubic", "spline".
mr	Memory resource.

Returns

Interpolated values.

See also

interp1

2-D data interpolation on a uniform grid (interp2(V, Xq, Yq, method)).

Parameters

V	Matrix of values on default integer grid.
Xq	Query points x-coordinates.
Yq	Query points y-coordinates.
method	Interpolation method ("linear", "nearest", "cubic", "spline").
mr	Memory resource.

Returns

Interpolated values.

interp2() [2/2]

Value nk::builtin::interp2	(	const Value &	x,
		const Value &	y,
		const Value &	v,
		const Value &	xq,
		const Value &	yq,
		const std::string &	method = "linear",
		std::pmr::memory_resource *	mr = nullptr )

2-D data interpolation on a grid (interp2(x, y, v, xq, yq, method)).

Parameters

x	Grid coordinates along columns.
y	Grid coordinates along rows.
v	Matrix of values on grid.
xq	Query points x-coordinates.
yq	Query points y-coordinates.
method	Interpolation method: "linear", "nearest", "cubic", "spline".
mr	Memory resource.

Returns

Interpolated values.

See also

interp1

interp3() [1/2]

Value nk::builtin::interp3	(	const Value &	V,
		const Value &	Xq,
		const Value &	Yq,
		const Value &	Zq,
		const std::string &	method = "linear",
		std::pmr::memory_resource *	mr = nullptr )

3-D data interpolation on a uniform grid (interp3(V, Xq, Yq, Zq, method)).

Parameters

V	3-D array of values.
Xq	Query points x-coordinates.
Yq	Query points y-coordinates.
Zq	Query points z-coordinates.
method	Interpolation method.
mr	Memory resource.

Returns

Interpolated 3-D array.

interp3() [2/2]

Value nk::builtin::interp3	(	const Value &	X,
		const Value &	Y,
		const Value &	Z,
		const Value &	V,
		const Value &	Xq,
		const Value &	Yq,
		const Value &	Zq,
		const std::string &	method = "linear",
		std::pmr::memory_resource *	mr = nullptr )

3-D data interpolation on a grid (interp3(X, Y, Z, V, Xq, Yq, Zq, method)).

Parameters

X	Grid coordinates along X.
Y	Grid coordinates along Y.
Z	Grid coordinates along Z.
V	3-D array of values.
Xq	Query points x-coordinates.
Yq	Query points y-coordinates.
Zq	Query points z-coordinates.
method	Interpolation method.
mr	Memory resource.

Returns

Interpolated 3-D array.

makima()

Value nk::builtin::makima	(	const Value &	x,
		const Value &	y,
		const Value &	xq,
		std::pmr::memory_resource *	mr = nullptr )

Modified Akima cubic Hermite interpolation (makima(x, y, xq)).

Parameters

x	Sample grid points.
y	Sample values.
xq	Query points.
mr	Memory resource.

Returns

Interpolated values with reduced overshoots.

See also

spline, pchip

matchpairs()

MatchpairsResult nk::builtin::matchpairs	(	const Value &	C,
		double	cU,
		const std::string &	mode = "min",
		std::pmr::memory_resource *	mr = nullptr )

Solves linear sum assignment problem with cost threshold (matchpairs(C, cU, mode)).

Parameters

C	Cost matrix.
cU	Unmatched cost threshold.
mode	Optimization mode ("min" or "max").
mr	Memory resource.

Returns

Matchpairs result containing matched indices and unmatched rows/cols.

mkpp()

Value nk::builtin::mkpp	(	const Value &	breaks,
		const Value &	coefs,
		std::pmr::memory_resource *	mr = nullptr )

Constructs a piecewise polynomial structure (ppform).

Parameters

breaks	Vector of break points.
coefs	Matrix of local polynomial coefficients.
mr	Memory resource.

Returns

Piecewise polynomial struct.

See also

unmkpp, ppval

padecoef()

PadeCoef nk::builtin::padecoef	(	double	T,
		int	N,
		std::pmr::memory_resource *	mr = nullptr )

Padé approximation of time delay ([num, den] = padecoef(T, N)).

Computes order-N Padé rational approximation of time delay exp(-s*T).

Parameters

T	Time delay in seconds.
N	Approximation order.
mr	Memory resource for allocations (nullptr for default).

Returns

PadeCoef containing numerator and denominator polynomials.

pchip()

Value nk::builtin::pchip	(	const Value &	x,
		const Value &	y,
		const Value &	xq = Value(),
		std::pmr::memory_resource *	mr = nullptr )

Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) (pchip(x, y, xq)).

Parameters

x	Sample grid points.
y	Sample values.
xq	Query points (or empty to return piecewise polynomial structure).
mr	Memory resource.

Returns

Shape-preserving interpolated values or piecewise polynomial struct.

See also

spline, makima, ppval

poly()

Value nk::builtin::poly	(	const Value &	r,
		std::pmr::memory_resource *	mr = nullptr )

Polynomial with specified roots or characteristic polynomial of a matrix.

Parameters

r	Vector of roots or square matrix.
mr	Memory resource.

Returns

Row vector of polynomial coefficients in descending powers.

See also

roots, polyval

poly_of_matrix()

Value nk::builtin::poly_of_matrix	(	const Value &	A,
		std::pmr::memory_resource *	mr = nullptr )

Characteristic polynomial of square matrix (poly_of_matrix(A)).

Computes coefficients of det(s*I - A).

Parameters

A	Square matrix.
mr	Memory resource for allocations (nullptr for default).

Returns

Row vector containing characteristic polynomial coefficients.

See also

poly, roots, polyvalm

polyarea()

Value nk::builtin::polyarea	(	const Value &	x,
		const Value &	y,
		std::pmr::memory_resource *	mr = nullptr )

Area of 2-D polygonal contour (polyarea(x, y)).

Parameters

x	Vertex x-coordinates.
y	Vertex y-coordinates.
mr	Memory resource.

Returns

Signed or unsigned polygon area.

polyder() [1/2]

std::tuple< Value, Value > nk::builtin::polyder	(	const Value &	b,
		const Value &	a,
		std::pmr::memory_resource *	mr = nullptr )

Computes derivative of polynomial quotient b / a ([q, d] = polyder(b, a)).

Parameters

b	Numerator polynomial coefficients.
a	Denominator polynomial coefficients.
mr	Memory resource.

Returns

Tuple containing numerator and denominator polynomial coefficients {q, d}.

polyder() [2/2]

Value nk::builtin::polyder	(	const Value &	p,
		std::pmr::memory_resource *	mr = nullptr )

Computes polynomial derivative (polyder(p)).

Parameters

p	Vector of polynomial coefficients.
mr	Memory resource.

Returns

Derivative polynomial coefficients.

See also

polyint, polyval

polydiv()

PolyDiv nk::builtin::polydiv	(	const Value &	b,
		const Value &	a,
		std::pmr::memory_resource *	mr = nullptr )

Polynomial deconvolution and division ([q, r] = polydiv(b, a)).

Parameters

b	Dividend polynomial coefficients.
a	Divisor polynomial coefficients.
mr	Memory resource for allocations (nullptr for default).

Returns

PolyDiv containing quotient q and remainder r.

See also

deconv, conv

polyfit()

Value nk::builtin::polyfit	(	const Value &	x,
		const Value &	y,
		int	n,
		std::pmr::memory_resource *	mr = nullptr )

Polynomial least-squares curve fitting (polyfit(x, y, n)).

Parameters

x	Independent variable vector.
y	Dependent variable vector.
n	Degree of the fitting polynomial.
mr	Memory resource.

Returns

Fitted polynomial coefficients of length n + 1.

polyint()

Value nk::builtin::polyint	(	const Value &	p,
		double	k = 0.0,
		std::pmr::memory_resource *	mr = nullptr )

Computes polynomial integral (polyint(p, k)).

Parameters

p	Vector of polynomial coefficients.
k	Constant of integration (default: 0.0).
mr	Memory resource.

Returns

Integrated polynomial coefficients.

See also

polyder, polyval

polyval()

Value nk::builtin::polyval	(	const Value &	p,
		const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Evaluates polynomial p at points x via Horner's scheme.

Parameters

p	Vector of polynomial coefficients.
x	Evaluation point(s).
mr	Memory resource.

Returns

Evaluated values with the same shape as x.

See also

roots, polyder, polyint

polyvalm()

Value nk::builtin::polyvalm	(	const Value &	p,
		const Value &	A,
		std::pmr::memory_resource *	mr = nullptr )

Matrix polynomial evaluation (polyvalm(p, A)).

Evaluates polynomial p at square matrix A using Horner's method with matrix multiplications: Y = p[0]*A^n + p[1]*A^(n-1) + ... + p[n]*I.

Parameters

p	Coefficient vector in descending powers of s.
A	Square matrix.
mr	Memory resource for allocations (nullptr for default).

Returns

Square matrix result of polynomial evaluation.

See also

polyval, poly_of_matrix

polyxpoly()

PolyxpolyResult nk::builtin::polyxpoly	(	const Value &	x1,
		const Value &	y1,
		const Value &	x2,
		const Value &	y2,
		std::pmr::memory_resource *	mr = nullptr )

Intersection points of polygonal lines or boundaries ([xi, yi] = polyxpoly(x1, y1, x2, y2)).

Parameters

x1	First line x-coordinates.
y1	First line y-coordinates.
x2	Second line x-coordinates.
y2	Second line y-coordinates.
mr	Memory resource.

Returns

Intersection coordinates struct.

ppval()

Value nk::builtin::ppval	(	const Value &	pp,
		const Value &	xq,
		std::pmr::memory_resource *	mr = nullptr )

Evaluates a piecewise polynomial structure at query points xq.

Parameters

pp	Piecewise polynomial structure.
xq	Query points.
mr	Memory resource.

Returns

Evaluated values.

See also

mkpp, unmkpp, spline, pchip

residue()

ResidueResult nk::builtin::residue	(	const Value &	b,
		const Value &	a,
		std::pmr::memory_resource *	mr = nullptr )

Partial fraction expansion (residues) in Laplace s-domain ([r, p, k] = residue(b, a)).

Converts transfer function B(s)/A(s) to partial fraction form: B(s)/A(s) = r(1)/(s-p(1)) + ... + r(n)/(s-p(n)) + k(s).

Parameters

b	Numerator polynomial coefficients in descending powers of s.
a	Denominator polynomial coefficients in descending powers of s.
mr	Memory resource for allocations (nullptr for default).

Returns

ResidueResult containing residues r, poles p, and direct terms k.

See also

residuez, poly, roots

residuez()

ResidueResult nk::builtin::residuez	(	const Value &	b,
		const Value &	a,
		std::pmr::memory_resource *	mr = nullptr )

Partial fraction expansion in Z-domain ([r, p, k] = residuez(b, a)).

Converts discrete transfer function B(z)/A(z) to partial fraction form in z^-1: B(z)/A(z) = r(1)/(1 - p(1)*z^-1) + ... + k(z^-1).

Parameters

b	Numerator polynomial coefficients in descending powers of z.
a	Denominator polynomial coefficients in descending powers of z.
mr	Memory resource for allocations (nullptr for default).

Returns

ResidueResult containing residues r, poles p, and direct terms k.

See also

residue, poly, roots

roots()

Value nk::builtin::roots	(	const Value &	p,
		std::pmr::memory_resource *	mr = nullptr )

Computes polynomial roots (roots(p)).

Parameters

p	Vector of polynomial coefficients in descending powers: p[0]*x^N + ... + p[N].
mr	Memory resource for allocations (nullptr for default).

Returns

Column vector of complex/real roots (eigenvalues of companion matrix).

See also

poly, polyval

spline()

Value nk::builtin::spline	(	const Value &	x,
		const Value &	y,
		const Value &	xq = Value(),
		std::pmr::memory_resource *	mr = nullptr )

Cubic spline data interpolation (spline(x, y, xq)).

Parameters

x	Sample grid points.
y	Sample values.
xq	Query points (or empty to return piecewise polynomial structure).
mr	Memory resource.

Returns

Interpolated values or piecewise polynomial struct.

See also

pchip, makima, ppval

tf2zp()

std::tuple< Value, Value, Value > nk::builtin::tf2zp	(	const Value &	b,
		const Value &	a,
		std::pmr::memory_resource *	mr = nullptr )

Transfer function to zero-pole-gain conversion ([z, p, k] = tf2zp(b, a)).

Parameters

b	Numerator polynomial coefficients.
a	Denominator polynomial coefficients.
mr	Memory resource for allocations (nullptr for default).

Returns

Tuple containing zeros z, poles p, and gain k.

See also

zp2tf, residue

trapz() [1/3]

Value nk::builtin::trapz	(	const Value &	x,
		const Value &	y,
		int	dim,
		std::pmr::memory_resource *	mr = nullptr )

Trapezoidal numerical integration along specified dimension (trapz(x, y, dim)).

Parameters

x	Coordinate points.
y	Function values.
dim	Dimension along which to integrate.
mr	Memory resource.

Returns

Integrated array.

trapz() [2/3]

Value nk::builtin::trapz	(	const Value &	x,
		const Value &	y,
		std::pmr::memory_resource *	mr = nullptr )

Trapezoidal numerical integration over coordinates x (trapz(x, y)).

Parameters

x	Coordinate points.
y	Function values.
mr	Memory resource.

Returns

Integral approximation.

See also

cumtrapz, integral

trapz() [3/3]

Value nk::builtin::trapz	(	const Value &	y,
		std::pmr::memory_resource *	mr = nullptr )

Trapezoidal numerical integration with unit spacing (trapz(y)).

Parameters

y	Function values vector or array.
mr	Memory resource.

Returns

Integral approximation.

See also

cumtrapz, integral

unmkpp()

Value nk::builtin::unmkpp	(	const Value &	pp,
		std::pmr::memory_resource *	mr = nullptr )

Extracts piecewise polynomial structure fields ([breaks, coefs, l, k, d] = unmkpp(pp)).

Parameters

pp	Piecewise polynomial struct.
mr	Memory resource.

Returns

Breaks vector from pp struct.

See also

mkpp, ppval

zp2tf()

std::tuple< Value, Value > nk::builtin::zp2tf	(	const Value &	z,
		const Value &	p,
		double	k,
		std::pmr::memory_resource *	mr = nullptr )

Zero-pole-gain to transfer function conversion ([b, a] = zp2tf(z, p, k)).

Parameters

z	Zeros vector.
p	Poles vector.
k	Scalar gain factor.
mr	Memory resource for allocations (nullptr for default).

Returns

Tuple containing numerator b and denominator a polynomials.

See also

tf2zp, residue