Polynomial Operations

poly

Return polynomial coefficients given a sequence of roots. The returned array represents coefficients in descending degree order, with a leading coefficient of 1.

function poly(seq_of_zeros: ArrayLike): NDArray

Name	Type	Default	Description
`seq_of_zeros`	`ArrayLike`	—	Array of polynomial roots.

Returns: NDArray — 1-D array of polynomial coefficients, highest degree first.

import * as np from 'numpy-ts';

// Polynomial with roots at 1, 2, 3  =>  (x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x - 6
const c = np.poly([1, 2, 3]);
// array([1, -6, 11, -6])

polyadd

Add two polynomials. Inputs are coefficient arrays in descending degree order. The result is trimmed of leading zeros.

function polyadd(a1: ArrayLike, a2: ArrayLike): NDArray

Name	Type	Default	Description
`a1`	`ArrayLike`	—	Coefficients of the first polynomial.
`a2`	`ArrayLike`	—	Coefficients of the second polynomial.

Returns: NDArray — Coefficients of the sum polynomial.

import * as np from 'numpy-ts';

// (x^2 + 2x + 3) + (4x + 5) = x^2 + 6x + 8
const sum = np.polyadd([1, 2, 3], [4, 5]);
// array([1, 6, 8])

polyder

Differentiate a polynomial one or more times. Coefficients are in descending degree order.

function polyder(p: ArrayLike, m?: number): NDArray

Name	Type	Default	Description
`p`	`ArrayLike`	—	Polynomial coefficients (highest degree first).
`m`	`number`	`1`	Number of times to differentiate.

Returns: NDArray — Coefficients of the m-th derivative.

import * as np from 'numpy-ts';

// d/dx (3x^3 + 2x^2 + x + 5) = 9x^2 + 4x + 1
const d = np.polyder([3, 2, 1, 5]);
// array([9, 4, 1])

// Second derivative: 18x + 4
const d2 = np.polyder([3, 2, 1, 5], 2);
// array([18, 4])

polydiv

Divide one polynomial by another. Returns quotient and remainder as a tuple.

function polydiv(u: ArrayLike, v: ArrayLike): [NDArray, NDArray]

Name	Type	Default	Description
`u`	`ArrayLike`	—	Dividend polynomial coefficients.
`v`	`ArrayLike`	—	Divisor polynomial coefficients.

Returns: [NDArray, NDArray] — Tuple of [quotient, remainder] coefficient arrays. Throws: Error if the divisor is the zero polynomial.

import * as np from 'numpy-ts';

// (x^3 - 1) / (x - 1) = x^2 + x + 1  remainder 0
const [q, r] = np.polydiv([1, 0, 0, -1], [1, -1]);
// q = array([1, 1, 1])
// r = array([0])

polyfit

Least-squares polynomial fit of degree deg to data points (x, y). Returns coefficients in descending degree order.

function polyfit(x: ArrayLike, y: ArrayLike, deg: number): NDArray

Name	Type	Default	Description
`x`	`ArrayLike`	—	x-coordinates of the sample points.
`y`	`ArrayLike`	—	y-coordinates of the sample points.
`deg`	`number`	—	Degree of the fitting polynomial. Must be less than the number of data points.

Returns: NDArray — Polynomial coefficients, highest degree first. Throws: Error if deg >= len(x).

import * as np from 'numpy-ts';

const x = np.array([0, 1, 2, 3, 4]);
const y = np.array([0, 1, 4, 9, 16]);

// Fit a degree-2 polynomial: should recover y = x^2
const coeffs = np.polyfit(x, y, 2);
// array([1, 0, 0])  (approximately)

polyint

Integrate a polynomial one or more times. Optionally specify integration constants.

function polyint(p: ArrayLike, m?: number, k?: number | number[]): NDArray

Name	Type	Default	Description
`p`	`ArrayLike`	—	Polynomial coefficients (highest degree first).
`m`	`number`	`1`	Number of times to integrate.
`k`	`number \| number[]`	`0`	Integration constant(s). A single number applies to all integrations; an array specifies the constant for each successive integration.

Returns: NDArray — Coefficients of the integrated polynomial.

import * as np from 'numpy-ts';

// Integrate 3x^2 + 2x + 1  =>  x^3 + x^2 + x + C
const p = np.polyint([3, 2, 1]);
// array([1, 1, 1, 0])

// Integrate with constant C = 5
const p2 = np.polyint([3, 2, 1], 1, 5);
// array([1, 1, 1, 5])

polymul

Multiply two polynomials. Equivalent to convolving their coefficient arrays.

function polymul(a1: ArrayLike, a2: ArrayLike): NDArray

Name	Type	Default	Description
`a1`	`ArrayLike`	—	Coefficients of the first polynomial.
`a2`	`ArrayLike`	—	Coefficients of the second polynomial.

Returns: NDArray — Coefficients of the product polynomial.

import * as np from 'numpy-ts';

// (x + 1) * (x - 1) = x^2 - 1
const prod = np.polymul([1, 1], [1, -1]);
// array([1, 0, -1])

polysub

Subtract two polynomials. Result is trimmed of leading zeros.

function polysub(a1: ArrayLike, a2: ArrayLike): NDArray

Name	Type	Default	Description
`a1`	`ArrayLike`	—	Coefficients of the first polynomial.
`a2`	`ArrayLike`	—	Coefficients of the second polynomial.

Returns: NDArray — Coefficients of the difference a1 - a2.

import * as np from 'numpy-ts';

// (x^2 + 2x + 3) - (2x + 1) = x^2 + 2
const diff = np.polysub([1, 2, 3], [2, 1]);
// array([1, 0, 2])

polyval

Evaluate a polynomial at one or more points using Horner’s method.

function polyval(p: ArrayLike, x: ArrayLike | number): NDArray | number

Name	Type	Default	Description
`p`	`ArrayLike`	—	Polynomial coefficients (highest degree first).
`x`	`ArrayLike`	—	Points at which to evaluate the polynomial. Can be a single number.

Returns: NDArray | number — Scalar if x is a single number, otherwise an NDArray of evaluated values.

import * as np from 'numpy-ts';

// p(x) = x^2 + 2x + 3
const p = [1, 2, 3];

// Evaluate at a single point
console.log(np.polyval(p, 2));  // 11

// Evaluate at multiple points
const vals = np.polyval(p, [0, 1, 2, 3]);
// array([3, 6, 11, 18])

roots

Find the roots of a polynomial with given coefficients. Uses the companion matrix eigenvalue method, the same algorithm used by NumPy.

function roots(p: ArrayLike): NDArray

Name	Type	Default	Description
`p`	`ArrayLike`	—	Polynomial coefficients (highest degree first). Leading zeros are stripped.

Returns: NDArray — 1-D complex128 array of roots, sorted by descending magnitude.

import * as np from 'numpy-ts';

// Roots of x^2 - 5x + 6 = (x-2)(x-3)
const r = np.roots([1, -5, 6]);
// array([3+0j, 2+0j])

// Roots of x^2 + 1 (complex roots)
const r2 = np.roots([1, 0, 1]);
// array([0+1j, 0-1j])

Overview

NDArray

Array Creation

Math

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Reductions & Statistics

Linear Algebra

Logic & Comparison

Sorting & Searching

Set Operations

Bitwise

Indexing

Gradient & Differences

Polynomials

Random (np.random)

FFT (np.fft)

I/O

Utilities

Polynomial Operations

poly

polyadd

polyder

polydiv

polyfit

polyint

polymul

polysub

polyval

roots

​poly

​polyadd

​polyder

​polydiv

​polyfit

​polyint

​polymul

​polysub

​polyval

​roots

poly

polyadd

polyder

polydiv

polyfit

polyint

polymul

polysub

polyval

roots