# polynomial interpolation

Published articles for polynomial interpolation.

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## Polynomial Multiplication Using the FFT

DevFeed: [Polynomial Multiplication Using the FFT](<https://devfeed.tech/articles/polynomial-multiplication-using-the-fft-40459.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2022/11/16/polynomial-multiplication-using-the-fft/>)

Published: 2022-11-16T08:00:00Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [polynomials](<https://devfeed.tech/topics/polynomials.md>), [Algorithm](<https://devfeed.tech/topics/algorithm.md>), [NumPy](<https://devfeed.tech/topics/numpy.md>), [Mathematics](<https://devfeed.tech/topics/mathematics.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [arrays](<https://devfeed.tech/tags/arrays.md>), [efficiently](<https://devfeed.tech/tags/efficiently.md>), [fft](<https://devfeed.tech/tags/fft.md>), [fourier-transform](<https://devfeed.tech/tags/fourier-transform.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [polynomial-interpolation](<https://devfeed.tech/tags/polynomial-interpolation.md>), [polynomials](<https://devfeed.tech/tags/polynomials.md>), [programming](<https://devfeed.tech/tags/programming.md>), [python](<https://devfeed.tech/tags/python.md>)

### AI overview

This tutorial explains how to multiply two polynomials efficiently using the Fast Fourier Transform. It contrasts the naive O(n^2) approach with polynomial interpolation, pointwise multiplication, and carefully chosen roots of unity that enable reusable computations.

### Source excerpt

Problem: Compute the product of two polynomials efficiently. Solution: import numpy from numpy.fft import fft, ifft def poly_mul(p1, p2): """Multiply two polynomials. p1 and p2 are arrays of coefficients in degree-increasing order. """ deg1 = p1.shape[0] - 1 deg2 = p1.shape[0] - 1 # Would be 2*(deg1 + deg2) + 1, but the next-power-of-2 handles the +1 total_num_pts = 2 * (deg1 + deg2) next_power_of_2 = 1 << (total_num_pts - 1).

## The Welch-Berlekamp Algorithm for Correcting Errors in Data

DevFeed: [The Welch-Berlekamp Algorithm for Correcting Errors in Data](<https://devfeed.tech/articles/the-welch-berlekamp-algorithm-for-correcting-errors-in-data-40387.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2015/09/07/welch-berlekamp/>)

Published: 2015-09-07T11:02:00Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Algorithm](<https://devfeed.tech/topics/algorithm.md>), [Code](<https://devfeed.tech/topics/code.md>), [data](<https://devfeed.tech/topics/data.md>)

Tags: [berlekamp-welsh](<https://devfeed.tech/tags/berlekamp-welsh.md>), [code](<https://devfeed.tech/tags/code.md>), [error-correcting-codes](<https://devfeed.tech/tags/error-correcting-codes.md>), [error-correction](<https://devfeed.tech/tags/error-correction.md>), [finite-fields](<https://devfeed.tech/tags/finite-fields.md>), [graphics](<https://devfeed.tech/tags/graphics.md>), [image-analysis](<https://devfeed.tech/tags/image-analysis.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [polynomial-interpolation](<https://devfeed.tech/tags/polynomial-interpolation.md>), [polynomials](<https://devfeed.tech/tags/polynomials.md>), [programming](<https://devfeed.tech/tags/programming.md>), [python](<https://devfeed.tech/tags/python.md>), [reed-solomon-codes](<https://devfeed.tech/tags/reed-solomon-codes.md>), [resilience](<https://devfeed.tech/tags/resilience.md>)

### AI overview

This tutorial explains the intuition behind Reed-Solomon error-correcting codes and implements them using polynomial-based encoding. It describes how adding redundant points can provide resilience to corrupted data while limiting the extra information required.

### Source excerpt

In this post we'll implement Reed-Solomon error-correcting codes and use them to play with codes. In our last post we defined Reed-Solomon codes rigorously, but in this post we'll focus on intuition and code. As usual the code and data used in this post is available on this blog's Github page. The main intuition behind Reed-Solomon codes (and basically all the historically major codes) is Error correction is about adding redundancy, and polynomials are a really efficient way to do that.

## Learning a single-variable polynomial, or the power of adaptive queries

DevFeed: [Learning a single-variable polynomial, or the power of adaptive queries](<https://devfeed.tech/articles/learning-a-single-variable-polynomial-or-the-power-of-adaptive-queries-40370.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2014/11/18/learning-a-single-variable-polynomial-or-the-power-of-adaptive-queries/>)

Published: 2014-11-18T09:00:18Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Computing](<https://devfeed.tech/topics/computing.md>), [Algorithm](<https://devfeed.tech/topics/algorithm.md>), [Query (disambiguation)](<https://devfeed.tech/topics/query.md>)

Tags: [adaptive-queries](<https://devfeed.tech/tags/adaptive-queries.md>), [algorithm](<https://devfeed.tech/tags/algorithm.md>), [github](<https://devfeed.tech/tags/github.md>), [implementation](<https://devfeed.tech/tags/implementation.md>), [learning-theory](<https://devfeed.tech/tags/learning-theory.md>), [number-theory](<https://devfeed.tech/tags/number-theory.md>), [polynomial-identity-testing](<https://devfeed.tech/tags/polynomial-identity-testing.md>), [polynomial-interpolation](<https://devfeed.tech/tags/polynomial-interpolation.md>), [polynomials](<https://devfeed.tech/tags/polynomials.md>)

### AI overview

This tutorial shows how adaptive queries can recover a secret single-variable polynomial with nonnegative integer coefficients using only two queries: p(1) and p(p(1)+1). It explains how modular arithmetic extracts the coefficients and contrasts this with the larger number of non-adaptive queries required by polynomial interpolation.

### Source excerpt

Problem: Alice chooses a secret polynomial $ p(x)$ with nonnegative integer coefficients. Bob wants to discover this polynomial by querying Alice for the value of $ p(x)$ for some integer $ x$ of Bob's choice. What is the minimal number of queries Bob needs to determine $ p(x)$ exactly? Solution: Two queries. The first is $ p(1)$, and if we call $ N = p(1) + 1$, then the second query is $ p(N)$.

## The Mathematics of Secret Sharing

DevFeed: [The Mathematics of Secret Sharing](<https://devfeed.tech/articles/the-mathematics-of-secret-sharing-40360.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2014/06/23/the-mathematics-of-secret-sharing/>)

Published: 2014-06-23T09:00:10Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Algorithm](<https://devfeed.tech/topics/algorithm.md>), [Haskell](<https://devfeed.tech/topics/haskell.md>), [Programming](<https://devfeed.tech/topics/programming.md>), [Programming language](<https://devfeed.tech/topics/programming-language.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [cryptography](<https://devfeed.tech/tags/cryptography.md>), [haskell](<https://devfeed.tech/tags/haskell.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [polynomial-interpolation](<https://devfeed.tech/tags/polynomial-interpolation.md>), [programming](<https://devfeed.tech/tags/programming.md>), [secret-sharing](<https://devfeed.tech/tags/secret-sharing.md>)

### AI overview

This tutorial introduces the secret sharing problem and describes a protocol in which a secret can be reconstructed from any required number of participants' pieces, while fewer pieces cannot reliably recover it. It presents the protocol in Haskell.

### Source excerpt

Here's a simple puzzle with a neat story. A rich old woman is drafting her will and wants to distribute her expansive estate equally amongst her five children. But her children are very greedy, and the woman knows that if he leaves her will unprotected her children will resort to nefarious measures to try to get more than their fair share. In one fearful scenario, she worries that the older four children will team up to bully the youngest child entirely out of his claim!