Fast Numerical Multivariate Multipoint Evaluation

We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both exact and approximate versions of the algorithm. The input to the algorithms are (1) coefficients of an m-variate polynomial f with degree d in each variable, and (2) points $\mathbf{a}_{1}, \ldots, \mathbf{a}_{N}$ each of whose coordinate has absolute value bounded by one. Approximate version: Given additionally an accuracy parameter t, the algorithm computes rational numbers $\beta_{1}, \ldots, \beta_{N}$ such that $\left|f\left(\mathbf{a}_{i}\right)-\beta_{i}\right| \leq 1 / 2^{t}$ for all i, and has a running time of $\left(\left(N m+d^{m}\right) t\right)^{1+o(1)}$ for all m and all sufficiently large d. Exact version (when over rationals): Given additionally a bound s on the bit-complexity of all the rational numbers in the input and output, the algorithm computes the rational numbers $f\left(\mathbf{a}_{1}\right), \ldots, f\left(\mathbf{a}_{N}\right)$, in time $\left(\left(N m+d^{m}\right) s\right)^{1+o(1)}$ for all m and all sufficiently large d. Our results also naturally extend to the case when the input is over the field of real or complex numbers under an appropriate standard model of representation of field elements in such fields. Prior to this work, a nearly-linear time algorithm for multivariate multipoint evaluation (exact or approximate) over any infinite field appears to be known only for the case of univariate polynomials, and was discovered in a recent work of Moroz [Proc. 62nd FOCS, 2021]. In this work, we extend this result from the univariate to the multivariate setting. However, our algorithm is based on ideas that seem to be conceptually different from those of Moroz [Proc. 62nd FOCS, 2021] and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar & Umans [Proc. 63rd FOCS, 2022] for multivariate multipoint evaluation over finite fields, and known efficient algorithms for the problems of rational number reconstruction and fast Chinese remaindering in computational number theory.

Authors

• Sumanta Ghosh
• Prahladh Harsha
• Simao Herdade
• Mrinal Kumar 0001
• Ramprasad Saptharishi

Keywords

• Computer science
• Computational modeling
• Approximation algorithms
• Finite element analysis
• Computational efficiency
• Galois fields
• Standards
• Sufficiently Large
• Running Time
• Real Numbers
• Complex Numbers
• Algorithm For Problem
• Evaluation Of Algorithms
• Polynomial Coefficients
• Finite Field
• Rational Numbers
• Computational Theory
• Number Theory
• Field Of Complex Numbers
• Field Of Real Numbers
• Exact Version
• Denominator
• Loss Of Generality
• Fast Fourier Transform
• Time Complexity
• Coordinates Of Points
• Input Size
• Input Bits
• Part Of The Input
• Bitwise Operations
• Complex Output
• Output Bits
• Computer Algebra
• Deterministic Time
• Input Elements
• Root Of Unity
• Continued Fraction
• Multivariate Multipoint Evaluation
• Numerical Algorithm
• Chinese Remainder Theorem