# Accelerated Newton Iteration: Roots of Black Box Polynomials and Matrix Eigenvalues

@article{Louis2015AcceleratedNI, title={Accelerated Newton Iteration: Roots of Black Box Polynomials and Matrix Eigenvalues}, author={Anand Louis and Santosh S. Vempala}, journal={ArXiv}, year={2015}, volume={abs/1511.03186} }

We study the problem of computing the largest root of a real rooted polynomial $p(x)$ to within error $\varepsilon $ given only black box access to it, i.e., for any $x \in {\mathbb R}$, the algorithm can query an oracle for the value of $p(x)$, but the algorithm is not allowed access to the coefficients of $p(x)$. A folklore result for this problem is that the largest root of a polynomial can be computed in $O(n \log (1/\varepsilon ))$ polynomial queries using the Newton iteration. We give a…

## 4 Citations

### Fast Approximation of Polynomial Zeros and Matrix Eigenvalues

- MathematicsArXiv
- 2023

. Given a black box (oracle) for the evaluation of a univariate polynomial p ( x ) of a degree d , we seek its zeros, that is, the roots of the equation p ( x ) = 0 . At FOCS 2016 Louis and Vempala…

### New Progress in Classic Area: Polynomial Root-squaring and Root-finding

- Computer Science, MathematicsArXiv
- 2022

This work has found simple but novel reduction of the iterations applied for Newton’s inverse ratio − p ′ ( x ) /p (x ) to approximation of the power sums of the zeros of p ( x) and its reverse polynomial and proposed to apply it for fast black box initialization ofPolynomial root-ﬁnding by means of functional iterations such as Newton's, Ehrlich's, and Weierstrass's.

### Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles

- Computer Science, MathematicsCASC
- 2022

This work describes root exclusion, root counting, root radius approximation and a procedure for contracting a disc towards the cluster of root it contains, called ε-compression, and combines them in a prototype root clustering algorithm that competes advantageously with user’s choice for root finding, MPsolve.

### Low-SNR Speech Enhancement and Separation in Driving Environment

- Engineering, Computer Science2019 IEEE International Conference on Consumer Electronics - Taiwan (ICCE-TW)
- 2019

The low frequency suppression pretreatment of mixed signal with noise is proposed, and the first-order recursive smoothing noise estimation algorithm is improved, and bias compensation algorithm is added to reduce the estimation error.

## 33 References

### On the complexity of computing determinants

- Mathematics, Computer Sciencecomputational complexity
- 2004

New baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems that deterministically compute the determinant, characteristic polynomial and adjoint of A with n3.2+o(1) and O(n2.697263) ring additions, subtractions and multiplications are presented.

### Exact solution of linear equations usingP-adic expansions

- Mathematics
- 1982

SummaryA method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA…

### The Quasi-Random Perspective on Matrix Spectral Analysis with Applications

- Computer Science, MathematicsArXiv
- 2015

This work analyzes the discrepancy of an n-dimensional sequence formed by taking the fractional part of integer multiples of the vector of eigenvalues of the input matrix, and gives rise to a conceptually new algorithm to compute an approximate spectral decomposition of any n x n Hermitian matrix.

### A basic family of iteration functions for polynomial root finding and its characterizations

- Mathematics
- 1997

### The complexity of the matrix eigenproblem

- Computer Science, MathematicsSTOC '99
- 1999

The bound O(n’ log n + (n log’ n) log b) on the randomized arithmetic complexity of the eigenproblem for generic matrices of the classes of n x n Toeplitz, Hank& ToePlitzlike, Hank &like and Toe Plits-likeplus-Hank&like matrices is proved.

### An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problems

- MathematicsMath. Program.
- 2009

Newton’s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second Newton step to a nonlinear-equations reformulation of the nonlinear complementarity problem (NCP) whose derivative is strongly semismooth when the function f arising in the NCP is sufficiently smooth.

### Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD

- Computer ScienceSIAM J. Sci. Comput.
- 2013

New spectral divide and conquer algorithms for the symmetric eigenvalue problem and the singular value decomposition that are backward stable, achieve lower bounds on communication costs recently derived by Ballard, Demmel, Holtz, and Schwartz, and have operation counts within a small constant factor of those for the standard algorithms.

### Fast linear algebra is stable

- MathematicsNumerische Mathematik
- 2007

It is shown that essentially all standard linear algebra operations, including LU decompositions, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in O(nω+η) operations.