# Research interests

I am generally interested in matrix eigenvalues and rational approximation theory, and their applications.- Algorithms for matrix decompositions (SVD, eigenvalue decomposition)
- Rational and polynomial approximation theory
- Eigenvalues and approximation theory in statistics (Monte Carlo integration, PCA)
- Eigenvalue perturbation theory, matrix analysis
- Numerical optimization

# Publications

- Evan S. Gawlik, Yuji Nakatsukasa and Brian D. Sutton,
**A backward stable algorithm for computing the CS decomposition via the polar decomposition.***SIAM Journal on Matrix Analysis and Applications,*39(3), 1448-1469, 2018. pdf - Yuji Nakatsukasa and Lloyd N. Trefethen,
**Rational approximation of x^n**, to appear in*Proceedings of the American Mathematical Society,*pdf. - Behnam Hashemi and Yuji Nakatsukasa,
**On the spectral problem for trivariate functions,**to appear in*BIT Numerical Mathematics,*pdf - Silviu-Ioan Filip, Yuji Nakatsukasa, Lloyd N. Trefethen, and Bernhard Beckermann,
**Rational minimax approximation via adaptive barycentric representations,***SIAM Journal on Scientific Computing,*, 40(4), A2427-A2455, 2018. pdf - Shinji Ito and Yuji Nakatsukasa,
**Stable polefinding and rational least-squares via eigenvalues**, pdf,*Numerische Mathematik,*Vol. 139 (3), pp 633-682, 2018. - Zhening Li, Yuji Nakatsukasa, Tasuku Soma, and André Uschmajew,
**On orthogonal tensors and best rank-one approximation ratio,***SIAM Journal on Matrix Analysis and Applications,*39(1), pp. 400-425, 2018. pdf - Yuji Nakatsukasa, Olivier Sete and Lloyd N. Trefethen,
**The AAA algorithm for rational approximation,***SIAM Journal on Scientific Computing,*40(3), A1494-A1522, 2018. pdf - Yuji Nakatsukasa, Leo Taslaman, Francoise Tisseur and Ion Zaballa,
**Reduction of matrix polynomials to simpler forms,***SIAM Journal on Matrix Analysis and Applications,*Vol. 39-1, pp. 148-177, 2018. pdf -
Satoru Adachi and Yuji Nakatsukasa,
**Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint,**to appear in*Mathematical Programming,*pdf. - Hadrien Montanelli and Yuji Nakatsukasa,
**Fourth-order time-stepping for stiff PDEs on the sphere,***SIAM Journal on Scientific Computing,*Vol. 40 (1), pp. A421--A451, 2018. pdf - Yuji Nakatsukasa,
**Accuracy of singular vectors obtained by projection-based SVD methods,***BIT Numerical Mathematics,*Vol. 57 (4), pp. 1137-1152, 2017. pdf - Satoru Adachi, Satoru Iwata, Yuji Nakatsukasa and Akiko Takeda,
**Solving the trust-region subproblem by a generalized eigenvalue problem,***SIAM Journal on Optimization,*Vol. 27 (1), pp. 269-291, 2017. pdf, MATLAB code

- Yuji Nakatsukasa, Vanni Noferini and Alex Townsend,
**Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach,***SIAM Journal on Matrix Analysis and Applications.*Vol 38-1, pp. 1-29, 2017.pdf - Yuji Nakatsukasa, Tasuku Soma and André Uschmajew,
**Finding a low-rank basis in a matrix subspace,***Mathematical Programming,*Vol. 162(1), pp. 325-361, 2017. pdf - Shinsaku Sakaue, Yuji Nakatsukasa, Akiko Takeda and Satoru Iwata,
**Solving generalized CDT problems via two-parameter eigenvalues,***SIAM Journal on Optimization,*Vol 26-3, pp. 1669-1694, 2016. pdf

- Yuji Nakatsukasa and Roland W. Freund,
**Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions,***SIAM Review,*Vol 58(3), pp. 461-493, 2016. pdf MATLAB codes

- Yuji Nakatsukasa and Vanni Noferini,
**On the stability of computing polynomial roots via confederate linearizations,**,*Mathematics of Computation*, Vol 85, pp.2391-2425, 2016. pdf - Stefan Guettel and Yuji Nakatsukasa,
**Scaled and squared subdiagonal Pade approximation for the matrix exponential,***SIAM Journal on Matrix Analysis and Applications,*Vol. 37(1), pp. 145-170, 2016. pdf MATLAB codes - Satoru Iwata, Yuji Nakatsukasa and Akiko Takeda,
**Computing the signed distance between overlapping ellipsoids,***SIAM Journal on Optimization,*Vol. 25(4), pp. 2359-2384, 2015. pdf (with corrections) -
Yusaku Yamamoto, Yuji Nakatsukasa, Yuka Yanagisawa and Takeshi Fukaya,
**Roundoff Error Analysis of the CholeskyQR2 Algorithm,***Electronic Transactions on Numerical Analysis,*Vol. 44, pp. 306-326, 2015. pdf -
Yuji Nakatsukasa, Vanni Noferini and Alex Townsend,
**Computing the common zeros of two bivariate functions via Bezout resultants,***Numerische Mathematik,*Vol. 129, pp. 181-209, 2015. pdf MATLAB code - Patrizio Neff, Yuji Nakatsukasa and Andreas Fischle,
**A logarithmic minimization property of the unitary polar factor in the spectral and Frobenius norms,***SIAM Journal on Matrix Analysis and Applications,*Vol. 35(3), pp. 1132-1154, 2014. pdf -
Johannes Lankeit, Patrizio Neff and Yuji Nakatsukasa,
**The minimization of matrix logarithms - on a fundamental property of the unitary polar factor,***Linear Algebra and its Applications,*Vol. 449, pp. 28-42, 2014. pdf -
Fernando de Teran, Ross Lippert, Yuji Nakatsukasa and Vanni Noferini,
**Flanders' theorem for many matrices under commutativity assumptions,***Linear Algebra and its Applications,*Vol. 443, pp. 120-138, 2014. pdf - Yuji Nakatsukasa and Nicholas J. Higham,
**Stable and efficient spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD,***SIAM Journal on Scientific Computing,*Vol. 35(3), pp. A1325-A1349, 2013. pdf. Codes available at MATLAB Central File Exchange - Yuji Nakatsukasa, Naoki Saito and Ernest Woei,
**Mysteries around the graph Laplacian eigenvalue 4,***Linear Algebra and its Applications,*Vol. 438 (8), pp. 3231-3246, 2013. pdf - Yuji Nakatsukasa and Nicholas J. Higham,
**Backward stability of iterations for computing the polar decomposition,***SIAM Journal on Matrix Analysis and Applications,*Vol. 33, No. 2, pp. 460-479, 2012. pdf - Ren-Cang Li, Yuji Nakatsukasa, Ninoslav Truhar and Wei-guo Wang,
**Perturbation of multiple eigenvalues of Hermitian matrices,***Linear Algebra and its Applications,*Vol. 437, pp. 202-213, 2012. pdf - Yuji Nakatsukasa,
**On the condition numbers of a multiple eigenvalue of a generalized eigenvalue problem,***Numerische Mathematik,*Vol. 121, No. 3, pp. 531-544, 2012. pdf - Yuji Nakatsukasa, Kensuke Aishima and Ichitaro Yamazaki,
**dqds with aggressive early deflation,***SIAM Journal on Matrix Analysis and Applications,*Vol. 33, No. 1, pp. 22-51, 2012. Fortran Code pdf - Yuji Nakatsukasa,
**Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices,***Applied Numerical Mathematics,*Vol.62, No. 1, pp. 67-78, 2012. pdf - Yuji Nakatsukasa,
**The tan theta theorem with relaxed conditions,***Linear Algebra and its Applications,*Vol. 436, pp. 1528-1534, 2012. pdf - Ren-Cang Li, Yuji Nakatsukasa, Ninoslav Truhar and Shufang Xu,
**Perturbation of partitioned Hermitian definite generalized eigenvalue problems,***SIAM Journal on Matrix Analysis and Applications,*Vol.32, No.2, pp. 642-663, 2011. pdf Erratum - Yuji Nakatsukasa,
**Gerschgorin's theorem for generalized eigenvalue problems in the Euclidean metric,***Mathematics of Computation,*Vol.80, pp. 2127-2142, 2011. pdf - Yuji Nakatsukasa, Zhaojun Bai and
Francois Gygi,
**Optimizing Halley's iteration for computing the matrix polar decomposition,***SIAM Journal on Matrix Analysis and Applications,*Vol.31, No.5, pp. 2700-2720, 2010. pdf - Yuji Nakatsukasa,
**Perturbation behavior of a multiple eigenvalue in generalized Hermitian eigenvalue problems,***BIT Numerical Mathematics,*Vol. 50, No. 1, pp. 109-121, 2010. pdf - Yuji Nakatsukasa,
**Absolute and relative Weyl theorems for generalized eigenvalue problems,***Linear Algebra and its Applications,*Vol. 432, No. 1, pp. 242-248, 2010. pdf

# conference proceedings

- Takeshi Fukaya, Ramaseshan Kannan, Yuji Nakatsukasa, Yusaku Yamamoto, Yuka Yanagisawa,
**Performance evaluation of the shifted Cholesky QR algorithm for ill-conditioned matrices,***Supercomputing SC18 poster, accepted.* - Yuji Nakatsukasa,
**Off-diagonal perturbation, first-order approximation and quadratic residual bounds for matrix eigenvalue problems,**to appear in*Lecture Notes in Computer Science, EPASA 2015, pp 233-249.*, Preprint. -
Yusaku Yamamoto,
Yuji Nakatsukasa,
Yuka Yanagisawa
and Takeshi Fukaya,
**Roundoff error analysis of the CholeskyQR2 algorithm in an oblique inner product,***JSIAM Letters,*Vol. 8, pp.5-8, 2016. -
Satoru Iwata, Yuji Nakatsukasa and Akiko Takeda,
**Global optimization methods for extended Fisher discriminant analysis,***AISTATS 2014,*pp. 411-419. link Related codes at MATLAB Central File Exchange -
Takeshi Fukaya, Yuji Nakatsukasa, Yuka Yanagisawa and Yusaku Yamamoto,
**CholeskyQR2: A Simple and communication-avoiding algorithm for computing a tall-skinny QR factorization on a large-scale parallel system,***ScalA 2014,*pp. 31-38. link

# preprints

- Yuji Nakatsukasa,
**Sharp error bounds for Ritz vectors and approximate singular vectors.**Preprint. -
Takeshi Fukaya, Ramaseshan Kannan, Yuji Nakatsukasa, Yusaku Yamamoto, Yuka Yanagisawa,
**Shifted CholeskyQR for computing the QR factorization of ill-conditioned matrices.**Preprint. - Yuji Nakatsukasa,
**Approximate and integrate: Variance reduction in Monte Carlo integration via function approximation.**Preprint. - Hatem Ltaief, Dalal E. Sukkari, Aniello Esposito,
Yuji Nakatsukasa and David E. Keyes,
**Massively Parallel Polar Decomposition on Distributed-Memory Systems, Preprint.** - Yuji Nakatsukasa and Vanni Noferini,
**Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems, Preprint.** - Yuji Nakatsukasa and Francoise Tisseur,
**Eigenvector error bound and perturbation for polynomial and rational eigenvalue problems,**Preprint.

# thesis

**Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition,**PhD dissertation, University of California at Davis, 2011. pdf

# awards

- Householder award, 2014.
- 15th Leslie Fox prize, 2011.
- SIAM UKIE annual meeting best poster prize (second prize), January 2018.
- JSIAM annual meeting outstanding talks, September 2014.
- SIAM UKIE Best Student Talks, Biennial NA Meeting, July 2011.
- Yueh-Jing Lin Fund, UC Davis Department of Mathematics, June 2011.