EDUCATION
M.Sc. in Applied Mathematics, Technion (Israel Institute of Technology)
Ph.D. in Applied Mathematics, Caltech (California Institute of Technology)
PUBLICATIONS
SELECTED NEWER PUBLICATIONS
[1] A. Melman (2023): “Matrices whose eigenvalues are those of a quadratic matrix polynomial”, Linear Algebra and its Applications, 676, 131—149.
[2] A. Melman (2022): “Rootfinding techniques that work”, The Teaching of Mathematics, XXV, 38—52.
[3] A. Melman (2022): “Polynomials whose zeros are powers of a given polynomial's zeros”, American Mathematical Monthly, 129, 276—280.
[4] A. Melman (2021): “Zero exclusion sectors for some polynomials with structured coefficients”, American Mathematical Monthly, 128, 322—336.
[5] S. Friedland and A. Melman (2020): “A note on Hermitian positive semidefinite matrix polynomials”, Linear Algebra and its Applications, 598, 105—109.
[6] A. Melman (2020): “A unifying framework for generalizations of the Eneström-Kakeya theorem”, Elemente der Mathematik, 75, 1--14.
[7] A. Melman (2020): “Directional bounds for polynomial zeros and eigenvalues”, Journal of Mathematical Analysis and Applications, 482, no. 2, 123571, 11pp.
[8] A. Melman (2019): “Polynomial eigenvalue bounds from companion matrix polynomials”, Linear and Multilinear Algebra, 67, 598—612.
[9] A. Melman (2019): “Extension of a theorem by Hayashi”, Journal of Mathematical Analysis and Applications, 470, 1046-1055.
[10] A. Melman (2018): “Eigenvalue bounds for matrix polynomials in generalized bases”, Mathematics of Computation, 87, 1935—1948.
[11] A. Melman (2018): “Optimality of a polynomial multiplier”, American Mathematical Monthly, 125, 158—163
[12] A. Melman (2018): “Improved Cauchy radius for scalar and matrix polynomials”, Proceedings of the American Mathematical Society, 146, 613-624.
SELECTED OLDER PUBLICATIONS
[1] A. Melman (2016): “Improvement of Pellet's theorem for scalar and matrix polynomials”, Comptes Rendus Math. Acad. Sci. Paris, 8, 859-863.
[2] A. Melman (2012): “Geometry of trinomials”, Pacific Journal of Mathematics, 259, 141-159.
[3] A. Melman (2012): “Ovals of Cassini for Toeplitz matrices”, Linear and Multilinear Algebra, 60, pp.189-199.
[4] A. Melman and W.B. Gragg (2006): “An optimization framework for polynomial zerofinders”, American Mathematical Monthly. 113, pp. 794-804.
[5] A. Melman: (2004): “Computation of the smallest even and odd eigenvalues of a symmetric positive-definite Toeplitz matrix”, SIAM J. on Matrix Analysis and Applications, 25, No. 4, pp. 947-963.
[6] A. Melman (2001): "The even-odd split Levinson algorithm for Toeplitz systems”, SIAM J. on Matrix Analysis and Applications, 23, No. 1, pp. 256-270.
[7] A. Melman and G. Rabinowitz (2000): "Efficient methods for a class of continuous nonlinear knapsack problems", SIAM Review, Vol. 42, No. 3, pp. 440-448.
[8] A. Melman (1998): "Spectral functions for real symmetric Toeplitz matrices", Journal of Computational and Applied Mathematics, 98, pp. 233-243.
[9] A. Melman (1997): "Analysis of higher-order methods for secular equations", Mathematics of Computation, Vol. 67, No. 221, pp. 271-286.
[10] A. Melman (1997): "Geometry and convergence for Euler's and Halley's methods", SIAM Review, 39, No. 4, pp. 728-735.
[11] A. Melman (1997): "A unifying convergence analysis of second-order methods for secular equations", Mathematics of Computation, Vol. 66, No. 217, pp. 333-344.
[12] A. Melman (1995): "Numerical Solution of a Secular Equation", Numerische Mathematik, 69, pp. 483-493.
[13] A. Ben-Tal, A. Melman and J. Zowe (1990): "Curved search methods for unconstrained optimization", Optimization, 21, No.5,pp. 669-695.
ALL PUBLICATIONS
[1] A.Ben-Tal, A.Melman and J.Zowe, "Curved search methods for unconstrained optimization", Optimization, 21 (1990), 669-695.
[2] A.Melman, "A new linesearch method for quadratically constrained convex programming", Operations Research Letters, 16 (1994), 67-77.
[3] A.Melman, "Numerical Solution of a Secular Equation", Numerische Mathematik, 69 (1995), 483-493.
[4] A.Melman and R.Polyak, "The Newton modified barrier method for QP problems", Annals of Operations Research, 62 (1996), 465-519.
[5] A.Melman, "A linesearch procedure in barrier methods for some convex programming problems", SIAM J. of Optimization, 6 (1996), 283-298.
[6] A.Melman, "A unifying convergence analysis of second-order methods for secular equations", Mathematics of Computation, 66 (1997), 333-344.
[7] A.Melman, "Geometry and convergence for Euler's and Halley's methods", SIAM Review, 39 (1997), 728-735.
[8] A.Melman, "Analysis of higher-order methods for secular equations", Mathematics of Computation, 67 (1997), 271-286.
[9] A.Melman, "A numerical comparison of methods for solving secular equations", Journal of Computational and Applied Mathematics, 86 (1997), 237-249.
[10] A.Melman, "Spectral functions for real symmetric Toeplitz matrices", Journal of Computational and Applied Mathematics, 98 (1998), 233-243.
[11] A.Melman, "Bounds on the extreme eigenvalues of real symmetric Toeplitz matrices", SIAM J. on Matrix Analysis and Applications, 21 (1999), 362-378.
[12] A.Melman, "A symmetric algorithm for Toeplitz systems", Linear Algebra and its Applications, 301 (1999), 145-152.
[13] A.Melman and G. Rabinowitz, "Efficient methods for a class of continuous nonlinear knapsack problems", SIAM Review, 42 (2000), 440-448.
[14] A.Melman, "A recurrence relation for real symmetric Toeplitz matrices", IEEE Transactions on Signal Processing, 48 (2000), 1829-1831.
[15] A.Melman, "Symmetric centrosymmetric matrix-vector multiplication", Linear Algebra and its Applications, 320 (2000), 193-198.
[16] A.Melman, "Extreme eigenvalues of symmetric positive definite Toeplitz matrices", Mathematics of Computation, 70 (2001), 649-669.
[17] A.Melman, "The even-odd split Levinson algorithm for Toeplitz systems", SIAM J. on Matrix Analysis and Applications, 23 (2001), 256-270.
[18] A.Melman, "A two-step even-odd split Levinson algorithm for Toeplitz systems", Linear Algebra and its Applications, 338 (2001), 219-237.
[19] A.Melman, "Computation of the smallest even and odd eigenvalues of a symmetric positive-definite Toeplitz matrix", SIAM J. on Matrix Analysis and Applications, 25 (2004), 947-963.
[20] A.Melman, "Computation of the Newton step for the even and odd characteristic polynomials of a symmetric positive-definite Toeplitz matrix", Mathematics of Computation, 75 (2006), 817-832.
[21] A.Melman and W.B.Gragg, "An optimization framework for polynomial zerofinders", American Mathematical Monthly, 113 (2006), 794-804.
[22] A.Melman, "A bug problem", College Mathematics Journal, May 2006 issue, 219-221.
[23] A.Melman, "Double-step Newton for polynomials with all real zeros", Applied Mathematics Letters, 20 (2007), 671-675.
[24] A.Melman, "Bounds on the zeros of the derivative of a polynomial with all real zeros", American Mathematical Monthly, 115 (2008), pp. 145-147.
[25] A.Melman, "Some properties of Newton's method for polynomials with all real zeros", Taiwanese Journal of Mathematics, 12 (2008), 2315-2325.
[26] A.Melman, "Overshooting properties of Newton-like and Ostrowski-like methods", American Mathematical Monthly, 116 (2009), 238-250.
[27] T.S.Carothers and A.Melman, "More bounds on the location of critical points of a polynomial with all real zeros", Pi Mu Epsilon Journal, 13 (2009), 13-20.
[28] A.Melman, "Spectral inclusion sets for structured matrices", Linear Algebra and its Applications, 431 (2009), 633-656.
[29] A.Melman, "An alternative to the Brauer set", Linear and Multilinear Algebra, 58 (2010), 377-385.
[30] A.Melman, "Fractional double Newton step properties for polynomials with all real zeros", Matematicki Vesnik, 62 (2010), 1-9.
[31] A.Melman, "Generalizations of Gershgorin disks and polynomial zeros", Proceedings of the American Mathematical Society, 138 (2010), 2349-2364.
[32] A.Melman, "Gershgorin disk fragments", Mathematics Magazine, 83 (2010), 123-129.
[33] A.Melman, "A pseudo Laguerre method", Matematicki Vesnik, 663 (2011), 295-304.
[34] A.Melman, "Modified Gershgorin disks for companion matrices", SIAM Review, 54 (2012), 355-373.
[35] A.Melman, "Ovals of Cassini for Toeplitz matrices", Linear and Multilinear Algebra, 60 (2012), 189-199.
[36] A.Melman, "Geometry of trinomials", Pacific Journal of Mathematics, 259 (2012), 141-159.
[37] A.Melman, "A single oval Oval of Cassini for the zeros of a polynomial", Linear and Multilinear Algebra, 61 (2013), 183-195.
[38] A.Melman, "Upper and lower bounds for the Perron root of a nonnegative matrix", Linear and Multilinear Algebra, 61 (2013), 171-181.
[39] A.Melman, "Comment on a result by Alpin, Chien, and Yeh", Proceedings of the American Mathematical Society, 141 (2013), 775-779.
[40] A.Melman, "The twin of a theorem by Cauchy", American Mathematical Monthly, 120 (2013), 164-168.
[41] A.Melman, "A somewhat unexpected concavity", The Teaching of Mathematics, 16 (2013), 18-21.
[42] A.Melman, "A geometric maximization problem", The Teaching of Mathematics, 16 (2013), 35-41.
[43] A.Melman, "Generalization and variations of Pelletís theorem for matrix polynomials", Linear Algebra and its Applications, 439 (2013), 1550-1567.
[44] A.Melman, "Inclusion disks for polynomial zeros in generalized bases", Linear Algebra and its Applications, 445 (2014), 326-346.
[45] A.Melman, "Implementation of Pellet's theorem", Numerical Algorithms, 65 (2014), 293-304.
[46] A.Melman, "Cauchy-type inclusion and exclusion regions for polynomial zeros", The Teaching of Mathematics, 17 (2014), 39-50.
[47] A.Melman, "Nonscalar matrix polynomial representation of some scalar polynomials", Linear Algebra and its Applications, 474 (2015), 141-157.
[48] A.Melman, "Geometric aspects of Pellet's and related theorems", Rocky Mountain Journal of Mathematics, 45 (2015), 603-621.
[49] A.Melman, "On Pellet's theorem for a class of lacunary polynomials", Mathematics of Computation, 85 (2016), 707-716.
[50] A.Melman, "Bounds for eigenvalues of matrix polynomials with applications to scalar polynomials", Linear Algebra and its Applications, 504 (2016), 190-203.
[51] A.Melman, "Cauchy-like and Pellet-like results for polynomials", Linear Algebra and its Applications, 505 (2016), 174-193.
[52] A.Melman, "Improvement of Pellet's theorem for scalar and matrix polynomials", Comptes Rendus Math. Acad. Sci. Paris, 8 (2016), 859-863.
[53] A. Melman: “Improved Cauchy radius for scalar and matrix polynomials”, Proceedings of the American Mathematical Society, 146 (2018), 613-624.
[54] A. Melman: “Optimality of a polynomial multiplier”, American Mathematical Monthly, 125 (2018), 158—163.
[55] A. Melman: “An alternative proof of Pellet's theorem for matrix polynomials”, Linear and Multilinear Algebra, 66 (2018), 785--791.
[56] A. Melman: “Eigenvalue bounds for matrix polynomials in generalized bases”, Mathematics of Computation, 87 (2018), 1935—1948.
[57] A. Melman: “Cauchy, Gershgorin, and matrix polynomials", Mathematics Magazine, 91 (2018), 274-285.
[58] A. Melman: “Optimization of a Cauchy radius improvement”, Pure and Applied Functional Analysis, 3 (2018), 639—651.
[59] A. Melman: “Extension of a theorem by Hayashi", Journal of Mathematical Analysis and Applications, 470 (2019), 1046-1055.
[60] A. Melman: “Polynomial eigenvalue bounds from companion matrix polynomials”, Linear and Multilinear Algebra, 67 (2019), 598--612.
[61] A. Melman: “Eigenvalue localization under partial spectral information”, Linear Algebra and its Applications, 573 (2019), 12—25.
[62] A. Melman: “Extensions of the Eneström-Kakeya theorem for matrix polynomials”, Special Matrices, 7 (2019), 304–315.
[63] A. Melman: “Directional bounds for polynomial zeros and eigenvalues”, Journal of Mathematical Analysis and Applications, 482 (2020), no. 2, 123571, 11pp.
[64] A. Melman: “A unifying framework for generalizations of the Eneström-Kakeya theorem”, Elemente der Mathematik, 75 (2020), 1--14.
[65] A. Melman: “A note on Brauer's theorem”, The Teaching of Mathematics, XXIII (2020), 17—19.
[66] S. Friedland and A. Melman: “A note on Hermitian positive semidefinite matrix polynomials”, Linear Algebra and its Applications, 598 (2020), 105—109.
[67] A. Melman (2020): “Refinement of Pellet radii for matrix polynomials”, Linear and Multilinear Algebra, 68 (2020), 2185—2200.
[68] A. Melman (2021): “Polynomials with no real zeros”, Mathematics Gazette, 105 (2021), 117—120.
[69] A. Melman (2021): “Zero exclusion sectors for some polynomials with structured coefficients”, American Mathematical Monthly, 128 (2021), 322—336.
[70] A. Melman (2022): “Polynomials whose zeros are powers of a given polynomial's zeros”, American Mathematical Monthly, 129 (2022), 276—280.
[71] A. Melman (2022): “An exclusion sector for zeros of polynomials with nonnegative coefficients”, College Mathematics Journal, 53 (2022), 312—314.
[72] A. Melman (2022): “Rootfinding techniques that work”, The Teaching of Mathematics, XXV (2022), 38—52.
[73] A. Melman: “Polynomial eigenvalue estimation: numerical radii versus norms”, Linear and Multilinear Algebra, 70 (2022), 7656—7671.
[74] A. Melman: “Matrices whose eigenvalues are those of a quadratic matrix polynomial”, Linear Algebra and its Applications, 676 (2023), 131--149.
