On the characterization of totally nonpositive matrices

The final publication is available at Springer via http://dx.doi.org/10.1007/s40324-016-0073-1[EN] A nonpositive real matrix $A= (a_{ij})_{1 \leq i, j \leq n}$ is said to be totally nonpositive (negative) if all its minors are nonpositive (negative) and it is abbreviated as t.n.p. (t.n.). In this work a bidiagonal factorization of a nonsingular t.n.p. matrix $A$ is computed and it is stored in an matrix represented by $\mathcal{BD}_{(t.n.p.)}(A)$ when $a_{11}< 0$ (or $\mathcal{BD}_{(zero)}(A)$ when $a_{11}= 0$). As a converse result, an efficient algorithm to know if an matrix $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(zero)}(A)$) is the bidiagonal factorization of a t.n.p. matrix with $a_{11}<0$ ($a_{11}= 0$) is given. Similar results are obtained for t.n. matrices using the matrix $\mathcal{BD}_{(t.n.)}(A)$, and these characterizations are extended to rectangular t.n.p. (t.n.) matrices. Finally, the bidiagonal factorization of the inverse of a nonsingular t.n.p. (t.n.) matrix $A$ is directly obtained from $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(t.n.)}(A)$).This research was supported by the Spanish DGI grant MTM2013-43678-P and by the Chilean program FONDECYT 1100029Cantó Colomina, R.; Pelaez, MJ.; Urbano Salvador, AM. (2016). On the characterization of totally nonpositive matrices. SeMA Journal. 73(4):347-368. doi:10.1007/s40324-016-0073-1S347368734Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)Alonso, P., Peña, J.M., Serrano, M.L.: Almost strictly totally negative matrices: an algorithmic characterization. J. Comput. Appl. Math. 275, 238–246 (2015)Bapat, R.B., Raghavan, T.E.S.: Nonnegative Matrices and Applications. Cambridge University Press, New York (1997)Cantó, R., Koev, P., Ricarte, B., Urbano, A.M.: $$LDU$$ L D U -factorization of nonsingular totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 30(2), 777–782 (2008)Cantó, R., Ricarte, B., Urbano, A.M.: Full rank factorization in echelon form of totally nonpositive (negative) rectangular matrices. Linear Algebra Appl. 431, 2213–2227 (2009)Cantó, R., Ricarte, B., Urbano, A.M.: Characterizations of rectangular totally and strictly totally positive matrices. Linear Algebra Appl. 432, 2623–2633 (2010)Cantó, R., Ricarte, B., Urbano, A.M.: Quasi- $$LDU$$ L D U factorization of nonsingular totally nonpositive matrices. Linear Algebra Appl. 439, 836–851 (2013)Cantó, R., Ricarte, B., Urbano, A.M.: Full rank factorization in quasi- $$LDU$$ L D U form of totally nonpositive rectangular matrices. Linear Algebra Appl. 440, 61–82 (2014)Fallat, S.M., Van Den Driessche, P.: On matrices with all minors negative. Electron. J. Linear Algebra 7, 92–99 (2000)Fallat, S.M.: Bidiagonal factorizations of totally nonnegative matrices. Am. Math. Mon. 108(8), 697–712 (2001)Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton University Press, New Jersey (2011)Gasca, M., Micchelli, C.A.: Total positivity and applications. Math. Appl. 359, Kluwer Academic Publishers, Dordrecht (1996)Gasca, M., Peña, J.M.: Total positivity, $$QR$$ Q R factorization and Neville elimination. SIAM J. Matrix Anal. Appl. 4, 1132–1140 (1993)Gasca, M., Peña, J.M.: A test for strict sign-regularity. Linear Algebra Appl. 197(198), 133–142 (1994)Gasca, M., Peña, J.M.: A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl. 202, 33–53 (1994)Gassó, M., Torregrosa, J.R.: A totally positive factorization of rectangular matrices by the Neville elimination. SIAM J. Matrix Anal. Appl. 25, 86–994 (2004)Huang, R., Chu, D.: Total nonpositivity of nonsingular matrices. Linear Algebra Appl. 432, 2931–2941 (2010)Huang, R., Chu, D.: Relative perturbation analysis for eigenvalues and singular values of totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 36(2), 476–495 (2015)Karlin, S.: Total Nonpositivity. Stanford University Press, Stanford (1968)Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27(1), 1–23 (2005)Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29(3), 731–751 (2007)Parthasarathy, T.: $$N$$ N -matrices. Linear Algebra Appl. 139, 89–102 (1990)Peña, J.M.: Test for recognition of total positivity. SeMA J. 62(1), 61–73 (2013)Pinkus, A.: Totally Positive Matrices. Cambridge Tracts in Mathematics, vol. 181. Cambridge University Press (2009)Saigal, R.: On the class of complementary cones and Lemke’s algorithm. SIAM J. Appl. Math. 23, 46–60 (1972

Accurate and Efficient Expression Evaluation and Linear Algebra

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High relative accuracy through Newton bases

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Bidiagonal decompositions of Vandermonde-type matrices of arbitrary rank

We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Vandermonde ones, among others. These results generalize the existing expressions for nonsingular matrices to matrices of arbitrary rank. For totally nonnegative matrices of the above classes, the new decompositions can be computed efficiently and to high relative accuracy componentwise in floating point arithmetic. In turn, matrix computations (e.g., eigenvalue computation) can also be performed efficiently and to high relative accuracy

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