Abstract. We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.

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Abstract. We consider the class of totally nonnegative (TN) matrices-matrices all of whose minors are nonnegative. Any nonsingular TN matrix factors as a product of nonnegative bidiagonal matrices. The entries of the bidiagonal factors parameterize the set of nonsingular TN matrices.We present new O(n 3 ) algorithms that, given the bidiagonal factors of a nonsingular TN matrix, compute its eigenvalues and SVD to high relative accuracy in floating point arithmetic, independent of the conventional condition number. All eigenvalues are guaranteed to be computed to high relative accuracy despite arbitrary nonnormality in the TN matrix.We prove that the entries of the bidiagonal factors of a TN matrix determine its eigenvalues and SVD to high relative accuracy.We establish necessary and sufficient conditions for computing the entries of the bidiagonal factors of a TN matrix to high relative accuracy, given the matrix entries.In particular, our algorithms compute all eigenvalues and the SVD of TN Cauchy, Vandermonde, Cauchy-Vandermonde, and generalized Vandermonde matrices to high relative accuracy.Key words. eigenvalue, singular value, high relative accuracy, totally positive matrix, totally nonnegative matrix, oscillatory matrix, sign regular matrix, bidiagonal matrix AMS subject classifications. 65F15, 15A18 DOI. 10.1137/S08954798034382251. Introduction. The matrices with all minors nonnegative are called totally nonnegative (TN). They appear in a wide area of problems and applications [6,16,19,21,22,31] and can be notoriously ill-conditioned-the Hilbert matrix and the Vandermonde matrix with increasing nonnegative nodes are two examples of many.When traditional algorithms are used to compute the eigenvalues or the singular values of an ill-conditioned TN matrix, only the largest eigenvalues and the largest singular values are computed with guaranteed relative accuracy. The tiny eigenvalues and singular values may be computed with no relative accuracy at all, even though they may be the only quantities of practical interest [8]. Their accurate computation using traditional algorithms is then only possible through an increase in the working precision, which may lead to a drastic increase in the computational time.As our first major contribution we present new O(n 3 ) algorithms that compute all eigenvalues and singular values of a nonsingular TN matrix to high relative accuracy. In particular, all computed eigenvalues and singular values, including the tiniest ones, must have correct sign and leading digits: In contrast, traditional SVD and symmetric eigenvalue algorithms guarantee only small absolute errors in the computed singular values and eigenvalues of TN matricesPreviously proposed nonsymmetric eigenvalue algorithms guarantee only the Perron root of a TN matrix to be computed to high relative accuracy [15]. The accuracy in the smaller eigenvalues is also affected by the angle between left and right eigenvectors.Relative accuracy is lost in the traditional eigenvalue and SVD algorithms because of certain round...

Abstract. We consider the problem of performing accurate computations with rectangular (m × n) totally nonnegative matrices. The matrices under consideration have the property of having a unique representation as products of nonnegative bidiagonal matrices. Given that representation, one can compute the inverse, LDU decomposition, eigenvalues, and SVD of a totally nonnegative matrix to high relative accuracy in O(max(m 3 , n 3 )) time-much more accurately than conventional algorithms that ignore that structure. The contribution of this paper is to show that the high relative accuracy is preserved by operations that preserve the total nonnegativity-taking a product, re-signed inverse (when m = n), converse, Schur complement, or submatrix of a totally nonnegative matrix, any of which costs at most O (max(m 3 , n 3 )). In other words, the class of totally nonnegative matrices for which we can do numerical linear algebra very accurately in O(max(m 3 , n 3 )) time (namely, those for which we have a product representation via nonnegative bidiagonals) is closed under the operations listed above.

Abstract. Vandermonde, Cauchy, and Cauchy-Vandermonde totally positive linear systems can be solved extremely accurately in O(n 2 ) time using Björck-Pereyra-type methods. We prove that Björck-Pereyra-type methods exist not only for the above linear systems but also for any totally positive linear system as long as the initial minors (i.e., contiguous minors that include the first row or column) can be computed accurately. Using this result we design a new O(n 2 ) Björck-Pereyra-type method for solving generalized Vandermonde systems of equations by using a new algorithm for computing the Schur function. We present explicit formulas for the entries of the bidiagonal decomposition, the LDU decomposition, and the inverse of a totally positive generalized Vandermonde matrix, as well as algorithms for computing these entries to high relative accuracy.