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11
Accurate SVDs of weakly diagonally dominant Mmatrices
, 2004
"... We present a new O(n³) algorithm which computes the SVD of a weakly diagonally dominant Mmatrix to high relative accuracy. The algorithm takes as an input the offdiagonal entries of the matrix and its row sums. ..."
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We present a new O(n³) algorithm which computes the SVD of a weakly diagonally dominant Mmatrix to high relative accuracy. The algorithm takes as an input the offdiagonal entries of the matrix and its row sums.
Accurate and efficient expression evaluation and linear algebra
, 2008
"... We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By ‘accurate ’ we mean that the computed answer has relative error less than 1, i.e., has some correc ..."
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We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By ‘accurate ’ we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: most of our results will use the socalled traditional model (TM), where the computed result of op(a, b), a binary operation like a + b, is given by op(a, b) ∗ (1 + δ) where all we know is that δ  ≤ε ≪ 1. Here ε is a constant also known as machine epsilon.
Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices
, 2011
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Accurate Solutions of MMatrix Sylvester Equations
, 2010
"... This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an Mmatrix Sylvester equation AX +XB = C by which we mean either both A and B are nonsingular Mmatrices or one of them and P = Im⊗A+B T⊗In are nonsingular Mmatrices, and C is e ..."
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This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an Mmatrix Sylvester equation AX +XB = C by which we mean either both A and B are nonsingular Mmatrices or one of them and P = Im⊗A+B T⊗In are nonsingular Mmatrices, and C is entrywise nonnegative. It is proved that small relative perturbations to the entries of A, B, and C introduce small relative errors to the entries of the solution X. Thus the smaller entries of X do not suffer bigger relative errors than its larger entries, unlikely the existing perturbation theory for (general) Sylvester equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute X as accurately as the input data deserve. 1
MMatrix Algebraic Riccati Equations
, 2011
"... A new doubling algorithm – AlternatingDirectional Doubling Algorithm (ADDA) – is developed for computing the unique minimal nonnegative solution of an MMatrix Algebraic Riccati Equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two ..."
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A new doubling algorithm – AlternatingDirectional Doubling Algorithm (ADDA) – is developed for computing the unique minimal nonnegative solution of an MMatrix Algebraic Riccati Equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two existing doubling algorithms – SDA of Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 393–412) and SDAss of Bini, Meini, and Poloni (Numer. Math., 116 (2010), pp. 553–578) for the same purpose. Also demonstrated is that all three methods are capable of delivering minimal nonnegative solutions with entrywise relative accuracies as warranted by the defining coefficient matrices of an MARE. 2000 Mathematics Subject Classification. 15A24, 65F30, 65H10. Key words and phrases. Matrix Riccati equation, MMatrix, minimal nonnegative solution, doubling algorithm
by which we mean the following conformally partitioned matrix
, 2010
"... This paper is concerned with the relative perturbation theory and its entrywise relatively accurate numerical solutions of an Mmatrix Algebraic Riccati Equations (MARE) XDX − AX − XB + C = 0 ..."
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This paper is concerned with the relative perturbation theory and its entrywise relatively accurate numerical solutions of an Mmatrix Algebraic Riccati Equations (MARE) XDX − AX − XB + C = 0
ACCURATE AND EFFICIENT LDU DECOMPOSITIONS OF DIAGONALLY DOMINANT MMATRICES ∗
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A POSITIVITY PRESERVING INEXACT NODA ITERATION FOR COMPUTING THE SMALLEST EIGENPAIR OF A LARGE IRREDUCIBLE MMATRIX
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Numerische Mathematik manuscript No. (will be inserted by the editor) Accurate Solutions of MMatrix Sylvester Equations
"... Abstract This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an Mmatrix Sylvester equation AX + XB = C by which we mean both A and B have positive diagonal entries and nonpositive offdiagonal entries and P = Im⊗A+BT ⊗ In is a no ..."
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Abstract This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an Mmatrix Sylvester equation AX + XB = C by which we mean both A and B have positive diagonal entries and nonpositive offdiagonal entries and P = Im⊗A+BT ⊗ In is a nonsingular Mmatrix, and C is entrywise nonnegative. It is proved that small relative perturbations to the entries of A, B, and C introduce small relative errors to the entries of the solution X. Thus the smaller entries of X do not suffer bigger relative errors than its larger entries, unlikely the existing perturbation theory for (general) Sylvester equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute X as accurately as the input data deserve.