## Complete orthogonal decomposition for weighted least squares |

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accurate solution algorithm gives algorithm is stable back substitution step basis rows basis vectors coefficient matrix column of RT column pivoting columns of ATD~1'2 complete orthogonal decomposition computed solution condition number CORNELL UNIVERSITY Decomposition for Weighted defined definition of stability Dl/2Uiy electrical networks entries error bound exact upper triangular following theorem form a basis full-rank weighted least-squares gives an accurate gives an upper inequality Lemma linearly independent lower bound machine roundoff matrix with orthonormal n x n upper triangular nonbasis rows Notice NSH algorithm NSH method nullspace numerical tolerance orthogonal decomposition algorithm pivoting is necessary positive definite m x m pre-pivoted Proof QR factorization step QTAT Recall residual portions rigorous stability analysis rithm requires less rowspace RT is well-conditioned scaling second step solving least-squares problems submatrix Suppose system of equations tth column upper bound upper triangular matrix Vavasis Weighted Least Squares weighted least-squares problem Xa and xa