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COMPUTATION AND CONDITION
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active constraints Algorithm artificial variable basic variables calculation components computed condition number Consider constraints Ax coordinate vector corresponding defined denote descent direction dimension dual eigenvalues equality constraints equations error exact solution example Exercise feasible direction feasible point feasible region Gaussian elimination given Householder transformations i-th ill-conditioned inequality constraints infeasible interchanges inverse iteration j-th Lagrange multiplier least-squares problem least-squares solution Lemma linear combination linear program linear system linearly independent LU factorization m x n matrix n x n n-vector nonbasic variables nonnegative nonsingular matrix norm null space objective function optimal orthogonal orthogonal matrix orthonormal permutation permutation matrix perturbation primal pseudoinverse QR factorization rank-one rank-retaining reduced remaining matrix residual right-hand side satisfies scalar search direction Section Show simplex method singular value decomposition solving standard-form step subject to Ax symmetric system Ax transformation triangular two-norm unique updating upper triangle upper-triangular vertex zero