## Contributions to the solution of systems of linear equations and the determination of eigenvalues |

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### Contents

Practical solution of linear equations and inversion | 1 |

Punchedcard experiments with accelerated gradient methods | 55 |

Iterative methods of solving linear problems on Hilbert space | 71 |

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### Common terms and phrases

acceleration step back-substitution bounded coefficients column-by-column multiplication completely continuous complex numbers computation conjugate gradient consider convergence Corollary corresponding Crout defined denote diagonal elements diagonal matrix disks eigenvalues eigenvectors elimination equivalent error example exists expanding subspaces f(xk follows function geometric given gives Hence Hermitian matrix Hestenes Hilbert matrix Hilbert space implies inequality initial vector inner product integral inverse involving iterative procedures Legendre operator lemma linear equations linear operator linear subspace lower triangular matrix M. R. Hestenes Math minimizes F(x nonnegative nonsingular normal matrix nxn matrix obtained optimum gradient method optimum gradient steps orthogonal Ostrowski Ostrowski's theorem positive definite operator positive numbers PROOF properties proved recorded row-by-row multiplication satisfy selfadjoint sequence of expanding solves eq space 21 steps and accelerate symmetric matrix Taussky theorem 1.2 tion triangular resolution unit triangle unitary n-space upper triangle vector X0 Wielandt zero