### Abstract

To solve an ill-conditioned system of linear algebraic equations (LAEs): Bxb = 0, we define an invariant-manifold in terms of r := Bxb, and a monotonically increasing function Q(t) of a time-like variable t. Using this, we derive an evolution equation for dx=dt, which is a system of Nonlinear Ordinary Differential Equations (NODEs) for x in terms of t. Using the concept of discrete dynamics evolving on the invariant manifold, we arrive at a purely iterative algorithm for solving x, which we label as an Optimal Iterative Algorithm (OIA) involving an Optimal Descent Vector (ODV). The presently used ODV is a modification of the Descent Vector used in the well-known and widely used Conjugate Gradient Method (CGM). The presently proposed OIA/ODV is shown, through several examples, to converge faster, with better accuracy, than the CGM. The proposed method has the potential for a wide-applicability in solving the LAEs arising out of the spatial-discretization (using FEM, BEM, Trefftz, Meshless, and other methods) of Partial Differential Equations.

Original language | English |
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Pages (from-to) | 275-298 |

Number of pages | 24 |

Journal | CMES - Computer Modeling in Engineering and Sciences |

Volume | 80 |

Issue number | 3-4 |

State | Published - 2011 |

### Keywords

- Conjugate Gradient Method (CGM)
- Ill-conditioned linear system
- Invariant-manifold
- Linear PDE
- Linear algebraic equations
- Optimal Iterative Algorithm with an Optimal Descent Vector (OIA/ODV)