lecture 4 - CE 30125 - Lecture 4 LECTURE 4 ITERATIVE...

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CE 30125 - Lecture 4 p. 4.1 LECTURE 4 ITERATIVE SOLUTIONS TO LINEAR ALGEBRAIC EQUATIONS • As finer discretizations are being applied with Finite Difference and Finite Element codes: • Matrices are becoming increasingly larger • Density of matrices is becoming increasingly smaller • Banded storage direct solution algorithms no longer remain attractive as solvers for very large systems of simultaneous equations
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CE 30125 - Lecture 4 p. 4.2 Example • For a typical Finite Difference or Finite Element code, the resulting algebraic equations have between 5 and 10 nonzero entries per matrix row (i.e. per algebraic equation asso- ciated with each node) = A γ δ 0 0 0 ε 0 0 0 0 0 0 0 0 β γ δ 0 0 0 ε 0 0 0 0 0 0 0 σ β γ δ 0 0 0 ε 0 0 0 0 0 0 σ 0 β γ δ 0 0 0 ε 0 0 0 0 0 σ 0 0 β γ δ 0 0 0 ε 0 0 0 0 α 0 0 0 β γ δ 0 0 0 ε 0 0 0 0 α 0 0 β γ δ 0 0 0 ε 0 0 0 α 0 0 0 β γ δ 0 0 ε 0 0 0 0 α 0 0 β γ δ 0 0 ε 0 0 0 α 0 0 β γ δ 0 0 τ 0 0 0 0 α 0 0 β γ δ 0 τ 0 0 0 0 0 0 α 0 0 β γ δ τ 0 0 0 0 0 α 0 0 β γ δ 0 0 0 0 0 α 0 0 0 β γ
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CE 30125 - Lecture 4 p. 4.3 • Banded compact matrix density Storage required for banded compact storage mode equals NM where N = size of the matrix, and M = full bandwidth Total nonzero entries in the matrix assuming (a typical estimate of) 5 non-zero entries per matrix row = 5 N • Banded compact matrix density = the ratio of actual nonzero entries to entries stored in banded compact mode Banded compact matrix density • Thus with the increasing size of problems/applications and the decreasing matrix densi- ties, iterative methods are becoming increasingly popular/better alternatives! NM Compact Matrix Density 100 20 0.25 10,000 200 0.025 10 6 2,000 0.0025 25 × 10 6 10,000 0.0005 Actual nonzero entries Banded storage ------------------------------------------------------------- 5 N -------- 5 M ---- == =
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CE 30125 - Lecture 4 p. 4.4 (Point) Jacobi Method - An Iterative Method • Let’s consider the following set of algebraic equations • Guess a set of values for • Now solve each equation for unknowns which correspond to the diagonal terms in , using guessed values for all other unknowns: a 11 x 1 a 12 x 2 a 13 x 3 ++ b 1 = a 21 x 1 a 22 x 2 a 23 x 3 b 2 = a 31 x 1 a 32 x 2 a 33 x 3 b 3 = XX 0 [] A x 1 1 b 1 a 12 x 2 0 a 13 x 3 0 a 11 --------------------------------------------------- = x 2 1 b 2 a 21 x 1 0 a 23 x 3 0 a 22 = x 3 1 b 3 a 31 x 1 0 a 32 x 2 0 a 33 =
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CE 30125 - Lecture 4 p. 4.5 • Arrive at a second estimate • Continue procedure until you reach convergence (by comparing results of 2 consecutive iterations) • This method is referred to as the (Point) Jacobi Method • The (Point) Jacobi Method is formally described in vector notation as follows: • Define A as • Such that all diagonal elements of A are put into D • Such that all off-diagonal elements of A are put into • The scheme is now defined as: • Recall that inversion of a diagonal matrix (to find ) is obtained simply by
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lecture 4 - CE 30125 - Lecture 4 LECTURE 4 ITERATIVE...

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