L6_LinEqnIterative - MA2213 Lecture 6 Linear Equations...

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Unformatted text preview: MA2213 Lecture 6 Linear Equations (Iterative Solvers) Iteration Methods p. 303-305 any science and engineering applications require solutions of large linear systems 000 , 10 , , ≥ ∈ = × n R A b x A n n irect methods, e.g. Gaussian elimination, that produce the exact solution (except for roundoff error) in a fixed number of steps, require excessive computation time and omputer memory. Iteration Methods Such systems are usually solved by iterative methods which produce a sequence of approximations to the solution. Often, the matrix A is very sparse (most entries = 0), and iterative methods are very efficient for sparse matrices. Jacobi Method p.304 ] [ 3 2 1 9 1 1 x x b x − − = ] 3 2 [ 3 1 2 10 1 2 x x b x − − = ] 4 3 [ 2 1 3 11 1 3 x x b x − − = equivalent 1 3 2 1 b x x x = + + ystem 2 3 2 1 3 10 b x x x = + + 3 3 2 1 11 4 b x x x = + + his suggest the Jacobi iteration sequence ] [ ) ( 3 ) ( 2 1 9 1 ) 1 ( 1 k k k x x b x − − = + ] 3 2 [ ) ( 3 ) ( 1 2 10 1 ) 1 ( 2 k k k x x b x − − = + ] 4 3 [ ) ( 2 ) ( 1 3 11 1 ) 1 ( 3 k k k x x b x − − = + for ,... 2 , 1 , = k tarting from an initial estimate 3 2 1 , , x x x MATLAB Code for Jacobi Method >> b = [10 19 0]'; >> x_final = jacobi_special(x_init,b,20)' x_final = 0 0 1.1111 1.9000 0.9000 1.6778 -0.9939 1.0351 2.0182 -0.8556 0.9819 1.9496 -1.0162 1.0074 2.0085 -0.9768 0.9965 1.9915 -1.0051 1.0015 2.0022 -0.9960 0.9993 1.9985 -1.0012 1.0003 2.0005 -0.9993 0.9999 1.9997 -1.0003 1.0001 2.0001 -0.9999 1.0000 1.9999 -1.0001 1.0000 2.0000 -1.0000 1.0000 2.0000 -1.0000 1.0000 2.0000 -1.0000 function x_final = jacobi_special(x_init,b,num_iter) % function x_final = jacobi_special(x_init,b,num_iter) % % bla bla bla % x(:,1) = x_init; for k = 1:num_iter x(1,k+1) = (1/9)*(b(1)-x(2,k)-x(3,k)); x(2,k+1) = (1/10)*(b(2) - 2*x(1,k) - 3*x(3,k)); x(3,k+1) = (1/11)*(b(3) - 3*x(1,k) - 4*x(2,k)); end x_final = x; First pull down the File menu and choose New m-file or Open an existing m-file, then type and save your program below – I suggest using the same name as the name of the function Always compare your numerical results with results available either in the textbook or elsewhere. extbook Results Jacobi Method p. 305 T b ] 19 10 [ = or the solution 0232 . 9768 . 0085 . 2 0074 . 1 5 0506 . 0162 . 1 9496 . 1 9819 . 4 1440 . 8556 . 0182 . 2 0351 . 1 3 3220 . 9939 . 6778 . 1 9000 . 2 1000 0000 . 9000 . 1 1110 . 1 1 2 − − − − If T x ] 1 2 1 [ − = T T x x x x ] [ ] [ ) ( 3 ) ( 2 ) ( 1 ) ( = ≡ the Jacobi iteration gives ∞ − ≡ || || ) ( ) ( 3 ) ( 2 ) ( 1 k k k k x x Error x x x k atrix Splitting for Jacobi Method p.306 , U L D A − − = If with D diagonal ; L, U lower, upper triangular, then b D x U L D x 1 1 ) ( − − + + = f + = x B b D U L 1-1 f ), ( D B − ≡ + ≡ where Jacobi Method If certain conditions on the matrix B hold then the following iteration produces a sequence f + = Bx x ,... 3 , 2 , 1 , , ) ( = ∈ k R x n k to n R x ∈ that satisfies that converges n R x...
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This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.

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L6_LinEqnIterative - MA2213 Lecture 6 Linear Equations...

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