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gets large In some applications the transition probabilities pij have the property that for each state i the probability pij = 0 for j i 2 (or pij = 0 for j i + 2) Then the linear equations are of the Hessenberg type Linear equations of the Hessenberg type can be ef ciently solved by a special code using the very stable QR method In solving the Markov chain equations (341) and (342) by a direct method, one of the equilibrium equations is omitted to obtain a square system of linear equations Iterative method of successive overrelaxation Iterative methods have to be used when the size of the system of linear equations gets large In speci c applications an iterative method can usually avoid computer memory problems by exploiting the (sparse) structure of the application An iterative method does not update the matrix of coef cients each time In applications these coef cients are usually composed from a few constants Then only these constants have to be stored in memory when using an iterative method In addition to the advantage that the coef cient matrix need not be stored, an iterative method is easy to program for speci c applications The iterative method of successive overrelaxation is a suitable method for solving the linear equations of large Markov chains The well-known Gauss Seidel method is a special case of the method of successive overrelaxation The iterative methods generate a sequence of vectors x(0) x(1) x(2) converging towards a solution of the equilibrium equations (341) The normalization is done at the end of the calculations To apply successive overrelaxation, we rst rewrite the equilibrium equations (341) in the form
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The standard successive overrelaxation method uses a xed relaxation factor for speeding up the convergence The method starts with an initial approximation vector x(0) = 0 In the kth iteration of the algorithm an approximation vector x(k) is (k) found by a recursive computation of the components xi such that the calculation (k) (k) of the new estimate xi uses both the new estimates xj for j < i and the old estimates xj(k 1) for j > i The steps of the algorithm are as follows: Step 0 Choose a non-zero vector x(0) Let k := 1 Step 1 Calculate successively for i = 1, , N the component xi(k) from
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is satis ed with > 0, a prespeci ed accuracy number, then go to step 3 Otherwise k := k + 1 and go to step 1 Step 3 Calculate the solution to (341) and (342) from xi = xi(k) xj(k) , 1 i N
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The speci cation of the tolerance number typically depends on the particular problem considered and the accuracy required in the nal answers In addition to the stopping criterion, it may be helpful to use an extra accuracy check for the equilibrium probabilities of the underlying Markov chain An extra accuracy check may prevent a decision upon a premature termination of the algorithm when the tolerance number is not chosen suf ciently small Notice that the normalizing equation (342) is used only at the very end of the algorithm In applying successive overrelaxation it is highly recommended that all of the equilibrium equations (341) are used rather than omitting one redundant equation and substituting the normalizing equation (342) for it The convergence speed of the successive overrelaxation method may dramatically depend on the choice of the relaxation factor , and even worse the method may diverge for some choices of A suitable value of has to be determined experimentally Usually 1 2 The choice = 12 is often recommended The optimal value of the relaxation factor depends on the structure of the particular problem considered It is pointed out that the iteration method with = 1 is the well-known Gauss Seidel method This method is convergent in all practical cases The ordering of the states may also have a considerable effect on the convergence speed of the successive overrelaxation algorithm In general one should order the states such that the upper diagonal part of the matrix of coef cients is as sparse as possible In speci c applications the transition structure of the Markov chain often suggests an appropriate ordering of the states Krylov iteration method The Gauss Seidel iteration method can further be re ned to obtain orthogonal basis vectors for a so-called Krylov space The construction of an appropriate Krylov