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Getting Started. All matlab files you need can be found in /u/rvdg/class/CS383C.F04/QRalg/matlab/ You may want to copy these over into a directory of yours Start up matlab. The Power Method. The first demonstration centers around the Power Method Two M-script files are involved:

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## Getting Started

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**Getting Started**• All matlab files you need can be found in /u/rvdg/class/CS383C.F04/QRalg/matlab/ • You may want to copy these over into a directory of yours • Start up matlab**The Power Method**• The first demonstration centers around the Power Method • Two M-script files are involved: • PowerMethod.m • This is the main driver • ShowPowerMethod.m • This is a utility routine that prints out interesting stuff**Enter matrix size**A random diagonal matrix is created Hit “Return” A random orthogonal matrix is created A random matrix is created Hit “Return”**Keep an eye on this**The first element should become 1 (since q should eventually be the direction of the first column of U). The other elements should become 0 (since q should eventually be orthogonal To the other columns of U.) Current estimate of lambda(1) Actual lambda(1) Hit “Return”**Here I report the ratios | lj|/|l1| as**the i-th element of this vector Notice that the component of q in the direction of uj should decrease in every iteration by a factor roughly equal to | lj|/|l1| Notice that the jth component of UT * q equals the length of the component of q in the direction of uj. By looking at this ratio, we are tracking by what factor these components are decreasing in each step.**I hit return a few times**!!!!! Starting to look like we want!**I hit return a few more times**!!!!! Starting to look like we want!**Finally reply “0” (or anything else**but a return)**Subspace Iteration**• The second demonstration centers around the relation between the Power Method and Subspace Iteration**Hit “return” a few times to create**a random matrix, etc.**The Power Method and Subspace**Iteration are now both tracked**A few iterations later**Again, these are the factors by which we would predict that components in the directions of uj would decrease (see similar discussion for power method.)**QR algorithm**• The third demonstration centers around the relation between Subspace Iteration and the QR algorithm**Recall the QR algorithm:**A0 = A For i=0,… Ai – r I = QR Ai+1 = R Q + r I**The difference here simply**comes from a different number of digits being printed**For now, just use shift rho = 0**by hitting return every time**A few more iterations later…**All off-diagonal elements are starting to become small**QR algorithm**• The fourth demonstration centers around the effects of choosing shifts**Notice MUCH faster convergence**to zero!**A few iterations later…**A few iterations later…**Now these elements are**converging faster

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