Projection Tutorial (page 3)

• Step 3: Begin to introduce projection.

  The previous image is perfectly adequate for getting a rough idea of the behaviour of the pseudospectra near the imaginary axis, but it would be better to have a higher quality plot for inclusion in a publication or talk. The problem is that although this plot took only a few seconds to generate, it is only computed on a coarse 24 by 24 grid (with 576 gridpoints in total). To move to a grid of 200 by 200 points to give a high quality plot would entail a wait of nearly twelve minutes on my Sun Ultra 5. This may be feasible for this reasonably small matrix, but we can do better.

  The key is to project our matrix of dimension 200 onto a subspace spanned by the invariant subspace corresponding to the eigenvalues near the region of the complex plane we are interested in. The projected matrix will hopefully have dimension much less than 200, and so the computation will be faster.

  The following figures indicate how our projection algorithm operates. The green square represents the area of the complex plane visible in the GUI (the area our grid is defined over), while the red square represents the eigenvalues whose eigenvectors we project onto. In the left figure, the projection level (`Safety') is set to 2, while in the right one the projection level is set to 1. The red rectangle is defined as the region which is (1+2b) times as high and (1+2b) times as wide as the rectangle visible in the GUI, where b is the Safety value.

  When you click `Go!', the matrix is first projected onto the space spanned by eigenvectors whose eigenvalues are within the red square. This is a non-trivial operation, but the benefits will far outweigh this extra cost if we are computing on a fine grid. The pseudospectra of this smaller, projected matrix are then computed. As long as the effect of the eigenvalues we have left out of our projection is small, the plot of the pseudospectra will look essentially identical, and importantly, the computation will be faster.

  First, try reducing the `Safety' value to 2 and recomputing the pseudospectra. The plot you get should look exactly the same as the one obtained before (with infinite safety i.e. no projection). On the coarse grid used here, the computation time is worse (we have to take account of the time taken to project the matrix), but we will see real differences when we move to the fine grid. This takes about 25 seconds on my Sun Ultra 5 workstation. Notice that the matrix dimension displayed by the GUI is now N=126 not N=200 (if you can't see this, turn on `Show dimension' in the `Extras' menu).

  Since we saw no difference for Safety=2, now reduce Safety to 1 and recompute again. Once more, there is no change to the pseudospectra. This time the computation is quicker, since the matrix dimension has already been reduced during the first projection. Your Pseudospectra GUI should now look like this:

  

Pseudospectra GUI home page.