108 CHAPTER 3. FUNCTIONAL DESCRIPTION OF Xµ
The outputs of the µfunction are: the upp er and lower bounds for µ;theDmatrix for
the upper bound; the Qmatrix for the lower bounds; and a sensitivity estimate for the
part of the Dmatrix corresponding to each block in ∆. The sensitivity estimate is
essentially the gradient of the upper bound value with respect to the valueof the D
scale. It is useful in weighting the Dscale fitting procedure.
The function syntax is shown below.
[mubnds,D,Dinv,Delta,sens] = mu(M,blk)
The upper and lower bounds are returned in mubnds. The variable sens gives a measure
of the sensitivity of the upper bound to the Dmatrix. B oth DandQare returned as
the matrices (or pdms) Dand Delta. Note that the inverse of D, used in the calculation
of the upper bound, is also returned (as Dinv). This is done as, in the non-square ∆
block case, the dimensions of Dand D−1are different.
The block structure is a vector of dimension: number of blocks ×2. For each block the
output and input dimension is specified. To specify a scalar ×identity block, the input
dimension is set to zero.
A power iteration, with several random restarts, is used for the lower bound. Theupper
bound calculation uses an Osborne balancingmethod and enhances this with the Perron
vector method for problems with less than 10 blocks. These methods have been found to
be appropriate for the vastma jorityof practically motivated problems.
New algorithms for these calculations arecurrently under development. The most
significant enhancement is the ability to calculate µwith respect to structures which
include real valued blocks. Because of the development effort in this direction, a wide
range of calculation options werenot provided for the mu function.
The following example gives the simplest matrix with a gap betweenµand the D-scale
upper bound. It also illustrates the use ofthe mu function for constant matrices.
The following is the classic example showing that muis not equal to its upper b ound for
more than three full blocks. We include a random scaling here to give a non-trivial
D-scale.
gamma = 3 + sqrt(3); beta = sqrt(3) -1
a = sqrt(2/gamma); b = 1/sqrt(gamma)