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[v,vD] = SSV(M, {scaling})

The 

ssv( )

 function computes an approximation (and guaranteed upper 

bound) to the Scaled Singular Value of a complex square matrix M, where 
M can be a reducible matrix. The scaled singular value v(M) is defined by:  

Scaling can be accomplished with one of three algorithms:

•

Perron-Frobenius—If 

{scaling="PF"}

 Safonov’s 

Perron-Frobenius method [Saf82] is used. This method finds the scaled 
singular value for non-negative real matrices M. In general, it is 
suboptimal if M is complex. This algorithm is the default because 
empirical tests show that is the fastest of the three.

•

Osborne—If 

{scaling="OS"}

, Osborne’s Method [Osb60] is used. 

This method solves the problem of finding D

 such that

where D is diagonal and positive, and 

 is the Frobenius norm. 

Thus, the Osborne method minimizes the Frobenius norm, and is 
therefore suboptimal.

•

Optimal—If 

{scaling="OPT"}

, Boyd’s ellipsoid algorithm 

[BYB89] is used. This algorithm computes the scaled singular value 
to a guaranteed accuracy. It is, however, the most computationally 
expensive of the three algorithms.

Consider the complex matrix M: 

M = [–1, jay, 0; 0, 2*jay, 1+jay;1, 0, 1];

ssv( )

 can return the optimally scaled singular value of M using Osborne, 

Perron-Frobenius, or Boyd methods:  

VOS=ssv(M,{scaling="OS"})

VPF=ssv(M,{scaling="PF"})

( ) =

σ DMD

1

–

(

)

×

∈

( ) 0 dia

,

≠

,

1

–

1

–

⋅