DFT calculation of the generalized and drazin inverse of a polynomial matrix
N. Karampetakis, S. Vologiannidis
Department of MathematicsAristotle University of Thessaloniki
Thessaloniki 54006, Greece
http://anadrasis.math.auth.gr
Objectives• A new algorithm is presented for the
determination of the generalized inverse and the drazin inverse of a polynomial matrix based on the discrete Fourier transform.
• The above algorithms are implemented in the Mathematica programming language.
Discrete Fourier Transform
2
1 1 2 2
2
1 2 1 2
1 2 1 2 1 20
In order for the finite sequence ( , ) and the sequence ( , ) to constitute a
DFT pair the following relations should hold [Dudgeon, 1984]:
( , ) ( , )M
k r k r
k k
X k k X r r
X r r X k k W W
Definition 2.
1 1 2
1 1 2 2
1 1 2
1 2 1 2 1 20 0 0
2
11 2
1 2 1 2
1, ( , ) ( , )
where
, ( 1) ( 1)
and ( , ), ( , ) are discrete argument matrix-valued functions,
with dimensions .
i
M M Mk r k r
r r
j
M
X k k X r r W WR
W e R M M
X k k X r r
p m
2
In order for the finite sequence ( ) and the sequence ( ) to constitute a
DFT pair the following relations should hold [Dudgeon, 1984]:
1( ) ( ) , ( ) ( )
1
where
M Mkr kr
j
X k X k
X k X k W X k X k WM
W e
Definition 1.
1 and ( ), ( ) are discrete argument matrix-valued functions,
with dimensions .
M X k X k
p m
Generalized Inverse
For every matrix p mA R , a unique matrix m pA R , which is called generalized inverse, exists satisfying (i) AA A A (ii) A AA A
(iii) T
AA AA
(iv) TA A A A
where TA denotes the transpose of A . In the special case that the matrix A is square nonsingular matrix, the generalized inverse of A is simply its inverse i.e. 1A A . In an analogous way we define the generalized inverse ( ) ( )m pA s R s of the
polynomial matrix ( ) [ ]p mA s R s
Computation of the generalized inverse
[Karampetakis] Let ( ) [ ]p mA s R s and
a(s,z) = 10 1 1det ( ) ( ) ( ) ( ) ( ) ( )T p p
p p pzI A s A s a s z a s z a s z a s
,
0 ( ) 1a s , be the characteristic polynomial of ( ) ( )TA s A s . Let 1( ) 0 ( ) 0p ka s a s
while ( ) 0ka s and { ( ) 0}i k is R a s Then the generalized inverse ( )A s of
( )A s for s R is given by 1
11 1 0 1( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ... ( )
k
kT Tk k k pa sA s A s B s B s a s A s A s a s I
If 0k is the largest integer such that ( ) 0ka s , then ( ) 0A s . For those is
find the largest integer ik k such that ( ) 0ik ia s and then the generalized inverse
( )iA s of ( )iA s is given by 1
11 1 0 1( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ... ( )
i i ik ii
kT Ti i k i k i i i k i pa sA s A s B s B s a s A s A s a s I
Computation of the generalized inverse via DFTStep 1. (Evaluation of the polynomial a(s,z))
1 2
1 2
1 2
1 20 0
( ) det ( ) ( )n n
l l Tl l p
l l
a s z a s z zI A s A s
We use the following (2 1) ( 1)R pq p interpolation points
( ) 1 2 and 0 1jr
i j i j iu r W i r M
where W2
1
1 21 2 2j
Mii e i M pq M p
1 2
1 1 2 2
1 2 1 2
1 2
2 2 1 1 1 1 1 20 0
det[ ( ) ( ( )) ( ( )) ]n n
r l r lTr r p l l
l l
u r I A u r A u r a W Wa
1 2[ ]r ra and
1 2[ ]l la form a DFT pair.
1 2
1 1 2 2
1 21 2
1 2
1 20 0
1 n nr l r l
r rl lr r
a W WaR
Step 2. (Evaluate )
Find 1 2( ) ( ) ( ) 0k k pk a s a s a s and ( ) 0ka s or 1 1 10 1 1 0l l l ka a a 1l
and 1
0l ka for some k .
( )ka s
Step 3. (Evaluate ) 1( ) ( ) ( )TkB s A s B s
We use the following (2 1) 1R p q interpolation points
( ) ru r W ,2
( 2 1) 1j
p qW e
~
0
( ( ))n
lrr l
l
B B u r BW
[ ]iB and [ ]lB form a DFT pair
0
1, 0 1 (2 1)
nlr
rlr
B W l p qBR
Step 4. Evaluate the generalized inverse
( )( )
( )k
B sA s
a s
Drazin Inverse
For every matrix m mA R , a unique matrix D m mA R , which is called Drazin inverse, exists satisfying (i) 1k D kA A A for 1( ) min( )k kk ind A k N rank A rank A
(ii) D D DA AA A (iii) D DAA A A
Drazin Inverse[Staminirovic] Consider a nonregular one-variable rational matrix ( )A s . Assume that
10 1 1( , ) det ( ) ( ) ( ) ... ( ) ( )m m
m m ma z s zI A s a s z a s z a s z a s where
0 ( ) 1a s z C is the characteristic polynomial of A(s) consider the following
sequence of m m polynomial matrices B 0 1 0( ) ( ) ( ) ... ( ) ( ) ( ) ( ) 1 0j
j j j ms a s A s a s A s a s I a s j … m
Let 1( ) 0 ( ) 0 ( ) 0m t ta s … a s a s Define the following
set { ( ) 0}i t is C a s Also assume 1( ) 0 ( ) 0 ( ) 0m r rB s B s B s and k=r-t
.In the case that s C \ and 0k , the Drazin inverse of ( )A s is given by 1 1 1
1
11 0 2 1
( 1) ( ) ( ) ( )
( ) ( ) ( ) ... ( ) ( ) ( )
D k k k kt t
tt t t m
A a s A s B s
B s a s A s a s A s a s I
In the case s C \ and 0k , we get ( )DA s O .
Computation of the Drazin Inverse via DFTStep 1 (Evaluation of a(s,z))
1 2
1 2
1 2
1 20 0
( ) det ( )n n
l ll l m
l l
a s z a s z zI A s
We use the following (2 1) ( 1)R mq m points
( ) 1 2iri i iu r W i ,
21
1 21 2 2j
MiiW e i M mq M m
1 2
1 1 2 2
1 2 1 2
1 2
2 2 1 1 1 20 0
det[ ( ) ( ( ))]n n
r l r lr r m l l
l l
u r I A u r a W Wa
1 2[ ]r ra and
1 2[ ]l la form a DFT pair
1 2
1 1 2 2
1 21 2
1 2
1 2 1 20 0
1 , 0 1 2 , 0 1
n nr l r l
r rl lr r
a W W l mq l maR
Step 2
Find 1 2( ) ( ) ... ( ) 0t t mt a s a s a s
( ) 0ta s or 1 1 10 1 1 0r r r ta a a 1r and
10r ta for some t .
Step 3(Evaluate ( ) 0mr t B s ,..., 1( ) 0 ( ) 0r rB s B s )
j mDetermine the value of ( )jB s at the following 1jn points (or any other 1jn
distinct points)
2
1( )j
n jru r W W e
Do WHILE ( ( ) 0jB s ( ))u r
1j j
Determine the value of ( )jB s at the following 1jn points
2
1( )j
n jru r W W e
END DO r j
Step 4 (Evaluation of 11( ) ( )k k
tA s B s )
11 0
11 0 2 1
( ) ( ) ( ) = ,
( ) ( ) ( ) ... ( ) ( ) ( )
nk k lt ll
tt t t m
B s A s B s B s
B s a s A s a s A s a s I
We use the following (n+1) interpolation points 2
1( ) ,j
r nu r W W e
0
nlr
r ll
BWB
[ ]iB and [ ]lB form a DFT pair
0
1 nlr
llr
B WBR
Step 5 (Evaluation of 1( )kta s )
1
0
( ) ( )n
k lt l
l
a s a s a s
We use the following (n+1) interpolation points
2
1( ) ,j
r nu r W W e
0
( ( ))n
lrr l
l
a u r a Wa
[ ]la and [ ]la form a DFT pair
0
1, 0 1
nlr
rlr
a W l naR
Step 6. (Evaluation of the Drazin inverse)
( )( )
( )D B s
A sa s
Implementation The above algorithms have been implemented in
Mathematica.
The following graphs shows the efficiency of the DFT based algorithms compared to the algorithms described in [Karampetakis 1997, Staminirovic and Karampetakis 2000]. The red surface represents the DFT based algorithms.
Graphs
Conclusions Two new algorithms have been presented for the
computation of the generalized inverse and Drazin inverse of a polynomial matrix.
The proposed algorithms proved to be more efficient from the known ones in the case where the degree and the size of the polynomial matrix get bigger.
The proposed algorithms can be easily extended to the multivariable polynomial matrices.