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.