BULLETIN
OF THE
POLISH
ACADEMY
OFSCIENCES
SCIENCES
BULLETIN
OF THE
POLISH
ACADEMY
OF
TECHNICAL SCIENCES, Vol. 64, No. 4, 2016
TECHNICAL
SCIENCES, Vol. XX, No. Y, 2016
DOI: 10.1515/bpasts-2016-0093
DOI: 10.1515/bpasts-2016-00ZZ
Inversion of selected structures of block matrices of chosen
Inversion of selected
structures ofsystems
block matrices of chosen
mechatronic
mechatronic systems
Inversion
of
selected
structures
block
matrices
of Kurzyk
chosen
1∗
1 and Dariusz
2
Tomasz Trawiński , Adam Kochan 1of
, Paweł
Kielan
Inversion
of
selected
structures
of
block
matrices
of
chosen
T. TRAWIŃSKI1*, A. KOCHAN1, P. KIELAN1, and D. KURZYK2
1 Department
mechatronic
systems
of Mechatronics,
Silesian University of Technology,
Akademicka 10A, 44-100 Gliwice, Poland
2Department
of
Mechatronics,
Silesian
University
of
Technology,
10AKaszubska
Akademicka23,
St.,44-100
44-100 Gliwice,
Gliwice, Poland
mechatronic
systems
Institute of Mathematics, Silesian University of Technology,
Poland
Institute of Mathematics,
St., 44-100 Gliwice, Poland
1∗ Silesian University 1of Technology, 23 Kaszubska
1
2
BULLETIN OF THE POLISH ACADEMY OF SCIENCES
TECHNICAL
SCIENCES,
Vol. XX, No. Y, 2016
BULLETIN
BULLETIN OF
OF THE
THE POLISH
POLISH ACADEMY
ACADEMY OF
OF SCIENCES
SCIENCES
DOI:
10.1515/bpasts-2016-00ZZ
TECHNICAL
SCIENCES, Vol.
Vol. XX,
XX, No.
No. Y,
Y, 2016
2016
TECHNICAL
SCIENCES,
DOI:
DOI: 10.1515/bpasts-2016-00ZZ
10.1515/bpasts-2016-00ZZ
1
2
Tomasz Trawiński , Adam Kochan , Paweł Kielan and Dariusz Kurzyk
1∗
1∗, Adam Kochan 1
1 , Paweł Kielan1
1 and Dariusz Kurzyk2
2
Tomasz
1
ThisDepartment
article
describes
how to calculate
theUniversity
number ofofalgebraic
operations
necessary
implement
block
matrix inversion that occur,
ofTrawiński
Mechatronics,
Silesian
Technology,
Akademicka
10A, to
44-100
Gliwice,
Poland
Abstract.
Abstract.
This paper describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occurs,
11 Department
among others,
in2 mathematical
models of Silesian
modern
positioning
of mass
storage23,
devices.
The
inversion
method of block matrices is
Institute of
of Mechatronics,
Mathematics,
University
of
Technology,
Kaszubska
44-100
Gliwice,
Poland
Silesian
University
of
Technology,
Akademicka
10A,
44-100
Gliwice,
Poland
Department
of
Mechatronics,
Universitysystems
ofsystems
Technology,
Akademicka
Gliwice,
among others,
in mathematical
models of Silesian
modern positioning
of mass storage
devices.10A,
The 44-100
inversion
method Poland
of block matrices is pre22 Institute
presented
as
well.
Presented
form
of
general
formulas
describing
the
calculation
complexity
of
inverted
form
of
were prepared
of
Mathematics,
Silesian
University
of
Technology,
Kaszubska
23,
44-100
Gliwice,
Poland
of Mathematics,
Silesian
University
of Technology,
Kaszubska
23,of44-100
Gliwice,
sented as well.Institute
The presented
form of general
formulas
describing
the calculation
complexity
inverted
form ofPoland
blockblock
matrixmatrix
were prepared
for three
different
cases
of
their
division
into
internal
blocks.
The
obtained
results
are
compared
with
a
standard
Gaussian
method
for three different cases of division into internal blocks. The obtained results are compared with a standard Gaussian method and the and
“inv”the "inv"
Abstract.
This article
describes
how to calculate
thefor
number
ofinversion
algebraicisoperations
necessary
to in
implement
block
matrix
inversion
that
occur,
method
used
in
Matlab.
The
proposed
method
matrix
much
more
effective
comparison
in
standard
Matlab
matrix
inversion
method used in Matlab. The proposed method for matrix inversion is much more effective in comparison in standard Matlab matrix inversion
among
inarticle
mathematical
models
modern
positioning
systems
mass storage
devices.
The
inversion
block matrices
is
Abstract.
This
describes
how
to
calculate
number
algebraic
necessary
to
implement
block
matrix
inversion
that
Abstract.
This
article
describes
how
to of
calculate
the
number
of
algebraicofoperations
operations
necessary
toGauss
implement
blockmethod
matrix of
inversion
that occur,
occur,
"inv"others,
function
(almost
two two
times
faster)
andand
isthe
much
less
numerically
complex
than
standard
Gauss
method.
“inv”
function
(almost
times
faster)
is
much
lessof
numerically
complex
than
standard
method.
presented
as well.
Presented form
of general
formulas
describing
the calculation
complexity
of inverted
form of method
block matrix
were
prepared
among
in
models
of
positioning
systems
of
devices.
The
of
matrices
is
among others,
others,
in mathematical
mathematical
models
of modern
modern
positioning
systems
of mass
mass storage
storage
devices.
The inversion
inversion
method
of block
block
matrices
is
for
three
different
cases
of
their
division
into
internal
blocks.
The
obtained
results
are
compared
with
a
standard
Gaussian
method
and
the
"inv"
Key
words:
arrowhead
matrices,
mechatronic
systems,
matrix
inversion,
computational
complexity
presented
as
well.
Presented
form
of
general
formulas
describing
the
calculation
complexity
of
inverted
form
of
block
matrix
were
prepared
Key
words:
arrowhead
matrices,
mechatronic
systems,
matrix
inversion,
computational
complexity.
presented as well. Presented form of general formulas describing the calculation complexity of inverted form of block matrix were prepared
method
in Matlab.
The
proposed
for matrix
inversion
is muchresults
more are
effective
in comparison
in standard
Matlab
matrix
for
different
cases
their
division
into
blocks.
The
compared
with
Gaussian
method
and
the
for three
threeused
different
cases of
of
their
divisionmethod
into internal
internal
blocks.
The obtained
obtained
results
are
compared
with aa standard
standard
Gaussian
method
and inversion
the "inv"
"inv"
1.
Introduction
"inv"
function
(almost
two
times
faster)
and
is
much
less
numerically
complex
than
standard
Gauss
method.
method
used
in
Matlab.
The
proposed
method
for
matrix
inversion
is
much
more
effective
in
comparison
in
standard
Matlab
matrix
inversion
 inversion
method used in Matlab. The proposed method for matrix inversion is much more effective in comparisonin standard Matlab matrix
matrices
1. Introduction
"inv"
function
(almost
times
and
much
numerically
complex
than
Gauss
method.
Ls example,
Msr .an
. . electromechan"inv"words:
function
(almost two
two
times faster)
faster)
and is
is systems,
much less
lessmatrix
numerically
complexelementary
than standard
standard
Gauss(blocks),
method. for
Key
arrowhead
matrices,
mechatronic
inversion,
 matrix [5]
 T motors) inertia
ical systemcomplexity
(squirrel cage induction
Mathematical
models
of physical
objects
are formulated
using,computational
 M
Lr . . . 
Key words:
words:
arrowhead models
matrices,
mechatronic
systems,
matrix inversion,
inversion,
computational
complexity
(2)
Key
arrowhead
matrices,
systems,
matrix
complexity
Mathematical
ofmechatronic
physical
objects
are
formulated
using,computational
may take the
form of:D =  sr

among
others,
the
Lagrangian
formalism.
It
can
be
represented
1. Introduction
.
.
.


.
.
..
others,form
the Lagrangian
formalism.
It canwritten
be represented
by theamong
following
of differential
equations
in maLs M.sr . ... 

1. Introduction
by
the
following
form
of
differential
equations
written
in
matrix


 LsT Msr . . . 
trix form [1]:
Mathematical
models of physical objects are formulated using,
Lsr
M

Lrsr . . . of
M
form [1]:
s
denotes
matrices
inductances(2)of stator
where Ls , Lr ,DM=
(2)
sr 
self

Mathematical
of physical
objects are
formulated
using,

among
others, models
the Lagrangian
formalism.
It can
be represented


T
Mathematical models of physical objects are formulated using,
T

L..rr matrix
M..sr


=
L
.. ... ... 
M
windings andD
rotor
windings,
of mutual stator (2)
- rotor in(2)
D
=
sr


among
others,
the
Lagrangian
formalism.
It
can
be
represented
by
the
following
form
of
differential
equations
written
in
ma..

 ..
among others,
the Lagrangian
be represented (1)
q̇ + Kq + GIt=can
τ (1)
Dq̈ +Cformalism.
.
.
.
. .. . matrices in electromechanductances respectively.. The .inertia
by
following
form
trix
form
[1]:
by the
the
following
form of
of differential
differential equations
equations written
written in
in mama.
.
.
,
L
,
M
denotes
matrices
of
self
inductances
of stator
where
L
ical
systems
may
have
structural
features
allowing
them to be
s
r
sr
trixwhere
formwhere
[1]:
trix
form
[1]:
D - denotes
- denotes
matrix
of cen-where
D denotesinertial
inertial matrix,
matrix, CCdenotes
matrix
of centrifugal
matricesofofself
selfinductances
inductances of
of stator
stator
where
Ls,, LMr, Mdenotes
sr denotematrices
L
Lss ,, and
windings
rotor
windings,
matrix
of
mutual
stator
rotor
inrr , into
sr
divided
a
number
of
submatrices,
for
the
most
elementary
L
M
denotes
matrices
of
self
inductances
of
stator
where
L
andand
Coriolis
forces,
Kq̇denotes
G denotes
vectorG windings andsr rotor windings, matrix of mutual stator – rotor
trifugal
Coriolis
forces,
-Gdenotes
stiffness
matrix,
Dq̈ +C
+ KqK+stiffness
= τ matrix,
(1)
windings
and
rotor
windings,
matrix
of
mutual
stator
rotor
inductances
respectively.
The
inertia
matrices
in
electromechanwindings
and rotor
matrix
mutual
stator
rotorSymmetric
inmatrices
-respectively.
thewindings,
blocks have
the of
size
of 2×2
[2,- 5].
of gravitational
denotes
vector
forces,
inductances
- denotes
vectorD
gravitational
- denotes(1)
vector
q̈
+C
q̇
+
=
ττ ofτ generalized
Dof
q̈ forces,
+C
q̇ +
+τ Kq
Kq
+G
G forces,
=
(1)
ductances
respectively.
The
inertia
matrices
in
electromechanical
systems
may
have
structural
features
allowing
them
to
beof head
ductances
respectively.
The
inertia
matrices
in
electromechaninertia
matrices
encountered
in
mathematical
models
of generalized
The inertia matrices in electromechanical systems may have
where
Dq -denotes
denotesvector
inertial
C -displacements.
denotes
matrix
of cen- disof generalized
forces,
qmatrix,
- denotes
vector of
generalized
ical structural
systems
may
haveallowing
structural
features
allowing
them
to be
be
into amay
number
ofstructural
submatrices,
fordrives
the
most
elementary
ical
systems
have
features
allowing
them
to
positioning
systems
of hard
disk
[3,into
4],
have
very difThe
inertial
matrix,
(1),
predominant
cases,divided
features
them
to be
divided
a number
where
D
-- denotes
inertial
matrix,
C
--indenotes
matrix
of
trifugal
Coriolis
forces,
Kpresent
- denotes
stiffness
G
where
Dand
denotes
inertial
matrix,
Cpresent
denotes
matrix
of cencenplacements.
The
inertial
matrix,
inin(1),
inmatrix,
predominant
divided
into
a
number
of
submatrices,
for
the
most
elementary
matrices
the
blocks
have
the
size
of
2×2
[2,
5].
Symmetric
divided
into
a
number
of
submatrices,
for
the
most
elementary
may
be
regarded
as symmetrical,
but τstiffness
for- mathematical
models
offerent
submatrices,
the most on
elementary
matrices
the blocks chains.
forms, for
depending
structures
of its –kinematic
Coriolis
forces,
-- denotes
G
-trifugal
denotes
vector
gravitational
forces,
vector
trifugal
and
Coriolis
forces,
Ksymmetrical,
denotes
stiffness
matrix,
G matrices
cases,and
may
be of
regarded
asK
butdenotes
formatrix,
mathematical
-- the
blocks
have
the
of
2×2
[2,
5].
Symmetric
inertia
matrices
encountered
insize
mathematical
models
of
head
matrices
the
blocks
have
the
size
of
2×2
[2,
5].
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of
wide
set
of
physical
objects
its
elements
are
indirect
function
have
the
size
of
2£2
[2,
5].
Symmetric
inertia
matrices
encounExemplary
forms
of
these
matrices
are
as
follows:
-of
denotes
vector
of
gravitational
forces,
τ
denotes
vector
generalized
forces,
q physical
- denotesobjects
vector its
generalized
dis- denotes
vector
ofsetgravitational
forces,
τof -elements
denotes are
vector
models
of
wide
of
indirect
inertia
matrices
encountered
in
mathematical
models
of
positioning
systems
of
hard
disk
drives
[3,
4],
have
very
difof time.
The
elements
of inertial
matrix
ondisangularinertia
tered
in mathematical
models
of head positioning
systems
of
matrices
encountered
in mathematical
models
of head
head
of generalized
generalized
forces,
q matrix,
denotes
vector
ofmay
generalized
placements.
The
inertial
present
in (1),
in depend
predominant
of
forces,
q
--elements
denotes
vector
of
generalized
dis

function
of
time.
The
of
inertial
matrix
may
depend
positioning
systems
of
hard
disk
drives
[3,
4],
have
very
different
forms,
depending
on
structures
of
its
kinematic
chains.
or
linear
displacements.
It
happens
very
often
in
mathematical
hard
disk
drives
[3,
4],
have
very
different
forms,
depending
positioning
systems
of
hard
disk
drives
[3,
4],
have
very
difplacements.
The
inertial as
matrix,
present in
in
(1),
inmathematical
predominant
d11
0
0 0...
0
cases,
may be
regarded
symmetrical,
but
forin
placements.
The
inertial
matrix,
present
(1),
predominant
on angular
or
linear
displacements.
It
happens
very
often
in
ferent
forms,
depending
onmatrices
structures
of
its
kinematic
chains.
modelling
of advanced
mechanical
systems
such
as robot
maon forms,
structures
ofof
itsthese
kinematic
chains.
forms of
these
Exemplary
forms
areExemplary
as its
follows:
ferent
depending
on
structures
of
kinematic
chains.

cases,
may
be
regarded
as
symmetrical,
but
for
mathematical
models
of
wide
set
of
physical
objects
its
elements
are
indirect
cases,
may
be regarded
symmetrical,
but for
mathematical
d22 are
d23as follows:
0...
0 


mathematical
modelling
ofInadvanced
mechanical
systems
suchExemplary
nipulators
[1, 23, as
24].
mathematical
models
of
electro-mematrices
are
as
follows:
forms
of
these
matrices
Exemplary forms
 matrices are as follows:

models
of
of
objects
its
are
indirect
 of these
function
ofwide
time.set
The
elements
of inertial
may
depend
models
of
wide
set
of physical
physical
its elements
elements
aremutual
indirect
chanical
systems
matrix
parameters
ofmatrix
and
induc 0
0 
d.33. . 0 . .0.
as robot
manipulators
[1], objects
[23],
[24].
Inselfmathematical
modd
0
0
D
=
(3)
11



function
of or
time.
Thedisplacements.
elements
of inertial
inertial
matrix
may
depend
on
angular
linear
It happens
very
in dis d

function
of
time.
The
elements
of
matrix
may
depend
tances
of
stator
and rotorsystems
windings
also
depend
onoften
angular

 d00 d00 00 .. .. .. . . 00 

els
of
electro-mechanical
matrix
parameters
of
self11
d
11
on angular
angular
or modelling
linear
displacements.
It happens
happens
very
often
in

mathematical
advanced
mechanical
systems
such
 22 23

. d
on
or
linear
It
very
often
in
placements
of displacements.
the of
rotor
Similarly,
in mathematical
models

k−1,k
and mutual
inductances
of[2].
stator
and rotor
windings
also
ded
d
0
.
.
.
0
22
23
d
mathematical
modelling
of
advanced
mechanical
systems
such


0
.
.
.
0
d
as
robot
manipulators
[1],
[23],
[24].
In
mathematical
mod22
23
mathematical
of systems
advancedofmechanical
systems such
D=
(3)
of headmodelling
positioning
mass
devices
sym 33


dk,k
pend
on
angular
displacements
of modern
the
rotor
[2].storage
Similarly,
in
... .. ..
00 
dd33
as
manipulators
[1],
[23],
In
els
of electro-mechanical
systems
parameters
of modself.
33 0
(hard
disk drives),
ofmatrix
inertial
matrices (which
are rep- (3)
D=
(3)
=
0
as robot
robot
manipulators
[1],elements
[23], [24].
[24].
In mathematical
mathematical
modD
(3)



. dk−1,k 


mathematical
modelsofofsystems
head of
positioning
systemsalso
ofselfmodern




els
electro-mechanical
matrix
of
.. .
and
mutual
inductances
stator
and
rotorparameters
deresented
by mass moments
inertia
orwindings
by masses)
on


els of
of
electro-mechanical
systems
matrix
parameters
ofdepend
self sym
. . . d
d11 d12 .d.. 13 dddk−1,k

1,k
storage
devices
(hard
disk
elements
of
inertial
k−1,k
k,k
andmass
mutual
inductances
of
statorof
and
rotor
windings
also
despatial of
configurations
of thewindings
links
of its
kinematic
pend
ontemporary
angular
displacements
thedrives),
rotor
[2].
Similarly,
in
and
mutual
inductances
stator
and
rotor
also
de
matrices
(which
are
represented
by
mass
moments
of
inertia
or
d
0
.
.
.
0 
sym
d
22
k,k
and
therefore
the angular
(orrotor
linear)
displacement
ofin

dk,k
pend
angular
displacements
of
[2].
Similarly,
mathematical
models
of head
positioning
systems
of modern
pend on
onchain
angular
displacements
of the
the
rotor
[2].
Similarly,
injoints

sym 


by
masses)
depend
on
temporary
spatial
configurations
of
the
[3,
4].
A thorough
analysis
of
the
structures
of
the
inertia
mad
d
.
.
.
d
d


11
12
13
mathematical
models
of
head
positioning
systems
of
modern
1,k
.
.
.
0
d
mass
storage
devices
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chanical
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motors)
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[21]
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[21]
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phisics
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ing
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T. Trawiński, A. Kochan, P. Kielan, and D. Kurzyk
Square matrices with entries equal zero except for their of proposed method which exhibits the smallest increase of
main diagonal, one row and one column have many applica- number of algebraic operation due to block matrix dimension
tions e.g. in wireless communication systems [19], neural-net- increase.
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Trawiński,
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[21] or phisics [22]. In this paper we propose a method of
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aphysical
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propose
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[4]
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[4]
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systems of hard disk drives [4] and in this case, this
this division
analysis
of
the
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imis
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chapter
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plement
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positioning
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of We
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general.
propose
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Due
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general
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rowhead),
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rowhead),
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ter
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ter
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matrix
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the
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the
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describing
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the
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matrix,
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ininin
former
works,
In chapter
2 general
information
about
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struc-matrix
representation
of the
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representation
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the
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equais
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works,
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tion
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gated
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block
gated
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block
the
description
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operation
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wireless
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[9].
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the
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[9].One
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ofof
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[9].[9].
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the
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the
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and
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and
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derived
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presented.
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determiproblems
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dethe
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arrow
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the problems associated with arrow matrices is effective de-decomplexity
and
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times.
In
paper
[18]
the
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and
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times.
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paper
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the
authors
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investigated
mutual
interactions
between
nation of the
eigenvalues
of [7–9].
Additionally,
considered
are
termination
the
eigenvalues
[7],
[8],
[9].
Additionally,
termination
the
eigenvalues
[7],[8],
[8],[9].
[9].
Additionally,
complexity and resultant computation times. In paper [18] the termination
ofofof
the
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ofofof
[7],
Additionally,
number
of
algebraic
operation
has
been
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for
an
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for
an
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structure
of
block
matrices
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more
than
parallel
matrix
inversion
methods
as
described
in
[10].
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[11]
considered
are
parallel
matrix
inversion
methods
as
described
considered
are
parallel
matrix
inversion
methods
as
described
number of algebraic operation has been calculated for an in- considered are parallel matrix inversion methods as described
16
elements)
and itsdefined
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and
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a quick
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solving asystems
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(only
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type),
but inin[10].
[11]
isis
presented
method
solving
systems
[10].
In
[11]
aquick
quick
method
solving
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version
process
ofofof
strictly
defined
matrix
(only
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type),
but
in [10].
InIn
[11]
isof
presented
a quick
method
of solving
systems
computation
times.ofof
Ininput
paperblock
[18] matrix,
the
number
of
algebraic
arrow
matrix
coefficients
is
presented.
under
different
partition
i.e.
into
4,
6
under
different
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input
block
matrix,
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into
4,
6
linear
equations
the
arrow
matrix
coefficients.
linear
equationsofof
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coefficients.
under different partition of input block matrix, i.e. into 4, 6 ofofof
arrow
matrix
ofofof
coefficients.
operation
hasmatrices.
been calculated
for
an showing
inversion
process
Asequations
shown in of
[6]the
inverted
block matrix
(5) can be repreand
1616
elementary
Other
papers
the
relaand
elementary
matrices.
Other
papers
showing
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As
shown
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inverted
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can
As
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inverted
block
matrix
(5)
canbeberepresented
represented
and
16
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matrices.
Other
papers
showing
the
relashown
strictlyblock
defined
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(only one
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under
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sented
as: in [6] inverted block matrix (5) can be represented
tion
between
matrix
structure
and
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kinematic
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between
block
matrix
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and
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kinematic
as:
as:
tion between
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andi.e.
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16 posielemenchains
ofof
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robots
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
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posi−1
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tary
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papers
showing
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between
 c0c0 −c
a1−1
−c
a2−1
.. ..... . ξξ1ξ1
−c
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0bb
0bb
0 1b
0 2b
tioning
systems)
[3,
4],
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.
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tioning
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4],
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or
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block matrix structure and structure of kinematic chains of
−1
−1 T T 2 −1


electric
machines
[2]
and
[5].
General
formulas,
describing
electric
machines
[2]
and
[5].General
General
formulas,
describing
cb0 2baa2−1
a−1 . .... . ξ2ξ2



1a1bbT1bc1c00b
electric
machines
[2]
and
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formulas,
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aa a
cc11c1
aa−1
robots
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of head
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2 22 2 . . . ξ2 

1
1


relation
between
block
matrix’s
structure
and
dimension,
and
relation
between
block
matrix’sor
structure
and
dimension,electric
and

 (6)
relation
between
block
and
systems)
[3,
4, 6]matrix’s
and [18]structure
structure
ofdimension,
winding of and
r=
r==

cc22c2
. .. ..... . ξξ33ξ3(6)
DDrD
(6)(6)
the
number
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operations
have
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presented
in
the
number
of
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operations
have
not
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presented
in


.
.
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
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.
.
machines
[2] and operations
[5]. Generalhave
formulas,
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a relation
the number
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not been
presented
in
.
.
.
.
.

. . . . . . .
authors’s
former
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chapter
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authors’s
former
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chapter
3ofof
this
paper
methods

. . 
between
block
matrix's
structure
and
dimension,
and the
authors’s
former
papers.
InInIn
chapter
33of
this
paper
methods
of
accounting
the
number
of
algebraic
operations,
necessary
of
accounting
the
number
of
algebraic
operations,
necessary
sym
sym
numberthe
of algebraic
not beennecessary
presented in
of accounting
number ofoperations
algebraichave
operations,
sym
cckkck
to
make
during
inversion
process
are
described.
Three
difto
make
during
inversion
process
are
described.
Three
difformer papers.
In chapter
3 of this paper
to makeauthors's
during inversion
process
are described.
Threemethods
dif−1
−1 T T−1
−1
k k −1
−1
T(c(c
where
(a(a
(a(a
bTTib(c
++
where
ferent
cases
of
block
matrices
internal
portioning
are
considferent
cases
of
block
matrices
internal
portioning
are
considj a−1
i−
0 0−−∑k∑
ja
i ==(a
i−−
of
accounting
the
number
of
algebraic
operations,
neces-where
wherecc00c0===(a
jbb
jj b
j))−1
j ) , ,cci, ic=
i −1
j jbbTb
00+
a
b
ferent cases of block matrices internal portioning are
considi
0 k− ∑
j
j
i
j
0
–1 T –1
T –1
–1 T –1 −1
–1−1
−1
−1
T
−1
−1
T
−1
−1
sary
to make
during
inversion
process
areofof
Three biba−1
c = (a
¡ bfor
ci{1,
= (a
¡ b
aj −c
b−c
)kak , ,
ered.
Inversion
times
for
allall
chosen
structures
block
matriered.Inversion
Inversion
times
for
chosen
structures
block
matri{1,
for
.k},
, k},
ja
i=
j bji)i ∈,∈
i (c0 ξ + b
j )0 b
i ) j for
0b
kbia
k−1
j ) bbi0)ib)−1
ered.
times
for
all
chosen
structures
of described.
block
matrib a i aj jbbTjb0j))−1
i ∈ {1,
. .. ...i.,.,k},
ξ11ξ1==
,
–1
–1 −c
T 0 bk ak–1
different
cases
of
block
matrices
internal
portioning
are ξi ξj==afor
1, …, kg,
ξ
= ¡
,
ξ
= a
b
c
b
a
,
−1
−1
−1
−1i 2 f
−1
−1 TcT0 bk a k−1
−1
ces
are
compared
with
times
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inversion
using
standard
inverT
ces
are
compared
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times
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using
standard
inverT
1
2
0
k
1
1
k
b
c
b
a
,
ξ
=
a
b
c
b
a
.
a
b
c
b
a
,
ξ
=
a
b
c
b
a
.
−1
−1
−1
ces are compared with times of inversion using standard inver- ξ 2=
T
T
0
3
0
2 a−1
0
3
0
k
k
k
k
1 1b 1c10–1bbkTac kbk, aξ–1
22 22
kk
considered.
Inversion
times
for The
all
chosen
structures
1 2 2 k0 k k3 .= a2 b2 c0 bk ak .
1ξ3 = a
sion
method
inin
Matlab
(“inv”
function).
presented
method
sion
method
Matlab
(“inv”
function).
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presented
methodof 2 The
inverted
block
matrix
of
inertia
(6)
consist
(as
can
The
inverted
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matrix
of
inertia
(6)
consist
(as
can
sion
method
in
Matlab
(“inv”
function).
The
presented
method
The inverted block matrix of inertia (6)
consist
ofofof
(as
can
block
matrices
are compared
with effective
times
of than
inversion
using
of
block
matrix
inversion
is
much
more
the
one
of
block
matrix
inversion
is
much
more
effective
than
the
one
bebe
observed)
elementary
matrices,
which
are
calculated
onon
the
observed)
elementary
matrices,
which
are
calculated
the
of blockstandard
matrix inversion
ismethod
much more
effective
than
the oneThebe
in Matlab
(“inv”
function).
The inverted
block matrices,
matrix
of which
inertia
(6)
consist
of (as
can
observed)
elementary
are
calculated
on
the
used
Matlab.
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inin
this
article
the
number
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used
Matlab.inversion
Also,
this
article
the
number
of
algebraic basis
of
the
block
matrix
elements
(5)
before
the
inversion.
It
basis
of
the
block
matrix
elements
(5)
before
the
inversion.
It
used
ininin
Matlab.
Also,
in
this
article
the
number
of
algebraic
presented
method
of
block
matrix
inversion
is
much
morebasis
beof
observed)
matrices,
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calculated
on Itthe
the blockelementary
matrix elements
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before
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isisis
operations
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invert
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invert
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without
possible
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without
operations
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matrices)
has
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calcueffective than the one used in Matlab. Also, in this articlepossible
basis of
block matrix
(5) before
the inversion.
It is
to the
calculate
chosenelements
elementary
submatrix
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lated
and
compared
with
Gaussian
method
ofof
matrix
inversion,
lated
and
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matrix
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need
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blocks
the
need
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of
the
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the
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lated
and
compared
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Gaussian
method
of
matrix
inversion,
the
number of
algebraic
operations
(necessary
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possible
to
calculate
chosen
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without
need of calculation of the remaining elements of the blocks
and
has
shown
the
advantages
proposed
method
which
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hasshown
shownthe
theadvantages
advantagesofofofproposed
proposedmethod
methodwhich
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matrix.
and
matrices)
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with Gaussianmatrix.
the need of calculation of the remaining elements of the blocks
exhibits
the
smallest
increase
number
ofof
algebraic
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exhibits
the
smallest
increase
number
algebraic
operation
method
of matrix
inversion,
and itof
has
shown
the
advantages matrix.
exhibits
the
smallest
increase
ofofof
number
algebraic
operation
Σ
due
block
matrix
dimension
increase.
due
block
matrix
dimension
increase.
due
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block
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dimension
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854
The
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the
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inversionof
theblock
block
matrix
Bull.
Pol. Ac.: Tech. 64(4) 2016
the
matrix
Unauthenticated
For
the
sake
consequence
further
course
the
discussion,
For
the
sake
consequence
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courseofof
the
discussion,
For
the
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consequence
ofofof
further
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discussion,
Download
Date |course
7/28/17of
10:28
PM
definition
of
block
dimension
is
formulated.
definition
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definition of block dimension is formulated.
tion as self inertia matrices, have following forms:
−1
−1 T −1
ci = (ai − bTi (c−1
0 + bi a j b j ) bi )
(11)
Inversion of selected structures of block matrices of chosen mechatronic systems
for i ∈ {1, . . . , k}.
By introducing the above indications of elementary matri-
By introducing
the above
indications
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3. The number of algebraic operations during ces, inverted
block matrix
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submatrix ddii..to calculate the negative
Number
of
algebraic
operations
necessary
feedback
Bull.
Pol.
Ac.:
Tech.
XX(Y)
2016
with
different
internal
divisions
into
elementary
blocks
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matrices.
Allperformed
ofinversion,
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have
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n2totake
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Table
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.
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6 the 9negative
Sum
of algebraic
operations
loII
matrices.
All internal
of
thesuch
analyzed
cases
of
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block
matrix
will
of algebraic
operations
necessary
to3 calculate
have
a structure
as
(5),above,
it will
only
of Number
(5)), then
the
number
of algebraic
to12thefeedback
calwith
different
divisions
into
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blocks
-size
subof
elementary
matrices.
As
mentioned
above,
each
of
inSum
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operations
loII operations
of of
elementary
matrices.
As
mentioned
each
of
the
in326 639 9412 12
Sum
of algebraic
operations
loII
elementary
matrices.
As
mentioned
above,
each
of
the
inBlock
dimension
n
5 block
matrix
d
.
i
verse
elementary
matrices
(6)
can
be
calculated
individually.
A
have
aelementary
structure
such
as As
matrix
(5),
itofabove,
will
differ
only
size
elementary
matrices.
mentioned
each
of
thein
inverse
culation of the leading element c0 (7) is determined by:
matrices.
All
of
the
analyzed
cases
the
block
matrix
will
verse
matrices
(6)
can
be
calculated
individually.
A
verse
elementary
matrices
(6) (6)
cancan
be calculated
individually.
A A
verse
elementary
matrices
becalculated
calculated
individually.
3 36 4operations
9 512
Sum
of algebraic
loII
detailed
analysis
of the
theasstructure
structure
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(6)
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Blockoperations
dimension
n 2process,
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matrices
(6)
can
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individually.
size n of the
block
matrix,
the inversion
of
of
elementary
matrices.
As
mentioned
above,
each
of in
theA deinhave
aanalysis
structure
such
matrix
(5),
it will
differ
only
size
detailed
analysis
of
of
the
matrix
(6)
reveals
that
detailed
of
the
structure
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the
matrix
(6)
reveals
that
–1 T - a partition
3.1.
Case
1
one-piece
elementary
matrices
detailed
analysis
of
the
structure
of
the
matrix
(6)
reveals
that
tailed
analysis
of
the
structure
of
the
matrix
(6)
reveals
that
addition
(subtraction)
in
triples
of
matrices
b
a
b
,
multipli3
6
9
12
Sum
of
algebraic
operations
l
3.1.
Case
1
one-piece
elementary
matrices
a
partition
there
are
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different
types
of
items
elementary
submatrices
verse
elementary
matrices
(6)
can
be
calculated
individually.
A
j
oII
j
j
of are
elementary
matrices.
As
mentioned
above, submatrices
each
of the in- 3.1.3.1.
there
are
four
different
types
of
items
- elementary
submatrices
Case
1 - 1one-piece
elementary
matrices
- a -partition
there
four
different
types
of
items
-items
elementary
–1 T matrix
Case
- one-piece
elementary
matrices
a partition
in
the
1-1-1-1
order
If the
the
block
can be
beinversion.
divided
into
there
are
four
different
ofcan
items
-calculation.
elementary
submatrices
there
are
four
different
types
ofof
–matrix
elementary
submatrices
cation
in
triples
of matrices
bj ablock
bj , and
aj matrix
in
the
1-1-1-1
order
If
matrix
can
divided
into
of
inverted
block
matrix
requiring
thethe
These
arethat
asin the
j matrix
detailed
analysis
of
thetypes
structure
(6)These
reveals
verse
elementary
matrices
(6)
be
calculated
individually.
A
of
inverted
block
matrix
requiring
the
calculation.
are
as
1-1-1-1
order
If
the
block
can
be
divided
intointo
of inverted
block
matrix
requiring
the
calculation.
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are
as
in
the
1-1-1-1
order
If
the
block
matrix
can
be
divided
one-piece
elementary
matrices
(itblock
will matrices
take
the
form
of aa mamaof
block
requiring
the
calculation.
These
are
By convention,
division
of the
m
atrix
forform
of
inverted
block
matrix
requiring
thethe
are
as one-piece
3.1.
Case
1 - such
one-piece
elementary
-one-piece
a partition
elementary
matrices
(it
will
take
the
of
follows:
there
areinverted
four
different
types
of
items
-calculation.
elementary
submatrices
detailed
analysis
of thematrix
structure
of
matrix
(6)These
reveals
that
one-piece
elementary
matrices
(it will
taketake
the the
form
of aof
mafollows:
follows:
one-piece
elementary
matrices
(it will
form
a maas
follows:
elementary
matrices
will
be
referred
as
division
in
1‒1–1‒1
trix
(5)),
then
the
number
of
algebraic
operations
related
to
follows:
in
the
1-1-1-1
order
If
the
block
matrix
can
be
divided
into
3.1.
Case
1
one-piece
elementary
matrices
a
partition
(5)),
then
the
number
of algebraic
operations
related
to
of
inverted
block
matrixtypes
requiring
the-calculation.
are as trixtrix
there
are four
different
of items
elementary These
submatrices
(5)),(5)),
then
the the
number
ofofalgebraic
operations
related
to to
trix
then
number
of
algebraic
operations
related
1.
first
elementary
matrix,
which
later
will
be
called
the
leading
order.
Calculated
number
algebraic
operations
for
various
the
calculation
of
the
leading
element
c
(7)
is
determined
by:
1.
first
elementary
matrix,
which
later
will
be
called
the
leading
one-piece
elementary
matrices
(it
will
take
the
form
of
a
mathe
1-1-1-1
order
If the
block
can
divided
the
calculation
of the
leading
element
c00 is
(7)
isbe
determined
by:
1.
elementary
matrix,
which
later
will
be called
the
leading
follows:
offirst
inverted
block
matrix
requiring
the
calculation.
are as the in
calculation
of the
the
leading
element
cmatrix
determined
by:into
1. 1.
first
elementary
matrix,
later
willwill
be called
theThese
leading
0 (7)
element,
has
thewhich
following
form:
dimensions
of
block
matrices
is the
shown
1. related
the
calculation
of
the
leading
element
ctake
(7)Table
is determined
by:
first
elementary
matrix,
which
later
be called
the
leading
0 in
block
size
n
of
the
block
matrix,
inversion
process,
operaelement,
has
the
following
form:
trix
(5)),
then
the
number
of
algebraic
operations
to
one-piece
elementary
matrices
(it
will
the
form
of
a
mablock
size
n
of
the
block
matrix,
the
inversion
process,
operaelement,
has
the
following
form:
follows: has the following form:
block
sizesize
n ofnthe
block
matrix,
the the
inversion
process,
operaelement,
−1 T
T,
block
ofthe
the
block
matrix,
inversion
process,
has thematrix,
following
form:
−1
bby:
tionscalculation
of addition
addition
(subtraction)
in triples
triples
of
matrices
b jj aaoperathe
of
the
leading
element
c
(7)
is
determined
1. element,
first elementary
which
later will be called the leading tions
−1
T
trix
(5)),
then
number
of
algebraic
operations
related
to
0
j,
b
tions
of
(subtraction)
in
of
matrices
b
j
of addition
(subtraction)
in triples
of matrices
b j a jb ba−1
Table
1 triples
jj , bTj ,
tions
of
addition
(subtraction)
in
of
matrices
−1
kk later will be called the leading
T
j
j
j
block
size
n
of
the
block
matrix,
the
inversion
process,
operahas thematrix,
following
k form:
−1
the
calculation
of
the
leading
element
c
(7)
is
determined
by:
T
1. element,
first elementary
which
−1 T
T −1
multiplication
in triples
triples
of matrices
matrices
and
matrix inina jcalculate
multiplication
in
of
bb jjto
aaleading
bbTjj ,, and
k b j a−1
Number of in
algebraic
operations
necessary
(7)multiplication
= (a
(a00 −
−∑
jj matrix
T b
−1)−1(7)
triples
of matrices
b j a−1
baa0T−1
a jthe
matrix
in−1 inT,
(7)
cc00 =
jj ,of
j ab−1
b j abb−1
(7) (7)
−∑
c0 =
multiplication
in
triples
ofelement
matrices
bj inversion
band
, and
a j matrix
matrix
jjj )bTjj )−1
a
b
tions
of
addition
(subtraction)
in
triples
matrices
b
block
size
n
of
the
block
matrix,
the
process,
operaelement, has the
following
form:
j
j
j
version.
By
convention,
such
division
of
the
block
for
j
j
j
j
c
a
b
)
=0(a
−
c0 (a
j
0 of the
version.
By
convention,
such
division
of
the
block
matrix
for
j j
0 ∑
j
j
version.
By
convention,
such
division
block
matrix
for
k
−1 for
j
T,
−1
Tmatrices
version.
By
convention,
such
division
of
the
block
matrix
a
b
tions
of
addition
(subtraction)
in
triples
of
b
j
one-piece
elementary
matrices
will
be
referred
as
division
in
j
−1 T −1
multiplication
in
triples
of
matrices
b
,
and
a
matrix
inb
a
jin j in
j jreferred
one-piece
elementary
will
be
as j5division
(a0 −in∑
b j a j b jinterpretation
)
(7)
c0 = that
3 j as4 division
Block
dimensionmatrices
n matrices
elementary
willwill
be 2referred
2.
elementary
matrices,
physical
can
beone-piece
−1
T , and
one-piece
elementary
matrices
be
referred
as
division
in
2.
elementary
matrices,
that
in kjphysical
interpretation
can
be
2. 2.
elementary
matrices,
that
in
physical
interpretation
can
be
1-1-1-1
order.
Calculated
number
of
algebraic
operations
for
version.
By
convention,
such
division
of
the
block
matrix
−1
T
−1
multiplication
in
triples
of
matrices
b
b
a
matrix
ina
2.
elementary
matrices,
that
in
physical
interpretation
can
be
j
j
1-1-1-1
order.
Calculated
number
of
algebraic
operations
for
j
j
order.
number
elementary
matrices,
can (7)
be 1-1-1-1
(a0feedback,
−in∑physical
b j a jhave
b jinterpretation
)following
c0 = that
responsible
for
negative
feedback,
have
following
forms:
4 algebraic
7 operations
10 operations
13 for for
Sum
of Calculated
algebraic
operations
loI of algebraic
1-1-1-1
order.
Calculated
number
of
responsible
for
negative
forms:
responsible
for negative
feedback,
have following
forms:forms: various
various
dimensions
of matrices
thesuch
block
matrices
is shown
shown
in
Table
1.
responsible
for negative
feedback,
have
following
one-piece
will
beisofreferred
division
in
version.
Byelementary
convention,
division
the
block
matrix
various
dimensions
of
the
block
matrices
is
in
Table
1.
dimensions
of the
block
matrices
shown
inasTable
1. for
negative
feedback,
have
following
forms:
2. responsible
elementary for
matrices,
that
in jphysical
interpretation
can be various
dimensions
ofmatrices
thebetween
block
matrices
is shown
inthe
Table
1.
General
relationship
the
dimension
of
block
1-1-1-1
order.
Calculated
number
of
algebraic
operations
for
one-piece
elementary
will
be
referred
as
division
in
General
relationship
between
dimension
of the
block
−1
relationship
between
the the
dimension
of the
block
−1
=in−c
−cphysical
b ahave
(8)
for
negative
feedback,
following forms:
2. responsible
elementary
matrices,
that
interpretation
can
be General
−1
General
relationship
between
the
dimension
ofinthe
block
General
relationship
between
the
dimension
of
the
block
ddii−c
=
00ibiia−1
(8) (8)
di =
matrix
block
dimension
n
and
the
number
of
algebraic
operavarious
dimensions
of
the
block
matrices
is
shown
Table
1.
1-1-1-1
order.
Calculated
number
of
algebraic
operations
for
ii (8)
0 bi a
matrix
block
dimension
n
and
the
number
of
algebraic
operamatrix
block
dimension
n
and
the
number
of
algebraic
opera= −c0 bi ahave
di feedback,
responsible for negative
following forms: (8) various
i
matrix
block
dimension
n
and
the
number
of
algebraic
operamatrix
block
dimension
n
and
the
number
of
algebraic
operations
that
are
needed
for
the
calculation
ofisleading
the
leading
element
dimensions
thebetween
block
matrices
shown
in
Table
1.
General
relationship
theofdimension
of
the
block
{1,
for
∈ {1,
k},
auxiliary
matrices
having
forms:
tions
that
are
needed
for
the
calculation
of
the
leading
element
that
are
needed
forof
the
calculation
element
∈
.. .. ,, k},
auxiliary
matrices
having
forms:
forfor
i ∈ii{1,
. . . ,.. k},
auxiliary
forms:
= matrices
−c0 bmatrices
a−1
(8) tions
dauxiliary
having
forms:
tions
are
needed
for
calculation
ofthe
thethe
leading
element
imatrices
ihaving
tions
that
are
needed
forthe
the
calculation
of
leading
element
i having
{1,i 2 f
for i ∈for
. . . ,1, …, kg,
k}, auxiliary
forms:
(7),
is
as
follows:
c
matrix
block
dimension
n
and
the
number
of
algebraic
operaGeneral
relationship
between
the
dimension
of
the
block
0
isfollows:
as follows:
follows:
cc0 (7),
is asis
−1
T
T
(7),
=
bTi ab−1
deeiii−a
=−c
−a0b−1
(9)c0 (7),
−1
is as
as
ctions
00 (7),that
(9)
=
−a
matrix
block
dimension
and
the number
operaei =
(9) (8)
arefollows:
needed fornthe
calculation
of of
thealgebraic
leading element
ii having forms:
−1
iii biT
fori ∈ {1, . . . , k}, auxiliary
matrices
i
ei = −ai bi (9)
(9)
tions
that
are
needed
for
the
calculation
of
the
leading
element
(7),
is
as
follows:
c
= 3(n
3(n
−+1)
1)1(13)
+ 11
(13)
0
matricesT having forms:
oI3(n
{1,
for
∈ {1,
− auxiliary
1}.
=
+
lloI
− 1)−
(13)(13)
∈
,, kkk},
−
1}.
forfor
i ∈ii{1,
. . . ,.. k.. ..−
1}.
ei = −a−1
(9) c0 (7), is as follows:loI =
(13)
loI = 3(n − 1) + 1
i bi
{1,i 2 f1, …, k ¡ 1g.
for
i ∈for
. .matrices,
. , k − 1}. which
3.
elementary
in
physical
interpretation
can
be
−1 T interpretation
elementary
matrices,
which
in
physical
interpretation
3. 3.
elementary
matrices,
which
cancan
be be
The
calculations
effort
necessary
to perform
perform
the designation
designation
eiin=physical
−aphysical
i
calculations
effort
necessary
to
the
i bare
calculations
effort
necessary
to1)
perform
designation
3. elementary
matrices,
which
in
interpretation
can(9)
be TheThe
3(n −
+perform
1 the the
(13)
loI =
responsible
for
positive
couplings,
forms
of
the
products
Thecalculations
calculations
effort
necessary
to
designa{1,
for
i
∈
.
.
.
,
k
−
1}.
responsible
for
positive
couplings,
are
forms
of
the
products
The
effort
necessary
to
perform
the
designation
responsible for positive couplings, are forms of the products
(8),
is
associated
with
the(13)
imof
negative
feedback
matrix
d
i
(8),
is
associated
with
the
imof
negative
feedback
matrix
d
(8),
is
associated
with
the
imof
negative
feedback
matrix
d
responsible
for
positive
couplings,
are
forms
of
the
products
i
=
3(n
−
1)
+
1
l
i
matrices,
in physical
interpretationcan
canbe
be of
3.
tion
of negative
feedback
d(8),
isassociated
associatedwith
with the
the imoI matrix
of
the
matrices
(8)
and
(9)which
{1, . .matrices,
forthe
i ∈elementary
. , k(8)
−
1}.
3.
elementary
which
in physical
interpretation
i (8),
of
matrices
and
(9)
is
negative
feedback
matrix
d
of the
matrices
(8)
and
(9)
i
plementation
of
algebraic
operations
necessary
to:
calculate
The calculations
effort
necessary
tonecessary
perform
theto:
designation
plementation
of algebraic
operations
necessary
calculate
of algebraic
operations
necessary
to:to:
calculate
theresponsible
matrices
(9)
forand
positive
couplings,
forms
theproducts
products
implementation
algebraic
operations
calculate
3. of
elementary
matrices,
which
in physical
interpretation
can be plementation
responsible
for (8)
positive
couplings,
areare
forms
ofofthe
plementation
ofofalgebraic
operations
necessary
to:
calculate
The
calculations
effort
necessary
to
perform
the
designation
and
b
,
calculate
the
inverse
form
of imthe
the
matrix
product
c
(8),
is
associated
with
of
negative
feedback
matrix
d
e
d
(10)
i
0
i
i
j
and
b
,
calculate
the
inverse
form
of
the
the
matrix
product
c
bi ,bcalculate
the
product
c0 cand
ei d j
the matrices
(8) (9)
and
the matrix
product
theinverse
inverseform
form of
of the
0and
ei d(9)
(10)(10) the matrix
i, icalculatethe
responsible
for (8)
positive
couplings,
of
theofmatrices
and
and
boperations
,dcalculate
the
inverse
form
ofimthe
the
matrix
product
c0 0matrix
ej i d j are forms of the products
(10) matrix
i
(8),
is
associated
with
the
of
negative
feedback
(in
present
case
dividing
by
an
element
of
the
mamatrix
a
plementation
of
algebraic
necessary
to:
calculate
i
i
(inpresent
present
case
dividing
by
an
element
of mathe mamatrix
matrix
aaii (in
case
– -dividing
anelement
element
of the
the
present
case
- dividing
bybyan
of
ai (in
of
(8)
and
(9)
{1,
{1,
for
∈matrices
− 1},
1},
∈ {1,
, k}.
k}.
present
case
-bof
dividing
by
an inverse
element
of
theofmamatrix
ai (in
{1,
ii{1,
∈
,, kk1},
−
∈
.. ..j ,(10)
forfor
i ∈the
. . . ,.. k.. ..−
j ∈ jj{1,
. . e. ,i..dk}.
plementation
of
algebraic
operations
necessary
to:
calculate
trix).
Calculated
numbers
algebraic
operations
for
different
and
,
calculate
the
form
the
the
matrix
product
c
(10)
trix).
Calculated
numbers
of
algebraic
operations
for
different
i
0
trix).
Calculated
numbers
of
algebraic
operations
for
different
trix).
Calculated
numbers
of
algebraic
operations
for
different
{1, . . . , k −
for
i ∈matrices,
1}, j can
∈ {1,be. .called
. , k}. in physical interpreta- trix). Calculated numbers of algebraic operations for different
4.
block
which
block
matrices,
which
called
in
physical
interpreta4. 4.
block
matrices,
which
cancan
be be
called
in
physical
interpretaand
b
,
calculate
the
inverse
form
of
the
the
matrix
product
c
block
dimension
n
of
the
block
matrix,
the
negative
feedback
ei dcalled
(10)
case
-i dividing
by
element
offeedback
the mamatrix
ai (in present
block
dimension
nnof
block
matrix,
thean
negative
feedback
0 the
j
block
dimension
of
the
block
matrix,
the
negative
dimension
n of
the
block
matrix,
the
negative
feedback
4. block
matrices,
which
be
in physical
tion
as
self
matrices,
have
following
forms:interpreta- block
{1,
{1,
for
∈for
. inertia
.inertia
. , kmatrices,
−
1},
j can
∈have
. .following
. , k}.
block
dimension
nshown
ofcase
the
block
matrix,
theelement
negative
feedback
as
self
matrices,
have
following
forms:
tiontion
asiself
inertia
forms:
(in
present
dividing
by
an
of
the
matrix
a
i 2 f1, …, k ¡ 1g,
j 2 f1, …, kg.
matrix
d
(8),
are
shown
in
Table
2.
It
should
be
noted,
that
in-mamatrix
d
(8),
are
in
Table
2.
It
should
be
noted,
that
intrix).
Calculated
numbers
of
algebraic
operations
for
different
i
i
matrix
d
(8),
are
shown
in
Table
2.
It
should
be
noted,
indi (8),
ini are shown in Table 2. It should be noted, that that
self
matrices,
have
following
forms:interpreta- matrix
{1, .inertia
for ias
∈matrices,
..,k −
1},
j can
∈ {1,
. .called
. , k}.
4. tion
block
which
be
in physical
matrix
di size
(8), of
arethe
shown
inof
Table
2.
Itone,
should
be noted,
that intrix).
Calculated
numbers
algebraic
operations
for
different
creasing
block
matrix
by
results
in
occurrence
block
dimension
n
of
the
block
matrix,
the
negative
feedback
creasing
of the
block
matrix
by one,
results
in occurrence
creasing
sizesize
of the
block
matrix
by one,
results
in occurrence
Tcan−1
−1
−1
T physical
−1−1 −1
T
−1
4. tion
block
matrices,
which
be
called
in
T bT
−1
T b
−1
=i (a
(a
−
(ccan
+
bcalled
a−1
)physical
bforms:
)−1 interpreta(11) block
self
matrices,
have
i following
creasing
of
matrix
bymatrix
results
inthe
occurrence
4.asblock
matrices,
which
be
in
Table
2matrix,
=
bTii−1
ccinertia
(11)(11)
−
ci =
nshown
ofblock
the
the
negative
-the
innoted,
first
row
of additional
an dimension
additional
negative
feedback
din
ii (a
iib−
i ) bii)−1 interpretation
i ab
matrix
disize
(8),negative
arethe
inblock
Table
2.
Itone,
should
thatrow
ini (c
i be
−1
jjj )bTjj )b−1
j i ab−1
00 b+
0 (c+
in
thefeedback
first
of
an
additional
negative
feedback
matrix
d
first
row
of
an
feedback
matrix
d
i
i
− bi (c0 have
+ bfollowing
bforms:
(11) ofNumber
tion asasself
have
following
i = (aimatrices,
ia
i)
of
algebraic
operations
necessary
to
calculate
the
negative
j b j ) forms:
selfcinertia
inertia
matrices,
in
the
first
row
an
additional
negative
feedback
matrix
d
i
matrix
d
(8),
are
shown
in
Table
2.
It
should
be
noted,
that
inand
first
column
of
the
inverted
block
matrix.
i column
creasing
size of
of the
the
block
matrix
by
one,
results in occurrence
andand
firstfirst
column
inverted
block
matrix.
{1,
for
∈ {1,
k}.
of the
inverted
block
matrix.
−1
T −1
−1 T −1
forfor
i ∈ii{1,
. . . ,.. k}.
∈
.. .. ,, k}.
matrix
di matrix.
(11) creasing
and
first
column
of block
thefeedback
inverted
block
size
of
the
matrix
by
one,
results
inthe
occurrence
j bTj )−1 bi )−1
for i ∈ {1, . .c.i ,=
k}.(ai − biT (c0−1 + bi a−1
General
relationship
between
block
dimension
of
the
block
the
firstblock
row
of
an
additional
negative
feedback
matrix
di -ofin
relationship
between
block
dimension
block
General
relationship
between
block
dimension
of
the
ci = (a
bi (c0indications
+ bi a j b jof
) elementary
bi ) (11)
(11) General
i−
2operations
3 di - 4in
5are
Block
dimension
n inverted
General
relationship
between
block
dimension
of
the
block
the
first
row
of
an
additional
negative
feedback
matrix
By
introducing
the
above
matrimatrix
and
the
number
of
algebraic
that
needed
and
first
column
of
the
block
matrix.
Byfor
introducing
above
indications
of elementary
matriBy
above
indications
of elementary
matri- matrix
andand
the the
number
of algebraic
operations
that that
are needed
matrix
number
of algebraic
operations
are needed
iintroducing
∈ {1, . . . ,the
k}.the
By
the
above
indications
of elementary
matrix
and
the
number
of
algebraic
are
needed
first
column
ofthe
the
inverted
block
matrix.
ces,
inverted
block
matrix
(6)
takes
the
following
form: matri- for and
3operations
6 matrix
9that
12
Sum
ofrelationship
algebraic
operations
loIIfeedback
(8)
is as
as
for
the
calculation
of
the
negative
feedback
matrix
General
between
block
dimension
of
block
{1,
forinverted
iintroducing
∈for
.
.
.
,
k}.
ces,ces,
inverted
block
matrix
(6)
takes
the
following
form:
block
matrix
(6)
takes
the
following
form:
(8)
is
as
the
calculation
of
negative
matrix
d
(8)
is
for
the
calculation
of
the
negative
feedback
ddiithe
i
i 2 f1, …, kg.
ces,Byinverted
block
matrix
(6)
takes
the
following
form:
(8)
is
as
for
the
calculation
of
the
negative
feedback
matrix
d
General
relationship
between
block
dimension
of
the
block
i
follows:and the number of algebraic operations that are needed
introducingthe above indications of elementary
matri- follows:
matrix
follows:

 
 the above indications of elementary
follows:
introducing
matrimatrix
the number
of algebraic
operations
that are
needed
d
d
.
.
.
d
cc00 d1 (6)
ces,Byinverted
block
takes
the
following
form:
is as
for
the and
calculation
of the
negative feedback
matrix
di (8)
1
2
k
d
.
.
.
d

c0matrix
d1 2 d2 . . . k dk
d
d
.
.
.
d
c


ces, inverted
block
matrix
(6)
takes
the
following
form:
(8)
is
as
for
the
calculation
of
the
negative
feedback
matrix
d
0
1
2
k


i


follows:
= 3(n
3(n1)−
− 1)
1)
(14)
loII
e d . . . ek 11ddkk 
Bull. Pol. Ac.: 
Tech. 64(4)
855(14)
oII3(n
−
(14)
loII l=


c1 cc2016
=
 
11e1 de211d22. . . . . .e1 de



follows:
=
3(n
−
1)
(14)
l
c
e
d
.
.
.
e
d

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 c0 dc 1 c2edccd2222 . . . ... ... ...e2 deekd222ddk


Download
Date
|
7/28/17
10:28 PM


=
3(n
−
1)
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l
oII
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
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to
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 
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The
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Dr = 
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
is
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with
mine
the
of
positive
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for
n=
2,matrix
3, . of
. .of
.positive
i d j (10),
(10),
isin
related
with
the
matrix
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

c0
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c1
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c2
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a-
1)
ri-
2)
ed
ix
ut
the matrix product c0 and bi , calculate the inverse form of thegebraic operations for different block dimensions n of block
gebraic operations for different block dimensions n of block
matrix ai (in present case - dividing by an element of the ma-matrix, necessary for calculation of self inertia matrices, are
matrix, necessary for calculation of self inertia matrices, are
trix). Calculated numbers of algebraic operations for differentshown in Table 4.
shown in Table 4.
block dimension n of the block matrix, theT.negative
feedback General
relationship
between block dimension of the block
Trawiński, A. Kochan, General
P. Kielan, and
D. Kurzyk between block dimension of the block
relationship
matrix di (8), are shown in Table 2. It should be noted, that in-matrix and the number of algebraic operations that are needed
matrix and the number of algebraic operations that are needed
creasing size of the block matrix by one, results in occurrencefor the calculation of the self inertia matrices ci (11) is as folfor the calculation of the self inertia matrices ci (11) is as foltheoccurrence
first rowlows:
of an
additional
negative
feedback
matrix
di - in in
creasing
size of
the block
matrix by
one, results
for the calculation of the self inertia matrices ci (11) is as follows:
and of
first
of negative
the inverted
blockmatrix
matrix.
ancolumn
additional
feedback
di – in the first row lows:
General
between
(16)
loIV = 8(n − 1)
and firstrelationship
column of the
invertedblock
blockdimension
matrix. of the block
(16)
loIV = 8(n − 1)(16)
General
between block
dimension
block
matrix and
the relationship
number of algebraic
operations
that of
arethe
needed
for n = 1, 2, . . . .
matrix
and the number
of algebraic
operations
that dare
needed
1, 2, . . . .
is asfor nfor=n = 1, 2, …
for the
calculation
of the negative
feedback
matrix
i (8)
total
amount
of algebraic
operations
needed
to calculate
for the calculation of the negative feedback matrix di (8) is as The
The total
amount
of algebraic
operations
needed
to calcuThe total
amount
of algebraic
operations
needed
to calculate
follows:
the
inverted
form
of
block
matrix
of
the
block
dimension
n
follows:
the inverted
of block
matrix
of the
block
dimension
the late
inverted
form ofform
block
matrix
of the
block
dimension
n
and nassuming
that
all
matrices
are
one-piece,
is
given
by
the
and
assuming
that
all
matrices
are
one-piece,
is
given
by
the
and assuming that all matrices are one-piece, is given by the
(14)following
loII = 3(n − 1)(14)
formula:
following
formula:
following
formula:
2
Tomasz Trawiński, Adam Kochan, Paweł Kielan and Dariusz n
Kurzyk
for nfor
=n = 2, 3, …
2, 3, . . . .
n
2
n
l
=
3
+ 19 n − 10 (17)
(17)
The
number
of
algebraic
operations
needed
in
order
to
de
o
The number of algebraic operations needed in order to deterl
=
3
+
19
(17)
2
2 − 10
o
Table 3feedback ei dj (10), is related
termine
the
matrix
of
positive
2
2
d
(10),
is
related
with
mine
the
matrix
of
positive
feedback
e
i
j
Number
algebraic
operations
necessary
to calculate
=n = 1, 2, …
1, 2, . . . .
withofthe
calculation
of the
auxiliary
matrixthe
e , positive
and itsfeedback
productfor nnfor
= 1, 2, . . . .
the calculation of the auxiliary
ei , and iits product withforRelationship
matrixmatrix
ei d j .
showing
thethe
number
algebraic
with negative feedback matrix dj. It should be emphasized that Relationship
Relationship
showing
numberof
algebraic operations
operations
showing
the number
ofofalgebraic
operations
negative feedback matrix
d
.
It
should
be
emphasized
that
this
j
to be
to to
calculate
thetheelementary
Block
dimension
n 3 dimension
4
5 n ¸ 3.
6
this type of matrices
occurs
for block
Theseneeded
needed
to done
be done
calculate
elementarymatrices
matrices (with
(with
needed to be done to calculate the elementary matrices (with
typecalculations
of matrices
occursoperations
for blockl of
dimension
n 18
≥, multiplication
3. 30
These cal-the
require:
inversion
the block
dimension
growth
blockmatrix),
matrix),can
canbe
be reprerepreSum of algebraic
9 a
block
dimension
growth
of of
thetheblock
oIII the3 matrix
i
the block dimension growth of the block matrix), can be repreT
ofsented
culations
require:
of the by
matrix
i , multiplication
of matrices
a–1binversion
, multiplication
(¡1) aand
multiplication by
sented
graphically
in Fig. 1.
graphically
in Figure
1.
−1 T i i
and operations
multiplication
bysented graphically in Figure 1.
matrices
matrixaidj.bCalculated
numbersby
of(−1)
algebraic
for vari , multiplication
eleme
eleme
agona
agona
symm
symm
trix in
trix in
matric
matric
lar ma
lar ma
sion o
sion o
dimen
dimen
the 1the 1tioned
tioned
1-2-21-2-2The
The
the lea
the lea
of inp
of inp
b j a−1
j
b j a−1
j
culate
culate
multip
multip
matrix
matrix
calcul
calcul
calcul
calcul
Tablematrix,
4
ious dimensions of the block
for the positive feedback3.2. Fig.
1. 2The
number of algebraic
required
to be
imple-ments
Case
- elementary
matricesoperations
in partition
in the
1-2ments
Case 2in- order
elementary
matrices
in partition
in -the
Number
operations necessary to calculate the self inertia 33.2. mented
matrix of
ei dalgebraic
to
calculate
the matrix
matrix
inverse:
a)
for 1-2a single- Gen
j are shown in Table 3.
2-2 order Assume
that
the block
can be divided
into
matrices ci .
2-2 order
Assume
that
the
block
matrix
can
be
divided
into
element matrix (special case) the number of operations - 1, b) - for Gen
Block dimension
3
4
5
Tablen 3 2
Sum of
of algebraic
algebraic operations
8 to
16calculate
24 the
32 positive 4
Number
operationsloIV
necessary
4
feedback matrix ei dj
3
Block dimension n
4
5
6
matrix d j . Calculated numbers of algebraic operations for var9
18
30
Sum of algebraic operations loIII
ious dimensions
of the block matrix, for3 the positive
feedback
matrix ei d j are shown in Table 3.
General
relationship
between
block
dimensionofofthe
the block
block
General
relationship
between
block
dimension
matrix
number
algebraicoperations
operationsthat
thatare
are needed
needed
matrix
andand
the the
number
of of
algebraic
for the
calculation
of the
positive
feedbackmatrix
matrixeedi dj (10)
(10) is
is
for the
calculation
of the
positive
feedback
i j
as
follows
(assuming
that
matrix
d
was
calculated
in
previous
i
as follows (assuming that matrix di was calculated in previous
step):
step):
(n − 2)(n − 1)
(15)
loIII = 3
(15)
2
for nfor=n = 3, 4, …
3, 4, . . ..
The number
of algebraic
operations
The number
of algebraic
operationsneeded
neededininorder
order to
to determine
selfself
inertia
matrices
(11)
termine
inertia
matrices
(11)isisassociated
associatedwith
with following
following
−1 T T
calculations:
productofofmatrices
matrices bbiai–1
b
;
matrix
a
inversion
b
;
matrix
ai invercalculations:
thethe
product
i
i
i
i
case
by division
by thebyelement
of this matrix);
the sumsion(in
(inthis
this
case
by division
the element
of this matrix);
mation in thein
inner
inversioninversion
of the internal
expression
the summation
thebrackets;
inner brackets;
of the
internal
(in parentheses)
and multiplying
it by the matrices
bi and
aiT ,
expression
(in parentheses)
and multiplying
it by the
matriexpressions
in external
parentheses
ces subtraction
bi and aTi , of
subtraction
of contained
expressions
contained
in exterand
its
inversions.
Calculated
numbers
of
algebraic
operations
nal parentheses and its inversions. Calculated numbers
of alfor different block dimensions n of block matrix, necessary for
gebraic operations for different block dimensions n of block
calculation of self inertia matrices, are shown in Table 4.
matrix, necessary for calculation of self inertia matrices, are
shown in Table 4.
Table 4
General
relationship
dimension
of the
block
Number
of algebraicbetween
operationsblock
necessary
to calculate
the self
matrix and the number ofinertia
algebraic
operations
that are needed
matrices
ci
for the calculation
of the
2
3 ci (11)
4 is 5as folBlock dimension
n self inertia matrices
lows:
Sum of algebraic operations loIII
8
loIV = 8(n − 1)
16
24
32
(16)
General relationship between block dimension of the block
for nmatrix
= 1, 2,
. . .the
. number of algebraic operations that are needed
and
The total amount of algebraic operations needed to calculate
the inverted form of block matrix of the block dimension n
and 856
assuming that all matrices are one-piece, is given by the
following formula:
lo = 3
n2
n
+ 19 − 10
2
2
(17)
block matrices with block dimension 2 the number of operations - 15,
c) - for matrices with block dimension 3 the number of operations 32, d) - for matrices with block dimension 4 the number of operations
- 52
Fig. 1. The number of algebraic operations required to be implemented
in order to calculate the matrix inverse: a) – for a single-element matrix
(special case) the number of operations – 1, b) – for block matrices
with block dimension 2 the number of operations – 15, c) – for matrices with block dimension 3 the number of operations – 32, d) – for
matrices with block dimension 4 the number of operations – 52
3.2. Case 2 – elementary matrices in partition in the 1‒2–2‒2
order. Assume that the block matrix can be divided into elementary
matrices, so could be present at the block main diagonal as
Fig. 2. Matrix partitioning into an 1-2-2-2 order: a) - block dimento:
one-piece
matrix, 2£2
square
sions of submatrices,
b) - dimensional
signs assignment
toand
the symmetric
submatricesmatrices. The effect of such division of input matrix into elementary
matrices, is that in the first line rectangular matrices (1£2 dimenTable 5
sional) and in the first column rectangular
matrices (with dimenNumber
of
algebraic
operations
necessary
calculate
the leading
sion 2£1) are located. Such a divisionto of
the block
matrixelement
into c0 .
n symmetric
2
3
4
5
one-piece elementary andBlock
2 by 2dimension
dimensional
matrices
will Inversion
be calledofthe
division insubmatrices
the 1‒2–2‒2
block45matrix
4-elementary
a−1
15The30
60
j order.
T and
3) fulfils
the aboveofmentioned
conditions,
partition
Multiplication
triple matrices
b j a−1
9 its 18
27 into
36
j bj
elementary matrices Additions
in 1‒2–2‒2
order is shown
/ Subtractions
1 in Fig.
2 2. 3
4
a)
Inversions / Divisions
Sum of algebraic operations loI
b)
1
26
1
51
1
76
1
101
elementary matrices, so could be present at the block main diagonal as to: one-piece matrix, 2 × 2 dimensional square and
symmetric matrices. The effect of such division of input matrix into elementary matrices, is that in the first line rectangular
matrices (1 × 2 dimensional) and in the first column rectangular matrices (with dimension 2 × 1) are located. Such a division2.ofMatrix
the block
matrix
into
one-piece
elementary
2 by 2
Fig.
partitioning
into
an 1‒2–2‒2
order:
a) – block and
dimensions of submatrices,
b) –matrices
signs assignment
the submatrices
dimensional
symmetric
will betocalled
the division in
the 1-2-2-2 order. The block matrix (3) fulfils the above mentioned conditions, and its partition into elementary matrices in
1-2-2-2 order is shown in FigureBull.
2. Pol. Ac.: Tech. 64(4) 2016
Unauthenticated
The number of algebraic operation
necessary to calculate
Download Date | 7/28/17 10:28 PM
the leading element c0 (7) is determined by block dimension n
of input matrices, but also by dimensions of matrices in triple
for n = 1, 2, . . . .
Calculated numbers of algebraic operations needed for a determination of the negative feedback matrix di (8), in relation
Inversion of selected structures of block matrices of chosen mechatronic systems
to different block dimensions of input matrices is presented in
Table 6. It is worth to underline, that increase of block dimenonecalculated
results innumber
appearing
of additional
negative
feedThe number of algebraic operation necessary to calculatesion byThe
of algebraic
operations
needed
to
(1
×
2
and
2
×
1
dimensional)
in
the
first
row
back
matrices
d
i
the leading element c0 (7) is determined by block dimension n obtain the positive
couplings matrices ei dj (10), upon different
first column
of the
block
matrix.
of input matrices, but also by dimensions of matrices in tripleand block
dimensions
of inverted
input block
matrix,
is shown in Table 7.
–1 T
Inversion
of
selected
structures...
bj aj bj . The number of algebraic operations necessary to cal- The overall relationship between the block dimension n of a
culate matrices bj aj–1bjT results from two rectangular matricesblock matrix (partitioned into submatrices
Table 7
in the 1-2-2-2 order)
Table 6
Table 8 to calculate the positive
Number
of algebraic
operations
necessary
multiplications (with dimensions 1£2 and 2£1) with squareand the
number
of
algebraic
operations
needed
to calculate
the
Number of algebraic operations necessary to calculate the negative feedback
Number of algebraic operations necessary to calculate
the self inertia
feedback matrix ei dj
matrix (with dimension 2£2).
Furthermore
it
is
necessary
to
(8),
is
as
follows:
negative
feedback
matrix
d
matrix di .
matrices ci .
i
Fig. 3.
inverse
ber of
the nu
sion 3
dimen
and th
matric
for n =
calculate the inverse form of 4 elementary aj–1 matrix. The calBlock dimension n
Block dimension n
2
3
4
5
Block dimension n 3 2 4 3 5 4 6
5
The
culation effort which should be carried
out for leading elements
23(n −bi1)
(19) 96
oII =
Inversion
of 4-elementary
submatrices
aj–1bTi 1524 45 48 90 72150
Inversion of 4-elementary submatrices a−1
15
30 45 60
Multiplication
of ltriple
matrices
a−1
j
i
verse
c0 calculations is summarized in Table
5. Inversion of selected structures...
−1
−1
Addition with c0
1
2
3
4
Multiplication of triple matrices −c0 bi ai
9 Inversion
18 27 of selected
36
forstructures...
n Matrix
= 2, 3,multiplication
... .
8 24 48
80
¡ai–1biT −1
2-2 or
−1 T
Inverting
of
matrix
(c
1
2
3
4
+
b
a
b
)
Sum of algebraic operations
l
23
46
69
92
i
oII
i
i
0
Table 5
TheMatrix
calculated
number
of
algebraic
operations
needed
to
obTable
6
Table
8
T −1 b 4 8 12 16 24 2440
dj −1Table
multiplication
Table 6
Calculation
of matrix
beTii (c
+ bi a8−1
i
Number ofoperations
algebraic operations
necessary to negative
calculatefeedback
the leading tainNumber
i bi ) dcalculate
0matrices
Number
of
operations
necessary
self
the
positive
couplings
upon
different 32
j (10), the
Inversion
structures...
Number of
of algebraic
algebraic operations necessary
necessary to
to calculate
calculate the
the negative
feedbackof selected
Number
of algebraic
algebraic
operations
necessary eto
toi calculate
the
self inertia
inertia
Subtractions
4
8
element
c
matrix
cci ..
27 81in 162
of algebraic
loIII matrix,
0
blockSum
dimensions
ofoperations
input matrices
block
is shown
Table12270
7. 16
matrix ddii ..
matrices
i
Table 7
Inversion
of
4-elementary
submatrices
15
30
45
Block
22
33 2 44 3 55 4
dimension
nn8
22 of 33a block
44 matrix
55 60
Table n6n to calculate
Table
5
Block
dimension
n dimension
Overall relationship Block
between
dimension
Rel
Block
dimension
Block
dimension
Number
of algebraic
operations
necessary
the positive feedback
Sum
of
algebraic
operations
l
53
106
159
212
−1dimension
T oIV
Overall
relationship
between
of48the
a block
matrix
Number of
algebraic operations
necessary
negative
feedback
Number of
algebraic
operations
to calculate
the
self
inertia
Inversion
of
submatrices
aa−1
15
60
Multiplication
of
triple
matrices
bbi aanecessary
24
72
96
−1
−1 border)
–1 30 the 45
T
matrix
ej i d to
. calculate
(partitioned
into
following
1-2-2-2
and
number
of
i
i
neede
j
Inversion
of 4-elementary
4-elementary
submatrices
15
30
45
60
Multiplication
of
triple
matrices
b
24
48
72
96
15
30
45
60
Inversion
of 4-elementary
submatrices
a
i i
j
i corder)
j di .
matrix−1
matrices
i.
(partitioned into following
1‒2–2‒2
and2 the number
of
Addition
cc−1
1the
33
Multiplication
−c
99
18 27 36
i a −1
0b
algebraic operations needed
towith
calculate
e44i d j verted
0−1
Addition
with
1 submatrices
2
Multiplication of
of triple
triple matrices
matrices
−c
–1
T 3 18 4 27 536
Block
dimension
n
6
0 bi aii
0
algebraic
operations
needed
to
calculate
the
submatrices
e
9
18
27
36
Multiplication
of
triple
matrices
b
a
b
−1
−1
T
Block
dimension
n
2
3
4
5
Block
dimension
n
2
3
4
j
i
j
j
of
+ bbi aai−1 bbiT ))
1
22
33
44 dj 5 the in
Sum
of algebraic
lloII −123
is asInverting
follows:
Inverting
of matrix
matrix (c
(c0−1
Sum
algebraic operations
operations
23 1546
46 4569
69 9092
92 150 (10),(10),
i
Inversion
of of
4-elementary
submatrices
oII a j −1
i −1 T1
0 +
is
as
follows:
−1
−1 Ti −1
T
Inversion
of
4-elementary
submatrices
a
15
30
45
60
Multiplication
of
triple
matrices
b
a
b
24
48
72
96
j
16
24
32
+ bi ai−1 biT ) −1 bi i i i8
AdditionsMatrix
/ Subtractions
124 2 48 3 80 4 Calculation
sented
Calculation of
of matrix
matrix bbiTi (c
(c0−1
16
24
32
multiplication −a−1 bT
8
0 + bi ai bi ) bi −18
Addition
with
c
1
2
3
4
Multiplication of triple matrices −c0i bi ai−1
9
18
27
36
(n − 2)(n −01)
Subtractions
44
88
12
16
i
Subtractions
12
16
matrix
Inversions / Divisions
Matrix multiplication ei d j
4 112 1 24 1 40 1
loIII
= 27 (c−1 + bi a−1 bT ) (20)
Inverting
of matrix
130 2 45 3(20)
4
Sum of algebraic
operations
loII
23
46 69 92
Table
77
i 15
i
0
Inversion
of
4-elementary
submatrices
60
2
Table
Inversion
of
4-elementary
submatrices
15
30
45
60
mensi
of algebraic
operationscalculate
loIII
27 positive
81 51feedback
162 76 270
−1
−1 T −1
Number
algebraic
operations
necessary
ofSum
algebraic
operations
26
101
Calculation
ofofmatrix
bTi (coperations
8106 16159 24212 32
bi ) b53
i
Number of
of Sum
algebraic
operations
necessaryloIto
to calculate the
the positive
feedback
0 + bi ai loIV
Sum
algebraic
for
n
=
3,
4,
.
.
.
.
Sum
of
algebraic
operations
l
53
106
159
212
oIV
for n = 3, 4, …
matrix
matrix eeii dd jj ..
Subtractions
4
8
12
16
The
number
of
algebraic
operations
forfordifferent
3.3. C
The
number
of
algebraic
operations
differentblock
block diTablen7 3
Block
dimension
4
5
6
Inversion
of
4-elementary
submatrices
15
30
45
60
Block dimension
n
3block4 dimension
5
Generally,
relationship
between
n6 feedback
of
blockmensions
mensions
n block
of block
matrix
(resultingfrom
fromthe
thepartitioning
partitioning in
n
of
matrix
(resulting
order
matrix
with
its
partition
into
elementary
matrices
with
dimenNumber
of
algebraic
operations
necessary
the
positive
−1 to calculate
Inversion
of
submatrices
aa−1
15
45
90
150
Sum of algebraic operations loIV
53 106 159
212
j
Inversion
of 4-elementary
4-elementary
submatrices
45matrices
90 with
150 dimen-1-2-2-2
j eiand
matrix
d j . 15numbers
with its
partition
into
1‒2–2‒2
order)
of self
the inertia
self inertia
matrices
ci (11),
are
shown
(11),
are
shown
in
order)
of
the
matrices
c
menta
sionsmatrix
according
to
1-2-2-2
order,
of
algebraic
op−1 elementary
i
T
Matrix multiplication
−a
biT
88
24
48
multiplication
−ai−1
24
48 of80
80
sionsMatrix
according
to calculate
1‒2–2‒2
and
numbers
algebraic
in
Table
8.
i
i border,
Table
8.
and sq
erations
necessary
to
the
leading
elements
c
(7),
is
Block dimension
n4
312 424 5 400 6
Matrix
multiplication
eei dd j
Matrix
multiplication
4 leading
12
24
40 c0 (7),
i j
operations
necessary
to
calculate
the
elements
−1
onal.
The
overall
relationship
between
the
block
dimension
n
of
as follows:
Inversion
of
4-elementary
submatrices a j 27 15
45
90
150
Sum
of
Sum
of algebraic
algebraic operations
operations lloIII
27 81
81 162
162 270
270
is as
follows:
Table
8
oIII −1 T
rical.
Matrix multiplication −ai bi
8
24
48
80 the block matrix (partitioned in the following 1-2-2-2 order)
Number of algebraic operations necessary to calculate the self
(18)
=
25(n
−
1)
+
1
(18)
Matrixlmultiplication
e
d
4
12
24
40
oI
i j
inertia matrices ci
Sum
of algebraic
operations loIIImatrices
27
81
162
270 Bull. Pol. Ac.: Tech. XX(Y) 2016
matrix
with
its
partition
into
elementary
with
dimenmatrix
its2,partition
into elementary matrices with dimenfor with
nfor
=n = 1, 2, …
1,
... .
3 to4 calculate
5
Block
n of algebraic operations2required
Fig.
3. dimension
The number
the
sions
to
1-2-2-2
order,
and
of
opCalculated
numbers
of algebraic
operations
neededfor
for
sions according
according
tonumbers
1-2-2-2
order,
and numbers
numbers
of algebraic
algebraic
op-aa deCalculated
of algebraic
operations
needed
de- inverse
matrix: of
a)triple
- for amatrices
single-element
case) 96
the num24 (special
48
72
Multiplication
bi ai–1biT matrix
erations
necessary
to
calculate
the
leading
elements
c
(7),
is
termination
of calculate
the
negative
feedback
matrix
di i(8),
(8),
in relation
relation
erations
necessary
to
the
leadingmatrix
elements
c00 in
(7),
is
termination
of
the
negative
feedback
d
ber
of
operations
1,
b)
for
block
matrices
with
block
dimension
2
matrix
with its block
partition
into elementary
matricesiswith
dimenas
to different
dimensions
of
inputmatrices
matrices
presented
in
1
2
3
4
Addition with c0–1
as follows:
follows:
to
different
block
dimensions
of
input
is
presented
in
the
number
of
operations
102,
c)
for
matrices
with
block
dimensionsTable
according
1-2-2-2
order, and
of
algebraic
opis to
worth
to underline,
thatnumbers
increaseof
ofblock
block dimendimenTable 6. necessary
It6.isItworth
to calculate
underline,
that
increase
sion
3 theofnumber
of0–1operations
2
3 with
4 block
+bi ai–1biT ) - 230, d) -1 for matrices
Inverting
matrix (c
erations
to
the
leading
elements
c
(7),
is
0
sion by one results
in
appearing
of
additional
negative
feedback
l
=
25(n
−
1)
+
1
(18)
of
operations
385
loI
= 25(n
− 1) +of1 additional negative
(18)feed- dimension 4 the number
sion
by
one
results
appearing
oI in
as follows:
16
24
32
Calculation of matrix biT (c0–1 +bi ai–1biT )–1bi 8
matrices di (1£2 and 2£1 dimensional) in the first row and first
(1
×
2
and
2
×
1
dimensional)
in
the
first
row
back
matrices
d
i
for
n
=
1,
2,
.
.
.
.
column
for n = 1,
2, . . . .of the inverted block matrix.
Fig.
3.
The
number
of
algebraic
operations
required
to
calculate
the
4needed
8 calculate
16
Fig. and
3.Subtractions
The
of of
algebraic
operations
required
to
the c
and first column ofofthe
block
thenumber
number
algebraic
operations
to12the
calculate
Calculated
i
loIinverted
= 25(noperations
− 1) +matrix.
1 needed
matrix:
a)
for
a
single-element
matrix
(special
case)
numCalculated numbers
numbers of algebraic
algebraic
operations
needed for
for aa dede-(18)inverse
inverse
matrix:
a)
for
a
single-element
matrix
(special
case)
the
numThe overall
relationship
between
the
block
dimension
n
of
a
matrices
(11),
is
as
follows:
Inversion
of
4-elementary
submatrices
15
30
45
60
termination
of
the
negative
feedback
matrix
d
(8),
in
relation
Table
6
termination of the negative feedback
matrix
dii (8), in relation ber
ber of
of operations
operations -- 1,
1, b)
b) -- for
for block
block matrices
matrices with
with block
block dimension
dimension 22
for
nNumber
=matrix
1, 2, of
. (partitioned
.dimensions
.algebraic
.
block
into
submatrices
the
1-2-2-2
order)the
operations
necessary
toincalculate
the negative
to
different
block
of
input
matrices
is
presented
in
Fig.
3.
The
number
of
algebraic
operations
required
to
number
of
operations
102,
c)
for
matrices
with
block
dimento different block dimensions of input matrices is presented in the number
53 with
106block
159calculate
212 the
Sum ofof
algebraic
operations
operations
- 102, loIV
c) - for matrices
dimenCalculated
numbers
of feedback
algebraic
operations
needed
for a dematrix
dneeded
and
the
number
ofunderline,
algebraic
operations
to calculate
thesion inverse
i of block
=
53(n
−
1)
(21)
l
matrix:
a)
for
a
single-element
matrix
(special
case)
the numTable
6.
It
is
worth
to
that
increase
dimen3
the
number
of
operations
230,
d)
for
matrices
with
block
oIV
Table 6. It is worth to underline, that increase of block dimen- sion 3 the number of operations - 230, d) - for matrices with block
termination
of the
negative
matrixnegative
di (8),
infeedrelation
(8),
is
as follows:
negative
feedback
matrix
di feedback
ber
of
operations
1,
b)
for
block
matrices
with
block
dimension
2
dimension
4
the
number
of
operations
385
sion
by
one
results
in
appearing
of
additional
2
3
4
5
Block
dimension
n
4 the
- 385
sion by one results in appearing of additional negative feed- dimension
for The
n=
1, 2,number
. . .relationship
. of operations
tomatrices
differentdiblock
dimensions
of input matrices
is presented in the
overall
between
thematrices
block dimension
of
number
of operations
- 102,
c) - for
with blockn dimen(1
×
2
and
2
×
1
dimensional)
back
–1 in the first row
2 and 2 × 1submatrices
dimensional)
in15the30first45
row 60
back matrices
d (1of×4-elementary
Inversion
aj
The
sum
of algebraic
operations
needed
to
calculate
inblock
matrix
(partitioned
in the
following
order)
Table
6. It isi of
worth
to underline,
increase
of block dimension
3 the
number
of operations
- 230,
d) - for 1‒2–2‒2
matrices
withan
loII =block
23(nthat
− 1)
(19)and the
and
first
column
the
inverted
matrix.
the
number
of
algebraic
operations
needed
to
calculate
ccblock
ii 1-2and first column
of
the
inverted
block
matrix.
verse
block
matrix
with
block
dimension
n
(partitioned
in
–1
and
the
number
of
algebraic
operations
needed
to
calculate
dimension
4 the number
of operations
- 385
and
the number
of algebraic
operations
needed to calculate ci
9 negative
18 27 afeed36
Multiplication
of triple
matrices ¡c
sion
by
one
results
in appearing
of0badditional
i ai
The
relationship
(11),
is
as
follows:
The
overall
relationship
between the
the block
block dimension
dimension nn of
of a matrices
foroverall
n=
2, 3,
. . .d. (1 × 2between
2-2 order)
can
be
obtained
by the following expression:
matrices
(11),
is
as
follows:
matrices
(11),
is
as
follows:
and
2
×
1
dimensional)
in
the
first
row
back
matrices
i
block
into
in
order)
23 46
69 to92
Sum(partitioned
of algebraic
operations
loII
block matrix
matrix
(partitioned
into submatrices
submatrices
in the
the 1-2-2-2
1-2-2-2
order)
number
of algebraic
operations
needed
obandThe
firstcalculated
column
of
the inverted
block
matrix.
and the number of lalgebraic
ci
and
the
number
of
algebraic
operations
needed
to
calculate
the
= 53(n
−
1)
(21)
n needed to calculate
n2 operations
and tain
the number
of algebraic
operations
needed
to
calculate
the
the
positive
couplings
matrices
e
d
(10),
upon
different
53(n
−121
1)(21)
(21)(22)
lloIV
i block
j
oIV==27
The
overall
relationship
between
the
dimension
n
of
a
+
−
73
matrices
(11),
is
as
follows:
o
is
as
follows:
negative
matrix
ddii (8),
(8),block
is between
as matrix,
follows:
negative
feedback
matrix
2
2
blockfeedback
dimensions
of
input
shown
in Table
The overall
relationship
theisin
block
dimension
n7.offor n = 1, 2, . . . .
block
matrix
(partitioned
into submatrices
the 1-2-2-2
order)
for n = 1, 2, . . . .
a blockrelationship
matrix (partitioned
into
submatrices
inblock
the 1‒2–2‒2
for
n = 1, 2, …
Overall
between
dimension
of
a
matrix
Relations
showing
the
number
of algebraic
operations
of
operations
needed
an
and the number of algebraic
operations
needed to calculate
the The
The sum
sum
of algebraic
algebraic
operations
needed
to calculate
calculate
an inin= 53(n
−needed
1) to
(21)
loIV
lloII
= 23(n
−
(19)
order) andinto
the following
number
algebraic
operations
needed
to calcuThe
sum
of
algebraic
operations
to
calculate
an
23(n
− 1)
1) order)
(19)
(partitioned
1-2-2-2
and
the
number
of
needed
to
be
done
to
calculate
the
elementary
matrices
of inoII = of
verse
block
matrix
with
block
dimension
n
(partitioned
in
1-2negative feedback matrix di (8), is as follows:
verse
block
matrix
with
block
dimension
n
(partitioned
in
1-2late the
negative feedback
di (8), the
is assubmatrices
follows:
block
with block
dimension
n (partitioned
in n of
for
n =block
1, 2,
. matrix
.matrix
..
algebraic
operations
needed matrix
to calculate
ei d j2-2 inverse
verted
(with
an increase
ofexpression:
block
dimension
for
order)
be
by
for nn =
= 2,
2, 3,
3, .. .. .. ..
2-2 1‒2–2‒2
order) can
can
be obtained
obtained
by the
the following
following
expression:
order)
can
be
obtained
by
the
following
expression:
The
sum
of
algebraic
operations
needed
to
calculate
in(10), is as follows:
be represented graphically, asanpreThe
algebraic
needed
= 23(noperations
− 1)(19)
The calculated
calculated number
number of
ofloII
algebraic
operations
needed to
to obob-(19) the input block matrix) can
2
verse
matrix
n that
(partitioned
1-2nn
nn2blockbedimension
sentedblock
in Figure
3. with
It should
considered
the inputinblock
tain
positive
couplings
eiidd jj (10),
upon different
tain the
the for
positive
couplings matrices
matrices
(10),
2-2 order) can be
(22)
n = 2, 3, …
lloo =
+
121
−
73
(22)
27
(n −e2)(n
− 1) upon different
+
121
−
73
(22)
=
27
for
n
=
2,
3,
.
.
.
.
obtained
by
the
following
expression:
matrix
has
been
partitioned
into
elementary
matrices
with
di22
22
block
loIIIblock
= 27 matrix,
block dimensions
dimensions of
of input
input
block
matrix,2is
is shown
shown in
in Table
Table 7.
7. (20)
The
calculated
number
of
algebraic
operations
needed
to
obmensions
resulting
from
following
1-2-2-2
order.
Overall
Relations
of
Overall relationship
relationship between
between dimension
dimension of
of aa block
block matrix
matrix
Relations showing
showing the
the number
number
of nalgebraic
algebraic operations
operations
n2
tain
the
positive
couplings
matrices
ei dand
upon different
for nBull.
= 3,
4,
.
.
.
.
j (10),
(partitioned
into
following
1-2-2-2
order)
the
number
of
needed
to
be
done
to
calculate
the
elementary
of
inlo = 27 the+elementary
121 − 73 matrices
Pol. Ac.:
Tech. 64(4)
2016 order) and the number of
(partitioned into
following
1-2-2-2
needed
to
be
done
to
calculate
matrices
of857
in-(22)
2
2
block
dimensions
of
inputtoblock
matrix,
issubmatrices
shown
in Table
7.
Theoperations
number ofneeded
algebraic
operations
for
different
block
di3.3.
Case
3
elementary
matrices
in
partition
in
the
1-2-1-2
d
algebraic
calculate
the
e
block
matrix
(with
an
increase
of
block
dimension
nn of
algebraic operations needed to calculate the submatrices eii d jj verted
Unauthenticated
verted
block
matrix
(with
an
increase
of
block
dimension
of
Overall
between
dimension
of apartitioning
block matrix
Relations
showing
the
number
ofcan
algebraic
operations
mensions
nrelationship
of block matrix
(resulting
from the
inthe input
order
Assume
thatcan
the
block
matrix
be divided
into
ele(10),
Download
Date
|
7/28/17
10:28
PM
block
matrix)
be
represented
graphically,
as
pre(10), is
is as
as follows:
follows:
the input block matrix) can be represented graphically, as pre(partitioned
into
1-2-2-2
order)ciand
needed
to
be
done
to
calculate
the
elementary
matrices
of
in(11),thearenumber
shown of
insented
1-2-2-2 order)
of following
the self inertia
matrices
mentary
matrices
in
such
a
way
that
1-element
matrices
(1
×
1)
in
in Figure
Figure 3.
3. It
It should
should be
be considered
considered that
that the
the input
input block
block
(n − to
2)(n
− 1) the submatrices ei d jsented
algebraic
calculate
verted
blockmatrices
matrix (with
an increase
of block
n of
Table 8. operations needed
and square
(2 × 2)occur
alternately
ondimension
the main diag-
Fig. 4. Matrix partitioning into 1-2-1-2 order: a) - block dimensions
of submatrices, b) - marks assignment to the submatrices
Inv
T. Trawiński, A. Kochan, P. Kielan, and D. Kurzyk
Table 9
Number of algebraic operations necessary to calculate the leading element c0 .
Block dimension n
a)
Inversion of 4-elementary submatrices a−1
j
T
Multiplication of triple matrices b j a−1
j bj
Additions / Subtractions
Inversions / Divisions
Sum of algebraic operations loI
2
15
3
4
b)
16 31
5
32
9
10
19
20
1
1
26
2
1
29
3
1
54
4
1
57
in the first row are, alternately, rectangular matrices (of dimension 1 × 2) and 1-element matrices (1 × 1), and the first column
is the block transposition of the first row. Such partitioning of
input block matrix will be referred to the partition in the 1-21-2 order. Such conditions correspond to an exemplary block
Fig.shown
4. Matrix
into the
1‒2–1‒2
order:
a) –by
block
matrix
inpartitioning
Figure 4 and
matrix
given
(3).dimensions
Number
of submatrices, b) – marks assignment to the submatrices
of algebraic operations necessary to be implemented in order
Fig. 3. The number of algebraic operations required to calculate the in-to calculate the form of the leading element c0 (7), using the
verse matrix: a) – for a single-element matrix (special case) the numberpartitioning of input matrices into elementary matrices in the
Table 9
of operations – 1, b) – for block matrices with block dimension 2 the
1-2-1-2
order,
the block
dimension
n ofthe
input
maNumber
of depend
algebraic on
operations
necessary
to calculate
leading
number of operations – 102, c) – for matrices with block dimension 3
n = 2,c0the input block matrix is
the number of operations – 230, d) – for matrices with block dimensiontrix. If a block dimension iselement
4 the number of operations – 385
composed of four matrices (Fig.4.a) and the number of alge2
3
4
5
Block dimension n
braic operations necessary to calculate –1the leading element c0
15 16
32
4-elementary submatrices ai
(7) isInversion
the sameofas
in the case analyzed in Chapter
3.2 -31formula
Relations showing the number of algebraic operations(18). Multiplication of triple matrices bi ai–1biT
9
10
19
20
needed to be done to calculate the elementary matrices of in- Total algebraic operations will be different for larger block
Additions / Subtractions
1
2
3
4
verted block matrix (with an increase of block dimension ndimensions n of the input block matrix, and will depend on the
Inversions
/ Divisions
1 on
1 the 1main 1diof the input block matrix) can be represented graphically, asnumber
of 1- and
4-element submatrices, lying
presented in Fig. 3. It should be considered that the input blockagonal
ofofthe
input block
matrix.
computation
29
54 effort
57
Sum
algebraic
operations
loI Algebraic 26
matrix has been partitioned into elementary matrices with di-associated with the leading element c (7) is shown in Table 9.
0
mensions resulting from following 1‒2–2‒2 order.
In general, the relationship between the block dimension n
In general, the relationship between the block dimension
of
the block matrix and the number of algebraic operations,
3.3. Case 3 – elementary matrices in partition in the 1‒2–1‒2 n of the block matrix and the number of algebraic operations,
the partitionto calculate
the leading
element
c0 , using
order. Assume that the block matrix can be divided into ele-needed
needed
to calculate
the leading
element
c0, using
the partiing
into
elementary
matrices
following
the
1-2-1-2
order,order,
is as
mentary matrices in such a way that 1-element matrices (1£1) tioning into elementary matrices following the 1‒2–1‒2
follows:
and square matrices (2£2) occur alternately on the main di- is as follows:
agonal. All matrices appearing on the main diagonal are symloI = n + 24d + 2(n − d − 1)(23)
(23)
metrical. The result of this partition is that the elementary matrices in the first row are, alternately, rectangular matrices
n ∧ n = 2, 3, . . . . where
d -dnumbers
(of dimension 1£2) and 1-element matrices (1£1), and thefor dfor<d < n ^ n = 2, 3, …
; where
– numbersofof4-element
4-element subsubmatrices
on
input
block
matrix
diagonal.
The
calculation
effirst column is the block transposition of the first row. Such matrices on input block matrix diagonal.
fort
necessary
to
be
performed
in
order
to
determine
the
negapartitioning of input block matrix will be referred to the parThe calculation effort necessary to be performed in order
feedback
matrices
di (8),feedback
also in matrices
this casediwill
on
tition in the 1‒2–1‒2 order. Such conditions correspond to antive to
determine
the negative
(8), depend
also in this
dimension
the
input
matrix and
theinput
number
1exemplary block matrix shown in Fig. 4 and the matrix giventhe block
case will
depend onofthe
block
dimension
of the
matrixofand
by (3). Number of algebraic operations necessary to be imple-and the
number of
1- and 4-element
located
on itsThe
maincaldi4-element
matrices
located onmatrices
its main
diagonal.
mented in order to calculate the form of the leading element agonal. The calculated numbers of algebraic operations needed
c0 (7), using the partitioning of input matrices into elementary6 to obtain the negative feedback matrices di (8), are shown in
matrices in the 1‒2–1‒2 order, depend on the block dimension Table 10.
n of input matrix. If a block dimension is n = 2, the input
block matrix is composed of four matrices (Fig. 4a) and the
Table 10
Number of algebraic operations necessary to calculate the negative
number of algebraic operations necessary to calculate the
feedback matrix di
leading element c0 (7) is the same as in the case analyzed in
Chapter 3.2 – formula (18).
2
3
4
5
Block dimension n
Total algebraic operations will be different for larger block
–1
15 16 31 32
Inversion of 4-elementary submatrices aj
dimensions n of the input block matrix, and will depend on the
–1
number of 1- and 4-element submatrices, lying on the main
8 10 18 20
Multiplication of triple matrices ¡c0 bi ai
diagonal of the input block matrix. Algebraic computation effort
23 26 49 52
Sum of algebraic operations loII
associated with the leading element c0 (7) is shown in Table 9.
858
Bull. Pol. Ac.: Tech. 64(4) 2016
Unauthenticated
Download Date | 7/28/17 10:28 PM
M
culate
negati
The
(partit
numb
tive fe
for d <
matric
algebr
back m
the in
subma
tain th
2-elem
matric
form s
lation
part, w
kind o
trices
two 4
subma
mensi
ces ei
are sh
The
dimen
follow
The
occurr
under
trix (p
der),
so-cal
numb
matrix
eleme
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nmn
of
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er
er
he
he
ais
ec0
la
ck
he
diort
9.
n
ns,
nas
3)
befaon
1al-
Multiplication of triple matrices −c0 bi ai
Sum of algebraic operations loII
8
23
10
26
18
49
20
52
of selected
of block
culated numbers of algebraic Inversion
operations
neededstructures
to obtain
the matrices of chosen mechatronic systems
negative feedback matrices di (8), are shown in Table 10.
The relationship between the dimension of the block matrix
The relationship between the dimension of the block matrix under increasing block dimension n of the input block matrix
(partitioned into blocks in the following 1-2-1-2 order) and the
(partitioned into blocks in the following 1‒2–1‒2 order) and the (partitioning into elementary matrices in the 1‒2–1‒2 order),
number
of algebraic
tocalculate
calculatethethe
nega- increases accordingly with sequence described by the so-called
number
of algebraicoperations
operationsneeded
needed to
negative
tive feedback
feedbackmatrices
matricesdid(8),
is
as
follows:
i (8),
is as follows:
third diagonal of Lozanic triangle [15]. Successive numbers
(24)
loII = 23d + 3(n − d − 1)(24)
for dfor<d < n ^ n = 2, 3, …
n ∧ n = 2, 3, . . . . where
d -dnumbers
; where
– numbersofof4-element
4-element subsubmatrices
on
the
input
block
matrix
diagonal.
The
number of
matrices on the input block matrix diagonal.
algebraic
determine
the positive
feedTheoperations
number ofrequired
algebraicto
operations
required
to determine
backthematrices
block
dimension
n of
positiveefeedback
matriceson
ei dthe
depend
on the block
i d j (10) depend
j (10)
n ofmatrix
the input
andof
the1-numbers
of 1- and
the dimension
input block
andblock
the matrix
numbers
and 4-element
4-elementlying
submatrices
the main As
diagonal.
As a result,
submatrices
on the lying
mainon
diagonal.
a result,
we obobtain
threeoftypes
of calculated
matrices,
namely:1-element,
1-element,
tain we
three
types
calculated
matrices,
namely:
2-element
(rectangular),
4-element(square
(squareand
andsymmetrical)
symmetrical)
2-element
(rectangular),
4-element
matrices.
Analyzing
the
relationship
(10)
and
the
block matrix
matrix
matrices. Analyzing the relationship (10) and the block
form
shown
in
Fig.
4,
it
can
be
concluded
that
in
the
calculation
form shown in Figure 4, it can be concluded that in the calcuof the 1-element matrices only 1-element matrices take part,
lation of the 1-element matrices only 1-element matrices take
while in the calculation of the 2-element matrices two kind
part,ofwhile
in the calculation of the 2-element matrices two
elementary matrices: 1-element and 4-element submatrices
kindare
of involved.
elementary
1-element
4-element
submaIn matrices:
the calculation
of the and
4-element
matrices
two
trices
are
involved.
In
the
calculation
of
the
4-element
matrices
4-element submatrices, one 1-element and two 2-element subtwo matrices
4-element
1-element between
and two the
2-element
aresubmatrices,
involved. Theone
relationships
dimensubmatrices
are
involved.
The
relationships
between
the disions of the matrices creating the positive feedback matrices
mensions
of the
the positive
feedback
matriei dj (10),
andmatrices
a numbercreating
of its internal
elements
generated
are
in Table
ces shown
ei d j (10),
and a11.
number of its internal elements generated
are shown in Table 11.
Table 11
The number of algebraic operations,
with a different block
Relationship
between
dimensions
–
numbers
elements
of positive
dimension n of block matrix (resulting fromof the
partitioning
in
feedback matrices ei dj and dimensions of input matrices
following 1-2-1-2 order), are shown in Figure 5.
2 and
3 columns,
4
5
n
TheBlock
sumdimension
of all 2-element
matrices (in rows
–1
15 block
16 31matrix)
32
Inversion
of 4-elementary
submatrices
occurring
above
the main diagonal
ofainverted
j
underMultiplication
increasing of
block
dimension n of the 8input
ma10 block
18 20
triple matrices ¡c0 bi ai–1
trix (partitioning into elementary matrices in the 1-2-1-2 or26 49 by 52
of algebraic
operationswith
loII sequence 23
der), Sum
increases
accordingly
described
the
so-called third diagonal of Lozanic triangle [15]. Successive
numbers
which
represents
block
dimension
n a different
(of input block
The
number
of algebraic
operations,
with
block
dimension
n of block
matrix (resulting
fromthe
the sum
partitioning
in
matrix)
correspond
a number,
representing
of all 2following
1‒2–1‒2
order),
are shown
Fig. 5. to the funcelement
positive
feedback
matrices
ei d j in
(according
The sum of all 2-element matrices (in rows and columns,
occurring above the main diagonal
of inverted block matrix)
Bull. Pol. Ac.: Tech. XX(Y) 2016
which represents block dimension n (of input block matrix)
correspond a number, representing the sum of all 2-element
positive feedback matrices ei dj (according to the function f 2(n))
are shown in Table 12.
Table 12
Number of algebraic operations necessary to calculate the positive
feedback matrix ei dj
Block dimension n
Fig. 5, case 1 (1-element)
ai–1biT c0 bj aj–1
Fig. 5, case 2 or 3 (2-element) ai–1biT c0 bj aj–1
Fig. 5, case 4 (4-element)
ai–1biT c0 bj aj–1
3
4
5
6
0
0
6
6
27
54
108 162
0
51
51
153
27 105 165 321
Sum of algebraic operations loIII
The sum of all 2-element matrices (in rows and columns,
occurring above the main diagonal of inverted block matrix)
under increasing block dimension n of the input block matrix
(partitioning into elementary matrices in the 1‒2–1‒2 order),
increases accordingly with sequence described by the so-called
third diagonal of Lozanic triangle [15]. Successive numbers
which represents block dimension n (of input block matrix)
correspond to a number representing the sum of all 2-element
positive feedback matrices ei dj (according to the function f2(n))
are shown in Table 13.
Table 13
Number of 2-element positive feedback matrices eidj
n
3
4
5
6
7
8
9
10
11
12
13
…
f 2(n)
1
2
4
6
9
12
16
20
25
30
36
…
The sum of all 4-element submatrices (lying above the main
diagonal) due to a increase of the block dimension n of the input
block matrix (partitioning in 1-2-1-2 order), increases following
a numerical sequence described by repeated triangular numbers
[12-14]. Successive numbers which represents block dimension
n corresponds a number, representing the sum of all 4-element
positive feedback matrices ei dj , according to function f4(n)
shown in Table 14.
Table 14
Number of 4-element positive feedback matrices eidj
Fig. 5. Relationship between dimensions and number of elements of
positive feedback matrices ei dj and dimensions of input matrices
n
3
4
5
6
7
8
9
10
11
12
13
…
f 4(n)
0
1
1
3
3
6
6
10
10
15
15
…
The sum of all 1-element matrices due to increase of the
block dimension n of the input block matrix (partitioning in
859
Bull. Pol. Ac.: Tech. 64(4) 2016
Unauthenticated
Download Date | 7/28/17 10:28 PM
Table 15
Number of 1-element positive feedback matrices ei d j .
n
f1 (n)
3
0
4
0
5
1
6
1
7
3
8
3
9
6
10
6
11 12 13 . . .
T.
P. Kielan, and D. Kurzyk
10 Trawiński,
10 15A. Kochan,
...
1‒2–1‒2 order), increases
accordingly
with a numerical seTable
16
Number
algebraic operations
necessary
to calculate
the self inertia
quenceofdescribed
by repeated
triangular
numbers
[12‒14].
matrices ci .the block dimension n correSuccessive numbers representing
a)
b)
c)
sponds a number,
representing
Block dimension
n
2 the sum
3 of4all 1-element
5
6 positive
7
of selected
structures...
feedback
matricesmatrices
ei dj , according
to function
f1(n) are
4-element
53 53
106
106
159 shown
159
of
nfofselected
ofselected
selected
structures...
structures...
structures...
in Table 15.
1-element matrices
8
8
16
16
24
ck
Table 15
Sum of algebraic
operations
loIV Table
53
114 matrices
122 e175
183
Number
of 1-element
positive
Table
Table
151561
15feedback
id j.
Table
15feedback
Number
Number
Number
ofof1-element
of1-element
1-element
positive
positive
positive
feedback
feedback
matrices
matrices
matrices
eiedi jde.ijd. j .
n Number
3 4 of 1-element
5 6
7positive
8 feedback
9 10 matrices
11 12e d 13 . . .
i j
n n 3 3 03 4 4 04 5 5 15 6 6 16 7 7 73 8 8 83 9 9 96 101010
f1n(n)
6 111111
10121212
10131313
15. .... . . . .
sivef1numbers
representing
the
block
dimension
n
corresponds
f(n)
f
(n)
(n)
0
0
0
0
0
0
1
1
1
1
1
1
3
3
3
3
3
3
6
6
6
6
6
6
10
10
10
10
10
10
15
15
15
... . . .
3
4
5
6
7
8
9 10 11 12 13 . .…
1 1 n
a number, representing the sum of all 1-element positive feed1
1
3 to function
3
6
6f (n)
10 are
10 shown
15 …in
f 4(n) 0 e d0 , according
back matrices
Table 16
i j
1
Number of algebraic operations
necessary
to calculate the self inertia
Table
Table
Table
161616
Table 15.
2
matrices
c
.
Number
Number
Number
of
of
algebraic
of
algebraic
algebraic
operations
operations
operations
necessary
necessary
necessary
to
to
calculate
to
calculate
calculate
the
thethe
self
self
self
inertia
inertia
inertia
i
The The
overall
relationship
between
the block
3
overall
relationship
between
blockdimension
dimension nn of
of
matrices
matrices
matrices
c2ic. i .ci .the
Block
dimension
n
3
4
5
6
7
the the
input
block
matrix
(partitioning
1
input
block
matrix
(partitioninginin1-2-1-2
1‒2–1‒2order)
order) and
and the
the
Block
Block
Block
dimension
dimension
dimension
n
n
n
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
4-element
matrices
53
53
106
106
159
159
ents of number
of algebraic
operations
needed
number
of algebraic
operations
neededtotocalculate
calculatethe
the positive
positive
4-element
4-element
4-element
matrices
matrices
matrices
5353follows:
53
535353
106
106
106
106
106
106
159
159
159
159
159
nts
ents
s ofofoffeedback matrices
es
1-element
8
8
16
16 159
24
e
d
(10),
is
as
feedback matricesi eji dj (10), is as follows:
ces
s
d)
Tomasz Trawiński, Adam Kochan, Paweł K
1-element
1-element
1-element
matrices
matrices
matrices
- 8 8 61
8 8 8114
8 1616122
16 161616
Sum of algebraic
operations
loIV - - 53
175 242424
183
Sum
Sum
Sum
ofofalgebraic
ofalgebraic
algebraic
operations
operations
operations
loIV
loIV
loIV 535353 616161 114
114
114 122
122
122 175
175
175 183
183
183
(25) Fig. 6. The number of algebraic operations required to calculate the
loIII = 51 f4 (n) + 27 f2 (n) + 6 f1 (n)(25)
inverted matrix: a) – for an input block matrix with block dimension
dback
.
nfor=n = 3, 4, …
3,
4, . . . . representing the block dimension n correspondsn = 2 the number of algebraic operations – 102, b) – for block matrices
numbers
edback
ack
back for sive
.
Calculated
number
ofof the
algebraic
operations
for
different
sive
sive
numbers
numbers
numbers
representing
representing
representing
the
the
the
block
block
block
dimension
dimension
dimension
ncorresponds
ncorresponds
corresponds
asive
number,
representing
sum
of
all
1-element
positive
feed-with n = 3 the number of operations – 143, c) – for matrices with n = 4
Calculated
number
algebraic
operations
forndifferent
block
6
the number of operations – 322, d) – for matrices n = 5 the number
dimensions
ofj , input
block
(partitioning
inshown
1-2a anumber,
adimensions
number,
number,
representing
representing
representing
the
the
the
sum
sum
sum
ofof
of
allall
all
1-element
1-element
1-element
positive
positive
positive
feedfeedfeedaccording
tomatrix
function
f1 (n)
are
in
back
matrices
n ofeni dinput
block
matrix
(partitioning
in
1‒2–1‒2
6 666 block
of operations – 396
order),
required
obtain
inertia
cshown
,djto
,according
,according
according
toself
toself
to
function
function
function
f1f(n)
f(n)
are
are
shown
shown
ininin
back
back
back
matrices
matrices
matrices
eiediedjito
i (11),
jobtain
1matrices
1 (n)
order),
required
thethe
inertia
matrices
care
are
Table
15.
6 162
6 6 1-2
i (11),
are
shown
in Table
Tablerelationship
16.
Table
Table
Table
15.
15.
15.
shown
in
16.
The
overall
between the block dimension n of
The
overall
relationship
between
the
block
dimension
The
The
The
overall
overall
overall
relationship
relationship
relationship
between
between
between
the
the
the
block
block
block
dimension
dimension
dimension
nof
nofthe
of
the
input
block
matrix (partitioning
in
1-2-1-2
order) nand
verted input block matrix (according to the growth of block
Table
16
the
input
block
matrix
(partitioned
in
1-2-1-2
order)
and
the
the
the
the
input
input
input
block
block
block
matrix
matrix
matrix
(partitioning
(partitioning
(partitioning
in
in
in
1-2-1-2
1-2-1-2
1-2-1-2
order)
order)
order)
and
and
the
the
number
of
algebraic
operations
needed
to
calculate
the
positive
21
321
321
Number
of
algebraic
operations
necessary
to
calculate
the
self
dimension) can be represented graphically in Fig. 6. It should
number
of
algebraic
operations
needed
calculate
self
inertia
number
number
number
of
of
algebraic
of
algebraic
algebraic
operations
operations
operations
needed
needed
to
to
calculate
to
calculate
calculate
the
the
the
positive
positive
positive
feedback matrices ei d j (10), is as follows:
inertia matrices ci
be considered that the input block matrix is partitioned into
matrices
cimatrices
(11),
is as
feedback
feedback
feedback
matrices
matrices
eiedifollows:
edji dj(10),
(10),
(10),
is
is
as
is
as
follows:
as
follows:
follows:
j
elementary matrices with dimensions consistent with the result
2
3
4
5
6
7
Block dimension
n
Fig.
(25)6. The number of algebraic operations required to calculate the
nloIII =n51 f 4 (n) + 27 f 2 (n) + 6 f 1 (n)
+
8(
−
1)
for
n
=
2,
4,
6,
8,.
.
.
53
of 1‒2–1‒2
partition
2
2
inverted
matrix: a)
- for anorder.
input block matrix with block dimension
==51
=5151
f4f(n)
++27
+2727
f2f(n)
++6+6106
f16f(n)
(25)
(25)
(25)
loIII
loIII
loIII
4f(n)
4 (n)
2f(n)
2 (n)
1f(n)
1 (n)
l4-element
matrices
53
53
106 159 (26)
159
oIV =
ns,. . .
for
n = 3, 4,53. .n−1
. . + 8 n−1
for
n
=
1,
3,
5,
7,.
.
.
n = 2 the number of algebraic operations - 102, b) - for block matrices
2
2
.x)
.... .. .. .
for
for
for
n1-element
n=n=3,
=3,4,
3,4,.4,
....... .number
.
matrices
–
8operations
8
16 for
16 different
24 with n = 3 the number of operations - 143, c) - for matrices with n = 4
Calculated
of algebraic
.a.... . . .
The
sum
of algebraic
operations
needed
to (partitioning
calculate
the
in-the
Calculated
Calculated
Calculated
number
number
number
of
of
of
algebraic
algebraic
algebraic
operations
operations
operations
for
for
for
different
different
different
4. Theof number
algebraic
operations
block
dimensions
n
of
input
block
matrix
in
1-2-number
operations of
- 322,
d) - for matrices
n = 5 the number of
Sum of algebraic operations loIV 53 61 114 122 175 183
input
block
matrix
(partitioned
in
1-2-1-2
order)
may
be
or- verse
block
block
block
dimensions
dimensions
dimensions
n
n
of
n
of
of
input
input
input
block
block
block
matrix
matrix
matrix
(partitioning
(partitioning
(partitioning
in
in
1-2in
1-21-2and
computation
times
operations - 396
1-2 order), required to obtain the self inertia matrices ci (11),
by
the required
following
expression:
he given
1-2
1-2
1-2
order),
order),
order),
required
obtain
toobtain
obtain
the
the
the
self
self
self
inertia
inertia
inertia
matrices
matrices
matrices
cici(11),
c(11),
i (11),
are
shown
inrequired
Tabletoto
16.

The
overall
relationship
between
the
block
dimension
n
of Numerical experiment was carried out in two stages. The Gauss
ve
are
are
are
shown
shown
shown
ininTable
inTable
Table
16.
16.
16.
The
overall
relationship
between the for
block
dimension

102
n=
2 and theitn of
...
is method
also known
its computational

is wellfor
known
and described incomplexity,
the literature,but
it isthere
also
the
input
block
matrix
(partitioned
in
1‒2–1‒2
order)
ck
The
The
The
overall
overall
overall
relationship
relationship
between
between
between
the
the
the
block
block
dimension
dimension
dimension
nnof
nofthe
of
input
matrix (partitioned
inblock
1-2-1-2
order) and
n block
nrelationship
... . .. . lo =the
(27)
known
for
its
computational
complexity,
but
there
is
no
innumber
of
algebraic
operations
needed
to
calculate
self
inertia
is
no
information
on
the
method
"inv"
(Matlab
function)
in
53
+
8(
−
1)
+
l
+
l
+
l
for
n
=
2,
6.
.
.
oIII
oII
oI
22of
2 matrix
the
the
the
input
input
input
block
block
block
matrix
matrix
(partitioned
(partitioned
(partitioned
ininin
1-2-1-2
1-2-1-2
1-2-1-2
order)
order)
order)
and
and
and
the
the
the

number
algebraic
operations
needed
to calculate
self
inertia

... . . .
n−1
n−1
formation
on
the
method
"inv"
(Matlab
function)
in
terms
of
matrices
c
(11),
is
as
follows:
terms
of
the
computational
complexity.
So
it
was
decided
to
i
+ 8 2isoperations
+
lfollows:
+ loI totocalculate
n = 1,
3.self
.inertia
.inertia
53
coIII + loII
number
number
number
ofofalgebraic
of
operations
operations
needed
needed
needed
tofor
calculate
calculate
self
self
inertia
2calgebraic
matrices
(11),
as
i algebraic
the computational
complexity.
Soalgorithms
it was decided
to light
present
the
present
the
effectiveness
of
shown
in
the
of
the
matrices
matrices
matrices
cic(11),
(11),
isisas
isasfollows:
asfollows:
follows:
ic(11),
i
Relationships
showing
the number of algebraic operationscomputational
effectiveness
of shown algorithms
in Gasuss
the lightmethod,
of the compucomplexity
compared to
and in
53 n2 + 8( n2 − 1) for n = 2, 4, 6, 8,. . .
in needed
n nton calculate
nn n
the
elementary
matrices
of
in=done
(26)
tational
complexity
compared
to
Gasuss
method,
and in
the
lto
(26)
oIVbe
umns,
+
+
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+
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8(
−
−
1)
−
1)
1)
for
for
for
n
n
=
n
=
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=
2,
4,
2,
4,
6,
4,
6,
8,.
6,
8,.
8,.
.
.
.
.
.
.
53
53
53
n−1
n−1
the
light
of
time
consumed
during
the
direct
calculation
using
2 2 2 2+2 8
2
n- vertedloIV
for
nthe
= growth
1, 3, 5, 7,.
.block
. (26)
53
=
=
=
(26)
(26)
l
l
2
2
light
of
time
consumed
during
the
direct
calculation
using
"inv"
oIV
oIV
input
block
matrix
(according
to
of
dimns,
mns,
umns,
n−1
n−1
n−1
n−1
n−1
n−1
atrix)
"inv" method and block inversion method.
for
for
nn=n=1,
=1,3,
1,3,5,
3,5,7,.
5,7,.7,.
. .. .. .
5353532 2 2++8+8 82 2 2 for
olmethod and block inversion method.
mension)
graphically
in Figure
6. It should
matrix)
atrix)
The can
sumbeofrepresented
algebraic operations
needed
to calculate
the inkrix)
maThe number
of algebraic
operations
that must
performed
ar be considered
The number
of algebraic
operations
that be
must
be perThe
sum
of
algebraic
operations
needed
to
calculate
the
that
the
input
block
matrix
is
partitioned
into
elThe
The
The
sum
sum
sum
of
of
of
algebraic
algebraic
algebraic
operations
operations
operations
needed
needed
needed
to
to
calculate
to
calculate
calculate
the
the
the
inininmakmama- verse input block matrix (partitioned in 1-2-1-2 order) may
be
-2
in
order
to
calculate
the
inverse
of
an
input
block
matrix
using
ck or- ementary
formed
in
order
to
calculate
the
inverse
of
an
input
block
mainverse
input
block
matrix
(partitioned
in
1‒2–1‒2
order)
may
matrices
with
dimensions
consistent
with
themay
result
verse
verse
input
input
input
block
block
block
matrix
matrix
matrix
(partitioned
(partitioned
(partitioned
inin1-2-1-2
in1-2-1-2
1-2-1-2
order)
order)
order)
may
may
bebebe
2y-2
ororor- verse
given
by
the
following
expression:
the
its
partitioning
into
blocks,
has
been
compared
with
the
numall ofgiven
trix
using
its
partitioning
into
blocks,
has
been
compared
with
be
given
by
the
following
expression:

1-2-1-2
partition
order.
given
by
byby
the
the
the
following
following
following
expression:
expression:
expression:
yby
the
the
the given
essive
ber of
performed
using standard
Gauss
the algebraic
number ofoperations
algebraic operations
performed
using standard
cfor n = 2

 102

sive
ssive
essive
block
The number
of algebraic
operations
performed
by
Gauss methods.
The number
of algebraic
operations
performed

102
102
102n + 8( n − 1) + l + l + l for
for
for
nn=n=2=
2 22, 6. . . methods.



(27)
53
=
o=
oIII
oII
oI for n
block
ock
block
number
operations
and
com-Gauss
all
2- 4. lThe
2n n nof
2n algebraic
by
Gauss
method,
can
be
represented
as
a formula
[16]:

method,
can
be
represented
as
a
formula
[16]:
n
n
he
(27)
(27)
lo== 5353
53
8(
+8(28(2−82−1)
−1)+
1)+l+
loIII
loIII
++l+
loII
loII
++l+
loIloIfor
for
for
nn=n=
=
2,6.
2,
6..6.
.3.
.(27)
.....(27)
n−1
n−1
oIII
oII
oI
all
lfuncall
2-2-2- lolo=
2 2+
2+
for
n2,
=
1,
putation
times

2 +n−1
2 + loIII + loII + loI


 53
1n−1
n−1
n−1
n−1
n−1
loIII
loIII
++l+
loII
loII
++l+
loIloI for
for
for
nn=n=1,
=1,3.
1,3..3.
.. .. .
5353532 2 2++8+8 82 2 2++l+
encuncfuncoIII
oII
oI
2
3
7
experiment
was carried
out inof two
stages.operations
The ce Numerical
Relationships
showing
the number
algebraic
lo = n3 + n2 − n (28)
(28)
main
3
2
6
Relationships
Relationships
Relationships
showing
showing
showing
the
the
the
number
number
number
ofofof
algebraic
algebraic
algebraic
operations
operations
operations
method
well
known
and
described
in thematrices
literature,
es- Gauss
needed
to beisdone
to
calculate
the
elementary
of inmain
emain
Relationships
showing
the the
number
of algebraic
operations
hemain
in- needed
needed
needed
toto
be
tobebe
done
done
done
to
to
calculate
tocalculate
calculate
the
the
elementary
elementary
elementary
matrices
matrices
matrices
ofofof
inininverted
input
block
matrix
(according
to the growth
of block
di- n - block dimension of the input matrix.
needed to be done to calculate the elementary matrices of in-where
where n – block dimension of the input matrix.
the
esininin- verted
folverted
verted
input
input
input
block
block
block
matrix
matrix
matrix
(according
(according
(according
to
to
the
to
the
the
growth
growth
growth
of
of
block
of
block
block
dididimension) can be represented graphically in Figure 6. It should
7 Results of algebraic calculation, showing the comparison
s
es
folfolfolngular mension)
mension)
mension)
can
can
can
bebebe
represented
represented
represented
graphically
graphically
ininFigure
inisFigure
Figure
6.6.It
6.Itshould
Itshould
should
be
considered
that
the inputgraphically
block
matrix
partitioned
intobetween
elthe methods employing the Gauss methods and of the
ular
gular
ngular
block bebe
be
considered
considered
considered
that
that
that
the
the
the
input
input
input
block
block
block
matrix
matrix
matrix
is
is
partitioned
is
partitioned
partitioned
into
into
into
elelelementary
matrices
with
dimensions
consistent
with
the
result
860
Bull. Pol.
Tech. 7-9.
64(4) 2016
inversion using block matrices, are shown
inAc.:
Figures
block
ock
block
of
all ementary
ementary
ementary
matrices
matrices
matrices
with
with
with
dimensions
dimensions
dimensions
consistent
consistent
consistent
with
with
with
the
the
the
result
result
result
of
1-2-1-2
partition
order.
In
order
to
compare
the
calculation
times
of
matrix
inversion
Unauthenticated
mfof
all
ofallall ofof1-2-1-2
funcof1-2-1-2
1-2-1-2
partition
partition
partition
order.
order.
order.
Download
Date | 7/28/17
10:28
PM
performed by block inversion
methods
and the
methods
used
ncuncfunc4. The number of algebraic operations and comin commercial software, the implementation of above menof the 4.4.4.The
The
The
number
number
number
ofofof
algebraic
algebraic
algebraic
operations
operations
operations
and
and
and
comcomcom-
62
162
162
153
53
153
153
321
carryi
to det
stream
ber of
of per
cal ex
no ev
the ca
partiti
The
proce
of cal
The r
the m
metho
than 1
5. S
Inversion of selected structures of block matrices of chosen mechatronic systems
Results of algebraic calculation, showing the comparison
between the methods employing the Gauss methods and of the
inversion using block matrices, are shown in Figs. 7‒9.
Fig. 7. Number of algebraic operations performed during matrix
inversion versus block dimensions: squares – numbers of algebraic
operations for Gauss method, circles – numbers of algebraic operations
for inversion using block matrices (partitioned in 1‒1–1‒1 order)
Fig. 8. Number of algebraic operations performed during matrix
inversion versus block dimensions: squares – numbers of algebraic
operations for Gauss method, circles – numbers of algebraic operations
for inversion using block matrices (partitioned in 1‒2–2‒2 order)
Fig. 9. Number of algebraic operations performed during matrix
inversion versus block dimensions: squares – numbers of algebraic
operations for Gauss method, circles – numbers of algebraic operations
for inversion using block matrices (partitioned in 1‒2–1‒2 order)
In order to compare the calculation times of matrix inversion performed by block inversion methods and the methods
used in commercial software, the implementation of above
mentioned algorithm using MATLAB software has been done.
In the programming environment the comparison with the "inv"
a standard function of the matrix inversion in MATLAB (under
the following conditions: forced CPU affinity for MATLAB
with single core processor) has been made. The priority of
performed tasks had been set to high.Calculations were made
on computer equipped with CPU AMD Phenom (tm) II X4 850
3.30 GHz frequency on operating system Windows 7 Professional 64-bit. Calculations has been performed in two loops: an
internal loop performing 10,000 times the calculation of matrix inversion and calculating the average time value of single
pass through the loop; and outer loop repeated 1000 times and
calculating the total computation time. This way of carrying
out numerical experiment, using a double loop, alows to determine if indeed calculations are carried out in a single stream
of processor unit. Linear relationship between the number of
passes the internal loop and the outer loop and the times of
performed calculation indicates properly conducted numerical
experiment. In other words, in the course of the experiment
no events inside the operating system does not interfere with
the calculations. For the block inversion we have used matrix
partitioned in the 1‒1–1‒1 order.
The calculation results of the matrix inversion times of the
procedures are summarized in Tables 17‒20, linear increase
of calculation time indicates properly conducted experiments.
861
Bull. Pol. Ac.: Tech. 64(4) 2016
Unauthenticated
Download Date | 7/28/17 10:28 PM
T. Trawiński, A. Kochan, P. Kielan, and D. Kurzyk
Table 17
Inversion of the matrices of block dimension n = 3
time of internal loop [s]
time of outer loop [s]
Block inversion
method – "inv"
0.0416(37)
0.0791(86)
41.637
79.186
Table 18
Inversion of the matrices of block dimension n = 5
time of internal loop [s]
time of outer loop [s]
Block inversion
method – "inv"
0.0663(32)
0.131(38)
66.332
131.38
Table 19
Inversion of the matrices of block dimension n = 7
time of internal loop [s]
time of outer loop [s]
Block inversion
method – "inv"
0.0932(57)
0.1648(0)
93.257
164.8
Table 20
Inversion of the matrices of block dimension n = 10
time of internal loop [s]
time of outer loop [s]
Block inversion
method – "inv"
0.1621(6)
0.2317(4)
162.16
231.74
The results presented times of calculations testify in favor
of the method of the block matrix inversion compared to the
method of “inv” used in Matlab. Computation times are shorter
than 1.5 to 2 times.
5. Summary
Number of elementary operations required to determine the
inverse matrix using the described algorithm depend on the
structure of the matrix and block size of block matrix. The
described algorithm is most effective for matrices with large
block size and divided into blocks of small size. This is particularly noticeable if you compare arrays block divided in the
order 1‒1–1‒1 and 1‒2–2‒2 in order, the computational complexity of the array divided into blocks of one-piece require
a minimum number of algebraic operations during the matrix
inversion. Computational complexity in the above two cases,
which is obvious, is of the order O(n2). The computational
complexity of matrix inversion divided in order 1‒2–1‒2 may
at first glance appear to be O(n), but if we make a thorough
analysis of the increase in the complexity of the component
ei dj , see Table 12, it turns out that it grows as well O(n2).
862
However, in the case of matrix inversion divided in order
1‒2–1‒2 computational complexity is between 1‒1‒1‒1 and
1‒2‒2‒2. In all of the analyzed cases, the elements causing
the most demanding calculations are the positive feedback
matrices eidj, here the complexity is of the order of O(n2). As
shown by formulas (15), (20) and in Table 12, the computational complexity O(n) are linked with the other elements of
the matrix like ci, c0 and di. Interesting results were obtained
by juxtaposing formulas describing the computational complexity of the block matrix inversion and the computational
complexity resulting from the Gauss method. Inverting block
matrix of small size block near n = 5 (and in this case smaller
then 4, 5 or 7) have a slightly more complex computation than
the Gauss method. However, the block matrix size greater
than three (in the case of partitioning in the order 1‒1–1‒1)
for presented method is more effective, as well in the case
of the matrix partitioned in order 1‒2–2‒2, this method is
more effective for matrices with greater block dimension then
n = 7. The calculation of the matrix inverse, divided in the
order 1‒2–1‒2 become more efficient than the Gauss method
on the block dimension greater then n = 5. The exact values of
computational complexity of the algorithm for block matrices
of the presented structures are given by formulae (17, 22) and
(27). Presented method is also characterized by shorter calculation times compared to the standard method of inverting
used in MATLAB/Simulink, computation times are shorter
than 1.5 to 2 times.
References
[1] J. de Jesus Rubio, C. Torres, C. Aguilar, “Optimal control based
in a mathematical model applied to robotic arms”, International
Journal of Innovative Computing, Information and Control 7(8),
5045‒5062, (2011).
[2] T. Trawiński, K. Kluszczyński, W. Kolton, “Lumped parameter
model of double armature VCM motor for head positioning
system of mass storage devices”, Przegląd Elektrotechniczny
87(12b), 184‒187, [in Polish] (2011).
[3] D. Słota, T. Trawiński, R. Wituła, “Inversion of dynamic matrices
of HDD head positioning system”, Appl. Math. Model. 35(3),
1497‒1505, (2011).
[4] T. Trawiński, “Kinematic chains of branched head positioning
system of hard disk drives”, Przegląd Elektrotechniczny 87(3),
204‒207,(2011).
[5] T. Trawiński, “Block form of inverse inductance matrix for poliharmonic model of an induction machine”, XXVIII International
Conference on Fundamentals of Electrotechnics and Circuit
Theory, IC-SPETO’2005, II, p.371, (2005).
[6] T. Trawiński, Modeling of Driving Lay-out of Mass Storage
Head Positioning System, Wydawnictwa Politechniki Śląskiej,
Gliwice, [in Polish] (2010).
[7] N. Jakovcevic Stor, I. Slapnicar, J. L. Barlow, “Accurate eigenvalue decomposition of arrowhead matrices and applications”,
Linear Algebra and its Applications, 464, 62–89, (2015).
[8] F. Diele, N. Mastronardi, M. Van Barel, E. Van Camp, “On
computing the spectral decomposition of symmetric arrowhead
matrices”, Computational Science and Its Applications – ICCSA
2004 Lecture Notes in Computer Science, 3044, 932–941,
(2004).
Bull. Pol. Ac.: Tech. 64(4) 2016
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Download Date | 7/28/17 10:28 PM
Inversion of selected structures of block matrices of chosen mechatronic systems
[9] L. Shen, B. W. Suter, “Bounds for eigenvalues of arrowhead matrices and their applications to hub matrices and wireless communications”, EURASIP Journal on Advances in Signal Processing,
2009, (2009).
[10] G. A. Gravvanis, K. M. Giannoutakis, “Parallel exact and approximate arrow-type inverses on symmetric multiprocessor
systems”, Computational Science – ICCS 2006 Lecture Notes
in Computer Science, 3991, 506‒513, (2006).
[11] G. A. Gravvanis, “Solving symmetric arrowhead and special
tridiagonal linear systems by fast approximate inverse preconditioning”, Journal of Mathematical Modelling and Algorithms
1(4), 269‒282, (2002).
[12] J. Tanton, “A dozen questions about triangular numbers”, Math
Horizons 13(2), 6‒7, 21‒23 (2005).
[13] P. A. Piza, “On the squares of some triangular numbers”, Mathematics magazine 23(1), 15‒16, (1949).
[14] J. A. Ewell, “A trio of triangular number theorems”, American
Mathematical Monthly 105(9), 848‒849, (1998).
[15] “Lozanic triangle”, The On-Line Encyclopedia of Integer Sequences, (2014).
[16] K. A. Atkinson, An Introduction to Numerical Analysis, John
Wiley and Sons (2nd ed.), New York, 1989.
[17] W. Hołubowski, D. Kurzyk, T. Trawiński, “A fast method for
computing the inverse of symmetric block arrowhead matrices”,
Appl. Math. Inf. Sci. 9(2L), 1‒5, (2015).
[18] T. Trawiński, “Inversion method of matrices with chosen structure with help of block matrices”, Przegląd Elektrotechniczny
85(6), 98‒101, [in Polish] (2009).
[19] H. T. Kung, B. Wilsey Suter, “A hub matrix theory and applications to wireless communications”. EURASIP Journal on Advances in Signal Processing 2007(1), 1‒8, (2007).
[20] E. Mizutani, J. W. Demmel, “On structure-exploiting trustregion
regularized nonlinear least squares algorithms for neuralnetwork
learning”, Neural Networks 16(5), 745‒753, (2003).
[21] M. Bixon, J. Jortner, “Intramolecular radiationless transitions”,
The Journal of Chemical Physics 48(2), 715‒726, (1968).
[22] J. W. Gadzuk, “Localized vibrational modes in Fermi liquids.
General theory”, Physical Review B 24(4), 1651, (1981).
[23] A. Ratajczak, “Trajectory reproduction and trajectory tracking
problem for the nonholonomic systems”, Bull. Pol. Ac.: Tech.
64(1), 63‒70 (2016).
[24] A. Mazur, M. Cholewiński, “Implementation of factitious force
method for control of 5R manipulator with skid-steering platform
REX”, Bull. Pol. Ac.: Tech. 64(1), 71‒80 (2016).
863
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