Code_Aster
®
Version
2.0
Titrate:
Algorithms of resolution for the quadratic problem
Date:
19/06/92
Author (S):
D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
Page:
1/26
Manual of Reference
R5.01 booklet: Modal analysis
HI-75-7816/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R5.01 booklet: Modal analysis
Document: R5.01.02
Algorithms of resolution for the problem
quadratic with the eigenvalues
Summary:
In this document, we fix the theoretical framework of the methods of search for eigenvalues of the problem
quadratic which is developed in the code of mechanics
Aster
.
Code_Aster
®
Version
2.0
Titrate:
Algorithms of resolution for the quadratic problem
Date:
19/06/92
Author (S):
D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
Page:
2/26
Manual of Reference
R5.01 booklet: Modal analysis
HI-75-7816/A
Contents
1 Introduction ............................................................................................................................................ 4
1.1 Position of the problem ....................................................................................................................... 4
1.2 Properties of the matrices ................................................................................................................... 4
1.3 Problem with the eigenvalues associated ........................................................................................... 4
1.4 Some conventional particular cases. ............................................................................................. 5
2 Reductions with a linear form ............................................................................................................ 6
2.1 Reductions with a linear form ...................................................................................................... 6
2.1.1 A particular choice for matrix Z: ±M ............................................................................. 7
2.1.2 Particular case of the positive matrix M definite ...................................................................... 7
2.1.3 Case of the symmetrical reduction for a singular matrix M ............................................. 8
2.2 Property of orthogonality of the clean vectors ............................................................................... 9
3 Method of determinant ...................................................................................................................... 10
3.1 General. ................................................................................................................................... 10
3.2 Method of Muller ......................................................................................................................... 10
3.2.1 Development of the method ............................................................................................ 10
3.2.2 Convergence of the method ................................................................................................ 12
3.2.3 Application of the method for search of eigenvalues ........................................ 12
3.2.3.1 Development ........................................................................................................ 12
3.2.3.2 Cost of the method in term of factorization ......................................................... 13
4 Methods of iteration reverses ................................................................................................................ 14
4.1 Method of opposite iteration suggested by Wilkinson ..................................................................... 14
4.2 Alternative developed by Jennings ................................................................................................ 14
4.3 The algorithm of iteration reverses
Aster
.......................................................................................... 15
4.3.1 Implementation ..................................................................................................................... 15
4.3.2 Criterion of stop ....................................................................................................................... 15
5 Lanczos Method applied to the quadratic problem ....................................................................... 16
5.1 Choice of a problem to approach ................................................................................................. 16
5.1.1 Form reverses standard ....................................................................................................... 16
5.1.2 Strategies of shift ......................................................................................................... 16
5.2 Method of approximation .............................................................................................................. 17
5.2.1 Approximate problem and algorithm of Lanczos ..................................................................... 17
5.2.2 Choice of a pseudo scalar product ...................................................................................... 19
5.3 Application to the quadratic Problem ............................................................................................ 19
5.3.1 Spectral operator ................................................................................................................ 19
5.3.2 Operator of pseudo scalar product .................................................................................. 20
5.3.3 Cost of the Lanczos phase .................................................................................................... 21
5.4 Implementation in Aster ............................................................................................................ 21
5.4.1 Parameters of the implementation ......................................................................................... 22
Code_Aster
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Titrate:
Algorithms of resolution for the quadratic problem
Date:
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Author (S):
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Key:
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5.4.2 Under space of approximation ............................................................................................... 22
5.4.3 Strategy of reorthogonalisation ........................................................................................... 22
5.4.4 Implementation of the phase Lanczos ................................................................................... 22
5.4.5 Restoration of the approximations for the quadratic problem ........................................... 23
6 Bibliography ......................................................................................................................................... 24
Appendix 1 Interpretation of the complex eigenvalues ........................................................................ 25
Appendix 2 linear Reductions ................................................................................................................ 26
Code_Aster
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Titrate:
Algorithms of resolution for the quadratic problem
Date:
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Author (S):
D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
Page:
4/26
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R5.01 booklet: Modal analysis
HI-75-7816/A
1 Introduction
1.1
Position of the problem
Dynamic analysis or the study of the stability of the balance of a mechanical structure led, in
tally of the linearized theory, to solve the matric differential equation of the second command
M
D
2
U
(
T)
D
T
2
+
(
.
G
+ C)
D
U
(
T)
D
T
+ (K + E) U (T) = F (T)
where
M
is the matrix of mass and inertia of the structure,
G
is the matrix induced by the gyroscopic effect (case of the revolving machines),
is a significant real parameter rotational speed,
C
is the matrix of damping induced by dissipative forces.
K
is the matrix of rigidity of the structure,
E
is the matrix of viscous damping interns structure,
F
is the external force (which is null in the case of the search for balance).
1.2
Properties of the matrices
The matrices considered are with real coefficients.
Classically, one considers:
M
is symmetrical (semi) definite positive,
G
is antisymmetric,
C
is symmetrical,
K
is symmetrical not necessarily definite positive,
E
is antisymmetric.
Consequently, one realizes that the simultaneous presence of the matrix of damping and the matrix
of gyroscopic effect destroys the symmetry or the antisymetry of the term speed; pareillement
internal damping producing an antisymmetric matrix, destroyed the property of symmetry of
stamp rigidity.
In addition, the introduction of linear relations modifies the character of positivity of the matrices:
stamp K is indefinite (with positive or negative eigenvalues).
1.3
Problem with the eigenvalues associated
The sought solutions are form (separation of the variables of space and time).
U (T) = E
t.x with
IC and X
IC
NR
What leads us to the quadratic problem with the eigenvalues according to:
2
M
+
(
.
G
+ C) + (K + E)
X
= 0
the solution U (T) can be rewritten in the form:
U
(
T) =
K
J
J
=
1
2n
X
J
E
J
T
Code_Aster
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Titrate:
Algorithms of resolution for the quadratic problem
Date:
19/06/92
Author (S):
D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
Page:
5/26
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R5.01 booklet: Modal analysis
HI-75-7816/A
that one can put in matric form:
U (T) = [X] [E
T] [K]
where
[X]
is the modal matrix
[N X 2n]
[E
T] is a diagonal matrix
[2nx2n]
[K]
is a matrix unicolonne
[2nx1]
Proposal:
The modal matrix [X] perhaps used like stamps transformation to uncouple N
equations of the quadratic problem of origin, its 2n columns are not linearly
independent.
Notice
:
One determines [X], using the two following identities:
U (0) = [X] [K]
D
U
dt
(
0
) =
X
K
1.4
Some conventional particular cases.
These particular cases were lengthily studied (cf for example [MEI.67], [ROS.84])
M
G
C
K
E
values
clean
vectors
rights
clean
lefts
conservative
0
0
0
IR
X
D
=
X
G
IR
conservative
gyroscopic
0
0
I IR
X
D
=
X
G
IC
dissipative
0
0
IC
X
D
=
X
G
IC
Code_Aster
®
Version
2.0
Titrate:
Algorithms of resolution for the quadratic problem
Date:
19/06/92
Author (S):
D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
Page:
6/26
Manual of Reference
R5.01 booklet: Modal analysis
HI-75-7816/A
2
Reductions with a linear form
One is interested in the possibilities of reduction of the quadratic problem in a generalized problem
equivalent.
The principles of reduction are applicable to unspecified matrices, and if we consider it
problem in (M, C, K) is only to simplify the talk.
2.1
Reductions with a linear form
There are several conventional methods to transform the quadratic problem into a problem
generalized with the eigenvalues.
We will develop the method which consists in introducing speed like auxiliary variable:
For that we introduce an additional equality:
Z
U
(T) - Z
U
(T) = 0
where Z is a matrix not identically null.
Our initial system can then be rewritten in the matric form in a space of dimension
double of initial space
0
Z
-
M
-
C
U
U
=
Z
0
0
K
U
U
And the problem with the eigenvalues associated is in its opposite form
With
. Z =
1
B
. Z
By supposing K and Z regular, the problem with the eigenvalues generalized thus obtained can
formally to put itself in the standard form:
0
-
K
- 1
M
-
K
- 1
C
X
X
=
1
X
X
Notation: one poses y =
X
The standard form associated the quadratic problem is independent of regular matrix Z
chosen.
Definition
One will call linear reduction of the quadratic problem, any generalized problem of which all them
clean elements (
,
(X, y)) check:
·
(
,
X) is clean solution of the quadratic problem,
·
and My =
MX (condition of coupling)
Code_Aster
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Algorithms of resolution for the quadratic problem
Date:
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Author (S):
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Key:
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Proposal
:
If (
,
(X, y)
T
) is a solution of the linearized equation and if Z is regular then (
, X) is solution of
quadratic problem.
This result is immediate.
2.1.1 A particular choice for matrix Z: ±M
·
one takes for matrix Z the matrix - M of the initial system, consequently the system linearized (A.Z =
µ
B.z)
is written:
0
M
M
C
y
X
=
1
M
0
0
-
K
y
X
If the matrices M, C and K are symmetrical and Si the matrix M is regular, this linear reduction
allows to build a symmetrical generalized problem.
·
one takes for matrix Z the matrix M of the initial system, consequently the system linearized (A.Z =
µ
B.z)
is written:
0
M
-
M
-
C
y
X
=
1
M
0
0
K
y
X
If M and K are definite positive then B is also, but matrix A is not symmetrical.
Proposal
:
If one chooses Z = ± M (matrix not identically null) and if (
, (X, y)
T
), with X not no one, is solution
clean of the equation generalized then (
, X) is clean solution of the quadratic problem.
2.1.2 Particular case of the positive matrix M definite
If the matrix of mass is definite positive then it a decomposition of Choleski admits
M = Q
T
.Q
If one introduces the linear transformation U (T) = Q
- 1
.q in the basic equation of the quadratic problem
after pre-multiplyhaving multiplied it by Q
- T
= (Q
- 1
)
T
, one obtains:
Q
(
T
) +
Q
T
.
C
.
Q
1
Q
(
T
) +
Q
T
.
K
.
Q
1
Q
(T) = 0
and while defining v =
(
Q
,
Q)
T
the preceding equation, leads us to the standard problem with the values
clean A. v =
.v with
With
=
QT
C
Q1
QT
K
Q1
0
Code_Aster
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2.1.3 Case of the symmetrical reduction for a singular matrix M
In this paragraph we suppose the matrices M, C and K symmetrical.
If M is singular and if the eigenvalue
0
then there is semi simple is an orthonormal matrix Q
such as.
Q
T
M
Q =
M
=
M
11
0
0
0
Indeed, the eigenvalue
0
being semi-simple, the core of M admits a base made up of vectors
clean. This base is supplemented and orthonormalisée. The matrix Q admits for vectors columns them
vectors of this base.
Let us note that M
11
is symmetrical (bus M is) and regular.
One introduces then a regularization of the matrix M represented by the matrix M
R
and defined by:
Q
T
M
R
Q
= Q =
M
1
1
0
0
M
22
=
M
R
where M
22
is a regular symmetrical matrix (for example the matrix identity).
Property:
The generalized problem associated the matrices
With
=
K
0
0
-
M
R
B
=
C
M
- M
0
is a symmetrical linear reduction of the quadratic problem.
Demonstration:
·
symmetry is ensured by construction.
·
the “tildées” matrices check
M
M
R
1
M
=
M
thus
Q
M
Q
T
Q
M
R
1
Q
T
Q
M
Q
T
=
Q
M
Q
T
, that is to say still
M
M
R
1
M
= M
However if
X
y
is a clean vector associated the eigenvalue
generalized problem one has
M
R
y =
MX, therefore My =
M
M
R
1
MX. The condition of coupling is established.
·
Let us suppose that under-vector X of the clean vector
X
y
that is to say no one. Then the equation
M
R
y =
MX led to y = 0 bus M
R
is regular, which is absurd.
The condition X 0 is thus established.
Code_Aster
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Note: Interpretation of the regularized matrix M
R
:
This symmetrical linear reduction is distinguished from that presented previously by
substitution of M per M
R
in matrix A.
If M is singuliére, this substitution forces the vector to have a component there
null in the core Mr. Ainsi the symmetrical linear reduction led to a generalized problem
posed in under space of
X
y
IR2n such as y
ker M - {0}.
If K is regular, the preceding generalized problem admits a form reverses standard
equivalent.
M
1
C
K
1
M
- M
R
- 1
M
0
X
y
=
1
X
y
2.2
Property of orthogonality of the clean vectors
It is immediate to show that the complex clean modes check the properties of orthogonality
exits of the following generalized problem if the matrices A and B are symmetrical:
X
I
,
y
I
.
A.
X
I
y
I
=
ij
has
I
X
I
,
y
I
.
B.
X
I
y
I
=
ij
B
I
If one develops the preceding expressions, by taking account of the linear reductions used one
obtains the expressions:
I
+
J
X
I
T
M
X
J
+
X
I
T
C
X
J
=
has
I
ij
-
I
J
X
I
T
M
X
J
+
X
I
T
K
X
J
=
B
I
ij
Note:
·
The first equality is independent of the regularity of M,
·
the second equality is obtained directly if M is regular and is established if not in
using the basic change of the regularization of Mr.
The modes of the quadratic problem are thus not M, C or K orthogonal.
Code_Aster
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Titrate:
Algorithms of resolution for the quadratic problem
Date:
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Key:
R5.01.02-A
Page:
10/26
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HI-75-7816/A
3
Method of determinant
3.1 General.
The search for zero of the characteristic polynomial, a quadratic problem to the eigenvalues,
pose the problems inherent in the polynomials of the complex variable with complex values.
Not having relation of command in IC, usual methods, at two points, of dichotomy type
or secant are inapplicable.
We present here the “popular” method more for search of zero of polynomials of
variable complexes with complex values, the method of Muller.
3.2
Method of Muller.
The method suggested by Muller [MUL.56] is an iterative method using as curve
of interpolation a parabola with horizontal axis.
This method is relatively easy to implement but it lends itself badly to search
zeros of real functions with real roots, because it plunges the interpolation in the complex plan and this
even on the basis of actual values.
Its interest is related to the class of this method “the methods by curves of interpolation”, namely:
·
the safety of the method by dichotomy, since search is carried out in a ball “
reducing " gradually,
·
that only the calculation of the function is necessary (calculation of not derived as in
method of Newton),
·
convergence is connected with a quadratic convergence.
The most widespread method of the methods by curves of interpolation is the method at two points known as
secant.
3.2.1 Development of the method
Let us note F (Z) = has
0
Z
N
+ has
1
Z
n-1
+… + has
N
the algebraic equation which one seeks the zeros, A
I
are
complexes and we suppose has
0
not no one.
The quadratic formula of interpolation of Lagrange gives us:
L
I
(F (Z)) = B
0
Z
2
+ B
1
Z
+b
2
and we consider the curve which passes by the last three points (reiterated):
(Z
I
, F (Z
I
)), (Z
i-1
, F (Z
i-1
)), (Z
i-2
, F (Z
i-2
))
and thus the coefficients B
0
, B
1
and B
2
check:
B
0
Z
I
2
B
1
Z
I
B
2
F
(
Z
I
)
B
0
Z
I
-
1
2
+
B
1
Z
I
-
1
+
B
2
=
F (
Z
I
-
1
)
B
0
Z
I
-
2
2
+
B
1
Z
I
-
2
+
B
2
=
F (
Z
I
-
2
)
and while posing
=
Z
Z
I
Z
I
-
Z
I
-
1
and
I
=
Z
I
Z
I
1
Z
I
-
1
-
Z
I
-
2
and
I
=
1 +
I
Code_Aster
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Titrate:
Algorithms of resolution for the quadratic problem
Date:
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Key:
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HI-75-7816/A
one can rewrite the formula of interpolation of Lagrange
L
I
F
(
Z)
2
I
F (
Z
I
2
)
I
2
F
(
Z
I
1
)
I
I
F
(
Z
I
)
I
+
I
F (
Z
I
-
2
)
I
2
-
F (
Z
I
-
1
)
I
2
+
F (
Z
I
)
I
+
I
+ F (
Z
I
)
the new point is:Z
i+1
= Z
One can solve the quadratic equation in
, we obtain then:
=
I
+
1
=
Z
I
1
Z
I
Z
I
-
Z
I
-
1
by taking the reverse of the conventional solution of a quadratic equation:
I
+
1
2
F
(
Z
I
)
I
G
I
±
G
I
2
-
4 F (
Z
I
)
I
I
F (
Z
I
-
2
)
I
-
F (
Z
I
-
1
)
I
+ F (
Z
I
)
with
G
I
=
F (
Z
I
-
2
)
I
2
-
F (
Z
I
-
1
)
I
2
+
F (
Z
I
)
(
I
+
I
)
From
i+1
, one obtains:
H
i+1
=
i+1
H
I
,
Z
i+1
=
Z
I +
H
i+1
who is one zero of the equation.
The sign of the denominator is then taken of such kind that it is of larger possible module and thus of
such kind that
i+1
maybe of larger possible module, and finally Z
i+1
will be the root nearest to
Z
I
Note:
Muller [MUL.56] proposes a process of initialization while using:
“arbitrary” values Z
0
= - 1., Z
1
=1., Z
2
= 0.
and values
has
N
+ has
n-1
+ has
N2
for F (Z
0
)
has
N
- has
n-1
+ has
N2
for F (Z
1
)
has
N
for F (Z
2
)
This choice of values then results in considering:
L
2
(F (Z)) = = has
N
+ has
n-1
Z
+a
N2
Z
2
who is an approximation of F (X) in the vicinity of the origin.
The advantage of this process of starting is that it does not require any evaluation of the polynomial F (X) and
that it is thus fast.
Code_Aster
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Algorithms of resolution for the quadratic problem
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D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
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HI-75-7816/A
3.2.2 Convergence of the method
Proposal
: To consider that convergence is assured as soon as
Z
I
Z
I
1
Z
I
<
for
given is one
acceptable criterion.
Let us show initially that during the unfolding of the algorithm three the reiterated successive ones are distinct.
If it were not the case, by supposing that Z
I
=
Z
i-1
and that X
I
that is to say the 1
Er
reiterated for which one has
convergence and thus
Z
I
Z
I
1
Z
I
<
.
If it is supposed that Z
I
=
Z
i-2
, one would have then
I
and
i+1
identically null and thus there would be Z
i+1
=
Z
I
from where contradiction.
Then into constant that the difference between two reiterated can only decrease, one obtains the result
announced.
3.2.3 Application of the method for search of eigenvalues
3.2.3.1 Development
That is to say to determine the eigenvalues of the system (
2
M+
K+C) X = 0,
We thus seek the zeros of the characteristic polynomial which we note
F (Z) = dét (Z
2
M + zK + C)
To calculate in sequence the zeros of the polynomial, we use a technique of deflation.
Consequently the polynomial considered is:
F
K
(
Z
)
F
(
Z)
Z
Z
I
I
=
1
K
where
Z
I
I
=
1, K
are K zeros already calculated.
The use of the algorithm is immediate, and we benefit from this adaptation to formulate it
slightly different way.
Let us note
F
I
=
F
K
(
Z
I
)
the value of the function to be interpolated at point Z
I
at the time of the search of K
ième
zeros.
It is then practical, for the implementation, to reveal a few intermediate quantities:
H
I
=
Z
I
-
Z
I
-
1
L
I
=
H
I
H
I
-
1
=
Z
I
Z
I
1
Z
I
-
1
-
Z
I
-
2
T
I
=
1 +
L
I
=
Z
I
Z
I
2
Z
I
-
1
-
Z
I
-
2
Code_Aster
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reiterated the L
i+1
sought being solution of:
I
+
1
2
I
G
0
,
I
+
I
+
1
G
1
,
I
+
I
=
0
with
G
0
,
I
=
F
I
2
F
I
I
-
F
I
1
F
I
I
+
1
G
1
,
I
=
F
I
2
F
I
I
2
-
F
I
1
F
I
I
+
I
+
T
I
One deduces some:
I
+
1
2
T
I
G
1
,
I
±
G
1
,
I
2
- 4
G
0
,
I
L
I
T
I
the sign of the denominator being taken so that L
i+1
(and thus H
i+1
) is of smaller module
possible
consequently
Z
I
+
1
=
Z
I
+
L
I
+
1
.
H
I
To converge towards the eigenvalues with positive imaginary part (and thus at positive frequency), one
takes
Z
I
+
1
=
R
E
Z
I
+
1
,
Im
Z
I
+
1
Notice on deflation:
When the eigenvalues appear per combined pairs, it is also necessary to eliminate
combined found eigenvalues and which physically corresponds to a negative frequency
3.2.3.2 Cost of the method in term of factorization
With each iteration of the method one makes:
Matrix algebra part (evaluation of the characteristic polynomial)
·
a linear combination of three matrices
O (N
2
)
·
a factorization LDL
T
combination die
O (Nb
2
), B width of
bandage
·
the product of the diagonal terms divided by deflation
O (N)
Method part:
·
operations of the method with properly spoken
O (1)
Moreover it is advisable to add, as an assumption of responsibility (since it is a method at three points)
twice the evaluation part of the characteristic polynomial.
Broadly this method costs (i+2) factorizations, for a solution calculated out of I iterations.
The cost of this method is such, that it must be to hold for small systems (case of adjustment)
or when it is important to have a very high degree of accuracy on the sought frequency.
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4
Methods of opposite iteration
The method of iteration opposite extends immediately to the quadratic problem by using its form
“linearized”.
The linearization of the quadratic problem not being single, there exist several alternatives.
4.1
Method of iteration opposite suggested by Wilkinson
Wilkinson [WIL.65] proposes to bring back the quadratic problem to the following standard problem
0
-
M
- 1
K
-
M
- 1
C
X
y
=
X
y
By supposing M invertible, and by posing y =
X
Being given
an approximation of the sought eigenvalue, one can define the iterative process
according to:
-
M
- 1
K
-
M
- 1
C
-
X
S
1
y
S
+
1
=
X
S
y
S
Matric equation, we deduce a system from equations to two unknown (y
s+1
, X
s+1
) and by
combination of this system we deduce the expression from y
s+1
and of X
s+1
.
2
M
C
K
y
S
+ 1
K
X
S
M
y
S
X
S
+
1
=
y
S
+
1
-
X
S
4.2
Alternative developed by Jennings
When M and/or K are singular, one obtains a stable and equivalent quadratic equation in
introducing an auxiliary parameter
µ
=
-
for a spectral shift of
.
While replacing
by
.
1
µ
+
1
in the quadratic equation, one obtains the quadratic problem in
µ
:
µ
2
2
M
+
C
+ K
+ 2
µ
K
-
2
M
+
2
M
-
C
+ K
y = 0
While noting
D (
)
=
2
M
+
C
+ K
the dynamic matrix which is regular by construction, one
bring back to the standard problem such as it is proposed by Wilkinson while posing y =
µ
X
0
-
D (
)
- 1
.
D
(-
)
-
D (
)
-
1
.
(
2 K - 2
2
M)
X
y
=
µ
X
y
This equation is stable in the direction where
µ
<
1
for
strictly positive.
This process thus makes it possible “to regularize” the orders of magnitude of the matrices M, C, and K through
stamp dynamic D (
).
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4.3
The algorithm of iteration reverses Aster
4.3.1 Implementation
This algorithm is available in
Aster
by the operator
MODE_ITER_INV.
That is to say
0
an approximate value of the sought eigenvalue, one builds the dynamic matrix D (
) that
one factorizes in form LDL
T
.
One initializes the iterative process by the vectors according to:
X
0
= {(1, 0)}
y
0
=
0
X
0
the iterative process to obtain the nth one reiterated:
·
Standardization of X
n-1
and y
n-1
to avoid the overflows of capacity:
X
N
1
X
N
1
X
N
1
M
y
N
-
1
=
y
N
-
1
X
N
-
1
M
·
Resolution of:
D (
0
)
X
N
=
C.
X
N
-
1
+
0
M
.
X
N
-
1
+
Mr.
y
N
-
1
·
Calculation of y
N
starting from X
N:
y
N
=
-
0
X
N
+
X
N
-
1
·
Evaluation
of
N:
N
N
-
1
1
X
N
M
X
N
This diagram can be put in the matric form:
D (
0
)
0
-
0
.
-
X
N
1
y
N
+
1
=
C
0
MM
-
0
X
N
y
N
4.3.2 Criterion
of stop
We use a simple result [GOH. &al.86] relating to the polynomials of matrices of the form:
D (
) = M
2 + C
+K
That is to say
O an eigenvalue of the operator D (
) and xo an associated clean vector not no one checking D (
O) xo
= 0 then we have the equality:
0
=
X
0
T
C
X
0
±
X
0
T
C
X
0
2
4
X
0
T
M
X
0
X
0
T
K
X
0
2 X
0
T
M
X
0
The criterion of stop is done on the relative variation of
0
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5
Lanczos method applied to the quadratic problem
In this chapter we suppose the matrices M, C and K real symmetrical, so that the problem
quadratic associated can be tiny room to a linear form symmetrical Az =
Bz
where
With
=
K
0
0
- M
R
B =
C
M
- M
0
Z =
X
y
M
R
is a regular matrix deduced from M which coincides with M if the latter is regular.
One seeks to develop the arithmetic method preserving real one overall in order to obtain
a problem reduces real.
5.1
Choice of a problem to be approached
5.1.1 Form reverses standard
One seeks an approximation of the couples (
, X) of clean parts of the quadratic problem which
correspond to the eigenvalues
close relations of a complex shift
=
+i
given.
The approximation of Galerkin of the spectral problem of an operator S in a subspace of Krylov K
m
= span (R
0
, Sr
0
., S
M-1
R
0
) allows to approach the clean couples of elements of the operator S
corresponding to the eigenvalues of larger modules.
The passage to the standard form reverses after spectral shift of
generalized problem
preceding, provides the spectral problem:
(A -
B)
- 1
B Z =
Z
This problem admits the same clean vectors as the problem generalized and of the eigenvalues
bound to those of the problem generalized by the relation:
=
1
-
Thus, an approximation of Galerkin of the spectral problem of the operator:
S = (A -
B)
- 1
B =
With
1
B
in under space of Krylov K
m
= span (R
0
, Sr
0
., S
M-1
R
0
) provides the approximation of the couples
clean parts of the quadratic problem sought.
5.1.2 Strategies of shift
The operator S preceding being complex, it calls in a natural way the use of the arithmetic one
complex. It is nevertheless possible to use the arithmetic real one to approach the couples
clean elements in which one is interested. It is enough to use a real operator of the same vectors
clean and whose eigenvalues of larger module correspond to those of the problem
quadratic closest to
.
Technique of double spectral shift by
and
proposed by Francis within the framework of the method
QR makes it possible to build such an operator noted to him also S:
S = [(A -
B) (A
B)]
- 1
B = [AB
- 1
With - 2
WITH + I
I
2
B]
- 1
B
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The major disadvantage of this technique is the filling of the matrix
AB
- 1
With - 2
WITH + I
I
2
B if B is not diagonal.
Parlett and Saad [PAR. SAA.87] propose an alternative which uses the real part or the imaginary part
of operator (A -
B)
- 1
B:
S
+
= Re [(A -
B)
- 1
B]
S
-
= Im [(A -
B)
- 1
B]
of which respectively noted eigenvalues
µ
+
and
µ
-
are related to the eigenvalues
problem
quadratic by the relations:
1
2
1
1
µ
-
=
1
2i
1
-
-
1
-
µ
+
and
µ
carry out the maximum of their module in the vicinity of
and
.
This approach preserves the arithmetic real one overall and avoids the disadvantage of the technique of
Francis, if the calculation of a vector S± v, where v are real, is carried out into arithmetic complex.
By noting that:
Re [(A -
B)
- 1
B] =
1
2
[(A -
B)
- 1
+ (A
B)
- 1
] B
Im [(A -
B)
- 1
B] =
1
2i
[(A -
B)
- 1
- (A
B)
- 1
] B
This approach can be interpreted like a technique of double shift summons, by opposition
with the approach of the double produced shift suggested by J-C.F. Francis.
5.2 Method
of approximation
From now on S will indicate one of the real operators S
+
or S
-
,
µ
one of its eigenvalues and P
stamp of a scalar pseudo-product (i.e. a symmetrical bilinear form not necessarily
defined positive).
5.2.1 Approximate problem and algorithm of Lanczos
When S is car-assistant for the pseudo scalar product induced by P, i.e.
(U, front)
P
= (With, v)
P
U, v
IR
2n
,
the method of Lanczos is used to generate a base of under space of Krylov K
m
.
The spectral problem is then approached by projection P-orthogonal on K
m
and the problem reduces thus
obtained is represented in the base of the vectors of Lanczos by a real matrix tridiagonale of command
Mr.
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The method of Lanczos extended to the scalar pseudo-products is defined by the formulas of
recursion following cf [R5.01.01]:
·
R
O
= arbitrary
·
O
=
0
, Q
O
=
0
1
= Sign ((R
O
, R
O
)
P
)
Q
1
=
1
R
0
,
R
0
P
1
=
0
·
for J = 1, 2,…, m
J
=
(Q
J
, Sq
J
)
P
R
J
=
Sq
J
-
J
J
Q
J
-
j-1
J
Q
j-1
j+1
=
sign ((R
J
, R
J
)
P
)
j+1
=
R
J
, R
J
P
Q
j+1
=
J
1
J
+
1
R
J
If Q is noted
m
the matrix 2nxm of the vectors of Lanczos Q
J
these formulas are written in the form
matric real following:
Q
MT
P Q
m
= J
m
= diag (
1
,…,
m
)
S Q
m
- Q
m
J
m
T
m
=
m+1
m+1
R
m
E
m T
with E
m T
= (
0
,…,
0
, 1)
T
m
the real matrix tridiagonale symmetrical m X m:
T
m
=
1
2
2
\
\
\
\
m
m
m
The product J
m
T
m
is a nonsymetric matrix tridiagonale as soon as J
m
is not proportional to
identity.
The two preceding matric relations make it possible to write:
Q
MT
P [S Q
m -
Q
m
J
m
T
m
] =
0
The application of this relation to a couple (
µ
(m)
, S
(m)
)
IC X IC
m
clean elements of the operator
represented by the matrix J
m
T
m
give:
Q
MT
P [S Q
m
S
(m) -
µ
(m)
Q
m
S
(m)
] =
0
who characterizes the couple (
µ
(m)
, Z
(m)
= Q
m
S
(m)
)
IC X IC
2n
like approximation of Galerkin by
projection P-orthogonal on K
m
of a clean couple of element of the operator S.
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5.2.2 Choice of a pseudo scalar product
The symmetry of matrices A and B ensures
that bilinear forms associated the matrices Re [(A -
B)
- 1
]
- 1
, Im [(A -
B)
- 1
]
- 1
and B are
scalar pseudo-product,
that the operator left real S
+
(respectively imaginary part S
-
) is autoadjoint for
scalar pseudo-product induced by Re [(A -
B)
- 1
]
- 1
or by B (respectively by Im [(A -
B)
- 1
]
- 1
or by
B).
The pseudo-products scalar induced by Re [(A -
B)
- 1
]
- 1
and Im [(A -
B)
- 1
]
- 1
are extensions of
scalar product used in the alternative of Pipano-Neuman of the Lanzcos algorithm.
If the matrix M is singular, the pseudo scalar product induced by B makes vectors
Z
=
X
y
where y
Ker M, of the quasi-null vectors. This disadvantage exists them also in the case of
scalar pseudo-products of Pipano-Neuman type (in particular if the matrix of the scalar pseudo-product
admits eigenvalues of null sum), but the occurrence of such an event is rare in
practical.
5.3
Application to the quadratic Problem
The use of this approach requires, with the method of Lanczos, the calculation of the real vectors Sz and
Pz for Z
IR
2n
.
5.3.1 Operator
spectral
That is to say the dynamic matrix D (
) associated the spectral shift
=
+ I
defined by:
D (
) =
2 M +
C + K
If D (
) then the operator is regular complexes (A -
B)
- 1
B can be written in the form
With
-
B
1
B
=
0
0
M
R
- 1
M
0
-
D
1
0
0
M
R
- 1
M
D
- 1
C
M
M
C
+
M
M
The calculation of Sz for S = Re [(A -
B)
- 1
B] and S = Im [(A -
B)
- 1
B] can be carried out without destroying
hollow structure of the matrices if the arithmetic complex is partially used in the algorithm:
·
Preparation into arithmetic complex
-
to form
D (
) =
2 M +
C + K
-
to factorize
D (
) in form LDL
T
·
Calculation of Sz
-
U
1
= Cx U
2
= MX U
3
= My
in IR
-
U
4
= D (
)
- 1
U
1
+
U
2
+u
3
in IC
-
according to the choice of the operator one obtains:
S
+
Z = Re [(A -
B)
- 1
B] Z =
R
E (
U
4
)
M
R
- 1
M
(X - R E (
U
4
)
S
-
Z = Im [(A -
B)
- 1
B] Z =
I
m
U
4
-
M
R
- 1
M
I m
U
4
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5.3.2 Operator of pseudo scalar product
·
Choice P = B
This choice is valid for the operators obtained in real left approach or part
imaginary.
Calculations can be carried out without assembly of B and into arithmetic real.
·
Choice P = Re [(A -
B)
- 1
]
- 1
This choice corresponds to the partly real approach of the operator S.
If the dynamic matrix D (
), where
is the real part of
,
is regular then the real operator
(A -
B)
- 1
B is defined by:
With
-
B
1
B
=
0
0
M
R
- 1
M
0
-
D
1
0
0
M
R
- 1
M
D
- 1
C
M
M
C
+
M
M
and the scalar pseudo-product is written:
Re [(A -
B)
- 1
]
- 1
= (A -
B) +
2
B (A -
B)
- 1
B
Consequently the calculation of Pz can be carried out like that of Sz while using exclusively
the arithmetic real one.
This approach requires the use of an auxiliary real matrix to store factorized
the dynamic matrix D (
).
·
Choice P = Im [(A -
B)
- 1
]
- 1
This choice corresponds to the partly imaginary approach of the operator S.
Formally we have:
Im [(A -
B)
- 1
]
- 1
=
1
[(A -
B) B
- 1
(A
B) +
2
B B
- 1
B]
The matrix B is regular under the condition necessary and sufficient that M is it and
B
1
=
0
M
1
- M
- 1
- M
- 1
C
M
- 1
One obtains then:
Im [(A -
B)
- 1
]
- 1
=
1
2
K
2
C
2
M
K
-
2
M
+ K
- 2
M
+ C
If M is singular, one can establish this equality by defining pseudo-opposite
B
1
of B
by
B
B
1
B
Z = B Z
O
ù
Z =
X
y
X
, y
IR
N
,
y
K
E R M -
0
The calculation of Pz can then be carried out without assembling the matrix P explicitly and while using
exclusively the arithmetic real one.
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5.3.3 Cost of the Lanczos phase
The cost report corresponds to the allowance of additional vectors (3 realities and 1 complex or real)
and with the allowance of the dynamic matrices used for the calculation of the operators S and P.
The following table summarize these allowances
= 0
=
IR *
= I
I IR
=
+
I
IC
Approach
S
K
IR
nxn
D (
)
IR
nxn
D (
)
IC
nxn
D (
)
IC
nxn
part
P = B
-
-
-
-
real
P = Re (A
- 1
)
- 1
-
-
K
IRnxn
D (
)
IR
nxn
Approach
S
D (
)
IC
nxn
D (
)
IC
nxn
part
P = B
-
-
Imaginary
P = Im (A
- 1
)
- 1
-
-
The cost in operation is divided into a fixed blow and a blow depend on the number of vector of Lanczos
to calculate.
The fixed cost corresponds to factorization LDLT of the additional matrices (carried out in
the arithmetic associated one) and is worth
O (B
2
N) if B is the bandwidth common to the matrices M, C and K.
The calculation of a vector of Lanczos requires:
·
2 scalar products of vector of I
R
2n
:
2
O (2n)
·
1 linear combination of 3 vectors of I
R
2n
:
3
O (2n)
·
the calculation of Sz:
3 products matrix-vector in I
R
N
:
3
O (2bn)
1 linear combination of 3 vectors in
IC
N
:
3
O (2n)
1 descent-increase in I
C
N
:
O (2bn)
1 scalar product - vector in I
C
N
:
O (N).
·
The calculation of Pz
For P = b:
3 products matrix-vector in IR
N
:
O (2bn)
For P = Re (A
- 1
)
- 1
:
5 products matrix-vector in IR
N
:
5
O (2bn)
2 combinations linear of 3 IR vectors
N
:
6
O (N)
1 descent gone up in IR
N
:
O (2bn)
1 linear combinations of 5 IR vectors
N
:
5
O (N)
For P = Im (A
- 1
)
- 1
:
6 products matrix-vector in IR
N
:
6
O (2bn)
2 linear combinations of 4 IR vectors
N
:
8
O (N)
Broadly the cost of the Lanczos phase is in
O (B
2
N) + 10 m
O (2bn).
5.4
Implementation in Aster
The matrices M and C are symmetrical semi-definite positive and the matrix K is symmetrical regular
indefinite. The pseudo scalar product retained corresponds to the extension of that proposed by Neuman and
Pipano [R5.01.01].
This algorithm is available in
Aster
by the operator
MODE_ITER_SIMULT
.
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R5.01 booklet: Modal analysis
HI-75-7816/A
5.4.1 Parameters of the implementation
The spectral problem with real operator is parameterized by:
·
value of the spectral shift
IC,
·
the choice of Re (A
- 1
B) or of Im (A
- 1
B)
The pseudo scalar product is then Re (A
- 1
)
- 1
or Im (A
- 1
)
- 1
and allowances and factorizations of
additional matrices are carried out has minimum in type and a number, this in agreement with the table
paragraph [§5.3.3].
5.4.2 Under space of approximation
The vector R
0
IR
2n
generating under space of approximation breaks up into
R
0
=
R
0
H
R
0
B
where
R
0
B
,
R
0
H
R
N
; the choice selected consists in posing
R
0
H
= 0 and to fire by chance the components from
R
0
B
,
while imposing to him null components in ker Mr.
If the dimension of under space is not specified, it is calculated by the empirical formula:
m = 2 Min (max (p+7), 2p, N)
where p is the number of couples of elements suitable to approach.
the dimension of the subspace of approximation are doubled because the couples of clean elements
complexes arise per combined pairs.
5.4.3 Strategy of reorthogonalisation
The use of arithmetic with finished precision, deteriorates the properties of orthogonality of the vectors of
Lanczos and with them the rate of convergence of the approximate couples.
The strategy of complete reorthogonalisation ensures the orthogonality of all the vectors of Lanczos, and
the algorithm then has a behavior close to that into arithmetic exact.
This strategy of reorthogonalisation forces to preserve the vectors P.Q
J
j= 1,…, Mr.
The reorthogonalisation of the vectors is carried out by the process of Gram-Schmid modified cf.
[R5.01.01].
5.4.4 Implementation of the Lanczos phase
The selected implementation is that described in [R5.01.01] within the framework of the generalized problem.
It is summarized by:
·
Inputs
:
-
the matrices P and S: i.e. the matrices M, C and K and factorized the LDLT of the matrices
dynamic additional.
-
m the number of vectors to be generated,
-
R
0
the vector generating under space of Krylov,
-
precision of orthogonalization and the number maximum of authorized reorthogonalisation.
Code_Aster
®
Version
2.0
Titrate:
Algorithms of resolution for the quadratic problem
Date:
19/06/92
Author (S):
D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
Page:
23/26
Manual of Reference
R5.01 booklet: Modal analysis
HI-75-7816/A
·
Exits
:
-
vectors of Lanczos (Q
1
, Q
2
,…, Q
m
),
-
the diagonal (
1
,
2
,…,
m
) and the on-diagonal (
1
,
2
,…,
m
) of the matrix tridiagonale
T
m
-
the vector (
1
,
2
,…,
m
) of the pseudo scalar products of the vectors of Lanczos.
·
Algorithm
:
-
Generation of the first vector Q
1
and of the coefficients
1
,
1
and
1
-
Loop generation of Q
J
,
J
,
J
and
J
for J = 2,…, m
For J = 2,3,…, m to make
Calculation of the direction of Q
J
Standardization of Q
J
, calculation of
J
and storage of Pq
J
Réorthogonalisation so necessary compared to Q
J
for I = 1,…, j-1
Reactualization of Q
J
,
J
,
J
and
J
in the event of reorthogonalisation,
Calculation of
J
and
J
5.4.5 Restoration of the approximations for the quadratic problem
Approximations
J
(
m
)
,
X
J
(
m
)
clean couples of the quadratic problem result from
clean couples
µ
J
(
m
)
,
S
J
(
m
)
matrix J
m
T
m
by:
·
X
J
(
m
)
=
N
O
Q
m
S
J
(
m
)
extraction of the “high” part of the vector.
·
choice
of
J
(
m
)
, root of the flexible quadratic equation
J
(
m
)
with
µ
J
(
m
)
who checks the condition of
coupling M [O
N
] Qm
S
J
(
m
)
=
J
(
m
)
M
X
J
(
m
)
Code_Aster
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Titrate:
Algorithms of resolution for the quadratic problem
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D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
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Manual of Reference
R5.01 booklet: Modal analysis
HI-75-7816/A
6 Bibliography
[1]
[CUL.WIL.85] J.K.CULLUM & R.A. WILLOUGHBY, broad Lanczos algorithms for symetric
eigenvalue computations - VOL1 Theory, Birkhäuser, 1985.
[2]
[GOH.al.86.] I. GOHBERG, P. LANCASTER & L. RODMAN, Quadratic matrix polynomials
with has parameter, Advances in applied mathemetics 7, pp253-281, 1986.
[3]
[JEN.77] A. JENNINGS, Matrix computation for engineers and scientists, John Wisley & Sounds,
1977.
[4]
[MEI.67] L. MEIROVITCH, Analytical methods in vibrations, The MacMillan Co. N.Y., 1967.
[5]
[MUL.56] D.E. MULLER, A method for solving algebraic equations using year automatic
computer, Maths. . Wash. Flight 10, p 208-215,1956.
[6]
[PAR. SAA.87] B.N. PARLETT & Y. SAAD, Complex shift and invert strategies for real
matrices, Linear will algebra and its N°88 applications/89, pp575-595, 1987.
[7]
[ROS. 84] Mr. REED, Vibrations of the mechanical systems, methods analytical and
applications, Masson, 1984.
[8]
[WIL.65] J.H. WILKINSON, The algebraic eigenvalue problem, Oxford University Close,
London 1965.
[9]
[R5.01.01] D. SELIGMANN - Algorithms of resolution for the problem generalized with
eigenvalues. Note EDF HI-75/7815 - Documentation
Aster
[R5.01.01].
Code_Aster
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Titrate:
Algorithms of resolution for the quadratic problem
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D. SELIGMANN, R. MICHEL
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Manual of Reference
R5.01 booklet: Modal analysis
HI-75-7816/A
Appendix 1 Interpretation of the complex eigenvalues
In the case of a symmetrical damping and in internal absence of damping, relations
orthogonalities and owing to the fact that the clean elements appear per combined pairs, one has the relations
following:
I
T
C
I
I
*
T
M
I
=
C
I
m
I
= 2 R E
I
I
T
K
I
I
*
T
M
I
=
K
I
m
I
=
I
2
If one notes
I
=
I
±
I
I
, one can then define:
I
=
I
=
I
2
+
I
2
=
I
I
I
Re
I
I
I
I
2
+
I
2
I
and one can write the eigenvalue complexes in the following form:
I
=
-
I
I
±
I
I
1
I
2
for which one can give a physical interpretation of the eigenvalue
The imaginary part represents the oscillatory part of the solution
I
is the pulsation of the i-ème mode
the real term represents the dissipative character of the system
I
is the damping of the i-ème mode,
D
I
=
I
1
-
I
2
is the reduced damping of I
- éme
mode.
Physical interpretation of the clean vectors:
·
The physical significance of the existence of a clean vector complexes, lies in the fact that if
structure vibrates on a clean mode, its various degrees of freedom do not vibrate with the same one
phase ones compared to the others.
·
The modal bellies and nodes do not correspond of the stationary points, but
move during the movement.
Note:
·
one finds the conventional formulation of the deadened systems with 1 degree of freedom
K
I
=
m
I
I
2
E T
C
I
=
2
m
I
I
I
·
K
I
,
m
I
E T
C
I
are real and are many quantities intrinsic with a mode (modal quantities) and
dependant on the standardization of the mode.
We remind the meeting that the modes of the quadratic problem do not diagonalisent the matrices M, K and
C.
Notice on the real term of the eigenvalue:
·
If the real part of the eigenvalue is negative, then the clean mode is a movement
deadened periodical of pulsation
·
If the real part of the eigenvalue is positive, then the clean mode is a movement
periodical of amplitude increasing and thus unstable.
Code_Aster
®
Version
2.0
Titrate:
Algorithms of resolution for the quadratic problem
Date:
19/06/92
Author (S):
D. SELIGMANN, R. MICHEL
Key:
R5.01.02-A
Page:
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Manual of Reference
R5.01 booklet: Modal analysis
HI-75-7816/A
Appendix 2 linear Reductions
form
problem
quadratic
2
M
+
C
+ K
.
X = 0
generalized
(1)
M
0
0
- K
+
C
K
K
0
X
X
= 0
generalized
(2)
0
M
M
C
+
M
0
0
K
X
X
= 0
standard
(1)
1
2n
0
N
- K
- 1
M
- K
- 1
C
X
X
0
standard
(2)
2n
-
M
1
C
M
1
K
N
0
X
X
=
0
Notice
: to obtain the forms standards it is necessary to suppose:
·
M and K regular for the form (1)
·
M regular for the form (2)