On the Vibration and Buckling of Orthotropic Plates of Variable ThicknessPLATES OF VARIABLE THICKNESS
PLATES OF VARIABLE THICKNESS
in Partial Fulfilment of the Requirements
for the Degree
Master of Engineering
MASTER OF ENGINEERING (1972) McMASTER UNIVERSITY (Mechanical Enginee'ring) Hamilton, Ontario
TITLE: On the Vibration and Buckling of Orthotropic Plates of Variable Thickness
AUTHOR: Krishan Kumar, M. Tech. (I. I. T.)
SUPERVISOR: Professor M. A. Dokainish
NUMBER OF PAGES: vii, 67
SCOPE AND CONTENTS:
The problem of a thin, orthotropic skew plate of linearly
varying thickness for vibration and buckling analyses is formulated
under the assumptions of small-deflection theory of plates. Using the
dimensionless oblique coordinates, the deflection surface of the plate
is expressed as a polynomial series,- each term of which satisfying the
required polar symmetry conditions, and the natural_frequencies are
computed using Galerkin method. As is required in Galerkin method, the
assumed deflection function satisfies all the boundary conditions on
all the edges of the plate. For the skew plate, clamped on all the
four edges, numerical results for the first few natural frequencies
are presented for various combinations of aspect ratio, skew angle and
taper parameter. Convergence study has been made for typical configura­
tion of the plate and the limited available data is inserted therein
along with the computed results, for comparison.
(ii)
ACKNOWLEDGEMENT
The author wishes to record his gratitude to Dr. M.A. Dokainish
for his guidance and encouragement throughout the various phases of this
research.
The support by the National Research Council's Grant No. A-2726
is gratefully acknowledged.
OF Ofil'HOTROPIC SKEW PLATE OF VARIABLE THICKNESS
CHAPTER III VIBRATIONS OF ORTHOTROPIC CLAMPED PLATE OF 15
VARIABLE THICKNESS
3.2 Numerical Computations 18
3.4 Effect of Rotary Inertia in Plates of 29
Variabl~ Thickness
VARIABLE THICKNESS
CONCLUSIONS 33
REFERENCES 34
APPENDIX 36
Element of mass matrix
Constants in equation (2.16)
Element of stiffness matrix
12 (1-l>xy 'l>yx)
Gxy ho3
Thickness of the plate with taper
Thickness of the plate without taper
(v)
Mx, My, Mxy Moment resultants for the plate problem
Nx, Ny, Nxy Stress resultants for the plate problem
Po' mo Parameters relating rigidity constants
Qx, Qy Transverse shears in x- and y- directions respectively
t Time
w(x,y) Deflection of plate independent of time
x, y Cartesian rectangular coordinates of plate
x, y Dimensionless oblique coordinates of plate
Taper parameter
Skew angle
l) xy'
Oblique coordinates of plate
(vi)
parameter
parameter
(vii)
The parallelogrammic flat plates are oftenly used in structures
of high-speed air vehicles. As no exact solutions exist to the governing
differential equation for the vibration/buckling analysis of parallelo­
gramrnic plates of varying thickness and material properties, it requires
attention of the designer to analyse such plates. For a typical con­
figuration of the skew plate simply supported along all the four edges,
Seth[l]appears to be the first to have given the exact solution~ However, no other significant exact solutions exist for parallelogrammic
plates. Some solutions have been obtained by approximate methods for
a few of the various combinations of boundary conditions of these plates.
In the literature, particular emphasis exists on the vibration analysis
of cantilevered pla~e because of its importance in the aircraft industry.
As far as is known, data on the natural frequencies of isotropic as well
as orthotropic skew plates of variable thickness is not available in the
literature for different combinations of aspect ratio, skew angle and
taper parameter.
In recent years, several analyses of vibrations of isotropic as
well as orthotropic skew flat plates of uniform thickness have been
presented by various authors. These investigations were based on the
small-deflection theory of thin plates.
1
2
Kaul and Cadambe( 2Jhave used the Rayleigh's method to find
the natural frequencies of isotropic skew plates of uniform thickness for
various combinations of boundary conditions. The upper bound, as cal­
culated by Rayleigh's method, and the lower bound of the natural fre­
quencies become inaccurate with decrease in the skew angle. HasegawaL3]
has given the lowest natural frequency for isotropic clamped plate,
using the Ritz method. Hamada [ 4J has obtained the fundamental frequency
of rhombic plates by employing the Lagrangian multiplier method.
Conway and Farnham( SJ have found the values of the first natural
frequency of skew plates of uniform thickness, using the point-matching
technique. Argyris and BuckC6Jhave solved the problem by the finite­
element approach. Durvasula [ 7] has used the Galerkin method to get
the first 6 to 8 natural frequencies of isotropic clamped plates of
uniform thickness. Kumar and Pandalai [BJ have reported the values of
first six natural frequencies of orthotropic skew plates for three
combinations of boundary conditions. ' .
The present work is concerned with the study of free vibrations
of isotropic as well as orthotropic parallelogrammic plates of variable
thickness. In view of the fact that the values of natural frequencies
associated with higher modes of these plates are well above the exact
values, the effect of rotary inertia can be investigated in the
frequency analysis.
The analysis is based on assuming a simple polynomial represen­
tation for the deflection surface in dimensionless oblique coordinates
satisfying the required polar symmetry and the boundary conditions of a
3
skew plate, and then employing the Galerkin's method. Numerical results
for the parallelogrammic plates with different combinations of aspect
ratio, skew angle and taper parameter are presented. Both isotropic as
well as orthotropic plates are studied. The properties of orthotropic
material correspond to those of Maple 5-ply wood. The computer program
developed for the vibration problem generates the natural frequencies
of plates. The nodal patterns are not given here. However, for the
plates with skewness, orthotropy and linearly varying thickness, there
__
OF ORTHarROPIC SKEW PLATE OF VARIABLE THICKNESS
Based on the small-deflection classical plate theory, the
partial differential equation, that governs buckling and vibration with
the effect of rotary inertia, of the skew plate of variable thickness is
derived. Selecting the appropriate terms in the partial differential
equation, the problems of vibration as well as buckling can be dealt
with separately. The plate is considered to be made of orthotropic
material. The constants that belong to orthotropic plate refer to the
orthogonal system of axes. For the isotropic material, Poisson's ratio
is taken to be 0.3 •
Force equilibrium equation in the z - direction:
Referring to Figures 2.1 and 2.2, and assuming the deflections
and slopes to be small, summing up forces in the z - direction yields
dQx 0 Qy ~w dx dy + dy dx dy
dx ~ Nx ax
d Nx ~w ()2 w ~w + ( Nx + dx ) dy + dx) Ny dX-­
dx ( d x ~ x2 dY
~w ~ 2 w ~ w~Ny + ( Ny + dy) dx (- + dy) Nxy dx
~ dy d y2 d x
4
x
4-a
z
I
dy
~ r-------~---- 1
FIG. 2 ·I Forces acting on rectangular plate element.
aw+ o (aw)ax iJ y dx
ow+.2...r.aw)dx 1 ox \ax I
oW+L(oW)dyax oy ax
FIG. 2 ·2 Deformed middle surface of plate element.
aw+L(aw)d +..Q..[aw +..£..(aw)d] dox iJy ox Y ax ox iJy ox Y x
5
d w c)2w dW0 Nxy + ( Nxy + dy ) dx <- + dy ) - Nxy dy
() y d x d x~y ~
d N ~w d 2 w + ( Nxy +
xy dx ) dy <-· + dx ) c) x d y dxdy
c)2w : ~ h dx dy •••• (2.1)
-~
throughout by the elemental area dx dy leads to
d Qx c) Qy dW dW + + d ( Nx-- ) + d ( Ny )
~ x dY ax dx ~ 1Y
dW d w + _e_ ( Nxy-- ) +1- ( Nxy )ax J y d y dx
02 w •••• (2.2): f> h
d t2
Moment eguilibrium equations:
Referring to Figures 2.1 and 2.3, in summing the moments about
x - and y - axes, it is found that terms containing Nx, Ny and Nxy
yield differentials of higher order than the others and the equations
reduce to [ 9]
d Mxy 0 My f> h3 a3 w Qy - =
d x d y 12 ay ~ t2
5-a
z
6
The quantities on the right-hand side represent the effect of
rotary inertia. The moment equilibrium equation about z - axis is
identically satisfied.
M Dx ( w, xx + Vyx W, yy )x =
•••• (2.4)lJMy : ( w, yy + xy w, xx )
Mxy = 2 Dxy w, xy '
where subscripts after the comma refer to partial differentiation with
respect to the independent variable (s) •
In the above,
h3 Ex DX :
h3 Gxy Dxy :
( w, xx + )) yx
~2 c) < (i> I 12 ) - [­ ( h3 w, x ) + d
~ t 2 dx dY
c)w d wa d-- (Nx -) + ( N ) - dx dx c) y xy ~
0 w +_J_ ) •••• (2.6)( Nxy
dY d x
Assuming a harmonic solution
i tw ( x, y, t ) : w ( x, y ) e ~ • • • • (2. 7)
equation (2.6) takes' the form
+ 4 ( Dxy w, xy ) - (C> h cJ w
h3+ <f>~2 I 12)[.l_ ( h3 w, x ) + E_ ( w, y ) Jox dY
: a ( Nx w, x ) + d ( Ny w, y) + ~ ( Nxy w, y ) ~ x d-; Tx
8
Equation (2.8) is the differential equation for vibrations and
buckling of a rectangular orthotropic plate of uniform thickness.
The linear thickness variation of plate is assumed to be of the
form
b
( 1 )3 0
( 1 + o( y )3Dxy: DXo Yo b
where and are constants.Dxo' Dyo DXo Yo
Substituting equations (2.9) and (2.10) in equation (2.8) leads
to
y ( 1 + o( ( w, yy + '))xy
b
9
J2 y ( )3+ 4 [ DXo Yo 1 + c( - ( w, xy J -~ h0 t.U :
dxay b
y ('>ho 3 CU 2 y .< 1 + o{ - ) w + )31 + o( w,{-ix [< - x J
b 12 b
w, y]} dY b
:. d ( Nx w, x ) + _£_ ( Ny w, y ) + d ( Nxy w, y)
dX dy dx
Simplifying equation (2.11) reduces to the following
y )3( 1 + J... - [ D~~ ( w, + )Jyx w, x,{yy):-v xxxx
b
+ DYo ( w, yyyy + l) xy w, xxyy ) + 4 DXo Yo w, xxyy]
6o( y 2 [
+-- ( 1 +o{- ) Dyo ( w, yyy + ))xy w, xxy ) + 2 DXo Yo w, xxij b b
6 t:J•.2 y y + (l+o{_) ( w, yy + ))xyW) - f ho W 2 ( l+Q(.-)w
lT DYo b 'xx b
r11~ c..o2 y 3 3c( + [ ( 1 + o( - ) ( w, xx + w, yy ) +
12 b b
x=!/a ; y= 17/b
Fl G. 2·4 Coordinates of the plate in skew system.
10
and buckling of a rectangular orthotropic plate of variable thickness.
Transformation to oblique coordinates:
oblique coordinates are given by
x = ~ + '1 cos 0 y : '1l sin 0 •••• (2.13)
Using the transformation relations as given by equation (2.13)
in equation (2.12) and multiplying throughout by a4 I Dy , and also 0
with
we obtain
( Cot 2 e - 2 cot @ cosec 8 • w, + cosec2 e . w, )• • w,"jj"JJ ~~~~ ;j~~
+ ( cot4 e . w, ~~ 4 cot3 e cosec e . w, '? + 6 cot2 e cosec2 0 • 1~n 511
3 4 4 - 4 cot e cosec e . w, '1'?'>'/ + co sec e . w, ) ] a
1 '??1 '>') IY)
il
6o( 2 3+ - f [ ( - cot e . w, ~ + 3 cot2 e cosec e . w,
b 0
'~:> f$"1
2 3 3 cot e cosec 8 • w,'J'?"? + cosec 8 • w,'>?~IY) )
4+ ( ).)xy + p0 mo - )J xy ) ( - cot 8 • w, + cosec 0 • w, )J a 1ff ~~~
6 o(, 2
+-- f 0 [ w,"'t"f ( °lJxy + cot2 Q ) - 2 cot e cosec e . w, 'Y/ 1
2 4+ cosec 8 • w,'>')'Y} ]a
'l't"4 I'\ 2 . 2 { f 3 [ ( ) 2+ u AR a 0 w, + w, cosec 0 - 2 cot 8 cosec e . ~s 'Y/'Y]
J 3o( f 2.w, + (cosec 8 • w, cot e . w, ) } 01~ b ~ 'f
(cosec e • w, - cot 8 • w,~~ ) -sri
(cot2 8 • w, - 2 cot e . cosec e . w, + cosec2 e . w, >] ri i~ ·'~
4 a
·- ( cosec 8 • w,~ - 2 cot 8 • w,r )
2 JNy+ cosec 8 + ( cot 9 • w,f - cot 8 cosec e . w~ ) W
2+ ( cosec e . w,,, - cot 9 cosec e . w, ~
.... (2.14)
12
x • 1 /a y = 'Yj/b •••• (2.15)
equation (2.14) takes the form
w, xxXff + c3 w, xxy:Y-c4 w, xyyf
+ w, YYYY ) f 2 0 --­xxx C7 w, ixy + Cs w, xyy
( C10 w, .... xx - w, x:Y + w, -­yy )
(C13 w, x - C14 w, y ) : •••• (2.16)
c2
C3
In the above,
2 2 4 : P0 + 2 Po mo cot 8 + cot 9
: 't ( 4 p 0
: '1 2 ( 2 p0 mo cosec2 8 + 6 cot2
= '/ 3 ( 4 cot 6 cosec3 e)
e cosec Q )
e cosec2 9 )
: '{4 cosec4 8
'/2 ( p0 mo + 3 cot2 0 )
2 18 o( cot 0 cosec 0
6 o( 3 cosec e
o(. cot e
fo
f o
fo
y2 ( ))xy + cot2 8 ) 6 o( 2=
- i3 • 12 o(2 cot e cosec e-
i4 • 6 cl..2 cosec2 e . ;:
4 'A2. 2 cosec S: rr R
y ( 1T' 4
'J ( 4 2 3 ~ cot e )= Tr 'A R
2- '12 ( 1T4 • 3 o( cosec- 'A R
)
)
)
)
)
)
C23 : ( 2 Nxy cot e - Nx - Ny cot2 e ) DYo
a2
C24 ( 2 Nxy cosec 9 - 2 Ny cot e cosec e ) y= DYo
a2 2
a2
c26 = - ( Nx, -x - 2 cot 0 • Nxy, x .. + 'I cosec e . N xy, ... y
DYo
+ cot2 8 • Ny, ..,x - 'f cot 8 cosec 9 • Ny, y ) ;
: a 2 ( y cot 8 . cosec e • Ny, x - y cosec e . Nxy, x
- '12 cosec2 e . Ny, -y )
Equation (2.16) is the governing partial differential equation
for vibrations and buckling of orthotropic parallelogrammic plates of
linearly varying thickness.
The problem of free vibrations of orthotropic parallelogrammic
plate of variable thickness is governed by the partial differential
equation (2.16) in which the coefficients C1 to C9 remain unaltered
and C10 : C15, C11 : Cl6' C12 : C17, the other coefficients being
identically equal to zero.
Fot a skew plate, rigidly clamped with the axial tension con­
sidered to be negligible on all the four edges, the boundary conditions
are defined as
w = 0 and w,n : O, on all the edges (3.1)
In the above, n represents the direction of outward drawn
normal to the edge.
(Figure 2.4) are defined by
x : + 1 y : ± 1 •••• (3.2)
To satisfy the required conditions on the boundaries of the
plate, a deflection function is chosen in the form
15
16
. oa
L -2 2 -2 2 -m -n w <x, 'Y> • A (l-x ) (l-y ) x y , 1't't\
m : O, 1, • • • n : O, 1, .•. • ••• (3.3)
where Aunt are the undetermined parameters.
Let w (x, y) be represented in the form
w c"X, -Y> Autn p_ c-x, -Y> ,
m a 0 n a 0 '"" . r_ [_
•••• (3.4)
On substituting w, as defined in equation (3.4), into equation
(2.16), the equation reduces to a form
)..2 Fl c-x, "Y> - R, •••• (3.5)- where R represents the residual functional, and in general, has a
· non-zero value in the case of approximate .solution.
The Galerkin's method requires that the residual R is
minimized by orthogonalizing it, and thus we obtain the following resulting
equation:
- 1 - l • • • • (3. 6)
In a convenient form, the above equation is represented as
17
D(ik) B(ik)In the above, and represent the elements ofmn mn
stiffness and mass matrices of the system, re spec tively.
For example,
+ 1 + l (x, y)= J I [ F2 <fik] dx dy,
- 1 - 1
eous, homogeneous equations in the infinite undetermined parameters,
namely, Aoo' Ao1, •·• For computational task, only a definite
number of terms in the double series are considered and the validity
of this is tested from comparison with the limited available data. To
examine the convergence, different number of terms have been used in
the double series for typical configuration of the skew plate. The
characteristic equation is obtained by setting the determinant of the
coefficients of Arnn in the matrix equation (3.7) equal to zero.
18
3.2 Numerical Computations
Numerical work has been carried out to compute the first few
natural frequencies of rectangular and skew plates of uniform and
variable thickness for different combinations of aspect ratio, skew
angle and taper parameter. Natural frequencies for both isotropic and
orthotropic plates have been computed. As each one of the equations in
equation (3.7) in which (m + n) is even (odd) contains only those co­
efficients Arnn. in which (i + k) is also even (odd), it is easy to
notice that the matrix equation (3.7) splits into two sets, the even
set and the odd set. Thus, for the even (odd) set, (m + n) and (i + k)
are both even (odd), representing the modes that are doubly symmetric
or doubly antisymmetric (symmetric-antisymmetric or antisymmetric­
symmetric). Using 16 terms in equation (3.3) by taking both m and
n equal to 3, the matrices 0£ order 8 x 8 have been solved to get the
natural frequencies for the even set, using a CDC 6400 Digital Computer.
The even set includes the fundamental frequency of the plate configura­
tions considered herein. Due to the large amount of time being used up
in the numerical evaluation of the surface integrals, calculations
were restricted to solving matrices of size 8 and only for the doubly
symmetric and doubly antisymmetric modes. Convergence study has been
made by considering different number of terms in the double series for
typical configuration of the plate.
19
Numerical results for the fundamental as well as the next
higher natural frequency have been presented for isotropic and ortho­
tropic plates with aspect ratios equal to l and 1/2, skew angles equal
0 0 0to 90 , 75 and 60 , and taper parameters equal to O, 0.4 and 0.8. The
results are presented in terms of the dimensionless frequency parameter
).. • The natural frequencies are reported in the ascending order of their
magnitude along with the mode that is labeled accordingly. Tables 3.1
to 3.3 give the values of ).for skew isotropic clamped plates with some
finite taper and with no taper at all. The comparison of these results
with those of the other authors for the natural frequencies in the case
of plates of uniform thickness is also shown therein for few skew angles.
Tables 3.1 to 3.3 contain the results for various new configurations of
the skew plates of varying thickness. It may be seen that the results
of present investigation are probably more accurate than that of at
least some of the other investigators for the fundamental mode of skew
plates of uniform thickness.
Po : LO ; mo:l.0· ).): 0.3; a • 90°'
"/ ~ 1 2
4. 4385
10.969 10.970 *
11. 704
2.5699
2.7632
0
Po : 1. O mo • 1.0 )) = o. 3 ; 9 :: 750
"/ ~ 1 2
LO o.o
= 1. 0 ; mo = 1.0 ; l) = 0.3 ; e = 60oP0
y x o.o
* **
The results of numerical calculation for the orthotropic plates
are presented in Tables 3.4 to 3.6 for two aspect ratios, three skew
angles and three taper parameters. For the orthotropic plates, the
properties of the material used correspond to those of Maple 5-ply wood.
Although the results seem to be quite accurate for rectangular and skew
plates of uniform thickness, there is no indication, whatsoever, that
the results for other configurations of the plates are reasonably accurate.
Consequently, the study of convergence has been made for the isotropic
plate with '{ = 1/2 and 9 • 600 , using each time different number of
terms in the series expansion and solving different order matrices for
the eigenvalues. The results of this study are given in Table 3.7.
The convergence of double series used in representing the deflection
surface of the plate is seen to be satisfactory. It is also noticed
that the deflection function, when assumed in such a representative form,
is much more rapidly convergent in comparison with t.hat based on char­ . [ 7, s]
acteristic functions of a clamped~clamped uniform beam for the
uniform plate vibration problems. It is so because only 16 terms are
required to get better results with the deflection function assumed
herein as against the 36-term solution of the vibration problem making
use of the beam eigen functions in the case of uniform thickness skew
plates.
Table 3.7 shows that 16 terms, using m and n each equal to
3, are sufficient to obtain reasonably good estimates of the natural
frequencies up to the first two modes for plates with the skew angle
not below 60°. However, 16 terms may not be sufficient to obtain good
results for any skew angle. The accuracy of the results presented for
TABLE 3. 4
Po : 1. 7664 mo = o. 3668 lJxy = 0.1206 e • 90°
y A 1 2
7.3258
Po : 1. 7664 mo = o.3668 •' ).)xy • O. 1206 a = 75°
'i 1 2D;:J o.o1.0 4.9889 13.750
4.9891 * 13.516 * 0.4 5. 2043 14. 677
0.8 5.7418 16. 770
0.4 4.2859 6.2567
0.8 4.5151 7.5658
FREQUENCY PARAMETER OF SKEW ORTHOTROPIC
CLAMPED PLATES OF VARIABLE THICKNESS
Po : 1. 7664 mo = o.3668 l>xy • O. 1206 e = 600
y 1 2DSJ . o.o1. 0 5.6585 15.012
5.6590 * 14. 747 *
0.4 5.8823 15.856
7.17900.4 4.6391
CONVERGENCE sruoy FOR SKEW ISOTROPIC CLAMPED PLATES
: 1. 0 mo • 1.0 ).) = 0.3 o( = o. 0 "/. 0.5 e • 60°P0
1 I 2 I 3 I 4
3. 2796 ,'r 9.8631 * ­I4 x 4 - 6 x 6 3.2705 6.1343 9.5833 - 8 x 8 I 3. 2649 5.6668 9. 5446 13.877
12 x 12 3.2648 5.6664 9.5036 13.759
* Figures in the Table represent the frequency parameters.
N
plates with different configurations ·and material properties can be
seen to be reasonable for skew angle as low as 60°. However, the re­
sults of calculation of the natural frequencies associated with higher
modes of plates at low angles of skew are only rough estimates as a
consequence of which these are not included herein. The natural fre­
quency of plate tends to increase with the orthotropic property of the
material. The effect of the skew angle on the fundamental frequency of
these plates, whether isotropic or orthotropic, appear to be practically
the same. The frequency values for the fundamental mode of plates with
different skew angles but with same aspect ratios as well as the same
taper parameters differ almost by a constant quantity. In other words,
the effect of the skew angle on the lowest natural frequency of plates
with the same aspect ratios and same taper parameters is almost the
same. However, the frequency values continue to attain larger magnitude
for the orthotropic plates in comparison-with the isotropic ones. This
is probably attribu~ed to the difference of material properties in the
orthogonal directions.
3.4 Effect of Rotary Inertia in Plates of Variable Thickness
The problem of vibrat'ions of orthotropic skewed flat plate of
variable thickness, with the effect of rotary inertia considered, is
governed by the partial differential equation (2.16) in which all the
coefficients are defined as in Chapter II, the coefficients c23, C24,
C25, C26 and C27 being equal to zero.
In view of the fact that the values of the natural frequencies
associated with higher modes of these plates are well above the exact
values, the effect of rotary inertia needs to be explored further,
making use of equation (2.16) with the appropriate coefficients as
said earlier.
Numerical results for the natural frequencies can be computed
using the deflection function as defined in equation (3.3) and then
employing Galerkin's variational method.
theory, the buckling problems of thin, orthotropic, flat parallelo­
grarnmic plates of uniform or variable thickness under combined loading
are governed by the partial differential equation (2.16). In equation
(2.16), the eigen value parameters A and A are identically equal to R
zero, the coefficients being defined in the same manner as in Chapter II.
In the case of a skew plate, clamped on all its edges, the
boundary conditions are defined as
w = 0 and w, n : 0 , on all the edges, • • • • ( 4. 1 )
where n stands for the direction of outward drawn normal to the edge
of the plate.
The boundaries of the plate in the dimensionless oblique co­
ordinates (refer to Figure 2.4) are defined by
x = + 1 y I: + 1 • • • • ( 4. 2)
In order that the required boundary conditions on the edges of
the plate are satisfied, the deflection function is suggested to take
the form 00 oO
-2 2 -m-n Ill xw c-x, y) ( 1 - -x2 )2 (1 - y ) L [ '\no y '
m = o, 1 n : O, 1
.... ( 4. 3)30
Using the deflection function, as defined in equation (4.3),
in the governing differential equation for the buckling problems and
then employing Galerkin method, the coefficients of buckling in their
dimensionless forms can be computed for different combinations of plate
aspect ratio, skew angle and taper parameter under the action of uniform
normal and shear stresses individually as well as various loadings of
the mixed type. Both isotropic and the orthotropic plates can be
studied.
Furthermore, equation (2.16) can be made use of in the investi­
gation of the buckling behavior of isotropic as well as orthotropic
plates of variable thickness under the action of loadings that are of
non-uniform type. Numerical results for the buckling coefficients
(eigenvalues) can be obtained using the·deflection function as defined
in equation (4.3) and then employing Galerkfo's variational approach.
It is hoped that·the deflection function as represented in the
simple polynomial form should prove more rapidly convergent than the
characteristic functions of a clamped-clamped uniform beam in the solu­
tion of uniform plate buckling problems and this aspect needs to be
explored further.
Numerical results for the buckling coefficients of rectangular
or skew isotropic and orthotropic plates of uniform or variable thick­
ness can be computed for any combination of aspect ratio, skew angle
and taper parameter by the: method given herein. As each equation in
the set of equations represented by equation (3.7) in which (m + n)
is even (odd) contains only those coefficients Anm in which (i + k)
is also even (odd), it is easy to see that the matrix equation (3.7)
splits into two distinct sets, the even set and the odd set, for (m + n),
(i + k) even and (m + n), (i + k) odd, representing modes that are doubly
symmetric or doubly antisymmetric and symmetric-antisymmetric or anti­
symmetric-symmetric respectively. Using the appropriate number of terms
that are required for obtaining reasonably good results, the matrices
of half the size of the total number of terms considered in the double
series can be solved, for both even and odd cases. The appropriate
number of terms can be decided upon by the convergence study for a few
representative configurations of the plates.
33
CONCLUSIONS
The problem of free vibrations of thin orthotropic skewed flat
plates is studied using the Galerkin's variational method. The plates
considered herein are clamped on all the four edges and are of uniform
or variable thickness. A simple polynomial representation is chosen for
the deflection surface which satisfies the required polar symmetry and
the boundary conditions of the plate. The numerical computations were
carried out, in most of the cases, by using 16 terms in the double
series of the deflection function. Numerical results for plates with
various configurations are presented and the limited available data is
inserted therein, wherever possible. The results for various new con­
figurations of isotropic and orthotropic plates are also given. Conver­
gence study has been made for a typical configuration of the plate. It
appears that a 16-t~rm solution is quite satisfactory for obtaining
reasonably accurate results for plates clamped on all the edges. However,
for higher modes at low skew angles it appears that more than 16 terms
would probably be required to ensure better convergence. With the in­
crease in aspect ratio, the natural frequency of the plates increases.
Also, with the orthotropic property of the material, the frequency values
tend to increase for various configurations of the plates.
34
REFERENCES
1. SETH, B.R.; "Transverse Vibrations of Rectilinear Plates", Proc.
Indian Acad. Sci., Sec. A, Vol. 25, Jan. 1947, PP. 25-29.
2. KAUL, R.K. and CADAMBE, v.; "The Natural Frequencies of Thin Skew
Plates", The Aeronautical Quarterly, Vol. 7, 1956, PP. 337-52.
3. HASEGAWA, H.; "Vibration of Clamped Parallelogranunic Isotropic
Flat Plates", Journal of the Aeronautical Sciences, Vol. 24,
1957, PP. 145-46.
4. HAMADA, M.; "Compressive or Shearing Buckling Load and Fundamental
Frequency of a Rhomboidal Plate with All Edges Clamped", Bulletin
of Jfil1E, Vol. 2, 1959, PP. 520-26 •
. 5. CONWAY, H.D. and FARNHAM, K.A.; ·~he Free Flexural Vibrations of
Triangular, Rhombic and Parallelogrammic Plates and Some
Analogies", International Journal of Mechanical Sciences, Vol.
7, Dec. 1965, PP. 811-16.
6. AI{GYRIS, J.H. and BUCK, K.E.; "The Oscillations of Plates of
Arbitrary Shape", paper presented at the Symposium on Numerical
Methods for Vibration Problems, July 12-15, 1966, Institute of
Sound and Vibration Research, University of Southampton, England.
35
7. DURVASULA, s.; '~atural Frequencies and Modes of Clamped Skew
Plates (Tech. Note)", AIAA Journal, Vol. 7, No. 6, June 1969,
PP. 1164-67.
8. KUMAR, K. and PANDALAI, K.A.V.; '~ree Vibration of Orthotropic
Skew Plates", Studies in Structural Mechanics, Hoff's 6Sth
Anniversary Volume, Stanford University, U.S.A., 1970, PP. 219-41.
9. LEISSA, A·W·; "Vibration of Plates", NASA SP-160, 1969, National
Aeronautics and Space Administration, U.S.A.
APPENDIX
HVD7,CM60000,T200. KUMAR K RUN < S, ' , ' , ' ' , 1 ) SETINDF. REDUCF. l(i().
6400 END OF RECORD PROGRA~ TST <tNPUT,OUTPUT,TAPE5=INPUT,TAPF6=0UTPUT>
c c ********************************************************** c C THIS PROGRAM COMPUTES THE FIRST FEW NATURAL FREQUENCIES OF C ISOTROPIC AND ORTHOTROPIC SKEW PLATES OF ANY ASPECT RATIO ( AND ANY ANGLF OF SKr=-W , \NITH UN!f:"ORM OR LINf:"ARLY VARYING C THICKNFSS TN ONF DIRECTION c ( ********************************************************** c C DESCRIPTION OF PARAMETERS c C ELPHA •••••••• TAPER PARAMETER C GAMMA •••••••• ASPFCT RATIO(A/B) C THETA •••••••• ANGLE OF SKFW C Pn•••••••••••l/ATFR!AL CONST~NT C OO•••••••••••MATFPIAL CONSTANT C XNEU ••••••••• POISSON S RATIO C B••••••••••••MASS MATRIX C D••••••••••••STTFFNESS MATRIX C Rl•••••••••••TNVFRSF. OF MASS MATRIX C KOUNT••••••••A COUNTER USED FOR DATA CARDS C KT•••••••••••NUMPFR OF DATA CARDS PLUS 1 C MMAX ••••••••• MAXTMUM VALUE OF M IN THF FIRST SERIES C NMAX ••••••••• MAXIMUM VALUE OF N IN THE SECOND SERIES C NTX •••••••••• NUM~FR OF TERMS IN THE X-DIRECTION C NTY •••••••••• NUMRER OF TERMS IN THE Y-DIRFCTION C ZERO ••••••••• VALUE 8ELOW WHICH AN ELEMENT OF INVERTEQ MATRIX IS O C IERR ••••••••• INDICATOR IN SURROUTINF INVMAT FOR SUCCESS OF INVERSION C Nle••••••••••WORKING ARRAY OF N2 FLFMFNTS IN SURROUTINE INVMAT C N2•••••••••••SIZF OF MATRICES ~•D•Dl AND CKM ( FVR •••••••••• RFAL EIGENVALUE PARAMETER C EVI •••••••••• IMAGINARY EIGENVALUE PARAMETER C VECR ••••••••• REAL EIGENVECTOR C VECI ••••••••• IMAGINARY EIGENVECTOR C INDTC •••••••• ERROR INDICATOR IN SURROUTINE EIGENP C CKM •••••••••• FINhL MATRIX TO COMPUTF THE FIGFNVALUFS AND EIGENVECTORS C OMFGA •••••••• r>I~FNC,JONLESS FRFQUFNCY PARllMF.TF.R c C SUBPROGRA~S CALLED c C ELEMENT,JNVMAT,MULTI,EIGENP c
DIMFNSION R( 2, 2),D( 2' 2),81( 2, 2),CKM( ;>, 2),QMEGA( 2) DIMENSION NJ( 2l•EVR< 2),FVI( 2hVECR( 2, 2hVECI( 2, 2),YNDIC< 2) EXTERNAL FUNC1,FUNC2 COMMON MI.NI,xI.xK,ELPHA,GAMMA,THETA,PO,QO,xNEU,NTX,NTY K 0UNT= 1 KT=?. MMAX=l NMAX=l N2=2
38
NTX=MMAX+l NTY=NMAX+l PI=3.14159265358979 PI2=PI**2 ZERO=l.E-07
1 READ(5,2) ELPHA,GAMMA,THETA,PQ,QQ,XNEU 2 FOR~ATC6F10.4)
WRITFC6,5) FLPHA,GAMMA,THETA,PO,QQ,XNEU CALL FLEMENT (R,D,N2,KOUNT) DO 3 Kl=l,N2 DO 3 K2=1,N2
3 BlCKl,K2)=R(Kl,K2l CALL INVMAT (Bl,N2,N2,ZERO,IERR,Nll CALL MULTI (81,D,CKM,N2,N2tN2l CALL ETGFNP (N?,N?,CKM,48.0,FVR,FVI,VFCR,vEcI,INDICl DO 4 KK=l,N2 OMFGACKK)=4.*SORTCEVRCKKI )/PI2 WRITE(6,6l KK,OMEGA(KKl WRITEC6,7l CVECR(J,KK),J=l,N2)
4 CONTINUE KOUNT=KOUNT+l !FC~nUNT.FO.~T) GO TO 8 r,0 Tn 1
5 FORMATC1Hl,9X,*FLPHA=*,F7.4,5X,*GAMMA=*,F?.4,5X,*THETA=*,F7.4,5X'* 1PO=*,F7.4,5X,*OO=*'F7.4,~X,*XNEU~*'F7.4,////)
6 FORMATC//,10X,*OMEGA(*,I2,*)=*'E13.6,30X,*THE CORRESPONDING -EIGENV lECTOR IS ••••••••••*)
7 FORMAT(/,70X,E13.6) ~ STQP
c c SUBROUTINE FLFMFNT. c c PURPOSE c
TO COMPUTE THE ELEMENTS OF THE STIFFNESS A~D MASS MATRICES c c
( USAGF c
r C ~
c AL•••••••••••ARRAY SPECIFYING LOWER BOUNDS OF INTEGRATION c RU•••••••••••ARRAY SPECIFYING UPPER BOUNDS OF INTEGRATION c R••••••••••••MASS MATRIX c De•••••••••••STJFFNFS~ MATRIX
,C N2•••••••••••STZF OF THF MATRTCFS ~ AND n KOUNT •••••••• A cnUNTER USED FOR DATA CARDS. c
c ELPHA •••••••• TAPER PARAMFTER c NTX •••••••••• NUMRER OF TERMS IN THE X-DIRECTION c NTY••••••••••NUM9ER OF TERMS IN THE Y-DIRECTION c le•••••••••••ROW SURSCRIPT OF MATRICES R AND D c J••••••••••••COLUMN SU8SCRIPT OF MATRICES B AND D
.c ~,N,J,K •••••• PnsITIVF INTfGERS c IDS •••••••••• COUNT~R IN NDINT CONTROLLING INTFRMFDIATE OUTPUT
c
39
C IMAX•••••••••MAXIMUM NUMBER OF ITERATIONS IN NDINT C Rl•••••••••••FINAL VALUE OF THE INTEGRAL OF FUNCl C R2•••••••••••FINAL VALUE OF THE INTEGRAL OF FUNC2 C TZERO •••••••• SFCOND LAST ESTIMATE OF INTFGRAL OF FUNCI OR FUNC2 C INDICl•••••••tRROR INDICATOR IN EVALUATING INTEGRAL OF FUNCl C INDIC2 ••••••• FRROR INDICATOR IN EVALUATING INTEGRAL OF FLiNC2 ( C
NOFl•••••••••TOTAL NOF2 ••••••••• TOTAL
NUMBER NUMBER
OF OF
CALLS CALLS
MADE MADE
TO TO
SU8PROGRAM SUBPROGRAM
FUNCl FUNC2
C ACCR ••••••••• ROUND-OFF INDICATOR FOR EXTRAPOLATION c C SUBPROGRA~S CALLED c C NDINT,FUNC1,FUNC2 c
DIMENStON AL(2),RU(2),p(N2,N2l,D(N2,N2l EXTERNAL FUNC1,FUNC2 COMMON MI,Nr,xr,xK,ELPHA,GAMMA,THETA,PO,QO,xNEU,NTX,NTY AL(l)=AU2l=-l.O BUC1l=BU(2)=+1.0 IDS=O tMAX=l5 Kl=l<'..?=1 M=4 WRJTE<6t9) DO 6 Il=l,NTX II=Il-1 XI=FLOAT<IIl DO 6 KA=ltNTY I( I.=K A-1 Xl<'..=FLOA T (KI l IK=II+KI IKODD=MOD(!Kt2l IF<IKODD.EQ.ll GO TO 6 DO 5 Ml=l,NTX Mt=MJ-1 DO 5 M?=J,f\ITY NI=M2-1 MN=MI+NI MNODD=MOO(MN,2) IF<MNODD.EQ.ll GO TO 5 · IFCK2.LT.Kll GO TO 4 IF<KOUNT.E0.1) GO TO 1
1 IF<KOUNT.GT.l) GO TO 2 CALL NDINT <FUNc1,2,AL,RLJ,1.o,o.5E-9,1,IMAX,Rl,TZERO,INDIC1,NOFlt
lMtIDStACCRl 2 CALL NDINT (FUNC2,2tALtRUtl.Ot0.5f-7,1,IMAX,R2,TZFRO,INDIC2•NOF2t
lMt!DS,ACCRl TF1KOUNT.GT.ll GO TO 3 R(Kl,K2)=Rl-l.O
3 DCK1,K2l=R2-1.0 WRITF(6,10l Kl,K2,B<Kl,K2),D(Kl,K2~,MI,NI,IT,KI
4 K2=K2+1 5 CONTINUE
Kl=Kl+l K?=l
6 CONTINUE DO 8 Kl=ltN2
40
DO 7 K2=ltN.?. IFCK2.LE.Kll GO TO 7 R(K2tK1 l=RCK1 tK2l DCK2tKll=DCKltK2l
7 CONTINUE 8 CONTINUE 9 FORMATC4Xt*L*t2Xt*J*•l1Xt*MASS MATRIX*tlOXt*STIFFNESS MATRIX*t4X•*
1M*t4Xt*N*t4Xt*l*t4Xt*K*t//l 10 FORMATC2Xt2I3t2F24.9t4I5l
RFTURN END SUBROUTINE NDINT CFUNCtN•A•BtSUMTtACCURtKSTARTtKSTOP,TONEltTZFROlt
1 INDICtNOFEVtMtIDStACCRl c c SUBROUTINE NDINT. c c IDENTIFICATION - CLPG00165 NDINT c c NDINT I N-DIMFNSIONAL INTEGRATION SUBROUTINF c c AUTHOR I DATE c A.VELD• NRC COMPUTATION CENTRE• NRCt JUNE lOt 1968 c c SOURCE LANGUAGE c CDC SYSTFM I
c c PURPOSE c CALCULATF THE c N-DIMENSIONAL c
6400 FORTRAN IV
c CALLING SEQUENCE c c CALL NDINT CFUNCtNtAtRtSUMTtACCURtKSTARTtKSTOP,TONEtTZEROt c 1 INDICtNOFEV,M,IDStACCRl (
c *** ON ENTRY *** c c FUNC - NAME OF USER SUPPLIED FUNCTION SUBPROGRAM c - FUNC MUST BE DECLARED EXTERNAL IN THE c N - THE DIMENSION OF THE INTEGRATION c A - AN ARRAY SP~CIFYING THE LOWER ROUNDS
·C B - AN ARRAY SPECIFYING THE UPDER ROUNDS NOTE THAT ACil AND RCIJ CORRESPOND TOc
CALLING PROGRAM
OF THE INTEGRATION. OF THE INTEGRATION.
THE ITH VARIABLE. IN THE I-DIMENSIONAL CASE A AND R NFED NOT BE DIMENSIONED
c c
SUMT - AMOUNT ADDED TO TONE AND TZERO AFTER THF CONVERGENCE c TEST HAS BEEN PASSED. THE CONVERGENCE TEST IS c JFCDARSCTONE-TZEROJ.LT.ACCUR*DABSCTZERO+SUMTl
·c ACCUR - ACCURACY REOUIRED KSTARTt KSTOP- SPECIFY THE ITERATION SEOUENrE. THF FIRST TIMFc
c NDINT rs CALLED KSTART MUST RE EQUAL TO 1 AND c KSTOP.GE.l• IN SUCCEEDING C~LLS KSTART=PREVIOUS
KSTOP+l AND KSTOP.GF.THIS KSTART.c c M - Ifl),J(2)•!<3lt REOUIRE M**Ntf2*M>**NtC3*MJ**N•••• c FUNCTION EVALUATIONS c IDS - CONTROLS TIMING AND INTERMEDIATE OUTPUT. ( :0 NO TIMER OR OUTPUT ( =L IMPLIES TI~ER rs USFD AND INTERMEDIATE OUTPUT GOES
jND INT -· LND INT -:ND INT
NDINT L. i;:ND INT ­
NDINT f. ND INT ND INT E
cND INT ND INT IC ND INT 11 ND INT 12 ND INT 13 ND INT 16 ND INT 17 ND INT 18 ND INT 19 ND INT 20 ND INT 21 ND INT 22 ND INT 23 NDINT 24 NDINT 25 ND INT 26 ND INT 27 ND INT 28 ND INT 29 ND INT 30 ND INT 31 ND INT 32 ND INT 33 ND INT 34 ND INT 35 ND INT 36 ND INT 37 ND INT 38 ND INT 39 ND INT 40 ND INT 41 ND INT 42 NDJNT 43
41
C 6N DATA SET L. NDINT 44 C NDINT 45 C *** ON EXIT *** NDINT 46 C NDINT 47 C TONF - LAST ESTIMATE OF THE INTFGRAL NDINT 48 ( TZERO - SECOND LAST ESTIMATE OF THE INTEGRAL NDINT 49 C NOTF THAT SUMT HAS BEEN ADDED TO ROTH TONE AND TZERO NDINT 50 C INDIC - AN FRROR INDICATOR NDINT 51 C INDIC=O NO ERROR NDINT 52 C =1 ACCURACY REQUIRED NOT OBTAINABLE IN KSTOP ITERATIONS NDINT 53 C =-1 IF ANY OF THE FOLLOWING CONDITIONS OCCUR NDINT 54 C KSTART.LE.O.OR.KSTOP.LT.KSTART.OR.GT.24 NDINT 55 C N.GT.10 NDINT 56 C ACCUR.LT.O.ODO NDINT 57 C =K IF ACCR*ACCUR.LT.0.50-15 FOR STEP K. TO PROCEED WITH NDINT 58 C A LARGER ACCUR VALUE SPECIFY KSTART=INDIC+l NDINT 59 C NOFEV - TOTAL NU~BER OF CALLS MADE TO THE FUNCTION SUBPROGRAM NDINT 60 C ACCR - ROUND-OFF INDICATOR FOR THE EXTRAPOLATION. NDINT 61 C NDINT 62 C THE PROGRAM CAN BE RE-ENTERED BY RESETTING KSTART AND KSTOP NDINT 63 C AS DESCRIBED NDINT 64 C NDINT 65 C ~ETHOD NDINT 66 C THF MIO-POINT FOR~ULA AND AN EXTFNDFD RICHARDSON EXTRAPOLATION NDINT 67 C TECHNIQUE IS EMPLOYFD. NO DUPLICATE FUNCTION CALLS ARE MADE. NDINT 68 C NDINT 69 C SUAPROGRAMS REQUIRED NDINT 70 C THE USER MUST SUPPLY THE FUNCTION SUBPROGRA~ FUNC (REPLACE BY NDINT 71 C USER SPECIFIFD NAME). THE FUNCTION SHOULD RF DECLARED AS FOLLOWS.NDINT 72 C NDINT 76 ( ON EXIT FUNC SHOULD CONTAIN THF FUNCTION FV~LUATFD FOR X WHERE X NDINT 77 c JS AN N-DIMFNSIONAL VECTOR. FOR THE 1-DIMENSIONAL CASE X NEED NOTNDINT 72 c BE DECLARED AN ARRAY. NDINT 79
NDINT SCc c THE PARAMETER I INDICATES THAT XCll• X(2l••••'X(Il• HAVE NDINT 81
c CHANGED SINCE THE LAST FUNCTION CALL. THE OTHER PARAMETERS NDINT 82 NDINT 8~'c HAVE NOT CHANGED IN VALUE. NDINT 84c
r--------------+--------------------------------------------------------MoJNT R~ C NDINT 90
c WORK AREAS NDINT 9? NDINT 94c
DIMENSION HC 10) ,LMBDA( 10) •X<lOl ,TC24l ,W1(25) •W2(49l NDINT 95 NDINT 9f::c
LOGICAL SIGN NDINT 97 NDINT 9f
c c
DEFINE THE MACHINE ACCURACY. SEE PROGRAMMERS GUIDE. NDINT 9S c NDINTlOC
BMACCUR=l.E-15 NDINT101 • NDINT102c
IF(KSTART.LF.O.oR.KSTOP.LT.KSTART.OR.KSTOP.r,T.24.0R.N.GT.10.oR. NDINTlOl 1 ACCUR.LT.0.0) INDIC=-1 NDINTlO~
c C c
NDINTlOIF<INDIC.EO~O> GO TO 112 NDINTlO·TONEl=SUMT NDINTlliTZEROl=SUMT
NOFEV=O NDINTll NDINTllRFTURN NDINTlL
ALL ERROR CONDITIONS PROCESSED NDINTll· NDINTll:
DO 100 K=KSTART,KSTOP NDINTllt NDINTll­
START WITH THE ACTUAL INTEGRATION. NDINTllf NDINTllc
IF(K.GT.ll GO TO 200 NDINT12C IF<IDS.EQ.0) GO TO 102 NDINT12: WR!TE(6,3000) SUMT,ACCUR,M,(J,A(I),8(J),I=l,Nl NDINT12~
FORMAT<*ISTART OF INTEGRATION USING NDINT*/ NDINT12: 1 *O*'* SUMT=•,E22.14,5x,• ACCUR=*'E9.?. ,5x,* M=*,I4// NDINT12L 2 * I*,14X,*A(l)*,23X,*B(Il* //(* *,I4,5x,E22.14,5x,E22.14) )NDINT12~
WRITE (6,3001) , FORMAT(*O*'*
INITIALIZE
TONE=O.O ACCR=l.O PBIMAI=l.O DO 101 I=l tN LMBl)A <I l =1 RTMAI=P (I l-A. (I l
K*,J4X,*I(K)*,23X,*J<Kl*•l4X,*NOFEV*'
VARIABLES FOR K=l
JF(RIMAI.F0.0.0) GO TO PBIMAI=PBIMAI*BIMAI Wl<ll=l.O W2Cll=J.O W2(2l=I.0 W2(3)=2.0 NOFFV=O
TZERO=TONE KM=K*M XJ.=2*KM KN=l DO 201 1=1 •N KN=KN*KM H C I l = ( R ( I l -,!! ( I l l IX 1 TONF=O.O
111
SUM OVER ALL FUNCTION VALUES
I=N DO 304 J=ltI X(Jl=A(J)+(?*LMRDACJl-ll*H(J) NOFEV=NOFcV+l TONF=TONF+FUNccx.T,N)
FIND THE NEXT PERMUTATION
DO 340 I=ltN
BX,*ACCR* /l NDINT12f NDINT12~ NDINT12t NDINT12c NDINT13C NDINT131 NDINT132 NDINTI33 NDINT134 NDINT135 NDINT13f NDJNT137 NDINT138 NDINT139 NDINT140 NDINT141 NDINT142 NDINT143 NDINT144 NDINT145 NDINT146 NDINT147 NDINT148 NDINT149 NDINT150 NDINT151 NDINT152 NDINT153 NDINT154 NDINT155 NDINT156 NDINT157 NnINT158 NDINT159 NDINT160 NnINT161 NDINT162 NDINT163 NDINT164
340 LMBDA(Il=l T(Kl=PRIMAI*TONE/KN
c C PERFORM THE RICHARDSON EXTRAPOLATION c
IFCK.GT.ll GO TO 509 TONF=TCll GO TO 513
c 509 TPOS=O.O
TNEG=O.O SIGN=.FALSE. Xl=K*K X?=Xl K~1=K-1
DO 5 1 2 I= 1 , K~A 1 WlCil=WlCil*(I*Il
512 X2=X2*Xl WlCKl=X2 KTM2=K*2 W2CKTM2l=W2CKTM2-l)*CKTM2-ll W2CKTM2+ll=W?CKTM2l*KTM2 DO 511 I= 1, K J=K-I+l GMKJ=2.0*WlCJl/CW2CK+J+ll*W2Cill*T(J) IFCSIGN) GMKJ=-GMKJ SIGN=.NOT.SIGN IF<GMKJ.LT.0.0) TNEG=TNEG+GMKJ IFCGMKJ.GT.O.Ol TPOS = TPOS +GMKJ
511 CONTINUE TONE=TNFG+ TPOS ACCR=AqSCTONF/CTPOS-TNFGl)
513 IF<IDS.FQ.Ol GO TO 514 WRtTEC6,3002l K,TCKJ,TONEtNOFEV,ACCR
3002 FORMAT(* *•t4,4x,F22.14,4X,E22.14,111,2x.El4.2) 514 IFCABSCTONE-TZEROl.LT.ACCUR*ABSCTZERO+SUMTl l GO TO
c c CONVERGFNCF NOT OqTAINA8LE IN KSTOP ITFRATIONS c
TNntC=l 999 TONEl=TONE+SUMT
TZEROl=TZERO+SUMT RFTURN
·DIMENSION ZC2) COMMON MI,NitXItXKtELPHA,GAMMA,THETA PT=3.1415026~358079
S=SJN(PI*THETA/lAO.) II=IFIXCXI) KI=IFIXCXK)
NDINT165 NDINT166 NDINT167 NDINT168 NDINT169 NDINT170 NDINT171 NDINT172 NDINT173 NDINT174 NDINT175 NDINT176 NDINT177 NDINT178 NDINT179 NDINT180 NDINT18J NDINT182 NDINT183 NDINT184 NDINT185 NDINT186 NDINT187 NDINT188 NDINT189 NDINT190 NDINT191 NDINT192 NDINT193 NDINT194 NDINT195 NDINT196 NDINT197 NDINT198 NDINT19c NDINT20C NDINT201
999 NDINT202 TO 990 NDINT203
NDINT2QL, NDINT205 NOINT20f NDINT207 NDINT20f NDINTZOS NDINT21C NDINT21J NDINT212 NDTNT21::'. NDINT214
1 C=O.O S=l.O
2 X=Z(l) Y=ZC?l FO=le+FLPHA*Y*S F02=F0**2 F03=F0**3 GG=GAMMA GG2=GG**2 GG.3=GG**3 GG4=GG**4 COT=C/S CSC=le/S COT2=COT**2 COT3=COT**3 COT4=COT2*COT2 CSC2=CSC**2 CSC3=CSC**3 CSC4=CSC2*CSC2 RO=PO*OO Cl=P0**2+2.*RO*COT2+COT4 C2=GG*C4.*RO*COT*CSC+4.*COT3*CSC) C3=GG2*(2e*RO*CSC2+6e*COT2*CSC2) C4=GG3*4.*COT*CSC3 C5=GG4*CSC4 C6=GG*6.*COT*(R0+COT2l*ELPHA C7=GG2*6.*CSC*(R0+3.*COT?l*fLPHA C8=GG3*1R.*COT*CSC2*ELPHA C9=GG4*6.*CSC3*FLPHA Cl5=Gli2*6•*<XNEU+COT2l*ELPHA**2 Cl6=GG3*12.*COT*CSC*ELPHA**2 Cl7=GG4*6•*CSC2*ELPHA**2 II=IFIXCXI> KI=TFIXCXK) XM=FLOAT(~Il
X"l=FLOATCNI l XX=l.-X*X
YY=l.-Y*Y XXX=XX*XX YYY=YY*YY XA=ARS(X) YA=ABS<Y> G=XXX*YYY GX=-4.*X*XX*YYY GXX=<-4.+12.*XA**2l*YYY GXXX=24.*X*YYY GXXXX=24.*YYY GY=-4.*Y*YY*XXX GYY=(-4.+12.*YA**2l*XXX GYYY=24.*Y*XXX GYYYY=24.*XXX GXY=l6.*X*Y*XX*YY GXXY=-4.*Y*YY*(-4.+12.*XA**2) GXXXY=-96.*X*Y*YY GXYY=-4.*X*XX*(-4.+12.*YA**2) GXYYY=-96.*X*Y*XX GXXYY=<-4.+1?.*YA**?l*f-4.+12.*XA**2> WXXXX=GXXXX*XA**4*Y~**4+4.*GXXX*XM*X*X*X*YA**4+6.*r,XX*XM*(XM-1.>*X lA**2*YA**4+4.*GX*XM*(XM-J.l*(X~-?.l*X*VA**4+G*XM*(XM-J.l*fXM-2.l*( 2XM-3.>*YA**4
WXXXY=GXXXY*XA**4*YA**4+r,XXX*XN*XA**4*Y*Y*Y+3.*GXXY*XM*X*X*X*YA**4 1+3.*GXX*XM*XN*X*X*X*Y*Y*Y+3.*GXY*XM*(XM-1.l*XA**2*YA**4+3.*GX*XM*X ?N*(XM-J.l*XA**2*Y*Y*Y+GY*XM*(XM-J.l*(XM-?.l*X*YA**4+G*XM*XN*fXM-l. 3)*(XM-?.J*X*Y*Y*Y
WXXYY=GXXYY*XA**4*YA**4+?.*GXYY*XM*X*X*X*YA**4+?.*tiXXY*XN*XA**4*Y* 1Y*Y+4.*GXY*XM*XN*X*X*X*Y*Y*Y+GXX*XN*<XN-1.>*XA**4*YA**2+GYY*XM*<XM 2-l.l*XA**2*YA**4+2.*GX*XM*XN*(XN-1.l*X*X*X*YA**2+2.*GY*XM*XN*(XM-l 3.l*XA**2*Y*Y*Y+G*XM*XN*(XM-1.l*(XN-l.l*XA**?*YA**2 WXYYY=GXYYY*XA**4*YA**4+GYYY*X~*X*X*X*YA**4+3•*GXYY*XN*XA**4*Y*Y*Y J+3.*GYY*XM*XN*X*X*X*Y*Y*Y+3.*GXY*XN*(XN-1.>*XA**4*YA**2~3.*GY*XM*X 2N*(XN-J.l*X*X*X*YA**2+GX*XN*<XN-J.)*(XN-?.l*XA**4*Y+G*XM*XN*_(XN-1. 3)*(XN-2e)*X*X*X*Y
WYYYY=GYYYY*XA**4*YA**4+4.*GYYY*XN*XA**4*Y*Y*Y+6.*GYY*XN*(XN-I.l*X 1A**4*YA**2+4.*GY*XN*(XN-1.l*(XN-2.>*XA**4*Y+G*XN*<XN-1.l*(XN-2.J*( 2XN-3.l*XA**4
WXXX=GXXX*XA**4*YA**4+3.*GXX*XM*X*X*X*YA**4+3.0*GX*XM*(XM-l.l*XA** 12*YA**4+G*XM*(XM-1.l*(XM-2.>*X*YA**4
WXXY=GXXY*XA**4*YA**4+GXX*XN*XA**4*Y*Y*Y+2.*GXY*XM*X*X*X*YA**4+2•* JGX*X~*XN*X*X*X*Y*Y*Y+GY*X~*fXM-l.l*XA**2*YA**4+G*XM*XN*(XM-l.l*XA* 2*?*Y*Y*Y WXYY=liXYY*XA**4*YA**4+\iYY*XM*X*X*X*YA**4+?.*~XY*XN*XA**4*Y*Y*Y+?.*
1GY*XM*XN*X*X*X*Y*Y*Y+GX*XN*fXN-1.l*XA**4*YA**2+G*XM*XN*fXN-l.l*X*X 2*X*YA**2
WYYY=GYYY*XA**4*YA**4+3.*GYY*XN*XA**4*Y*Y*Y+3.*GY*XN*(XN-1.l*XA**4 l*YA**2+G*XN*(XN-l.)*(XN-2el*XA**4*Y
WXX=GXX*XA**4*YA**4+2.*GX*XM*X*X*X*YA**4+G*XM*(XM-1.l*XA**2*YA**4 WXY=GXY*XA**4*YA**4+GX*XN*XA**4*Y*Y*Y+r,Y*XM*X*X*X*YA**4+G*XM*XN*X*
lX*X*Y*Y*Y WYY=GYY*XA**4*YA**4+2.*GY*XN*XA**4*Y*Y*Y+G*XN*(XN-I.l*XA**4*YA**2 FF=(Cl*WXXXX-C?*WXXXY+C3*WXXYY-C4*WXYYY+C5*WYYYYl*F03-(C6*WXXX-C7*
IWXXY+CR*WXYY-C9*WYYY)*FO?+(Cl5*WXX-Cl6*WXY+Cl7*WYYl*FO FUNC2=FF*G*(X**<MI-4l*Y**(NI-4ll*X**II*Y**KI RFTURN END
1
46
SUBROUTINE MULTI.
PURPOSE TO MULTIPLY TWO MATRICES OF GENERAL TYPE TO FORM A RESULTANT MATRIX
USAGF CALL MULTI (A,B,c.rr,JJ,KK>
DESCRIPTION OF PARAMETERS A•••••••• FIRST INPUT MATRIX R•••••••• SECONn INPUT MATRIX c •••••••• RFSULTANT OUTPUT MATRIX II ••••••• NUMRFR OF ROWS IN A AND C JJ ••••••• NUM8ER OF COLUMNS IN RAND C KK ••••••• NUMBER OF COLU~NS IN A AND ROWS IN B
DIMENSION AC II tKKl •R-<KK,JJ) •C< II ,JJ) DO 1 I=l•II DO 1 J=l,JJ ((!,J)=O.O DO 1 K=l,KK C<I•J>=CCJ,J)+A(I,Kl*P(K,J) cnNTINUF RETURN F:ND SURROUTINE EIGENP(N,NM,A,T,EVR•EVI,VECR,VECI,INDIC)
SURROUTINF EIGENP.
PURPOSF TO FIND ALL THF EIGENVALUES AND CORRESPONDING EIGENVECTORS OF A RFAL GFNERAL MATRIX.
USAGF CALL EIGENPCNtNM,A,TtEVRtEVJ,VECRtVECI,INDIC)
DESCRIPTION OF PARAMTERS. INPUT- N­ THE ORDER OF THE MATRIX.
NM- THE FIRST DIMENSION OF THF DOURLY SUBSCRJPTFD ARRAYS At VECR AND VECI. THF SECOND DIMENSION IS AT LEAST N. THE UPPER LIMIT FOR NM rs 100.
A­ A MATRIX OF ORDER N <DESTROYED>. T- THE NUMBER OF RITS IN THE MANTISSA OF A
FLOATING-POINT NUMBER. FOR THE CDC6400t T=48.0. OUTPUT- FVR- THE REAL PARTS OF THE N EIGENVALUES ARE
EVRCI), I=l•2•••••N• EVI- THE IMAGINARY PARTS OF THF N EIGENVALUES ARE
EVICT>• I=l•2•••••N• VECR- THE REAL COMPONENTS OF THF NORMALIZED
EIGENVECTOR I• I=l•2••••'N' WILL BE FOUND IN THE FIRST N PLACES OF COLUMN I OF VECR.
VECI- THE IMAGINARY (OMPONFNTS OF THE NORMALIZED EIGFNVFCTOR Tt I=l•2••••'N' WILL BE FOUN~ IN THE FIRST N PLACES OF COLUMN I OF VECI.
c c c c c c c c c c c
47
INDIC- ARRAY INDICt INDJC(l)t I=l,2••••'N' JNDICATFS THE SUCCESS OF THE SUBROUTINE AS FOLLOWS
VALUE OF INDIC<Il-EIGENVALUE I-EIGENVECTOR I 0 NOT FOUND NOT FOUND 1 FOUND NOT FOUND 2 FOUND FOUND
SUAPROGRAMS CALLFO ESCALE,HESQR,R~ALVE,CnMPVE
METHOD FIRST IN SURROUTINF ESCALE THF MATRIX IS SCALED SO THAT THE CORRESPONDING ROWS AND COLUMN ARE APPROXIMATELY RALANCED AND THFN THF MATRIX TS NORMALIZFD SO THAT TH~ VALUE OF THE EUCLIDEAN NORM OF THE MATRIX IS EQUAL TO ONE. THE EIGENVALUES ARE COMPUTED BY THE QR DOUBLE-STEP METHOO IN SURROUTINE HESQR. THF EIGENVECTORS ARE COMPUTED RY INVERSE ITERATION IN SUBROUTINE REALVEt FOR THE REAL EIGENVALUES, OR IN SUBROUTINE CO~PVE• FOR THE COMPLEX EIGENVALUES.
REFFRENCF WILKINSON,J.H.<1965}. THE ALGFRRAIC EIGFNVALUE PRORLFM. OXFORD,
CLARENDON PRFSS. J.GRAD AND M.A.BRERNER. COMMUNICATIONS OF THF ACM, VOL.llt
N0.12•DEC.tl968.
STORAGE RFOUIREMFNTS CDC6400- 13?4 WORDS PLUS AN ADDITIONAL 3*N*N+3*N
WORDS ARF REQUIRFD.
ACCURACY CDC6400- THF ACCURACY MAY BE CONTROLLFD RY T, US~D TO
COMPUTE EX. FOR T=48.0t MATRICES TESTED YIELDED EIGENVALUES AND EIGENVECTORS CORRECT TO AT LEAST TEN SIGNIFICANT FIGURES. THE ABOVE RESULTS ARE VALID FOR MULTIPLE EIGENVALUES ALSO.
TYPICAL TIMINGS CDC6400- MATRIX OF ORDER 10- 0.989 SECONDS, T=48.0
MATRIX OF ORDER 30- 16.686 SECONDSt T=48.0 MATRIX OF ORDER 50- 69.221 SECONDSt T=48.0
SOURCE LANGUAGE FORTRAN
AUTHOR J.GRAD AND M.A.BREBNER.
CHECKED BY w.WARDt NnV/69.
1EVRC1l,EVI<1 },INDICtll
48
DIMENSION JWORKClOQ),LOCALClOOl,PRFACTClOOl , 'SUeD r A ( 1 00 ) 'WORK 1 ( , 0 0) • WORK 2 ( 100 l 'WORK ( 1 no)
IFCN.NE.llGO TO 1 EVR < 1 ) = A ( 1 '1 ) FV I C 1 ) = 0 • 0 VECRCl.ll = 1.0 VECI<l•l> = o.o INDJC(l) = 2 Ci(") T0 ::>11
c 1 CALL FSCALF(N,NM,AtVECitPRFACTtENORMl
C THE COMPUTATION OF THE EIGENVALUES OF THE NORMALISED C ll.1ATRIX.
EX= EXPC-T*ALOG(2.0l) CALL HESQR(NtNM,AtVECI,EVRtEVItSUBDIAtINDIC,EPS,EX)
c C THE POSSIBLE DECOMPOSITION OF THE UPPER-HESSENBFRG MATRIX C INTO THE SUBMATRICFS OF LOWER ORDER IS INDICATED IN THE C ARRAY LOCAL. THF DFCO~POSITION OCCURS WHEN SOME C SURDIAGONAL FLEMFNTS ARE TN MODULUS LESS THAN A SMALL C POSITIVE NUMRER FPS DEFINEn IN THE SURROUTINF HFSQR • THE C AMOUNT OF WORK IN THE EIGENVECTOR PROBLEM MAY BE
- C DIMINISHED IN THIS WAY. J = N I = 1 LOCAL ( 1 ) = 1 JFCJ.FQ.l)GO TO 4
2 IF<ARS(SUBDIA(J-lll.GT.EPS)GO TO 3 I =: I+l LOCAL<Il=O
3 J = J-1 LOCAL(T)=LOCAL(J)+l IF<J.NE.l)GO TO 2
c ·c THF FIGENFVFCTOR PRORLFM.
4 K = 1 KON = 0 L = LOCAL(ll M = N DO 10 I=l,N
IVEC = N-1+1 JFCI.LE.LlGO TO 5 K = K+l M = N-L L = L+LOCALCKl
5 IFCINDICCIVECl.FQ.O)GO TO 10 IFCEVICIVEC>.NE.O.OlGO TO 8
.(
C TRANSFER OF AN UPPER-HESSENBERG MATRIX OF THE ORDER M FROM C THE ARRAYS VFCI AND SIJRDIA INTO THE .ARRAY A.
DO 7 K1=1tM DO 6 Ll=KltM
6 A<KltLll = VECICKltLll IFCKl.EQ.l)GO TO 7 A(Kl,Kl-ll = SUBDIA<Kl-1)
7 CONTINUE c
C THE COMPUTATION OF THF REAL EIGENVECTOR IVFC OF THF UPPER­ ( HESSENRERG MATRIX CORRESPONDING TO THE REAL EIG~NVALUE C EVRfIVEC).
CALL RFALVE(N,NMtMtJVECtAtVECRtEVRtEVItIWORKt 1 WORK,JNDICtEPS,FX)
GO TO 10 c C THF COMPUTATION OF THF COMPLFX FIGENVFCTOR IVEC OF THE C UPPFR-HESSFNRERG MATRIX CORRFSPOND!NG TO THE COMPLFX C EIGFNVALUF FVR<IVFC) + I*FVI<IVEC). IF THE VALUF OF KON JS C NOT EQUAL TO ZERO THEN THIS COMPLEX EIGENVECTOR HAS C ALREADY BEEN FOUND FROM ITS CONJUGATE.
8 IFCKON.NE.O)GO TO 9 KON = l CALL CO~PVFCNtNMtMt!VECtAtVFCR,VFCitFVRtEVItINDICt
1 IWORKtSURDIAtWORKltWORK2tWORKtFPStEX) GO TO JC
9 KON = 0 10 CONTINUE
c C THE RFCONSTRUCTION OF THE MATRIX USED IN THE REDUCTION OF C MATRIX A TO AN UPPER-HESSFNBERG FORM RY HOUSEHOLDER METHOD.
DO l 2 t =1 'N DO l l J= I' N
A(J,J) = O.O 11 ACJtil = o.o 12 ACitil = 1.0
IF<N.LE.2lGO TO 15 M = N-2 DO 14 K=ltM
L = K+l DO 14 J=2tN
Dl = O.O DO 13 I=LtN
D2 = VECICitK) 13 Dl = Dl+ D2*A<JtI)
DO 14 I=LtN 14 A(J,Jl = A<Jt!l-VECI(I,Kl*Dl
·c ( THF COMPUTATION OF THF FT\iFNVFCTORS OF THF ORIGINAL NON­ C SCALED MATRIX.
15 KON = 1 DO 24 I=ltN
L = 0 IF<EVI<Il.EQ.0.0) GO TO 16 L = 1 IF<KON.EO.O)GO TO 16 KON = 0 GO TO 24
16 DO 1 8 J =1 t "'
Dl = O.O D? = O.O DO 17 K.=ltN
D3 = A(J,Kl Dl = Dl+D3*VECR(K.,Il IF<L.FO.O)GO TO 17 D2 = D2+D3*VECR(K,I-l)
so
17 CONTINUE WORK(J) = Dl/PRFACT(JJ IFCL.EQ.O) GO TO 18 SURDTACJ)=D2/PRFACT(J)
18 CONTINUE c C THE NORMALISATION OF THF FIGFNVECTORS AND THE COMPUTATION C OF THE EIGENVALUES OF THE ORIGINAL NON-NORMLISED MATRIX.
IFCL.EQ.llGO TO 21 Dl = o.o DO 19 M=l,N
19 Dl = Dl+WORKCM>**2 Dl = DSQRTCD1> DO 20 M=l•l\l
VECI (M,I) = O.O 20 VECR(M,Il = WORKCM)/Dl
EVRCI) = EVRCil*ENORM GO TO 24
c 21 K0N = 1
EVRCT) = EVRCT)*FNOR~
EVR ( I -1 ) = EVR C I ) EVICT) = EV!(Il*ENORM EVICI-ll =-EVICil R = O.O DO 22 J=l•N
R1 = WORKCJl**? + SURDIACJ)**2 IFCR.GF.Rl)GO T0 22 R = Rl L = J
22 CONTINUE D3 = WORKCL>
R 1 = SUB DI AC L)
DO 23 J=l•N Dl = WORK(J) D2 = SURDIA(J) VECRCJ,Il = (Dl*D3+D2*Rll/R VECICJt!) = CD2*D3-Dl*Rll/R VECR(J,I-1) = VFCR(J,Il
23 VECICJ'1-ll =-VECI(J,I) 24 CONTINUE
. c 25 RETURN
END SURROUTINE FSCALFCN,NM,AtHtPRFACTtENORM)
C PURPOSE C TO SCALE A MATRIX. (
C U~Af-F ~C CALL FSCALFCNtNM,A,H,PRFACTtFNORMl .-c
C DFSCRIPTTON OF PARAMFTFRS C INPUT- N- THE ORDER OF THE MATRIX TO BE SCALED. C NM- THE FIRST DIMENSION OF DOURLY SUBSCRIPTED ARRAYS
51
c A AND H. THE SFCOND DIMENSION IS AT LEAST N. c OUTPUT- PRFACT-THE COMPONENT I OF THE EIGENVECTOR ORTAINED RY
·c USING THE SCALED MATRIX MUST BE DIVIDED BY THE c VALUE PRFACT(IJ, I=lt2'•••'N• IN THIS WAY, THE c EIGENVECTOR OF THE NON-SCALED MATRIX IS c ORTAINED. PRFACT IS A DOURLE PRECISION ARRAY. c ENORM- THE EIGENVALUES OF THE SCALED MATRIX MUST BE c MULTIPLIED BY THE SCALAR ENORM IN ORDER TO
r C OBTAIN THE EIG~NVALUES OF THF NON-SCALED MATRIX. c IF FNORM=I.O, THEN A HAS NOT BEEN SCALED. c IN/OUT- A- INITIALLY, A IS THE MATRIX TO BE SCALED. c ON RETURN, EITHER A IS UNCHANGED OR IS SCALED. c (SEE METHOD RELOWl. c WORK- H- A DOUBLY SUBSCRIPTED WORK ARRAY TO STORE A.
. c c SURPROGRAMS CALLED c NONE c c METHOD c FIRST A IS STORED IN H. THEN A IS SCALED SO THAT THE QUOTIFNT
" c OF THE ARSOLUTE SUM OF THE OFF-DIAGONAL ELEMENTS OF COLUMN I c AND THE ABSOLUTE SUM OF THE OFF-DIAGON~L ELEMENTS OF ROW I LIES c WITHIN THE VALUES POUNDI=0.75 AND BOUND2=1.33. AFT~R THE MATRIX c IS SCALEDt IT IS NORMALIZED SO THAT THE VALUE OF THE EUCLIDEAN c NORM IS I.O. IF THF PROCESS OF SCALING TAKES MORE THAN 30 c ITFRATIONS, THF PROCESS FAILS AND H IS STORED RACK IN A. r c RFF~RFNCF
c J.GRAD AND M.A.RREBNER. COMMUNICATIONS OF. THE ACM,VOL.11,N0.12, c DEC.1968. c c STORAGE REQUIRE~ENTS c CDC6400- 360 OCTAL STORAGE LOCATIONS PLUS AN ADDITIONAL c 2*N*N+2*N DECIMAL STORAGE LOCATIONS ARE REQ~IRED. c c TYP~CAL TIMINGS c MATRIX OF ORDER 10- 0.027 SECONDS c MATRIX OF ORDER 30- 0.231 SECONDS c MATRIX OF ORDER 50- 0.639 SECONDS c c SOURCE LANGUAGE c FORTRAN c c R~MARKS
c SUBROUTINE ESCALE IS CALLED BY SUBROUTINF ElGENP. c c AUTHOR c J.GRAD AND M.A.RRERNER. c c CHFCKFn RY c W.WARDt NOV/69. c
DOUBLE PRECISION COLUMN,FACTOR,FNORM,PRFACT,Q,ROW DIMENSION A(NM,lJ,H(NM,lltPRFACT(ll REAL ROUND1,ROUND2tFNOR~
FIOUNDl=0.75 ROUND2=1.3~
JTFR=O 3 NCOUNT=O
DO 8 I=l,N COLUMN=O.O ROW=O.O DO 4 J=l,N
IF<I.EQ.J) GO TO 4 COLUMN=COLUMN+ARSCA(J,J)) ROW=ROW+ARSCACI,J))
4 CONTINUE IF(COLUMN.FQ.O.Ol GO TO 5 JF<ROW.EQ.O.Ol GO TO 5 Q=COLUMN/ROW IF<O.LT.BOUNDl) GO TO 6 IFCO.GT.BOUND2l GO TO 6
5 NCOUNT=NCOUNT+l GO TO ·8
6 FACTOR=DSQRTCQl DO 7 J=l'N
IFCI.EQ.J) GO TO 7 A(J,J>=ACitJl*FACTOR ACJ,Il=A(J,Il/FACTOR
7 CONTINUE PRFACT(l)=PRFACTCil*FACTOR
8 (ONTINUF ITfR=ITFR+l IF(ITFR.GT.30) GO TO 11 IFCNCOUNT.LT.N) GO TO 3 FNORM=O.O DO 9 I=ltN
DO 9 J=l,N O=A(J,J)
9 FNORM=FN0RM+O*O FNORM=DSORTCFNORM) Dn 1 0 T= 1 , N
DO 10 J= 1 , ~I 10 ACI,Jl=A(I,Jl/FNORM
ENORM=FNORM GO TO 13
11 no l? T=ltN PRFACTCI)=l.O D0 12 J=l ,~1
12 A(J,J)=H(J,J) ENORM=l.O
13 RETURN END SURROUTINE HESQR(N,NMtA,HtEVR,EVI,SUFIDIAtINDIC,EPS,EX)
c C SUBROUTINE HESOR. c C C
PURPOSE TO FIND ALL THE FIGENVALUES OF A REAL GENERAL MATRIX. THIS
,,
(
53
c CORRESPONDING EIGENVECTORS ALSO. JF ONLY THE EIGENVALUES ARE c DESIRFD' USE SURROUTINE REIGEN.
·c J'. USAGF c CALL HESQR(N,N~'A'H,EVR,EVI,SURDIA,INDIC,EPS,EX> c c DF="SCR T PTT ON OF PARAMETERS c INPUT­ N­ THE ORDER OF THE MATRIX A. c NM­ THE FIRST DIMENSION OF THE DOUBLY SUBSCRIPTED c ARRAYS A AND H. THE SECOND DI~ENSION IS AT c LEAST N.
A­ ANN RY N MATRIX CDFSTROYED>. c EX­ EX=2**<-T>. T IS THE NUMPER OF BITS IN THE c MANTISSA OF A FLOATING POINT NUMBER. FOR THE c CDC6400, T=48. c OUTPUT­ H­ THE ORIGINAL MATRIX A IS REDUCED TO UPPER c HESSENBERG FORM H RY MEANS OF SIMILARITY c TRANSFORMATIONS CHOUSFHOLDER METHOD). THE c MATRIX H rs PRESERVED IN THE UPPER HALF OF
THF ARRAY H AND IN THF ARRAY SURDIA' SUBDIACI>' I=l,z, ••• ,N. THE SPECIAL VECTORS USED IN THE DFFINITION OF THE HOUSEHOLDER TRANSFORMATION MATRICES ARE STORED IN THE LOWER PART OF THE ARRAY H.
SUBDIA­ SEF DESCRIPTION OF H• EVR­ THF REAL PARTS OF THE N FIGENVALUES WILL BE
EVI­ FOUND IN EVR ( I l ' I= 1'2' • •• 'N • THF IMAGINARY PARTS OF THE N EIGF.NVALUFS WILL BE FOUND IN EVICil, I=1,z, ••• ,N.
INDIC­ THE ARRAY INDIC, INDICCJ), I=1,2, ••• ,N, INDICATES THE SUCCESS OF THE ROUTINE AS FOLLOWS
VALUE OF INDICCI> EIGENVALUE I 0 NOT FOUND 1 FOUND
FPS­ FPS IS A SMALL POSITIVE NUMRFR THAT NUMFRICALLY REPRESENTS ZERO IN THE PROG"RAM. EPS=CEUCLIDEAN NORM OF H>*EX.
SUBPROGRAMS CALLED ~10r..1F
MFTHOD THE MATRIX IS REDUCED TO UPPER HESSEN8ERG FORM BY MEANS OF SIMILARITY TRANSFORMATIONS !HOUSEHOLDER METHOD). THEN THE EIGENVALUES ARF COMPUTED BY THE QR DOUBLF-STEP METHOD.
REFERENCE GRAD,J. AND BRERNER,M.A.<1968). COMMUNICATIONS OF THE ACM,
VOL.11,N0.12. WILKINSON,J.H.Cl965l. THE ALGFBRAIC EIGENVALUF PRORLEM, OXFORD,
CLARENDON PRESS.
STORAGF RFQUIREMENTS CDC6400­ 1058 WORDS PLUS AN ADDITIONAL 2*N*N+4*N
WORDS ARE REQUIRED.
54
C SOURCE LANGUAGF C FORTRAN c C REMARKS C SUBROUTINF HESOR IS CALLED BY SUBROUTINE EIGENP. c C AUTHOR C J.GRAD AND M.A.RRERNER. (
C CHECKEO RY C W.WARD' NOV/69. c
DOURLF PRECISION s,sR,SR2,x,y,z DIMENSION A(NM,l),H(NM,l),EVR(l),EVI(l),5UBDIA(l),INDIC(l)
c C RFDlJCTION OF THE MATRIX A TO AN UPDER-HESSFNRFRr, FORM H. C THFRE ARE N-2 STEPS.
IF(N-2)14'1'2 J SUBDIA(ll = A(2,l)
GO TO 14 2 M = N-2
DO 12 K=l,M L = K+l ,c; = n.n DO 3 I=LtN
H(I,Kl = A(J,K) 3 S = S+ABS(A(J,Kl)
IF(S.NE.ABS(A(K+l,K>>>GO TO 4 SUBDIA(K) = A(K+l,K) H(K+l,K) = O.O GO TO 12
4 SR2 = O.O DO 5 I=L,N
SR= A(J,K) SR = SR/S A<I•K> =SR
5 SR2 = SR2+SR*SR SR = DSORT(SR2) IF(A(L,K).LT.O.O)GO TO 6 SR = -SR
6 SR2 = SR2-SR*A(L,Kl A<L•Kl = A(L,Kl-SR H(L,Kl = H(L,Kl-SR*S SUBDIA(K) = SR*S X = S*DSQRT(SR2l DO 7 T=L,N
HCJ,Kl = HCitKl/X 7 SUBDTh(J) = A(J,K)/SR2
C PREMULTTPLICAT!ON ~y THF ~ATRIX PR. DO 9 J=L•N
SR = o.o DO 8 I=LtN
R SR= SR+A(I,Kl*A(J,J) ~ DO 9 I=LtN
q A(I,Jl = A(J,J)-SURDlACil*SR C POSTMULTIPLTCATTON RY THE ¥ATRIX PR.
DO 11 J=l•N
SS
10 SR= SR+ACJtI)*ACitK) DO 11 I =L, N
11 A(J,J) = A(J,I>-SURDIACI)*SR 12 CONT H!UF
DO 13 K=ltM 13 ACK+ltK) = SURDIACK)
C TRANSFER OF THE UPPER HALF OF THE MATRIX A INTO THE .C ARRAY H AND THE CALCULATION OF THE SMALL POSITIVE NUMBER .C EPS.
SURDIACN-1) = ACN,N-1) 14 EPS = o.o
DO 15 K=l•N INDICCK) = 0 JFCK.NE.N>EPS = EPS+SURDIACK>**2 DO 15 I=K•N
HCK,Y) = ACKtI) 15 EPS = EPS + ACK,I)**2
EPS = EX*SORTCEPS) c C THF OR ITERATIVE PROCESS. THE UPPER-HESSENRERG MATRIX H IS C REDUCEO TO THE UPPER-MODIFIED TRIANGULAR FORM. ('
C DETERMINATION OF THE SHIFT OF ORIGIN FOR THE FIRST STEP OF C THE OR ITERATIVE PROCESS.
SHIFT = ACN,N-1> IFCN.LF.?.)SHIFT = O.O IFCACN,NJ.NF.O.O)SHIFT = O.O IFCACN-1,N>.NE.0.0)SHIFT = O.O IFCACN-l•N-1>.NF.O.OJSHIFT = O.O M = N NS= 0 MAXST = N*lO
c ( TESTING IF THE UPPER HALF OF THE MATRIX rs EQUAL TO ZERO. C iF IT JS FOUAL TO ZFRO TH~ QR PROCESS JS NOT NECESSARY.
DO 16 I=2tN DO 16 K=ItN
IFCACI-ltKl.NF.O.OlGO TO 18 16 CONTINUE
DO 117 I=l•N JNDICCI)=l Ev'RCI) = ACI,J)
17 FVICT> = O.O GO TO 37
c C START THF MAIN LOOP OF THE QR PROCESS.
18 K=M-1 Ml=K l=K
C F!Nn ANY DFCOMPOSITTONS OF THE MATRIX. ( JUMP TO 34 IF THF LAST SU9MATRTX OF THE DFCOMPOSITION IS ( nF THF OROFR ONF. C JUMP TO ?5 IF THE LAST SURMATRJX OF THE DECOMPOSITION IS C nF THE OPDFR TWO.
IFCKl37t34'19
19 IF(ARS(A(M,Kll.LE.EPSlGO TO 34 IFCM-2.EQ.O)GO TO 35
20 I = I-1 IF<AASCACKtil>.LE.EPS)GO TO 21 K = I IFCK.GT.l)GO TO 20
21 JF(K.EQ.MllGO TO 35 C TRANSFORMATION OF THE MATRIX OF THE ORDER GREATfR THAN TWO.
S = ACMtM)+A(MltMll+SHIFT SR= A(M,M)*A<MltMll-ACMtMll*A(MltMl+0.25*SHIFT**2 A(K+2tK) = O.O
c CALCULATE x1,v1,z1,FOR THE SUBMATRIX OBTAINED BY THF C DECOMPOSITION.
X = A(K,Kl*CACKtK>-Sl+A(KtK+l>*A<K+ltKl+SR Y = A<K+ltK)*{A(K,K)+A(K+l,K+l)-5) R = DARS<X>+DAB5CY) JF(R.FQ.O.O>SHIFT = A(M,M-1) JF(R.FQ.0.0)\,0 TO 21 Z = A(K+2tK+l>*A(K+1tK> SHIFT = o.o NS = NS+l
c C THE LOOP FOR ONE STEP OF THE QR PROCESS.
DO 33 I=KtMl IF<T.EO.KlGO TO 22
C CALCULATE XR,YR,ZR. X = A(!tT-ll Y = A<t+l•l-1> z = o.o IF(l+2.GT.M)G0 TO 22 Z = A<I+2.I-1>
22 SR2 = DARS<X>+DARS<Yl+DABS<Zl IF(SR2.Eo.o.O)GO TO 23 X = X/SP2 Y = Y/SR2 Z = Z/SR2
23 S = D5QRTCX*X + Y*Y + Z*Zl IFCX.LT.O.O>GO TO 24 s = -s
24 IF<I.EQ.KlGO TO 25 A<ItI-1) = S*SR2
25 IFCSR2.NE.O.OlGO TO 26 IFCI+3.GT.M>GO TO 33 GO TO 32
26 SR = 1.0-X/S s =.x-s X = Y/S
... Y = Z/5 ,.. ,, PRFMULTifLICATION BY THE MATRIX PR.
DO 28 J=·J tM S = A( I ,J)+A( I+J ,J)*X 1FCI+2.GT.M>GO TO 27 S = S+ACI+2,Jl*Y
27 S = S*SR A(J,J) = A(J,J)-5 A<I+l,Jl = ACI+l,J>-S*X IFCI+2.GT.M>GO TO 28
57
ACJ+2,Jl = ACJ+2,Jl-S*Y 28 CONTINUE
C POSTMULTIPLICATION RY THE MATRIX PR. L = !+2 JFCJ.LT.MllGO TO 29 l = M
;:>o !")(') ~1 J=K,L S = A(J,Jl+A(J,J+ll*X JF(I+2.GT.MlGO TO 30 S = S + A<J,I+2l*Y
":\O S = S*SR ACJtll = A(Jtil-S A(J,J+ll=A(J,I+ll-S*X IFCI+2.GT.MlGO TO 31 A(J,I+2l=A(Jtl+2l-S*Y
31 CONTINUE IFCI+3.GT.MlGO TO 33 S = -A<I+3tI+2l*Y*SR
32 ACI+3,Jl = S AC I+3'1+1 l = S*X A<I+3tI+2) = S*Y + ACI+3tI+2l
33 CONTINUE c
IFCNS.GT.MAXSTlGO TO 37 GO TO 18
c C COMPUTE THE LAST FIGENVALUF.
34 EVRCMl = ACMtM) EV ICM) = 0 • 0 TND IC (Ml = 1 M = K GO TO 18
c C COMPUTE THE FTGFNVALUES OF THE LAST 2X2 MATRIX OBTAINED RY C THE DECOMPOSITION.
35 R = 0.5*CA(K,Kl+A(M,Mll S = 0.5*CACM,Ml-A(K,Kll S = S*S + A(K,Ml*ACMtKl INDIC<Kl = 1 rr-mtc·<Ml = 1 IF ( S .• LT. 0. 0 l GO TO 36 T = 1DSQRT ( S) EVR Ck l = R'-T EVRCMl = R+T EV I ( K l = o.o EV I CM l = o.o M = M-2 GO TO 18
36 T = DSQRTC-Sl EVR(Kl = R EVICK) = T EVRCM) = R EV I CM l = -T M = M-? GO TO 1 8
c 37 RETURN
END SUBROUTINE REALVE(N,NM,M,IVEC•A•VECR•EVR,EVJ,
lIWORK,WORK,INDIC,EPS,EXJ c C SUAROUTTNE REALVF. c C PURPOSF C TO FIND THE REAL EIGENVECTOR OF THE REAL UPPER-HESSENRERG C MATRIX IN THF ARRAY A CORRESPONDING TO THE REAL EI\:ENVALUE C STORED IN FVR(IVECl. THIS SUBROUTINE SHOLJLD BE CALLED BY C SUBROUTINE FIGENP. ("
~
'C WILKINSON,J.H.(1965). THE ALGEBRAIC EIGENVALUE PROBLEM. OXFORD, C CLARENDON PRESS. C J.GRAD AND M.A.BREBNER. COMMUNICATIONS OF THE ACM, VOL.11'
~c No.12,DEc.,1968. c C STORAGE REQUIREMENTS C CDC6400­ 114 WORDS PLUS AN AnDITIONAL 2*N*N+5*N
~C WORDS ARE RFQUIRED. c C SOURCE LANGUAGE C FORTRAN c C REMARKS C SUAROUTINE REALVF IS CALLED RY SUBROUTINF EIGENP. c C AUTHOR C J.GRAD AND M.A.BREBNER. r C CHECKED RY C W.WARD, NOV/69. c
DOURLE PRECISION StSR DIMENSION ACNM,l),VFCR(NMtl),EVRCl) D JM FN S I ON FV I C 1 > ' I WORK C1 l '~JOR KC 1 ) ' IND IC C1 ) VECRCltIVECl = 1.0 JFCM.FQ.l)GO TO 24
C SMALL PERTURBATION OF EQUAL EIGENVALUES TO OBTAIN A FULL C SET OF FIGFNVECTORS.
EVALUE = EVR(IVEC> IECIVEC.EO.M)GO TO 2 K = IVEC+l R = o.o DO 1 I=K,M
IFCEVALUE.NF.EVRCJ))GO TO 1 IECEVJCI).NF.O.O>GO TO 1 R = R+3.0
1 CONTINUE EVALUE = EVALUE+R•EX
2 DO 3° K=l,M 3 AC~tKl = ACK,Kl-EVALUE
GAUSSIAN ELIMINATION OF THF UPPER-HESSFNBERG MATRIX A. ALL ROW INTERCHANGES ARE INnICATFD IN THF ARRAY !WORK.ALL THE MULTIPLIERS ARE STORED AS THE SUBDIAGONAL ELEMENTS OF A.
K = M-1 DO 8 I= 1 , K
L = I+l JWORKCil = 0 I F ( A C I + 1 , I l • NE' • 0 • 0 l GO T 0 4 IFCACitil.NF.0.0>GO TO 8 ACI,Y) = EPS GO TO 8
4 IFCARSCACitill.GF.ARSCACl+l,Tl>>GO TO 6 I \I/ORI( ( I ) = 1 DO 5 J=I ,M
60
5 A< l+ltJ) = R 6 R = -A<I+ltl)/ACl.I)
A(J+ltl) = R DO 7 J=LtM
7 A<I+l,J) = A(l+l•J>+R*ACJ,J) 8 CONTINUE
IFCACM,M>.NF.O.Ol GO TO 9 A(M,tvl) = EPS
- c c THE VECTOR c1,1 ••••• 1l rs STORED IN THE PLACE OF THE RIGHT C HAND SIDE COLUMN VECTOR.
9 DO 11 I=l•N JF(·I.GT.MlGO TO 10 WORK C I l = 1. 0 GO TO 11
10 WORKCil = O.O 11 CONTINUE
c c THE JNVFRSE ITERATION rs PFRFORMFD ON THE MATRIX UNTIL THF c INFTNITF NORM OF THF RJGHT-H~ND srnF VECTOR JS GREATER
· C THAN THF ROUND DFFJNFD AS 0.01/(N*FXJ. BOUND= 0.01/(EX * FLOATCNll MS = 0
ITER = l c C THE BACKSURSTITUTION.
12 R =. O. 0 D0.15 I=ltM
J = ~-I+l S = WORK(J) IFCJ.EQ.MlGO TO 14 L = J+l DO 13 K=L•~
SR = WORKCK) 11 S = ~ - SR*A(J,Kl 14 WORKCJ) = S/ACJtJ)
T = ARSCWORK(J)) IF<R.GE.T)GO TO 15 R = T
15 CONTINUE c C THF COMPUTATION OF THE RIGHT-HAND SIDE VECTOR FOR THE NEW C ITERATION STEP.
DO 16 I=ltM 16 WORKCIJ = WORKCil/R
•C C THE COMPUTATION OF THF RESIDUALS AN~ COMPARISON OF THE C RESIDUALS OF THE TWO SUCCFSSIVF STEPS OF THE INVERSE c ITERATION.IF THE INFINITE NORM OF THE RESIDUAL VFCTOR rs C GREATER THAN THE INFINITE NORM OF THE PREVIOUS RESIDUAL C VFCTOR THE COMPUTFD FICiFNVFCTOR OF THE- PRFVIOUS STFP IS C TAKFN AS THE FINAL EIGENVECTOR.
Rl = o.n !)(') 1 8 I= J 'M
T = o.o
T = ARS (T)
18 CONTINUE IF(JTER.FO.l)GO TO 19 JF(PREVIS.LE.Rl)GO TO 24
lQ DO ?0 T=l•M 20 VECR(J,IVFC) =WORK(!)
PREVIS = Rl IFCNS.FQ.l)GO TO 24 IFCITER.GT.6)G0 TO 25 ITER = ITER+l IF<R.LT.BOUND)GO TO 21 NS = 1
c C GAUSSIAN ELIMINATION OF THE RIGHT-HAND SIDE VECTOR.
21 K = M-J DO 2'3 T=l,K
R = WORK( I+l) IF(IWORK(I).EQ.O)GO TO 22 WORK( I+l )=ltJORKC I )+WORK( I+l )*AC I+ltl) WORK(l) = R GO TO 23
22 WORK(I+l)=WORK(I+l)+WORKCil*A(l+l,Il 23 CONTINUE
GO TO 1? c
26 VFCR(J,JVFC) = O.O ?7 RETURN
END SUBROUTINE COMPVE(N,NM.M.IVEc.A,vEcR.H,EVR,FVI.INDIC,
1IWORK•SU8DIA•WORK1,WORK2,WORK,EPS,EXl c
·C SUBROUTINE COMPVE. c
r C PURP0SF C TO FIND THF COMPLEX EIGENVECTOR OF THE RFAL UPPER-HESSENBERG
C MATRIX IN ARRAY A CORRESPONDING TO THE COMPLEX EIGENVALUE C WITH THE REAL PART IN EVRCIVEC) AND THE COMPLEX PART IN
'C EVICIVECl. THIS SUBROUTINE SHOULD BE CALLED RY SUBROUTINE C EIGENP. c C USAGE C CALL COMPVF(N,NM•~•IVFC,A•VFCR•H•EVR•FVI,INDIC,JWORK,SURDIA, C WORK1,WORK2,WORK,EPS,EX) c C DESCRIPTION OF PARAMETERS C INPUT- N- THF ORDER OF THF MATRIX. C NM- THE FIRST DIMENSION OF THE DOUBLY SUBSCRIPTED
62
c ARRAYS A, VECR AND H. THE SECOND DIMENSION IS AT c LEAST N. ­ c M- THE ORDER OF TYE UPPER-HESSENRERG MATRIX IF SOME c SUB-DIAGONAL ELEMENTS ARE EOUAL TO ZERO. THE
VALUE OF M IS CHOSEN SO THAT THE LAST N-Mc COMPONENTS OF THE EIGENVECTOR ARE ZERO.
c c
IVEC- GTVFS THE POSITION OF THE EIGENVALUE JN THE ARRAY FOR WHICH THE CORRESPONDING EIGENVECTOR ISc
( C0MPUTFD. c A- A MATRIX OF ORDER N. c H- THE· MATRIX ON WHICH THE INVERSE ITERATION IS c PERFORMED IS BUILT UP IN THE ARRAY A BY USING c THE UPPER-HESSENBERG MATRIX PRESERVED IN THE c ARRAY H AND IN THE ARRAY SUBDIA !SEE THE WRITEUP c FOR SURROUTINE HESQR)• c SUBDIA-SEE ~ESCRIPTION OF H. c EVR- THE REAL PARTS OF THE N EIGENVALUES ARE c EVRCIJ, 1=1,2, ••• ,N. c EVI- THE IMAGINARY PARTS OF THF N EIGENVALUES ARE c EVICIJ, I=l,z, ••• ,N.
EPS- A SMALL POSITIVE NU~BER THAT NUMERICALLYc c REPRESENTS ZERO IN THE PROGRAM. EPS=(EUCLIDEAN c NORM OF H FROM SURROUTINE HESQRJ*EX. c EX- FX=2.0**C-Tl WHFRE T IS THF NUMPFR OF RINARY c AITS IN THF MANTISSA OF A FLOATING POINT NUMBER. c FOR THE CDC6400, T=48• c OUTPUT- VECR- THE REAL PARTS OF THF FIRST M COMPONENTS OF THE c COMPUTED COMPLEX EIGENVECTOR WILL BF FOUND IN c THE FIRST M PLACES OF THE COLUMN WHOSE TOP
ELEMENT IS VECRCl•IVE-Cl AND THE CORRESPONDINGc IMAGINARY PARTS OF THE FIRST M COMPONENTS OFc THE COMPLEX EIGENVECTOR -wJLL RE FOUND IN THEc FIRST M PLACES OF THF COLlJMN WHOSE TOP ELFMFNTc
c IS VECR(l,IVEC-ll. c INDIC- THE ARRAY INDIC• INDIC(JJ, I~1,z, ••• ,N, c INDICATES THE SUCCESS OF THE SUBROUTINE AS c FOLLOWS c VALUE OF INDICCil EIGENVFCTOR c 1 NOT FOUND c 2 FOUND c WORK- IWORK- A SINGLY SUBSCRIPTED WORK ARRAY OF DIMENSION c AT LEAST N. c ·woRKl- A SINGLY SUBSCRIPTED WORK ARRAY OF DIMENSION c AT LEAST N. c WORK2- A SINGLY SUBSCRIPTED WORK ARRAY OF DIMENSION c AT LEAST N. c WORK- A SINGLY SUBSCRIPTED WORK ARRAY OF DIMENSION c AT LEAST N. c C SURPROGRAMS CALLEO C NONE c C METHOD C INVERSE ITERATION IS USED ON THE REAL UPPER-HFSSENRERG MATRIX C JN ARRAY A CORRESPONDING TO THE COMPLEX FIGENVALUF. c C RFFFRFN(F
63
C WILKINSON,J.H.(19651. THE ALGFBRAIC EIGENVALUE PROBLEM. OXFORD, C CLARENDON PRESS. C J.GRAD ANO M.A.PRERNFR. COMMUNICATIONS OF THE ACM, VOL.11, C N0.12,DFC.,1968. c C STORAGE RFQUIREMFNTS C CDC6400- 799 WORDS PLUS AN ADDITIONAL 3*N*N+8*N C WORDS ARF REQUIRED. c C SOURCE LANGUAGE C FORTRAN c C RFMARKS C SURROUTINF COMPVE IS CALLFD ~y SUBROUTINF EIGENP. c .( AUTHOR C J.GRAD AND M.A.BREBNER. c C CHECKED RY C WeWARD, NOV/69. c
DOURLF PRECT~TON D,Dl DIMENSION A(NM•ll,VECR(NM,ll•H<NM,ll,EVR(l),EVI(ll•
lINDJC(ll•IWORK(lJ,SUBDIA(ll,WORKl(ll•WORK2(1)• 2WORK <1) .
FKSI = EVRC IVEC) ETA = FVICIVFCl
C THE MODIFICATION OF THE EIGENVALUE CFKSI + l*ETA) IF MORE C FIGENVALUES ARE FOUAL.
IF(!VEC.EO.MlGO TO 2 K = IVEC+l R = O.O DO 1 I=K,M
IF<FKSI.NE.EVR(ll)GO TO 1 IF(ARSCETA).NE.ARS(FVI(llllGO TO 1 R = R + 3.0
1 CONTINUE R = R*FX FKSI = FKSI+R F.TA = ETA, +R
.C c THE MATRlrx ( (H-FKSI*I l*(H-FKSI*I) + CETA*ETA >*I) rs C STORED INTO THE ARRAY A.
? R = FKSI*FKSI + ETA*ETA S = ?eO*FKSI L = "'1-1 DO '5 I=J,M
D0 4 J=ItM D = o.o ACJ,Il = o.o DO 3 '<' = I,J
3 D = D+HCltKl*H(K,Jl 4 ACI,JJ = 0-S*H(J,J) '5 ACitTl = A(T,Il+R
DO 9 I=ltl R = SURDIAC I 1 AC I+ 1 'I J = -S*R
64
II = I+l DO 6 J=ltil
6 ACJtil = A(J,Il+R*HCJtI+l) IFCI.EO.llGO TO 7 ACI+l•I-ll = R*SURDIACI-ll
7 DO 8 J=ItM 8 A(l+ltJl = ACI+ltJl+R*H(J,Jl 9 CONTINUE
c C THE GAUSSIAN ELIMINATION OF THE MATRIX C ( (H-FKST*I l*(H-FKSI*T) + CETi\*FTAl*I) IN THE ARRAY A. THE C ROW INTFRCHANGES THAT OCCUR ARE INDICATED IN THF ARRAY C TWORK. ALL THF MULTIPLIFRS ARE STORED IN THE FIRST AND IN C THE SECOND SURDIAGONAL OF THE ARRAY A.
K = M-1 DO 18 I=ltK
I 1 = I+l I 2 = t +2 IWORKCil = 0 JF(l.E~.KlGO TO 10 IF<ACI+2•Il.NF.0.0JGO TO 11
10 IF<ACI+ltIJ.NE.Q.OlGO TO 11 IFCACitil.NE.O.O>GO TO 18 AClt!l = EPS (i() Tr) 18
r 11 IFCl.FQ.KlGn TO 1?
IFCARS(A(J+l,Jll.GE.ARSCACI+2d)llGO TO 12 I.FCAASCACit!ll.GE.ABSCA(J+2,Jl)lGO TO 16 L = !+2 IWORKCI> = 2 GO TO 13
12 IFCARS(A(Itlll.GF.ARS(A(l+l•IlllGO TO 15 L = T+l I WORK (I ) = 1
-c 1'3 DO 14 J=I•M
R = ACitJl A( I tJ) = ACL,J)
14 A<L•Jl = R 15 IFCI.NE.KlGO TO 16
I 2 = II 16 DO 17 L=il•l2
R = -ACLtl)/A(J,Il ACLtll = R DO 17 J=Il,M
17 ACLtJ) = A(L,Jl+R*A(J,Jl 18 CONTINUE
IFCACM,MJ.NF.O.OlGO TO 19 ACMtMl = EPS
c . C THE VECTOR <l•l•••••ll IS STORED INTO THE RIGHT-HAND SIDE
C VECTORS VFCRC ,JVFCl AND VECRC tlVEC-ll RFPRESENTING THE "C COMPLEX RIGHT-HAND SIDE VECTOR.
1 q DO 21 I = 1 , N IFCI.GT.MlGO TO 20 VFCR(I,YVECl = 1.0
65
20 VECRCI,IVEC) = O.O VECRCI,IVEC-1) = O.O
21 CONTINUE c C THE INVERSE ITERATION IS PERFORMED ON THE MATRIX UNTIL THE C TNFTNITF NORM OF THF RIGHT-HAND SIDE VECTOR IS GREATER C THAN THF. ROUND DEFINFD AS 0.01/(N*EXl.
BOUND= 0.01/CFX*FLOAT!N)) NS = 0 JTfR = 1 DO 22 T=1,M
22 WORK(!) = H(I,Il-FKSI c ( THF SFQUFNCF OF THF COMPLFX VECTORS Z(S) = P(Sl+I*O<Sl AND C W!S+ll+I*V!S+l) IS GIVEN BY THE RELATIONS C (A - <FKSI-T*ETA>*I>*W!S+l) = ZCSl AND C Z<S+ll = WCS+ll/MAXCW<S+ll). C THE FINAL W<Sl IS TAKFN AS THE COMPUTED EIGENVECTOR. c C THF COMPUTATION OF THF RIGHT-HAND SIOE VECTOR C (A-FKST*I>*PCSl-FTA*OCSl. A IS AN UPPER-HESSFN8FRG MATRIX.
23 DO 'l7 I=l•M D = WORKCil*VFCRCI•IVFCl IF<I.EO.llGO TO 24 D = D+SURDIA<I-ll*VFCRCI-1,IVECl
24 L = I +1 IF<L.GT.MlGO TO 26 DO 25 K=L ,~,,
25 D = D+HCJ,Kl*VFCRCK•IVEC) ?6 VFCRCI,IVFC-1) = D-FTA*VFCR(I,IVEC-ll 27 CONTINUE
c C GAUSSIAN ELIMINATION OF THE RIGHT-HAND SIDE VECTOR.
K. = ,...,_l DO 28 I =l 'K
L = I+ I WORK ( I ) R = VFCR(L,IVEC-1) VECRCL,IVFC-lJ = VECR(J,IVFC-ll VECR(I,IVFC-ll = R VPCR<I+l,JVEC-1> = VECRCI+l,IVFC-l)+ACI+l,Il*R IF<I.EQ.KlGO TO 28 VECRCI+2,IVEC-1) = VECRCI+2,IVEC-l)+A(I+2,Il*R
28 CONTINUE c C THF COMPUTATION OF THF REAL PART UCS+ll OF THE COMPLEX C VFCTOR WCS+ll. THE VFCTOR U(S+ll IS 08TAINFD AFTFR THF C RACKSURSTITUTION.
DO 31 I= l ,M J = M-I+l D = VECR(J,IVEC-1) IF(J.EQ.MlGO TO 30 L = J+l DO 29 K=L,M
Dl = ACJ,K) 29 D = D-Dl*VECR(K,IVFC-1)
66
30 VECRCJ,IVEC-1) = D/A(J,J) '.H CONTINUE
c C THF COMPllTATT0N ()F THF TMAG!MARY PART VC S+l) OF THF VECTOR c wis+1>. WHFRE VCS+l) = (P(S)-(A-FKSI*Il*UCS+l) l/FTA.
DO ? 5 I= 1 , ~ D = WORKCT>*VFCRCitIVEC-1) IFCI.EO.I>GO TO 32 D = D+SUBOIACI-l>*VFCRCI-1,IVEC-1>
32 L = T+1 IFCL.GT.MJ\;O TO 34 D('I ?3 K=L•~
33 D = D+H(J,Kl*VFCRCK,TVFC-1) 34 VECRCitIVEC) = CVECR(J,IVECl-Dl/ETA 35 CONTINUE
c C THE COMPUTATION OF CINFIN. NORM OF WCS+ll>**2.
L = 1 s = o.o DO 36 I=ltM
R = VFCRCTtIVFC>**2 + VECR<I•IVFC-11**2 IF<R.LE.SlGO TO 36 S = R L = I
36 CONTINUE C THE COMPUTATION OF THE VECTOR ZCS+lltWHERE ZCS+l)= WCS+l)/ C <COMPONENT OF WCS+ll WITH THE LARGEST ABSOLUTE VALUE) •
U = VFCRCLtlVFC-1) V =. VECRCLtlVEC) DO 37 I=ltM
B = VECR (I ti VFC) R = VECRCitIVEC-1> VECRCltIVECl = CR*U + B*V)/S
37 VFCRCI,IVEC-1> = CR*U-R*Vl/S C THF COMPUTATION OF THF RESIDUALS ANO COMPARISON OF THE ( RFSIDUALS OF THF TWO succ~SSIVF ST~Ps OF THE INVERSE C TTFRATTON. IF THF TNFINITF NORM OF THE RESIDUAL VECTOR IS C GRFATFR THAN THE INFINITE NORM OF THF PREVIOUS RESIDUAL C VECTOR THE COMPUTED VECTOR OF THE PREVIOUS STEP IS TAKEN C AS THE COMPUTED APPROXIMATION TO THE EIGENVECTOR.
B = O.O DO 41 I= 1 ,~
R = WORKCil*VFCRCitIVEC-ll - ETA*VECRC!tIVECl U = WORK<I>*VECRCitTVEC> + ETA*VECRCitIVEC-1> IF ( I • FQ. 1 l GO T 0 3 R R = R+SURDIA(!-ll*VECRCI-1,IVEC-ll U = U+SUBDIA<I-l>*VECR(I-1,IVEC)
38 L = I+l IFCL.GT.M>GO TO 40 DO 39 J=LtM
R = R+H(I,J)*VFCR(J,JVFC-1} 39 U = U+H(f,J}*VECR(J,JVFC} 40 U = R*R + U*U
IFCB.GE.U)GO TO 41 B = U
41 CONTINUE IF<ITER.EQ.llGO TO 42
67
WORKlCJl = VECRCJ,IVEC) 43 WORK2Cil = VECRCJ,JVFC-ll
PREVIS = B IF<NS.EQ.l)GO TO 46
ITFR = ITFR+l IFCITER.GT.6)GO TO 47
IF!BOUND.GT.SQRTCSl)Gn TO 23 NS = 1 Gn rn ?3
44 DO 45 I=I,N VECRCJ,IVECl = WORKlCil
45 VECRCI,IVFC-IJ=WORK?Cll 46 INDICC!VFC-1 l = 2
INDICCJVFCl = 2 4 7 RF TURI''
END 6400 END OF RECORD
0.0000 0.5000 90.0000 1.7664 0.3668 0.1206 END OF FILE
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