APPENDIX AEIGENVALUES

Eigenvalue, or characteristic-value, problems are a special class of problems that arecommon in engineering and scientific problem contexts involving vibrations and elasticity.In addition, they are used in a wide variety of other areas including the solution of lineardifferential equations and statistics.

Before describing numerical methods for solving such problems, we will present somegeneral background information. This includes discussion of both the mathematics and theengineering and scientific significance of eigenvalues.

A.l Morhemoticol Bockground

Chapters 8 through 12 dealt with methods for solving sets of linear algebraic equations ofthe general form

[A ] { r } : { b }

Such systems are called nonhontogeneoas because of the presence of the vector { b } on theright-hand side of the equality. If the equations comprising such a system are linearlyindependent (i.e., have a nonzero determinant), they will have a unique solution. In otherwords, there is one set of x values that will make the equations balance.

In contrast, a homogeneorrs linear algebraic system has the general form

l A l { . t } : 0

Although nontrivial solutions (i.e., solutions other than all x's:0) of such systems arepossible. they are generally not unique. Rather, the simultaneous equations establish rela-tionships among the .r's that can be satisfied by various combinations of values.

Eigenvalue problems associated with engineering are typically of the general form

( a 1 1 - ) " ) x 1 | a n x t * ' . ' * a l n x u : 0

{ t y x 1 I ( a 2 2 - } , ) r 2 + . . . + a 2 r F u : 0

Iq , 1 . 1 I - T - nr3x l * . - . -+ (e , , , , - ,1 , )x , , - 0

566 APPENDIX A EIGENVALUES

where )' is an unknown parameter called the eigenvalue, or characteristic value. A solution{,r} tbr such a system is referred to as an eigenvector. The above set of equations may alsobe expressed concisely as

l t a t - ^ t I ] ] { x } :o ( A . l )

The solution of Eq. (A.l) hinges on determining.l.. One way to accomplish this isbased on the fact that the determinant of the matrix ltal

* ),[1]] must equal zero for non-trivial solutions to be possible. Expanding the determinant yields a polynomial in )". whichis called the characteristic' pol ,-nomial.The roots of this polynornial are the solutions forthe eigenvalues. An example of rhis approach, called the poll,yorrro, methocl, will be pro-vided in Section A.3. Before describing the method, we will first describe how eisenvaluesar ise in engineer ing and sc ience.

A.2 Physicol Bockground

The mass-spring system in Fig. A.lo is a simple context to i l lustrate how eigenvalues occurin physical ploblem settings. It also will help to illustrate some of the mathernatical con-cepts introduced in Section A. l.

?l sintplity the analysis, assume that each rnass has no external or damping fbrces act-ing on it. ln addition, assume that each spring has the same natural length / and the samespring constant t. Finally. assume that the displacernent of each spring is rneasuled relativeto its own local coordinate system with an origin at the spring's equilibrium position(Fig. A. la). Under these assumprions, Newton's second law can be employed to develop aforce balance lbr each mass:

d - X rm r - , i - - k r r 1 k ( x : - - l 1 . 1

FIGURE A.IPosit ioning lhe mosses owoy from eqi, i l ibr ium creotes forces in the sprinos lhoi on rereose ieodto osci l lct ions of lhe nrosses. The posit ions of the mosses con be referenled fo locoJ coordinoteswi ih o r ig ins o t the i r respec t ive equ i l ib r ium pos i f ions

0 0 0 - l t t t t f 0 0 0 - l n t 2

| - _ r 0 0 07 f - - - 7 t ? T - , 7 f

APPENDIX A EIGENVALUES 567

and

' t ) - 'u . \ lt n : - : - k ( . r : - 1 1 ) - ( r 3

d t -

where ,t; is the displacement of mass i away from its equil ibrium position (Fig. A.lb). Bycollecting terms, these equations can be expressed as

d - r ' .

nr t '4 - f t (-2.r1 *.r ;1 : gd t -

t )0 - - X 1

n 1 : - - - - f t ( r t - 2 x ; ) : g- . t r :

From vibration theory, it is known that solutions to Eq. (A.2) can take the form

xi : Xi sin(a.rr)

where X; : the amplitude of the vibration of mass i and a: the frequency of the vibra-tion, which is equal to

2n

7,,

where {, is the period. From Eq. (A.3) it follows that

x'i : -Xia2 sin(a' 't)

Equations (A.3) and (A.5) can be substituted into Eq. (A.2),terms, can be expressed as

(4.2a)

(A.2b)

(A.3)

(#--')", -- !-x, . (# - ,')x, : o

k- X z - 0m 1

(A.4)

(A.5)

which. after collection of

(A.6ru)

(A.6b)

Comparison of Eq. (4'.6) with Eq. (A. l) indicates that at this point, the solution hasbeen reduced to an eigenvalue problem. That is, we can determine values of the eigenvalueol2 Ihat satisfy the equations. For a two-degree-of-freedom system such as Fig. A.l, therewill be two such values. Each of these eigenvalues establishes a unique relationshipbetween the unknowns X called an eigenvector. Section A.3 describes a simple approach todetermine both the eigenvalues and eigenvectors. lt also illustrates the physical signifi-cance of these quantities fbr the mass-spring system.

A.3 The Polynomiol Merhod

As stated at the end of Section A.1, the polynomictl nrcthod consists of expanding thedeterminant to generate the characteristic polynomial. The roots of this polynomial are thesolutions for the eigenvalues. The following example illustrates how it can be used todetermine both the eigenvalues and eigenvectors for the mass-sprinq system (Fie. A.1).

s68 APPENDIX A EIGENVALUES

EXAMPLE A. l The Polynomiol Method

I Problem Stotement. Evaluate the eigenvalues ancl the eigenvectors of Eq. (A.6) for thei case where rl | : tt12: 40 kg and li : 200 N/m.

Solution. Substituting the parameter values into Eq. (A.6) yields

( f 0 @ 2 ) x t - 5 X z

- 5 X 1 * ( 1 0 - r r . , r ; X 2

The determinant of this system is

( r ' ) t - 2oo2 +75 : o

which can be solved by the quadratic formula for o,f : l-5 ancl -5 s 2. Therefore, the fre-quencies for the vibrations of the masses are .D : 3.873 s I and 2.236 s-t, respectively.These values can be used to determine the periods fbr the vibrations with Eq. (A.4). For thefirst mode" Tp : 1.62 s. and fbr the second, 7,, :2.81 s.

As stated in Section A.l, a unique set of values cannot be clbtained for the unknownamplitudes X. However, their ratios can be specified by substituting the eigenvalues back

FIGURE A.2The pr inc ipo l modes o f v lb ro t lon o f two equo l mosses connected by th ree iden l i co l spr ngs

between f ixed wolls

T _

1 . 6 2

2.81

- 0

- o

(a) First mode (b) Second mode

APPENDIX A EIGENVALUES 569

into the equations. For example. for the flrst mode (ro2 : l5 s 2):

( 1 0 - 1 5 ) x r - - 5 X 2 : g- 5 X r * ( 1 0 - 1 5 ) X 2 : 0

Thus .weconc lude tha tX l - -X2 . l nas in r i l a r f ash ion fb r thesecondmode( r , . , 1 : 5s - r ) .

Xr : Xt. These relationships are the eigenvectors.This example provides valuable information regarding the behavior of the systent in

Fig. A.L Aside from its period, we know that if the system is vibrating in the first mode,the eigenvector tells us that the amplitucle of the second mass wil l be equal but of oppositesign to the amplitude of the first. As in Fig. A.2a, the masses vibrate apart and then togetherindefinitely.

In the second mode, the eigenvector specifies that the two mersses have equal ampli-tudes at all t imes. Thus, as in Fig. 4.2b, they vibrate back and forth in unison. We shouldnote that the configuration of the amplitudes provides guidance on how to set their initialvalues to attain pure motion in either of the two modes. Any other configurntion will leadto superposition of t lre rrodes.

We should recognize that MATLAB has built- in functions to facil i tate the polynomialmethod. For Exarnpie A.l, the poly function can be used to generate the characteristicpolynontial as in

> > A = t 1 0 - 5 ; - 5 I O ) ;- _ . - > p = p o t \ , ( A )

r - 2 0 7 a

Then, the roots function can be employed to compute the eigenvalues:

> > r o o t s ( p )

u t t t = ,

5

A.4 The Power Merhod

The power method is an iterative approach that can be employed to determine the largestor dominant eig,envctlue. With slight modification, it can also be employed fo determine thesmallest value. It has the additional benefit that the corlesponding ei-eenvector is obtainedas a by-product of the method.

To implement the power method, the system being analyzed is expressed in the form

[A l { . r } : ) " { r ) (4.7)

As illustrated by the following example, Eq. (A.7) forms the basis fbr an iterative solr.rtiontechnique that eventually yields the highest eigenvalue and its associated eigenvector.

APPENDIX A EIGENVALUES

EXAMPLE A.2 Power Method for Highest Eigenvolue

Problem Stotement. Using the same approach as in Section A.2, we can derive the fol-lowing homogeneous set of equations for a three mass-four spring system between two

: 0

t,̂ X , :om )

fr,, (* - ')x,: o

If all the masses m : I kg and all the spring constants k : 20 N/m, the system can beexpressed in the matrix format of Eq. (A. 1) as

[ 4 0 - 2 0 0 I

| - 2 0 4 0 - 2 0

| - r l i l : 0

L 0 - 2 0 4 0 )

where the eigenvalue ). is the square of the angular fiequency c,r2. Employ the powermethod to determine the highest eigenvalue and its associated eigenvector.

Solution. The system is first written in the lbrm of Eq. (A.7):

f ixed walls:

( ' * _- ' )x , - r r .\ l n r / h t t

k / 2 k , \- - - v - ( l?1r . ' ) r , -

40Xr -20X2 : ) , X t

- 2 0 X 1 + 4 0 X 2 - 2 0 X 2 : ) " X z

-20x2 - l40Xt : ) 'Xs

At this point, we can make initial values of the X's and use the lefrhand side to compute aneigenvalue and eigenvector. A good first choice is to assume that all the X's on the left-handside of the equation are equal to one:

4 0 ( 1 ) - 2 0 ( l ) : 2 0

- 2 0 ( 1 ) + 4 0 ( l ) - 2 0 ( 1 ) : 0

- 2 0 ( l ) - l 4 0 ( l ) : 2 0

Next, the right-hand side is normalized by 20 to make the largest element equal to one:

[ 20 ] [ t l{ 0 l : 20 {01lzo l l r l

Thus, the normalization factor is our first estimate of the eigenvalue (20) and the cone-sponding eigenvector is I I 0 I .l

r . This iteration can be expressed concisely in matrix

,'1'l {i}:ll{}:"{ilform as

[ 40 -20

| -20 40

I n a n

APPENDIX A EIGENVALUES 571

The next i terat ion consis ls of mul t ip ly ing lhe matr ix by I 0 1 l z t o g i v e

[1l' jl, -n] l?]:detenrine the error estimate:

1 4 0 - l 0 l' r , , , : l ^ l x l f ) 0 7 :

I +r,

The process can then be repeated

Tltird iterutiott:

[ 40 -20 0 l l l

| - 2 0 4 0 - l ( l I I - l

L;" -; r;ll

[1$ :;;n] { i,, }:FiJth iteratiort:

[ 4 0 - 2 0 0 l ] - 0 . 1 1 4 2 e

t - 2 0 4 0 - t 0 t { |L o _20 +o l [_ .0 .1142s

where le,, | : 2.08c/c .

-1ll:-'

60-8060

-0.75I

-0.75- _RO

rl:l : "

I- l

1

Therefore. the eieenvalue estimate for the second iteration is 40. which can be emploved to

where le,, | : l50c/o (which is high because of the sign change).

Fourtlt iteratiou:

'or'.;troo'"nul

where lc,, | : 2147o (wlrich is high because of the sign change).

:llfii'i^ 1 -0.70833

:68 .57141 It -0.70833

Thus, the eigenvarlue is convergir.rg. Afier several more iterations, it stabilizes on avalue of 68.28421 with a corresponding eigenvector of | -0.707 107 I -0.101 lU )r .

Note that there are some instances where the power method will converge to thesecond-largest eigenvalue instead of to the largest. James. Snrith, and Wolford ( 1985) pro-vide an i l lustration of such a case. Other special cases are discussed in Fadeev and Fadeeva( l 963) .

In addition, there are sometimes cases where we are interested in determining thesmallest eigenvalue. This can be done by applying the power method to the matrix inverseof [Al. For this case, the power method wil l converge on the largest value of l/ l- in otherwords, the smallest value of i. An application to find the smallest eigenvalire wil l be left asa problem exercise.

572 APPENDIX A EIGENVALUES

Finally, after finding the largest eigenvalue, it is possible to deternine the next highestby replacing the original matrix by one that includes only the remaining eigenvalues. Theprocess of removing the largest known eigenvalue is called deflation.

We shouid mention that aithough the power method can be used to iocate intermediatevalues, better methods are available for cases where we need to determine all the eigenval-ues as described in Section A.5. Thus, the power method is primarily used when we wantto locate the largest or the smallest eigenvalue.

A.5 MATLAB Function: ers

As might be expected, MATLAB has powerful and robust capabilities for evaluating eigen-values and eigenvectors. The function e:ig, which is used for this purpose, can be used togenerate a vector of the eigenvalues as in

- e e l g t ' 4 t

where e is a vector containing the eigenvalues of a square matrix a. Alternatively. it can beinvoked as

where Dis a diagonal rnatrix of the eigenvalues and I'is a full matrix whose columns arethe corresponding eigenvectors.

EXAMPLE A.3 Use of MATLAB to Determine Eigenvolues ond Eigenvectors

Problem Stotement. Use MAILAB to determine all the eigenvalues and eigenvectorsfor the system described in Example A.2.

Solution. Recall that the n.ratrix to be analyzed is

[ 40 -20 0 I

| -20 40 -20

|L 0 -20 40 _.1

The matrix can be entered as

> > A = 1 4 0 - 2 0 O ; - 2 A 4 A - 2 A ; 0 2 0 4 a l ;

If we just desire the eigenvalues we can enter

r e = e i q ( A l

e =

I I . 1 I 5 1

4 0 . O O O O

6 8 . 2 , 8 4 3

Notice that the highest eigenvalue (68.2843) is consistent with the value previously deter-mined with the power method in Example A.2.

APPENDIX A EIGENVALUES 573

If we want both the eigenvalues and eigenvectors, we can enter

0 . 7 4 1 1 - 0 . 5 0 0 0- 0 . 0 0 0 0 4 . 1 0 7 r

0 . 7 0 ? 1 - 0 . 5 0 0 0

C1 -

L I . 1 I 5 1 0 00 4 0 . 0 0 0 0 00 0 6 8 . 2 8 4 3

Again, although the results are scaled differently. the eigenvector conesponding to thehighest eigenvalue [-0.5 0.707 I -0.5]7 is consistent with the value previouslydeterrn ined wi th the power mcthod in Example A.2: | -0 .707 107 | -0.107 107 ) ' .

, . . EIGENVALUES AND EARTHAUAKES

Bockground. Engineers and scientists use mass-spring models to gain insight into thedynamics of structures under the influence of disturbances such as earthquakes. Figure A.3shows such a representation for a three-story building. Each floor mass is represented bynt,, and each floor stiffness is represented by ft, for i : 1 to 3.

For this case, the analysis is limited to horizontal motion of the structure as it is sub-jected to horizontal base motion due to earthquakes. Using the same approach as developedin Section A.2. dvnamic force balances can be developed for this svstem as

0 . 5 0 0 0a . 7 a l I0 . 5 0 0 0

/ t . I t . \

{ ^ ' - ^ t - r l . l x , -

\ l r r /t .

___: Xt Im z

K2

* t

/ k z * k z ' \l - - u t , l\ tTlt /

x2

L -X 2 -

mZ

L^ 3

* ,

- n

X : : 0

*,*(h-oi)x.:o

where X, represents horizontal floor translations (rn), and @,, is the natura[ or rcsonant,

frequency (r'adians/s). The resonant fiequency can be expressed in Hertz (cycles/s) bydividing itby 21r radians/cycle.

Use MATLAB to determine the eigenvalues and eigenvectors for this system. Graph-ically represent the modes of vibration for the structure by displaying the amplitudes ver-sus height for each of the eigenvectors. Normalize the amplitudes so that the translation ofrhe third floor is one.

574 APPENDIX A EIGENVALUES

cont inued

lar : 8,000 kg

t , -

t , -

t . -

| .....- U U U - . . - - ]

rh : 10,000 kg

l *- uuu - - l

'|1r : 12.000 kg

[*-

1,800 kN/m

2,400 kN/m

3.000 kN/m

FIGURE A.3

Solution. The parameters can be substituted into the force balances to give

(+so - . | , ) x r - 2oox2 -o

-240Xt+ (420 - rT )X , - 180x , : s225X2+(225-u ' l ) x . : g

A MAILAB session can be conducted to evaluate the eigenvalues and eigenvectors as

> > A = f 4 5 0 - 2 l l C 0 ; - ? . 4 0 4 2 i ) 1 8 C ; 0 - 2 2 5 2 2 5 1 ;' > : >

l v , c l r . - c i q ( ; r )

0 . 5 ! l l ! r - L r . 6 l 4 i 1 0 . 2 9 1 1O . r l 0 ' i - 0 . 3 5 0 6 0 . 5 7 2 50 . _ 1 4 1 1 0 . 5 8 9 0 0 . 1 6 5 4

c i =i r q 3 . ! 9 8 2 0 0

0 1 3 9 . 4 t ' i 9 0{ t 0 5 6 . 9 2 3 9

Therefore, the eigenvalues are 698.6, 339.5, and 56.92, and the resonant frequencies in

Hz are

: > L / : r - s . i r t { c i - i a q ( d ) ) ' i 2 i p i

4 . 2 t 6 5 2 . 9 3 2 4 1 . 2 0 0 8

The corresponding eigenvectors are (normalizing so that the amplitude for the third floor

is one)

{lt:'#:} {-3.33il; }0.38008e 10.746eee

I

APPENDIX A EIGENVALUES 575

continued

Mode 1la \ :1 ' 'OOU^r ,

FIGUR.E A.4

- 1 0 1

Mode 2(aL : 2 .9 t rO r r ,

- 1 0 1

Mode 3ko,, = 4.2066 Hzl

A graph can be made showing the three modes (Fig. A.4). Note that we have ordered them

fiom the lowest to the highest natural frequency as is customary in structural engineering.

Natural frequencies and mode shapes are characteristics of structures in terms of their

tendencies to resonate at these frequencies. The frequency content of an eafihquake typically

has the most energy between 0 to 20 Hz and is influenced by the earthquake magnitude, the

epicentral distance, and other factors. Rather than a single frequency, they contain a spec-

trum of all frequencies with varying amplitudes. Buildings are more receptive to vibration at

their lower modes of vibrations due to their simpler deformed shapes and requiring less

strain energy to deform in the lower modes. When these amplitudes coincide with the nat-

ural frequencies of buildings, large dynamic responses are induced creating large stresses

and strains in the structure's beams, columns, and foundations. Based on analyses like the

one in this case study, structural engineers can more wisely design buildings to withstand

earthquakes with a good factor of safety.

1 ilii;l

,' .i,ii'', ,iii'ii.il"

APPENDXB

fm insearch , 181,326f o r m a t b a n k , 2 3f o r m a t c o m p a c t , 2 lf crrmat 7onrJ,23f o r m a t t o n g e , 2 3f o r m a c i o n g e n g , 2 3f o r m a t t o o s e , 2 lf o r m a [ s h o r t , 2 3f o r m a t s h o r t e . 2 3f o r m a t s h o r t e n g , 2 3f p 1 o t , 6 7

fpr in t f , 48f z e r o , l - 5 1g r a d i e n t , 4 6 3

gr r lo ! -1J

he lp , 30h e l p e l f u n . 3 0h i s t , 2 9 l

n o r d o I t , J 4h o l d o n . 3 4humps, 78,440

i n l i n e , 6 7

i n p u t , 4 7

i n r - e r p 1 , 3 7 6

i n t e r p 2 , 3 8 l

in t -e rp3 , 3 [32inv, 201,203isempcr r , 1271e_qend, 385, 4151engt l - r .32

l i n s p a c e , 2 6

1 o a d , 5 0

r o q , 3 01 o q l 0 , 3 0Ios2 ,30 , 126los los , 40 ,106logspace ,26Lookfor ,36,44) . . ) L ' . )

m a x , 3 1 , 2 2 6 , 2 9 0mear r ,31 ,290median, 290mesh .62meshsr id, I 80. 466m i n . 3 1 , 2 9 0mode,290riargin, 57norm,25Jo d e 1 l 3 , 5 l 7o d e 1 5 s , 5 2 9o d e 2 3 , 5 l 6ode23s,529ode23L,529ode23 tb ,529o d e 4 5 , 5 1 7 , 5 4 6o d e s e t , 5 1 9ones ,25opt imset , 153, 179.326pause ,62pch ip , 374 ,377

p 1 o t , 3 3p lo t3 , 35 , -508po rv . 156 ,569

MATLAB BUILT.IN FUNCTIONS

abs, 30

a c o s , 3 0a s c i i , 5 la x i s s q u a r e , 3 5b e e p , 6 2

b , - s s e l j , 3 5 6c e 1 1 , 3 1 , I 8 0chol,245

c 1 a b e 1 , 1 8 0 , 4 6 6

c l e a r , 5 0cond, 257, 338c o n t o u r , 1 8 0 , 4 6 6con_,,, 157c u m t r a p z , 4 l 4

d b 1 q u a d , 4 l 8

d e c o n v , 1 5 6

d e t , 2 l 7

d taq ,574

d l f f . 4 l 3 , 4 6 0 , 4 6 2d i s p , 4 7

d o u b l e , 5 l

e i s , 5 7 2

e 1 f u n , 3 0

eps, 90

e r f , 4 4 5

error, -52e x p , 3 0

e1'e, 202

f a c t o r i a l . 4 0 , 5 9

f i x , 3 6 5

f 1 o o r , 3 1 , 1 8 0

fminbnd, 178

APPENDIX B MATLAB BUILT-IN FUNCTIONS 577

p o l y f i , r , 3 0 1 , 3 2 5 ,

P o f y \ / a l . 1 5 6 , 3 0 7 ,p r o d , 3 l

q u a d , 4 3 9

q r L a d i , 4 3 9

qu i r re r ,4 ( t5

r e a t m a x , 9 0

r e a l m i n , 9 0

r o o t s , 1 5 5 , 5 6 9

ror-rncl, 3 |

save, 50

s e m r i o g y , 4 0

s e t , 3 8 5

s i g n , 5 5s i n , 3 0size,202s o r t , 3 ls p 1 r n e , 3 7 4sq r t , 30s q r t m , 3 ls r d , 2 9 0S U r r p L O t ! - 1 4

surn, 3 1, 246s u r f c , 1 8 0tan, 30l -anh, 7

r r c ,62r r - o 1 1

t o c , 6 2

t r a p z , 4 l 4 , 4 2 2

t r i p l e q u a d , 4 l S' " ,a r .290

vararqJ in , T0

who, 25

whos, 25

) 4 1 4 D e 1 , - 1 J

y l a b e l , 3 3z e r o s , 2 5

: l a b e 1 . 1 8 0

339339

APPENDIX CMATLAB M.FILE FUNCTIONS

M-file Nome Descripfion Poge

b r s e c LeuLodeGaussNa i vet , a l l s s P r v o r

G a u s s S e i d e lgo ld r rL i n

incse.rr - . :h

1 rn reg rn - r f c n i i n a

NewL rntnerrtmu f tnewtraphrk4 sysrombergTabl eLo. ika r a pr rapuneLlT r i d i a g

Root locol ion wi th b isect ionnlegrol ion of o s ing e ordinory d l f ferent io equotLon wi lh Euler 's methodSo ving l inecr syslems wi lh Gouss e iminol ion w fhoul p ivolrngSoving inecr sysiems wi th Gouss e minal ion wi th port io l p ivot ingSolv ing ineor syslems wi th th-- Gouss-Seidel methodMinlmum of one-din,ensictnoi funct ion wi th golden-sect ion seorchRoot ocot ion r ,v i lh on incremenic i seorchInterpolo l ron wi th the Logronge po ynomioll i r q o I ' o g \ l l i r e r r ' r ' I ' r : o l e g - - , i o r

Cub i c sp i ne w i t h nc l u r c l end cond i t i onsnlerpolot ion wi th the Newion polynomiolRool ocol ion for non ineor systems o[ equoironsRoot locolion with lhe Ner,r,ion-Rophson methodIntegrc l ion ol system ol ODEs wl th 4th order RK methodIntegroi ion of o [uncl ion v" , i lh Romberg in legrot ionToble ookup wi th l ineor interpoJot ionlniegrolion cl c funclicn \\,rth lhe .')rlposit6 i16pa1616Jo 1ul-.Integrof ion of uneqr ispcced dcrtc wi th the i ropezoidc ru leSolv ing l r id icgoncl l inecrr systems

1 2 74862 2 22272691 7 6t 2 c3493063 8 33462761495424323644044 1 3'2')9