smallest egenvalue

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Numerical Methods

Course 5

Eigenproblems.The power method and the inverse power method.

these notes are available for download at

http://cemsig.ceft.utt.ro/astratan/didactic/nm

Eigenproblems in structural engineering

Earthquake engineering

Modal analysis (eigenproblem) is used to determine the

fundamental modes of vibration of a structu re

Earthquake forces are determined based on the results of the

modal analysis and the characteristic of ground motion

Buckling of structures or elements


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Consider a system of three

homogeneous linear algebraic

equations of the form Four unknowns: X1, X2, X3, and Solutions of the equation:

the trivial solution X=0

solutions 0 possible for special values of, called eigenvalues Finding eigenvalues represent an eigenproblem

Unique values ofXT=[X1 X2 X3] cannot be determined, but

for each eigenvalue i, relative values of X1, X2, X3 can beobtained

Xivectors corresponding to

ieigenvalues are called

eigenvectors , and they determine the mode of osci llation

of the physical system

The eigenproblem can be wri tten as:

Consider the following 22eigenproblem:

It can be rearranged as:

The two equations represent two

straight lines with slopes m1 and m2.

Trivial solution: when m1m2 Al ternat ive solu tion, when m1=m2

values of for which m1=m2 are calledeigenvalues

the relative values of x1 and x2 (given by

the slope m) are called eigenvectors

Eigenproblem: geometrical representation


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Organisation of Chapter 2

Mathematical characteristics of eigenproblems

A system of nonhomogeneous linear algebraic equationsof the form Cx=b can be solved by the Cramer's ru le as:

where Cj is the matrix C with the column j replaced by

vectorb. In general det(Cj)0, and unique solution arefound for xj.

A system of homogeneous l inear algebraic equations ofthe form Cx=0 can be solved by the Cramer's rule as:

In general det(C)0, and the only solution is the trivial one x=0 For special forms ofC that involve an unspecified arbitrary scalar

, values of can be chosen to force det(C)=0, so that solutionother than the trivial one is possible. The solution x is not unique

in this case, but relative values of xj can be determined


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Eigenproblems arise when the C matrix takes the form:

Values of determined so that

are the eigenvalues of the problem

The homogeneous system of equations can be written as:

In many problems B=I, so that it becomes:

IfBI, a matrix can be defined, and

multiplying the equation to the left by B-1, one

gets which is a classical eigenproblem.

An eigenvalue problem is most commonly stated as:

Finding eigenvalues

Consider the eigenproblem:

It can be solved by expanding the det(A-I)=0 and findingthe roots of the resulting n

th

-order polynomial, called thecharacteristic equation :

The eigenvalues are: =13.870585, 8.620434, 2.508981


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The eigenvectors are found for each eigenvalue in the

following way:

X1 is set to 1.0 X2 and X3 are found from two of the original equations

as: X3=(10-)/15- and X2=(8-)/2-X3 substituting 1 to 3 into the expressions of X2 and X3, it yields:

Eigenproblem summary

Eigenproblems arise from homogeneous systems of

equations that contain an unspecified parameter in thecoefficients

The characteristic equation is determined by expanding

the determinant

Eigenvalues i

(i=1,2,,n) are found by solving the n-th

order polynomial in Eigenvectors xi (i=1,2,,n) are found by subst ituting the

individual eigenvalues into the homogeneous set ofequations and solving for the relative values ofxi

For large systems of equations

expanding the determinant is difficul t

solving a n-th order polynomial is difficult

As a resul t, alternative procedures are required


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The power method

The power method is an iterative technique used todetermine the largest (in absolute value) eigenvalue of

an equation of the form and can be summarized

as follows:

1. Assume a trial vectorx(0) for the eigenvectorx. Choose one

component ofx to be unity, designating that component as the

unity component.

2. Perform the matrix multiplication: Ax(0)=y(1)

3. Scale y(1) so that the unity component remains unity: y(1)=(1)x(1)4. Repeat steps 2 and 3 to convergence. When the solution

converged, the value of is the largest (in absolute value)eigenvalue, and the vectorx is the corresponding eigenvector(scaled to unity on the unity component)

The general algorithm of the power method is:

Limitations of the power method:

when the iterations indicate that the unity component may be

zero, a different unit y component must be chosen

the method converges slowly when the magnitudes of the two

largest eigenvalues are close

when the largest eigenvalues are of equal magnitude, the power

method, as described, fails

Example of the power method: find the eigenvalue of

largest magnitude and the corresponding eigenvector forthe matrix

assume x(0)T=[1.0 1.0 1.0] and let the third component x3 be the

unity component

applying equation


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step 0

step 1

The results of

iterations continued

until changed by lessthan 0.000001 ispresented here

The largest eigenvalue

and the corresponding

eigenvector are

Basis of the power method

Assumptions:

Matrix A is of s ize nn and is nonsingular Its n eigenvalues verify the following:

The corresponding n eigenvectors

are linearly independent

Thus, any arbit rary vector x can be expressed as a

combination of the eigenvectors:

Multiplying both sides by A, A2,,Ak (superscript

denoting repetitive matrix mult iplication), and recalling

that , yields:


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Factoring the 1k from the last equation yields:

Since |1|>|i| for i=2,3,,n, the ratios (i/1)k 0 as k,and the above equation approaches the limit

which approaches 0 if |1|1.Therefore, it must be scaled between iterations.

Scaling can be accomplished by scaling any of thecomponents of vector y(k) by unity. Choose the y1 as that

component. Thus, x1 wil l become 1.0, and

In the next step:

Taking the ratios of the last to equations gives

Thus, if y1(k)=1, then y1

(k+1)=1. If y1(k+1) is scaled so thaty1

(k+1)=1, then y1(k+2)=1.

Consequently, scaling a particular component of vectory

each iteration, factors 1 out of vectory. In the limit , ask, the scaling factor approaches 1, and the scaledvectory approaches the eigenvectorx1.


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Several restric tions apply to the power method:

The largest eigenvalue must be distinct

The n eigenvectors must be independent The initial guess xi

(0) must contain some component of

eigenvectorxi, so that Ci0 The convergence rate is proport ional to the ratio

where i is the largest (in magnitude) eigenvalue and i-1 is thesecond largest (in magnitude) eigenvalue

The inverse power method

The inverse power method can be used to determine thesmallest (in magnitude) eigenvalue of matrix A.

The procedure essentially finds the largest (in magnitude)eigenvalue of the inverse matrix A-1, which is the smallest

(in magnitude) eigenvalue of matrix A.

Prove:

Consider the standard eigenproblem

Multiply the equation by A-1 to the left:

Rearranging, yields an eigenproblem forA-1:

The eigenvalues ofA-1 are the reciprocals of eigenvalues

of matrix A

The eigenvectors of matrix A-1 are the same as the

eigenvectors of matrix A


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The power method may be used to solve the problem:

However, to avoid computation of the inverse matrix, the

LU method can be employed:

The power method applied to A-1 is given by

Multiplying the above equation by A gives:

which can be written as

This equation is in the standard form Ax=b, where x=y(k+1) and

b=x(k). Therefore, for a given x(k), the y(k+1) can be found by theDoolittle LU method.

The procedure of the inverse power method is as follows:


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Example: find the smallest (in magnitude) eigenvalue of

the matrix

Assume x(0)T=[1.0 1.0 1.0], and choose the first component to be

unity

Solve for the L and U matrices using Doolit tle method:

Solve forx' using forward substitution Lx'=x(0)

Solve fory(1) using backward substitution Uy(1)=x'

Scale y(1) so that the unity component is unity


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The results of i terations continued until inverse changedby less than 0.000001 is presented in the table

The final solution for the smallest eigenvalue and the

corresponding eigenvector are

smallest egenvalue

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