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7/28/2019 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