Numerical Methods - Power Method for Eigen values
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Feb 10, 2016
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About This Presentation
Power method for eigen values and eigen vectors, largest eigen value
Slide Content
Numerical Methods
Power Method for Eigen values
Dr. N. B. Vyas
Department of Mathematics,
Atmiya Institute of Technology & Science,
Rajkot (Gujarat) - INDIA
[email protected]
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method for Eigen values
Dr. N. B. Vyas
Department of Mathematics,
Atmiya Institute of Technology & Science,
Rajkot (Gujarat) - INDIA
[email protected]
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method
Power method is particularly useful for estimating numerically
largest or smallest eigenvalue and its corresponding eigenvector.
The intermediate (remaining) eigenvalues can also be found. The power method, which is an iterative method, can be used
when
(i) nhasnlinearly independent eigenvectors.(ii)
j1j>j2j j3j : : : jnj
When this ordering is adopted, the eigenvalue1with the
greatest magnitude is called thedominanteigenvalue of the
matrix A
And the remaining eigenvalues2; 3; : : : ; nare called the
subdominant eigenvalues of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Working rules for determining largest eigenvalue.
LetA= [aij] be a matrix of ordernn. We start from any vectorx0(6= 0) withncomponents such that
Ax0=x
In order to get a convergent sequence of eigenvectors
simultaneously scaling method is adopted.
In which at each stage each components of the resultant
approximate vector is to be divided by its absolutely largest
component.
Then use the scaled vector in the next step. This absolutely largest component is known as numerically
largest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Accordinglyxin eq -(1) can be scaled by dividing each of its
components by absolutely largest component of it. Thus
Ax0=x=1x1;x1is the scaled vector ofx
Now scaled vectorx1is to be used in the next iteration to obtain
Ax1=x=2x2
Proceeding in this way, nally we getAxn=n+1xn+1; where
n= 0;1;2;3; :::Wheren+1is the numerically largest eigenvalue
upto desired accuracy andxn+1is the corresponding eigenvector.
NOTE :The initial vectorx0is usually taken as a vector with
all components equal to 1.
Characteristic:The main advantage of this method is its
simplicity. And it can handle sparse matrices too large to store
as a full square array. Its disadvantage is its possibly slow
convergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method:Determining smallest eigenvalue.
Ifis the eigenvalue of A, then the reciprocal
1
is the eigenvalue
ofA
1
.
The reciprocal of the largest eigenvalue ofA
1
will be the
smallest eigenvalue of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Power Method:Determining smallest eigenvalue.
Ifis the eigenvalue of A, then the reciprocal
1
is the eigenvalue
ofA
1
.
The reciprocal of the largest eigenvalue ofA
1
will be the
smallest eigenvalue of A.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
35
2 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
35
2 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Sol.:LetA=
35
2 4
andx0=
1
1
Ax0=
35
2 4
1
1
=
2
2
=2
1
1
=2x1
Ax1=
35
2 4
1
1
=
8
6
= 8
1
0:75
= 8x2
Ax2=
35
2 4
1
0:75
=
6:75
5
= 6:75
1
0:7407
= 6:75x3
)largest eigen value is 6:7015 and the corresponding eigen vector is
1
0:7403
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
1 2
3 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
1 2
3 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
2 3
5 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
2 3
5 4
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
4 2
1 3
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
4 2
1 3
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
2
4
21 0
1 2 1
01 2
3
5
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
2
4
21 0
1 2 1
01 2
3
5
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
2
4
31 0
1 2 1
01 3
3
5
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Ex:Use power method to estimate the largest eigen value and the
corresponding eigen vector ofA=
2
4
31 0
1 2 1
01 3
3
5
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
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