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Numerical MethodsPower Method for Eigen values
Dr. N. B. Vyas
Department of Mathematics,Atmiya Institute of Technology & Science,
Rajkot (Gujarat) - [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 numericallylargest 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 usedwhen
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|When this ordering is adopted, the eigenvalue λ1 with thegreatest magnitude is called the dominant eigenvalue of thematrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called thesubdominant 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 numericallylargest 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 usedwhen
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|When this ordering is adopted, the eigenvalue λ1 with thegreatest magnitude is called the dominant eigenvalue of thematrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called thesubdominant 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 numericallylargest 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 usedwhen
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|When this ordering is adopted, the eigenvalue λ1 with thegreatest magnitude is called the dominant eigenvalue of thematrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called thesubdominant 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 numericallylargest 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 usedwhen
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|When this ordering is adopted, the eigenvalue λ1 with thegreatest magnitude is called the dominant eigenvalue of thematrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called thesubdominant 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 numericallylargest 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 usedwhen
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|
When this ordering is adopted, the eigenvalue λ1 with thegreatest magnitude is called the dominant eigenvalue of thematrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called thesubdominant 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 numericallylargest 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 usedwhen
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|When this ordering is adopted, the eigenvalue λ1 with thegreatest magnitude is called the dominant eigenvalue of thematrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called thesubdominant 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 numericallylargest 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 usedwhen
(i) The matrix A of order n has n linearly independent eigenvectors.
(ii) The eigenvalues can be ordered in magnitude as|λ1| > |λ2| ≥ |λ3| ≥ . . . ≥ |λn|When this ordering is adopted, the eigenvalue λ1 with thegreatest magnitude is called the dominant eigenvalue of thematrix A
And the remaining eigenvalues λ2, λ3, . . . , λn are called thesubdominant 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.
Let A = [aij ] be a matrix of order n× n.
We start from any vector x0(6= 0) with n components such thatAx0 = x
In order to get a convergent sequence of eigenvectorssimultaneously scaling method is adopted.
In which at each stage each components of the resultantapproximate vector is to be divided by its absolutely largestcomponent.
Then use the scaled vector in the next step.
This absolutely largest component is known as numericallylargest 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.
Let A = [aij ] be a matrix of order n× n.
We start from any vector x0(6= 0) with n components such thatAx0 = x
In order to get a convergent sequence of eigenvectorssimultaneously scaling method is adopted.
In which at each stage each components of the resultantapproximate vector is to be divided by its absolutely largestcomponent.
Then use the scaled vector in the next step.
This absolutely largest component is known as numericallylargest 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.
Let A = [aij ] be a matrix of order n× n.
We start from any vector x0(6= 0) with n components such thatAx0 = x
In order to get a convergent sequence of eigenvectorssimultaneously scaling method is adopted.
In which at each stage each components of the resultantapproximate vector is to be divided by its absolutely largestcomponent.
Then use the scaled vector in the next step.
This absolutely largest component is known as numericallylargest 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.
Let A = [aij ] be a matrix of order n× n.
We start from any vector x0(6= 0) with n components such thatAx0 = x
In order to get a convergent sequence of eigenvectorssimultaneously scaling method is adopted.
In which at each stage each components of the resultantapproximate vector is to be divided by its absolutely largestcomponent.
Then use the scaled vector in the next step.
This absolutely largest component is known as numericallylargest 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.
Let A = [aij ] be a matrix of order n× n.
We start from any vector x0(6= 0) with n components such thatAx0 = x
In order to get a convergent sequence of eigenvectorssimultaneously scaling method is adopted.
In which at each stage each components of the resultantapproximate vector is to be divided by its absolutely largestcomponent.
Then use the scaled vector in the next step.
This absolutely largest component is known as numericallylargest 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.
Let A = [aij ] be a matrix of order n× n.
We start from any vector x0(6= 0) with n components such thatAx0 = x
In order to get a convergent sequence of eigenvectorssimultaneously scaling method is adopted.
In which at each stage each components of the resultantapproximate vector is to be divided by its absolutely largestcomponent.
Then use the scaled vector in the next step.
This absolutely largest component is known as numericallylargest 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.
Let A = [aij ] be a matrix of order n× n.
We start from any vector x0(6= 0) with n components such thatAx0 = x
In order to get a convergent sequence of eigenvectorssimultaneously scaling method is adopted.
In which at each stage each components of the resultantapproximate vector is to be divided by its absolutely largestcomponent.
Then use the scaled vector in the next step.
This absolutely largest component is known as numericallylargest eigenvalue.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of itscomponents by absolutely largest component of it. ThusAx0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; wheren = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalueupto desired accuracy and xn+1 is the corresponding eigenvector.
NOTE : The initial vector x0 is usually taken as a vector withall components equal to 1.
Characteristic: The main advantage of this method is itssimplicity. And it can handle sparse matrices too large to storeas a full square array. Its disadvantage is its possibly slowconvergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of itscomponents by absolutely largest component of it. ThusAx0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; wheren = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalueupto desired accuracy and xn+1 is the corresponding eigenvector.
NOTE : The initial vector x0 is usually taken as a vector withall components equal to 1.
Characteristic: The main advantage of this method is itssimplicity. And it can handle sparse matrices too large to storeas a full square array. Its disadvantage is its possibly slowconvergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of itscomponents by absolutely largest component of it. ThusAx0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; wheren = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalueupto desired accuracy and xn+1 is the corresponding eigenvector.
NOTE : The initial vector x0 is usually taken as a vector withall components equal to 1.
Characteristic: The main advantage of this method is itssimplicity. And it can handle sparse matrices too large to storeas a full square array. Its disadvantage is its possibly slowconvergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of itscomponents by absolutely largest component of it. ThusAx0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; wheren = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalueupto desired accuracy and xn+1 is the corresponding eigenvector.
NOTE : The initial vector x0 is usually taken as a vector withall components equal to 1.
Characteristic: The main advantage of this method is itssimplicity. And it can handle sparse matrices too large to storeas a full square array. Its disadvantage is its possibly slowconvergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Accordingly x in eq -(1) can be scaled by dividing each of itscomponents by absolutely largest component of it. ThusAx0 = x = λ1x1; x1 is the scaled vector of x
Now scaled vector x1 is to be used in the next iteration to obtain
Ax1 = x = λ2x2
Proceeding in this way, finally we get Axn = λn+1xn+1; wheren = 0, 1, 2, 3, ... Where λn+1 is the numerically largest eigenvalueupto desired accuracy and xn+1 is the corresponding eigenvector.
NOTE : The initial vector x0 is usually taken as a vector withall components equal to 1.
Characteristic: The main advantage of this method is itssimplicity. And it can handle sparse matrices too large to storeas a full square array. Its disadvantage is its possibly slowconvergence.
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Eigen values and Eigen vectors by iteration
Power Method: Determining smallest eigenvalue.
If λ is the eigenvalue of A, then the reciprocal1
λis the eigenvalue
of A−1.
The reciprocal of the largest eigenvalue of A−1 will be thesmallest 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 of A =
[3 −5−2 4
]
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
Example
Sol.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]
=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]
= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]
=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]
= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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.: Let A =
[3 −5−2 4
]and x0 =
[11
]Ax0 =
[3 −5−2 4
] [11
]=
[−22
]= −2
[1−1
]= −2x1
Ax1 =
[3 −5−2 4
] [1−1
]=
[8−6
]= 8
[1
−0.75
]= 8x2
Ax2 =
[3 −5−2 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 of A =
[1 23 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 of A =
[2 35 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 of A =
[4 21 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 of A =
2 −1 0−1 2 −10 −1 2
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 of A =
3 −1 0−1 2 −10 −1 3
Dr. N. B. Vyas Numerical Methods Power Method for Eigen values
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