gqz: Generalized Schur decomposition
In geigen: Calculate Generalized Eigenvalues, the Generalized Schur Decomposition and the Generalized Singular Value Decomposition of a Matrix Pair with Lapack
Description
Usage
Arguments
Details
Value
Source
References
See Also
Examples
Description
Computes the generalized eigenvalues and Schur form of a pair of matrices.
Usage
1
Arguments
A
left hand side matrix.
B
right hand side matrix.
sort
how to sort the generalized eigenvalues. See ‘Details’.
Details
Both matrices must be square.
This function provides the solution to the generalized eigenvalue problem defined by
A*x = lambda B*x
If either one of the matrices is complex the other matrix is coerced to be complex.
The sort argument specifies how to order the eigenvalues on the
diagonal of the generalized Schur form, where it is noted that non-finite eigenvalues never
satisfy any ordering condition (even in the case of a complex infinity).
Eigenvalues that are placed in the leading block of the Schur form
satisfy
Nunordered.
-negative real part.
+positive real part.
Sabsolute value < 1.
Babsolute value > 1.
Rimaginary part identical to 0 with a tolerance of 100*machine_precision as determined by Lapack.
Value
The generalized Schur form for numeric matrices is
(A,B) = (Q*S*Z^T, Q*T*Z^T)
The matrices Q and Z are orthogonal. The matrix S is quasi-upper triangular and
the matrix T is upper triangular.
The return value is a list containing the following components
Sgeneralized Schur form of A.
Tgeneralized Schur form of B.
sdimthe number of eigenvalues (after sorting) for which the sorting condition is true.
alpharnumerator of the real parts of the eigenvalues (numeric).
alphainumerator of the imaginary parts of the eigenvalues (numeric).
betadenominator of the expression for the eigenvalues (numeric).
Qmatrix of left Schur vectors (matrix Q).
Zmatrix of right Schur vectors (matrix Z).
The generalized Schur form for complex matrices is
(A,B) = (Q*S*Z^H, Q*T*Z^H)
The matrices Q and Z are unitary and the matrices S and
T are upper triangular.
The return value is a list containing the following components
Sgeneralized Schur form of A.
Tgeneralized Schur form of B.
sdimthe number of eigenvalues. (after sorting) for which the sorting condition is true.
alphanumerator of the eigenvalues (complex).
betadenominator of the eigenvalues (complex).
Qmatrix of left Schur vectors (matrix Q).
Zmatrix of right Schur vectors (matrix Z).
The generalized eigenvalues can be computed by calling function gevalues.
Source
gqz uses the LAPACK routines DGGES and ZGGES.
LAPACK is from http://www.netlib.org/lapack.
The complex routines used by the package come from LAPACK 3.8.0.
References
Anderson. E. and ten others (1999)
LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at
http://www.netlib.org/lapack/lug/lapack_lug.html.
See the section Eigenvalues, Eigenvectors and Generalized Schur Decomposition
(http://www.netlib.org/lapack/lug/node56.html).
See Also
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
# Real matrices
# example from NAG: http://www.nag.com/lapack-ex/node116.html
# Find the generalized Schur decomposition with the real eigenvalues ordered to come first
A <- matrix(c( 3.9, 12.5,-34.5,-0.5,
4.3, 21.5,-47.5, 7.5,
4.3, 21.5,-43.5, 3.5,
4.4, 26.0,-46.0, 6.0), nrow=4, byrow=TRUE)
B <- matrix(c( 1.0, 2.0, -3.0, 1.0,
1.0, 3.0, -5.0, 4.0,
1.0, 3.0, -4.0, 3.0,
1.0, 3.0, -4.0, 4.0), nrow=4, byrow=TRUE)
z <- gqz(A, B,"R")
z
# complexify
A <- A+0i
B <- B+0i
z <- gqz(A, B,"R")
z
Example output
$S
[,1] [,2] [,3] [,4]
[1,] 3.800912 -69.450536 50.3134928 -43.288415
[2,] 0.000000 9.203303 -0.2001382 5.988071
[3,] 0.000000 0.000000 1.4278921 4.445296
[4,] 0.000000 0.000000 0.9019321 -1.196199
$T
[,1] [,2] [,3] [,4]
[1,] 1.900456 -10.228464 0.8658216 -5.2133694
[2,] 0.000000 2.300826 0.7915033 0.4261764
[3,] 0.000000 0.000000 0.8101464 0.0000000
[4,] 0.000000 0.000000 0.0000000 -0.2822896
$sdim
[1] 2
$alphar
[1] 3.8009124 9.2033029 0.8571429 0.8571429
$alphai
[1] 0.000000 0.000000 1.142857 -1.142857
$beta
[1] 1.9004562 2.3008257 0.2857143 0.2857143
$Q
[,1] [,2] [,3] [,4]
[1,] 0.4642154 0.7886199 0.29148041 -0.2786068
[2,] 0.5001547 -0.5986435 0.56379490 -0.2713053
[3,] 0.5001547 0.0154060 -0.01073843 0.8657324
[4,] 0.5330990 -0.1395249 -0.77269604 -0.3150858
$Z
[,1] [,2] [,3] [,4]
[1,] 0.996056108 -0.001400016 0.08867638 -0.002601947
[2,] 0.005691749 -0.040374311 -0.09375821 -0.994759728
[3,] 0.062609241 0.719382130 -0.69083549 0.036273380
[4,] 0.062609241 -0.693438754 -0.71140159 0.095553937
$S
[,1] [,2] [,3] [,4]
[1,] 3.800912-0i -10.526945-68.64809i 23.568740-33.363185i 51.077080+11.308345i
[2,] 0.000000+0i 9.203303+ 0.00000i -1.276124+ 3.354217i 0.939029+ 4.704892i
[3,] 0.000000+0i 0.000000+ 0.00000i 1.999285- 2.665713i -1.464686- 2.802026i
[4,] 0.000000+0i 0.000000+ 0.00000i 0.000000+ 0.000000i 1.029500+ 1.372666i
$T
[,1] [,2] [,3]
[1,] 1.900456+0i -1.550376-10.11028i 3.0294927+0.1063502i
[2,] 0.000000+0i 2.300826+ 0.00000i -0.7277894+0.1710293i
[3,] 0.000000+0i 0.000000+ 0.00000i 0.6664283+0.0000000i
[4,] 0.000000+0i 0.000000+ 0.00000i 0.0000000+0.0000000i
[,4]
[1,] 4.1865850-1.1010167i
[2,] 0.4907230+0.0914670i
[3,] -0.3379006+0.2448629i
[4,] 0.3431666+0.0000000i
$sdim
[1] 2
$alpha
[1] 3.800912-0.000000i 9.203303+0.000000i 1.999285-2.665713i 1.029500+1.372666i
$beta
[1] 1.9004562+0i 2.3008257+0i 0.6664283+0i 0.3431666+0i
$Q
[,1] [,2] [,3]
[1,] -0.4594808-0.0661317i -0.00726767-0.78858646i 0.0388454+0.3080388i
[2,] -0.4950535-0.0712516i 0.00551691+0.59861808i 0.0117161+0.5692560i
[3,] -0.4950535-0.0712516i -0.00014198-0.01540535i -0.2041832-0.0962817i
[4,] -0.5276618-0.0759448i 0.00128582+0.13951896i 0.1467471-0.7119816i
[,4]
[1,] 0.2242458-0.1260935i
[2,] 0.1819758-0.1848415i
[3,] -0.8131595+0.1934388i
[4,] 0.3969078+0.1017345i
$Z
[,1] [,2]
[1,] -0.985897027-0.141897220i 1.2902e-05+1.399957e-03i
[2,] -0.005633697-0.000810841i 3.7208e-04+4.037260e-02i
[3,] -0.061970670-0.008919254i -6.6296e-03-7.193516e-01i
[4,] -0.061970670-0.008919254i 6.3905e-03+6.934093e-01i
[,3] [,4]
[1,] -0.00825948+0.06955706i -0.02922826-0.04593012i
[2,] -0.54940006-0.30734608i -0.74114512+0.22965060i
[3,] 0.07329972-0.53813510i 0.24008926+0.35491456i
[4,] 0.10804663-0.54051388i 0.29228258+0.35491456i
geigen documentation built on May 30, 2019, 5:03 p.m.
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Description Usage Arguments Details Value Source References See Also Examples
Description
Computes the generalized eigenvalues and Schur form of a pair of matrices.
Usage
1 |
Arguments
A |
left hand side matrix. |
B |
right hand side matrix. |
sort |
how to sort the generalized eigenvalues. See ‘Details’. |
Details
Both matrices must be square. This function provides the solution to the generalized eigenvalue problem defined by
A*x = lambda B*x
If either one of the matrices is complex the other matrix is coerced to be complex.
The sort argument specifies how to order the eigenvalues on the
diagonal of the generalized Schur form, where it is noted that non-finite eigenvalues never
satisfy any ordering condition (even in the case of a complex infinity).
Eigenvalues that are placed in the leading block of the Schur form
satisfy
Nunordered.
-negative real part.
+positive real part.
Sabsolute value < 1.
Babsolute value > 1.
Rimaginary part identical to 0 with a tolerance of 100*machine_precision as determined by Lapack.
Value
The generalized Schur form for numeric matrices is
(A,B) = (Q*S*Z^T, Q*T*Z^T)
The matrices Q and Z are orthogonal. The matrix S is quasi-upper triangular and the matrix T is upper triangular. The return value is a list containing the following components
Sgeneralized Schur form of A.
Tgeneralized Schur form of B.
sdimthe number of eigenvalues (after sorting) for which the sorting condition is true.
alpharnumerator of the real parts of the eigenvalues (numeric).
alphainumerator of the imaginary parts of the eigenvalues (numeric).
betadenominator of the expression for the eigenvalues (numeric).
Qmatrix of left Schur vectors (matrix Q).
Zmatrix of right Schur vectors (matrix Z).
The generalized Schur form for complex matrices is
(A,B) = (Q*S*Z^H, Q*T*Z^H)
The matrices Q and Z are unitary and the matrices S and T are upper triangular. The return value is a list containing the following components
Sgeneralized Schur form of A.
Tgeneralized Schur form of B.
sdimthe number of eigenvalues. (after sorting) for which the sorting condition is true.
alphanumerator of the eigenvalues (complex).
betadenominator of the eigenvalues (complex).
Qmatrix of left Schur vectors (matrix Q).
Zmatrix of right Schur vectors (matrix Z).
The generalized eigenvalues can be computed by calling function gevalues.
Source
gqz uses the LAPACK routines DGGES and ZGGES.
LAPACK is from http://www.netlib.org/lapack.
The complex routines used by the package come from LAPACK 3.8.0.
References
Anderson. E. and ten others (1999)
LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at
http://www.netlib.org/lapack/lug/lapack_lug.html.
See the section Eigenvalues, Eigenvectors and Generalized Schur Decomposition
(http://www.netlib.org/lapack/lug/node56.html).
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | # Real matrices
# example from NAG: http://www.nag.com/lapack-ex/node116.html
# Find the generalized Schur decomposition with the real eigenvalues ordered to come first
A <- matrix(c( 3.9, 12.5,-34.5,-0.5,
4.3, 21.5,-47.5, 7.5,
4.3, 21.5,-43.5, 3.5,
4.4, 26.0,-46.0, 6.0), nrow=4, byrow=TRUE)
B <- matrix(c( 1.0, 2.0, -3.0, 1.0,
1.0, 3.0, -5.0, 4.0,
1.0, 3.0, -4.0, 3.0,
1.0, 3.0, -4.0, 4.0), nrow=4, byrow=TRUE)
z <- gqz(A, B,"R")
z
# complexify
A <- A+0i
B <- B+0i
z <- gqz(A, B,"R")
z
|
Example output
$S
[,1] [,2] [,3] [,4]
[1,] 3.800912 -69.450536 50.3134928 -43.288415
[2,] 0.000000 9.203303 -0.2001382 5.988071
[3,] 0.000000 0.000000 1.4278921 4.445296
[4,] 0.000000 0.000000 0.9019321 -1.196199
$T
[,1] [,2] [,3] [,4]
[1,] 1.900456 -10.228464 0.8658216 -5.2133694
[2,] 0.000000 2.300826 0.7915033 0.4261764
[3,] 0.000000 0.000000 0.8101464 0.0000000
[4,] 0.000000 0.000000 0.0000000 -0.2822896
$sdim
[1] 2
$alphar
[1] 3.8009124 9.2033029 0.8571429 0.8571429
$alphai
[1] 0.000000 0.000000 1.142857 -1.142857
$beta
[1] 1.9004562 2.3008257 0.2857143 0.2857143
$Q
[,1] [,2] [,3] [,4]
[1,] 0.4642154 0.7886199 0.29148041 -0.2786068
[2,] 0.5001547 -0.5986435 0.56379490 -0.2713053
[3,] 0.5001547 0.0154060 -0.01073843 0.8657324
[4,] 0.5330990 -0.1395249 -0.77269604 -0.3150858
$Z
[,1] [,2] [,3] [,4]
[1,] 0.996056108 -0.001400016 0.08867638 -0.002601947
[2,] 0.005691749 -0.040374311 -0.09375821 -0.994759728
[3,] 0.062609241 0.719382130 -0.69083549 0.036273380
[4,] 0.062609241 -0.693438754 -0.71140159 0.095553937
$S
[,1] [,2] [,3] [,4]
[1,] 3.800912-0i -10.526945-68.64809i 23.568740-33.363185i 51.077080+11.308345i
[2,] 0.000000+0i 9.203303+ 0.00000i -1.276124+ 3.354217i 0.939029+ 4.704892i
[3,] 0.000000+0i 0.000000+ 0.00000i 1.999285- 2.665713i -1.464686- 2.802026i
[4,] 0.000000+0i 0.000000+ 0.00000i 0.000000+ 0.000000i 1.029500+ 1.372666i
$T
[,1] [,2] [,3]
[1,] 1.900456+0i -1.550376-10.11028i 3.0294927+0.1063502i
[2,] 0.000000+0i 2.300826+ 0.00000i -0.7277894+0.1710293i
[3,] 0.000000+0i 0.000000+ 0.00000i 0.6664283+0.0000000i
[4,] 0.000000+0i 0.000000+ 0.00000i 0.0000000+0.0000000i
[,4]
[1,] 4.1865850-1.1010167i
[2,] 0.4907230+0.0914670i
[3,] -0.3379006+0.2448629i
[4,] 0.3431666+0.0000000i
$sdim
[1] 2
$alpha
[1] 3.800912-0.000000i 9.203303+0.000000i 1.999285-2.665713i 1.029500+1.372666i
$beta
[1] 1.9004562+0i 2.3008257+0i 0.6664283+0i 0.3431666+0i
$Q
[,1] [,2] [,3]
[1,] -0.4594808-0.0661317i -0.00726767-0.78858646i 0.0388454+0.3080388i
[2,] -0.4950535-0.0712516i 0.00551691+0.59861808i 0.0117161+0.5692560i
[3,] -0.4950535-0.0712516i -0.00014198-0.01540535i -0.2041832-0.0962817i
[4,] -0.5276618-0.0759448i 0.00128582+0.13951896i 0.1467471-0.7119816i
[,4]
[1,] 0.2242458-0.1260935i
[2,] 0.1819758-0.1848415i
[3,] -0.8131595+0.1934388i
[4,] 0.3969078+0.1017345i
$Z
[,1] [,2]
[1,] -0.985897027-0.141897220i 1.2902e-05+1.399957e-03i
[2,] -0.005633697-0.000810841i 3.7208e-04+4.037260e-02i
[3,] -0.061970670-0.008919254i -6.6296e-03-7.193516e-01i
[4,] -0.061970670-0.008919254i 6.3905e-03+6.934093e-01i
[,3] [,4]
[1,] -0.00825948+0.06955706i -0.02922826-0.04593012i
[2,] -0.54940006-0.30734608i -0.74114512+0.22965060i
[3,] 0.07329972-0.53813510i 0.24008926+0.35491456i
[4,] 0.10804663-0.54051388i 0.29228258+0.35491456i
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