DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.
DGEEV computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugate-transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.
Usage
dgeev(
JOBVL = NULL,
JOBVR = NULL,
N = NULL,
A,
LDA = NULL,
WR,
WI,
VL = NULL,
LDVL = NULL,
VR = NULL,
LDVR = NULL,
WORK = NULL,
LWORK = NULL
)Arguments
- JOBVL
a character.
- = 'N':
left eigenvectors of A are not computed;
- = 'V':
left eigenvectors of A are computed.
- JOBVR
a character.
- = 'N':
right eigenvectors of A are not computed;
- = 'V':
right eigenvectors of A are computed.
- N
an integer. The order of the matrix A. N >= 0.
- A
a matrix of dimension (LDA,N), the N-by-N matrix A.
- LDA
an integer. The leading dimension of the matrix A. LDA >= max(1,N).
- WR
a vector of dimension (N). WR contain the real part of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
- WI
a vector of dimension (N). WI contain the imaginary part of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
- VL
a matrx of dimension (LDVL,N)
- If
JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues.
- If
JOBVL = 'N', VL is not referenced.
- If
the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL.
- If
the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1).
- LDVL
an integer. The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.
- VR
a matrix of dimension (LDVR,N).
- If
JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues.
- If
JOBVR = 'N', VR is not referenced.
- If
the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR.
- If
the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and v(j+1) = VR(:,j) - i*VR(:,j+1).
- LDVR
an integer. The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= N.
- WORK
a matrix of dimension (MAX(1,LWORK))
- LWORK
an integer. The dimension of the array WORK.LWORK >= max(1,3*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good performance, LWORK must generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
Examples
set.seed(4669)
A = matrix(rnorm(16),4)
WR= matrix(0,nrow=4,ncol=1)
WI= matrix(0,nrow=4,ncol=1)
VL = matrix(0,ncol=4,nrow=4)
eigen(A)
#> eigen() decomposition
#> $values
#> [1] 0.85730911+1.690442i 0.85730911-1.690442i -0.86120947+0.000000i
#> [4] -0.04044732+0.000000i
#>
#> $vectors
#> [,1] [,2] [,3]
#> [1,] -0.20131416-0.08051662i -0.20131416+0.08051662i -0.4396218+0i
#> [2,] -0.73251338+0.00000000i -0.73251338+0.00000000i 0.4547709+0i
#> [3,] -0.06644069-0.55880008i -0.06644069+0.55880008i -0.1693250+0i
#> [4,] 0.22714677+0.21942261i 0.22714677-0.21942261i -0.7558076+0i
#> [,4]
#> [1,] -0.9125418+0i
#> [2,] -0.2656930+0i
#> [3,] 0.2527695+0i
#> [4,] 0.1810590+0i
#>
dgeev(A=A,WR=WR,WI=WI,VL=VL)
#> [1] 0
VL
#> [,1] [,2] [,3] [,4]
#> [1,] -0.099694024 0.1149983 -0.2126753 -0.7057922
#> [2,] 0.657326284 0.0000000 -0.1918649 0.3331385
#> [3,] 0.007357843 0.5515939 -0.3234797 0.3139870
#> [4,] 0.451853982 -0.1904644 -0.9018414 0.5406369
WR
#> [,1]
#> [1,] 0.85730911
#> [2,] 0.85730911
#> [3,] -0.86120947
#> [4,] -0.04044732
WI
#> [,1]
#> [1,] 1.690442
#> [2,] -1.690442
#> [3,] 0.000000
#> [4,] 0.000000
rm(A,WR,WI,VL)
A = as.big.matrix(matrix(rnorm(16),4))
WR= matrix(0,nrow=4,ncol=1)
WI= matrix(0,nrow=4,ncol=1)
VL = as.big.matrix(matrix(0,ncol=4,nrow=4))
eigen(A[,])
#> eigen() decomposition
#> $values
#> [1] -0.7802559+0.6873447i -0.7802559-0.6873447i 0.6930662+0.4810716i
#> [4] 0.6930662-0.4810716i
#>
#> $vectors
#> [,1] [,2] [,3]
#> [1,] 0.1471182+0.4334710i 0.1471182-0.4334710i -0.2986975+0.07375248i
#> [2,] 0.2638302+0.1607608i 0.2638302-0.1607608i -0.4446219-0.18640256i
#> [3,] -0.4094915+0.0733955i -0.4094915-0.0733955i 0.7778787+0.00000000i
#> [4,] -0.7224532+0.0000000i -0.7224532+0.0000000i 0.2526100-0.06323534i
#> [,4]
#> [1,] -0.2986975-0.07375248i
#> [2,] -0.4446219+0.18640256i
#> [3,] 0.7778787+0.00000000i
#> [4,] 0.2526100+0.06323534i
#>
dgeev(A=A,WR=WR,WI=WI,VL=VL)
#> [1] 0
VL[,]
#> [,1] [,2] [,3] [,4]
#> [1,] -0.63149291 0.0000000 -0.35751797 -0.04531232
#> [2,] -0.22007417 0.1890068 0.81587447 0.00000000
#> [3,] -0.33976651 0.2889619 0.32444761 0.26761276
#> [4,] -0.08779556 -0.5571465 0.04124336 -0.16091204
WR[,]
#> [1] -0.7802559 -0.7802559 0.6930662 0.6930662
WI[,]
#> [1] 0.6873447 -0.6873447 0.4810716 -0.4810716
rm(A,WR,WI,VL)
gc()
#> used (Mb) gc trigger (Mb) limit (Mb) max used (Mb)
#> Ncells 1144085 61.2 2286118 122.1 NA 2286118 122.1
#> Vcells 2048446 15.7 8388608 64.0 65536 5285191 40.4