Computes selected eigenvectors of an upper (quasi-) triangular matrix computed by ?hseqr.
FORTRAN 77:
call strevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info)
call dtrevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info)
call ctrevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
call ztrevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
Fortran 95:
call trevc(t [, howmny] [,select] [,vl] [,vr] [,m] [,info])
C:
lapack_int LAPACKE_strevc( int matrix_order, char side, char howmny, lapack_logical* select, lapack_int n, const float* t, lapack_int ldt, float* vl, lapack_int ldvl, float* vr, lapack_int ldvr, lapack_int mm, lapack_int* m );
lapack_int LAPACKE_dtrevc( int matrix_order, char side, char howmny, lapack_logical* select, lapack_int n, const double* t, lapack_int ldt, double* vl, lapack_int ldvl, double* vr, lapack_int ldvr, lapack_int mm, lapack_int* m );
lapack_int LAPACKE_ctrevc( int matrix_order, char side, char howmny, const lapack_logical* select, lapack_int n, lapack_complex_float* t, lapack_int ldt, lapack_complex_float* vl, lapack_int ldvl, lapack_complex_float* vr, lapack_int ldvr, lapack_int mm, lapack_int* m );
lapack_int LAPACKE_ztrevc( int matrix_order, char side, char howmny, const lapack_logical* select, lapack_int n, lapack_complex_double* t, lapack_int ldt, lapack_complex_double* vl, lapack_int ldvl, lapack_complex_double* vr, lapack_int ldvr, lapack_int mm, lapack_int* m );
The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.
The routine computes some or all of the right and/or left eigenvectors of an upper triangular matrix T (or, for real flavors, an upper quasi-triangular matrix T). Matrices of this type are produced by the Schur factorization of a general matrix: A = Q*T*QH, as computed by ?hseqr.
The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w, are defined by:
T*x = w*x, yH*T = w*yH, where yH denotes the conjugate transpose of y.
The eigenvalues are not input to this routine, but are read directly from the diagonal blocks of T.
This routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input matrix.
If Q is the orthogonal/unitary factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the matrices of right and left eigenvectors of A.
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
CHARACTER*1. Must be 'R' or 'L' or 'B'.
If side = 'R', then only right eigenvectors are computed.
If side = 'L', then only left eigenvectors are computed.
If side = 'B', then all eigenvectors are computed.
CHARACTER*1. Must be 'A' or 'B' or 'S'.
If howmny = 'A', then all eigenvectors (as specified by side) are computed.
If howmny = 'B', then all eigenvectors (as specified by side) are computed and backtransformed by the matrices supplied in vl and vr.
If howmny = 'S', then selected eigenvectors (as specified by side and select) are computed.
LOGICAL.
Array, DIMENSION at least max (1, n).
If howmny = 'S', select specifies which eigenvectors are to be computed.
If howmny = 'A' or 'B', select is not referenced.
For real flavors:
If omega(j) is a real eigenvalue, the corresponding real eigenvector is computed if select(j) is .TRUE..
If omega(j) and omega(j+1) are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either select(j) or select(j+1) is .TRUE., and on exit select(j) is set to .TRUE.and select(j+1) is set to .FALSE..
For complex flavors:
The eigenvector corresponding to the j-th eigenvalue is computed if select(j) is .TRUE..
INTEGER. The order of the matrix T (n ≥ 0).
REAL for strevc
DOUBLE PRECISION for dtrevc
COMPLEX for ctrevc
DOUBLE COMPLEX for ztrevc.
Arrays:
t(ldt,*) contains the n-by-n matrix T in Schur canonical form. For complex flavors ctrevc and ztrevc, contains the upper triangular matrix T.
The second dimension of t must be at least max(1, n).
vl(ldvl,*)
If howmny = 'B' and side = 'L' or 'B', then vl must contain an n-by-n matrix Q (usually the matrix of Schur vectors returned by ?hseqr).
If howmny = 'A' or 'S', then vl need not be set.
The second dimension of vl must be at least max(1, mm) if side = 'L' or 'B' and at least 1 if side = 'R'.
The array vl is not referenced if side = 'R'.
vr (ldvr,*)
If howmny = 'B' and side = 'R' or 'B', then vr must contain an n-by-n matrix Q (usually the matrix of Schur vectors returned by ?hseqr). .
If howmny = 'A' or 'S', then vr need not be set.
The second dimension of vr must be at least max(1, mm) if side = 'R' or 'B' and at least 1 if side = 'L'.
The array vr is not referenced if side = 'L'.
work(*) is a workspace array.
DIMENSION at least max (1, 3*n) for real flavors and at least max (1, 2*n) for complex flavors.
INTEGER. The leading dimension of t; at least max(1, n).
INTEGER. The leading dimension of vl.
If side = 'L' or 'B', ldvl ≥ n.
If side = 'R', ldvl ≥ 1.
INTEGER. The leading dimension of vr.
If side = 'R' or 'B', ldvr ≥ n.
If side = 'L', ldvr ≥ 1.
INTEGER. The number of columns in the arrays vl and/or vr. Must be at least m (the precise number of columns required).
If howmny = 'A' or 'B', m = n.
If howmny = 'S': for real flavors, m is obtained by counting 1 for each selected real eigenvector and 2 for each selected complex eigenvector;
for complex flavors, m is the number of selected eigenvectors (see select).
Constraint: 0 ≤ m ≤ n.
REAL for ctrevc
DOUBLE PRECISION for ztrevc.
Workspace array, DIMENSION at least max (1, n).
If a complex eigenvector of a real matrix was selected as specified above, then select(j) is set to .TRUE. and select(j+1) to .FALSE.
COMPLEX for ctrevc
DOUBLE COMPLEX for ztrevc.
ctrevc/ztrevc modify the t(ldt,*) array, which is restored on exit.
If side = 'L' or 'B', vl contains the computed left eigenvectors (as specified by howmny and select).
If side = 'R' or 'B', vr contains the computed right eigenvectors (as specified by howmny and select).
The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.
For real flavors: corresponding to each real eigenvalue is a real eigenvector, occupying one column;corresponding to each complex conjugate pair of eigenvalues is a complex eigenvector, occupying two columns; the first column holds the real part and the second column holds the imaginary part.
INTEGER.
For complex flavors: the number of selected eigenvectors.
If howmny = 'A' or 'B', m is set to n.
For real flavors: the number of columns of vl and/or vr actually used to store the selected eigenvectors.
If howmny = 'A' or 'B', m is set to n.
INTEGER.
If info = 0, the execution is successful.
If info = -i, the i-th parameter had an illegal value.
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.
Specific details for the routine trevc interface are the following:
Holds the matrix T of size (n,n).
Holds the vector of length n.
Holds the matrix VL of size (n,mm).
Holds the matrix VR of size (n,mm).
If omitted, this argument is restored based on the presence of arguments vl and vr as follows:
side = 'B', if both vl and vr are present,
side = 'L', if vr is omitted,
side = 'R', if vl is omitted.
Note that there will be an error condition if both vl and vr are omitted.
If omitted, this argument is restored based on the presence of argument select as follows:
howmny = 'V', if q is present,
howmny = 'N', if q is omitted.
If present, vect = 'V' or 'U' and the argument q must also be present.
Note that there will be an error condition if both select and howmny are present.
If x i is an exact right eigenvector and yi is the corresponding computed eigenvector, then the angle θ(yi, xi) between them is bounded as follows: θ(yi,xi)≤(c(n)ε||T||2)/sepi where sepi is the reciprocal condition number of xi. The condition number sepi may be computed by calling ?trsna.
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