Visible to Intel only — GUID: GUID-BF5DAB23-821E-4BA1-B2A6-41D1F69A7DDE
Visible to Intel only — GUID: GUID-BF5DAB23-821E-4BA1-B2A6-41D1F69A7DDE
?trevc3
Computes selected eigenvectors of an upper (quasi-) triangular matrix computed by ?hseqr using Level 3 BLAS
Syntax
call strevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)
call dtrevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)
call ctrevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, rwork, lrwork, info)
call ztrevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, rwork, lrwork, info)
Include Files
- mkl.fi
Description
This routine computes some or all of the right and left eigenvectors of an upper triangular matrix T (or, for real flavors, an upper quasi-triangular matrix T) using Level 3 BLAS. 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 the following:
T*x = w*x, yH*T = w*yH
where yH denotes the conjugate transpose of y.
The eigenvalues are not passed to this routine but are read directly from the diagonal blocks of T.
This routine returns one or both of the matrices X and Y of the right and left eigenvectors of T, or one or both of the products Q*X and 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 the right and left eigenvectors of A.
Input Parameters
side |
CHARACTER*1 Must be 'R', 'L', or 'B'.
|
howmny |
CHARACTER*1 Must be 'A', 'B', or 'S'.
|
select |
Array with a size of at least max (1, n) If howmny = 'S', select specifies which eigenvectors are to be computed. If howmny = 'A' or howmny = 'B', select is not referenced. For real flavors:
For complex flavors:
|
n |
INTEGER The order of the matrix T (n≥ 0). |
t, vl, vr, work |
Arrays:
|
lwork |
INTEGER The size of the work array. Must be at least max(1, 3*n) for real flavors, and at least max(1, 2*n) for complex flavors. If lwork = -1, a workspace query is assumed; the routine calculates only the optimal size of the work array and returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla. For details, see "Application Notes" below. |
ldt |
INTEGER The leading dimension of t. It is at least max(1, n). |
ldvl |
INTEGER The leading dimension of vl.
|
ldvr |
INTEGER The leading dimension of vr.
|
mm |
INTEGER The number of columns in one or both of the arrays vl and vr. Must be at least m (the precise number of columns required).
Constraint: 0 ≤mm≤n. |
rwork |
The workspace array is used in complex flavors only. Its dimensionis max (1, lrwork). |
lrwork |
INTEGER The size of the rwork array. It must be at least max(1, n). If lrwork = -1, a workspace query is assumed; the routine calculates only the optimal size of the work array and returns this value as the first entry of the rwork array, and no error message related to lrwork is issued by xerbla. For details, see "Application Notes" below. |
Output Parameters
select |
If a complex eigenvector of a real matrix was selected as specified above, then select(j) is set to .TRUE. and select(j + 1) is set to .FALSE.. |
t |
COMPLEX for ctrevc3 DOUBLE COMPLEX for ztrevc3 ctrevc3 or ztrevc3 modifies the t(ldt,*) array, which is restored on exit. |
vl, vr |
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). Treated column-wise, the eigenvectors form a rectangular n-by-mm matrix.
For real flavors:
A real eigenvector corresponding to a real eigenvalue occupies one column of the matrix; a complex eigenvector corresponding to a complex eigenvalue occupies two columns. The first column holds the real part of the eigenvector, and the second column holds the imaginary part of the eigenvector. The matrix is stored in a one-dimensional array as described by matrix_layout (using either column major or row major layout).
|
m |
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 one or both of vl and vr actually used to store the selected eigenvectors. If howmny = 'A' or 'B', m is set to n.
|
work(1) |
On exit, if info = 0, work(1) returns the required optimal size of lwork. |
rwork(1) |
On exit, if info = 0, then rwork(1) returns the required optimal size of lrwork. |
info |
INTEGER If info = 0, the execution is successful. If info = -i, the ith parameter contained an illegal value. |
Application Notes
If xi is an exact right eigenvector and yi is the corresponding computed eigenvector, 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. You can compute the condition number sepi by calling ?trsna.