Visible to Intel only — GUID: GUID-9C4D95D6-73D4-4BCB-BCD5-B555D7E947F8
Visible to Intel only — GUID: GUID-9C4D95D6-73D4-4BCB-BCD5-B555D7E947F8
?laqr2
Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
Syntax
call slaqr2( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )
call dlaqr2( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )
call claqr2( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )
call zlaqr2( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )
Include Files
- mkl.fi
Description
The routine accepts as input an upper Hessenberg matrix H and performs an orthogonal/unitary similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been overwritten by a new Hessenberg matrix that is a perturbation of an orthogonal/unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries.
This subroutine is identical to ?laqr3 except that it avoids recursion by calling ?lahqr instead of ?laqr4.
Input Parameters
- wantt
-
LOGICAL.
If wantt = .TRUE., then the Hessenberg matrix H is fully updated so that the quasi-triangular/triangular Schur factor may be computed (in cooperation with the calling subroutine).
If wantt = .FALSE., then only enough of H is updated to preserve the eigenvalues.
- wantz
-
LOGICAL.
If wantz = .TRUE., then the orthogonal/unitary matrix Z is updated so that the orthogonal/unitary Schur factor may be computed (in cooperation with the calling subroutine).
If wantz = .FALSE., then Z is not referenced.
- n
-
INTEGER. The order of the Hessenberg matrix H and (if wantz = .TRUE.) the order of the orthogonal/unitary matrix Z.
- ktop
-
INTEGER.
It is assumed that either ktop=1 or h(ktop,ktop-1)=0. ktop and kbot together determine an isolated block along the diagonal of the Hessenberg matrix.
- kbot
-
INTEGER.
It is assumed without a check that either kbot=n or h(kbot+1,kbot)=0. ktop and kbot together determine an isolated block along the diagonal of the Hessenberg matrix.
- nw
-
INTEGER.
Size of the deflation window. 1 ≤ nw ≤ (kbot-ktop+1).
- h
-
REAL for slaqr2
DOUBLE PRECISION for dlaqr2
COMPLEX for claqr2
DOUBLE COMPLEX for zlaqr2.
Array, DIMENSION (ldh, n), on input the initial n-by-n section of h stores the Hessenberg matrix H undergoing aggressive early deflation.
- ldh
-
INTEGER. The leading dimension of the array h just as declared in the calling subroutine. ldh≥n.
- iloz, ihiz
-
INTEGER. Specify the rows of Z to which transformations must be applied if wantz is .TRUE.. 1 ≤ iloz ≤ ihiz ≤ n.
- z
-
REAL for slaqr2
DOUBLE PRECISION for dlaqr2
COMPLEX for claqr2
DOUBLE COMPLEX for zlaqr2.
Array, DIMENSION (ldz, n), contains the matrix Z if wantz is .TRUE.. If wantz is .FALSE., then z is not referenced.
- ldz
-
INTEGER. The leading dimension of the array z just as declared in the calling subroutine. ldz ≥ 1.
- v
-
REAL for slaqr2
DOUBLE PRECISION for dlaqr2
COMPLEX for claqr2
DOUBLE COMPLEX for zlaqr2.
Workspace array with dimension (ldv, nw). An nw-by-nw work array.
- ldv
-
INTEGER. The leading dimension of the array v just as declared in the calling subroutine. ldv ≥ nw.
- nh
-
INTEGER. The number of column of t. nh ≥ nw.
- t
-
REAL for slaqr2
DOUBLE PRECISION for dlaqr2
COMPLEX for claqr2
DOUBLE COMPLEX for zlaqr2.
Workspace array with dimension (ldt, nw).
- ldt
-
INTEGER. The leading dimension of the array t just as declared in the calling subroutine. ldt≥nw.
- nv
-
INTEGER. The number of rows of work array wv available for workspace. nv≥nw.
- wv
-
REAL for slaqr2
DOUBLE PRECISION for dlaqr2
COMPLEX for claqr2
DOUBLE COMPLEX for zlaqr2.
Workspace array with dimension (ldwv, nw).
- ldwv
-
INTEGER. The leading dimension of the array wv just as declared in the calling subroutine. ldwv≥nw.
- work
-
REAL for slaqr2
DOUBLE PRECISION for dlaqr2
COMPLEX for claqr2
DOUBLE COMPLEX for zlaqr2.
Workspace array with dimension lwork.
- lwork
-
INTEGER. The dimension of the array work.
lwork=2*nw) is sufficient, but for the optimal performance a greater workspace may be required.
If lwork=-1,then the routine performs a workspace query: it estimates the optimal workspace size for the given values of the input parameters n, nw, ktop, and kbot. The estimate is returned in work(1). No error messages related to the lwork is issued by xerbla. Neither H nor Z are accessed.
Output Parameters
- h
-
On output h has been transformed by an orthogonal/unitary similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries.
- work(1)
-
On exit work(1) is set to an estimate of the optimal value of lwork for the given values of the input parameters n, nw, ktop, and kbot.
- z
-
If wantz is .TRUE., then the orthogonal/unitary similarity transformation is accumulated into z(iloz:ihiz, ilo:ihi) from the right.
If wantz is .FALSE., then z is unreferenced.
- nd
-
INTEGER. The number of converged eigenvalues uncovered by the routine.
- ns
-
INTEGER. The number of unconverged, that is approximate eigenvalues returned in sr, si or in sh that may be used as shifts by the calling subroutine.
- sh
-
COMPLEX for claqr2
DOUBLE COMPLEX for zlaqr2.
Arrays, DIMENSION (kbot).
The approximate eigenvalues that may be used for shifts are stored in the sh(kbot-nd-ns+1)through the sh(kbot-nd).
The converged eigenvalues are stored in the sh(kbot-nd+1)through the sh(kbot).
- sr, si
-
REAL for slaqr2
DOUBLE PRECISION for dlaqr2
Arrays, DIMENSION (kbot) each.
The real and imaginary parts of the approximate eigenvalues that may be used for shifts are stored in the sr(kbot-nd-ns+1)through the sr(kbot-nd), and si(kbot-nd-ns+1) through the si(kbot-nd), respectively.
The real and imaginary parts of converged eigenvalues are stored in the sr(kbot-nd+1)through the sr(kbot), and si(kbot-nd+1) through the si(kbot), respectively.