# zlaqr3.f man page

zlaqr3.f —

## Synopsis

### Functions/Subroutines

subroutinezlaqr3(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)ZLAQR3performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

## Function/Subroutine Documentation

### subroutine zlaqr3 (logicalWANTT, logicalWANTZ, integerN, integerKTOP, integerKBOT, integerNW, complex*16, dimension( ldh, * )H, integerLDH, integerILOZ, integerIHIZ, complex*16, dimension( ldz, * )Z, integerLDZ, integerNS, integerND, complex*16, dimension( * )SH, complex*16, dimension( ldv, * )V, integerLDV, integerNH, complex*16, dimension( ldt, * )T, integerLDT, integerNV, complex*16, dimension( ldwv, * )WV, integerLDWV, complex*16, dimension( * )WORK, integerLWORK)

**ZLAQR3** performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

**Purpose:**

```
Aggressive early deflation:
ZLAQR3 accepts as input an upper Hessenberg matrix
H and performs an unitary similarity transformation
designed to detect and deflate fully converged eigenvalues from
a trailing principal submatrix. On output H has been over-
written by a new Hessenberg matrix that is a perturbation of
an unitary similarity transformation of H. It is to be
hoped that the final version of H has many zero subdiagonal
entries.
```

**Parameters:**

*WANTT*

```
WANTT is LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the triangular Schur factor may be
computed (in cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve
the eigenvalues.
```

*WANTZ*

```
WANTZ is LOGICAL
If .TRUE., then the unitary matrix Z is updated so
so that the unitary Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
```

*N*

```
N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
```

*KTOP*

```
KTOP is INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block
along the diagonal of the Hessenberg matrix.
```

*KBOT*

```
KBOT is INTEGER
It is assumed without a check that either
KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
determine an isolated block along the diagonal of the
Hessenberg matrix.
```

*NW*

```
NW is INTEGER
Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
```

*H*

```
H is COMPLEX*16 array, dimension (LDH,N)
On input the initial N-by-N section of H stores the
Hessenberg matrix undergoing aggressive early deflation.
On output H has been transformed by a unitary
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
```

*LDH*

```
LDH is integer
Leading dimension of H just as declared in the calling
subroutine. N .LE. LDH
```

*ILOZ*

`ILOZ is INTEGER`

*IHIZ*

```
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
```

*Z*

```
Z is COMPLEX*16 array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
```

*LDZ*

```
LDZ is integer
The leading dimension of Z just as declared in the
calling subroutine. 1 .LE. LDZ.
```

*NS*

```
NS is integer
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
```

*ND*

```
ND is integer
The number of converged eigenvalues uncovered by this
subroutine.
```

*SH*

```
SH is COMPLEX*16 array, dimension KBOT
On output, approximate eigenvalues that may
be used for shifts are stored in SH(KBOT-ND-NS+1)
through SR(KBOT-ND). Converged eigenvalues are
stored in SH(KBOT-ND+1) through SH(KBOT).
```

*V*

```
V is COMPLEX*16 array, dimension (LDV,NW)
An NW-by-NW work array.
```

*LDV*

```
LDV is integer scalar
The leading dimension of V just as declared in the
calling subroutine. NW .LE. LDV
```

*NH*

```
NH is integer scalar
The number of columns of T. NH.GE.NW.
```

*T*

`T is COMPLEX*16 array, dimension (LDT,NW)`

*LDT*

```
LDT is integer
The leading dimension of T just as declared in the
calling subroutine. NW .LE. LDT
```

*NV*

```
NV is integer
The number of rows of work array WV available for
workspace. NV.GE.NW.
```

*WV*

`WV is COMPLEX*16 array, dimension (LDWV,NW)`

*LDWV*

```
LDWV is integer
The leading dimension of W just as declared in the
calling subroutine. NW .LE. LDV
```

*WORK*

```
WORK is COMPLEX*16 array, dimension LWORK.
On exit, WORK(1) is set to an estimate of the optimal value
of LWORK for the given values of N, NW, KTOP and KBOT.
```

*LWORK*

```
LWORK is integer
The dimension of the work array WORK. LWORK = 2*NW
suffices, but greater efficiency may result from larger
values of LWORK.
If LWORK = -1, then a workspace query is assumed; ZLAQR3
only estimates the optimal workspace size for the given
values of N, NW, KTOP and KBOT. The estimate is returned
in WORK(1). No error message related to LWORK is issued
by XERBLA. Neither H nor Z are accessed.
```

**Author:**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

September 2012

**Contributors:**

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 266 of file zlaqr3.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

zlaqr3(3) is an alias of zlaqr3.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK