# dlaqr3.f man page

dlaqr3.f —

## Synopsis

### Functions/Subroutines

subroutinedlaqr3(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)DLAQR3performs the orthogonal 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 dlaqr3 (logicalWANTT, logicalWANTZ, integerN, integerKTOP, integerKBOT, integerNW, double precision, dimension( ldh, * )H, integerLDH, integerILOZ, integerIHIZ, double precision, dimension( ldz, * )Z, integerLDZ, integerNS, integerND, double precision, dimension( * )SR, double precision, dimension( * )SI, double precision, dimension( ldv, * )V, integerLDV, integerNH, double precision, dimension( ldt, * )T, integerLDT, integerNV, double precision, dimension( ldwv, * )WV, integerLDWV, double precision, dimension( * )WORK, integerLWORK)

**DLAQR3** performs the orthogonal 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:
DLAQR3 accepts as input an upper Hessenberg matrix
H and performs an orthogonal 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 orthogonal 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 quasi-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 orthogonal matrix Z is updated so
so that the orthogonal 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 orthogonal 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 DOUBLE PRECISION 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 an orthogonal
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 DOUBLE PRECISION array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the orthogonal
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.
```

*SR*

`SR is DOUBLE PRECISION array, dimension (KBOT)`

*SI*

```
SI is DOUBLE PRECISION array, dimension (KBOT)
On output, the real and imaginary parts of approximate
eigenvalues that may be used for shifts are stored in
SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
The real and imaginary parts of converged eigenvalues
are stored in SR(KBOT-ND+1) through SR(KBOT) and
SI(KBOT-ND+1) through SI(KBOT), respectively.
```

*V*

```
V is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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; DLAQR3
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 274 of file dlaqr3.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

dlaqr3(3) is an alias of dlaqr3.f(3).