# lahqr - Man Page

lahqr: eig of Hessenberg, step in hseqr

## Synopsis

### Functions

subroutine clahqr (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info)
CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine dlahqr (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info)
DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine slahqr (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info)
SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine zlahqr (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info)
ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

## Function Documentation

### subroutine clahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) w, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, integer info)

CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

```    CLAHQR is an auxiliary routine called by CHSEQR to update the
eigenvalues and Schur decomposition already computed by CHSEQR, by
dealing with the Hessenberg submatrix in rows and columns ILO to
IHI.```
Parameters

WANTT

```          WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.```

WANTZ

```          WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.```

N

```          N is INTEGER
The order of the matrix H.  N >= 0.```

ILO

`          ILO is INTEGER`

IHI

```          IHI is INTEGER
It is assumed that H is already upper triangular in rows and
columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
CLAHQR works primarily with the Hessenberg submatrix in rows
and columns ILO to IHI, but applies transformations to all of
H if WANTT is .TRUE..
1 <= ILO <= max(1,IHI); IHI <= N.```

H

```          H is COMPLEX array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO is zero and if WANTT is .TRUE., then H
is upper triangular in rows and columns ILO:IHI.  If INFO
is zero and if WANTT is .FALSE., then the contents of H
are unspecified on exit.  The output state of H in case
INF is positive is below under the description of INFO.```

LDH

```          LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).```

W

```          W is COMPLEX array, dimension (N)
The computed eigenvalues ILO to IHI are stored in the
corresponding elements of W. If WANTT is .TRUE., the
eigenvalues are stored in the same order as on the diagonal
of the Schur form returned in H, with W(i) = H(i,i).```

ILOZ

`          ILOZ is INTEGER`

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.```

Z

```          Z is COMPLEX array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the current
matrix Z of transformations accumulated by CHSEQR, and on
exit Z has been updated; transformations are applied only to
the submatrix Z(ILOZ:IHIZ,ILO:IHI).
If WANTZ is .FALSE., Z is not referenced.```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).```

INFO

```          INFO is INTEGER
= 0:  successful exit
> 0:  if INFO = i, CLAHQR failed to compute all the
eigenvalues ILO to IHI in a total of 30 iterations
per eigenvalue; elements i+1:ihi of W contain
those eigenvalues which have been successfully
computed.

If INFO > 0 and WANTT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the
eigenvalues of the upper Hessenberg matrix
rows and columns ILO through INFO of the final,
output value of H.

If INFO > 0 and WANTT is .TRUE., then on exit
(*)       (initial value of H)*U  = U*(final value of H)
where U is an orthogonal matrix.    The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.

If INFO > 0 and WANTZ is .TRUE., then on exit
(final value of Z)  = (initial value of Z)*U
where U is the orthogonal matrix in (*)
(regardless of the value of WANTT.)```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```     02-96 Based on modifications by
David Day, Sandia National Laboratory, USA

12-04 Further modifications by
Ralph Byers, University of Kansas, USA
This is a modified version of CLAHQR from LAPACK version 3.0.
It is (1) more robust against overflow and underflow and
(2) adopts the more conservative Ahues & Tisseur stopping
criterion (LAWN 122, 1997).```

Definition at line 193 of file clahqr.f.

### subroutine dlahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, integer info)

DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

```    DLAHQR is an auxiliary routine called by DHSEQR to update the
eigenvalues and Schur decomposition already computed by DHSEQR, by
dealing with the Hessenberg submatrix in rows and columns ILO to
IHI.```
Parameters

WANTT

```          WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.```

WANTZ

```          WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.```

N

```          N is INTEGER
The order of the matrix H.  N >= 0.```

ILO

`          ILO is INTEGER`

IHI

```          IHI is INTEGER
It is assumed that H is already upper quasi-triangular in
rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
ILO = 1). DLAHQR works primarily with the Hessenberg
submatrix in rows and columns ILO to IHI, but applies
transformations to all of H if WANTT is .TRUE..
1 <= ILO <= max(1,IHI); IHI <= N.```

H

```          H is DOUBLE PRECISION array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO is zero and if WANTT is .TRUE., H is upper
quasi-triangular in rows and columns ILO:IHI, with any
2-by-2 diagonal blocks in standard form. If INFO is zero
and WANTT is .FALSE., the contents of H are unspecified on
exit.  The output state of H if INFO is nonzero is given
below under the description of INFO.```

LDH

```          LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).```

WR

`          WR is DOUBLE PRECISION array, dimension (N)`

WI

```          WI is DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the computed
eigenvalues ILO to IHI are stored in the corresponding
elements of WR and WI. If two eigenvalues are computed as a
complex conjugate pair, they are stored in consecutive
elements of WR and WI, say the i-th and (i+1)th, with
WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
eigenvalues are stored in the same order as on the diagonal
of the Schur form returned in H, with WR(i) = H(i,i), and, if
H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).```

ILOZ

`          ILOZ is INTEGER`

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.```

Z

```          Z is DOUBLE PRECISION array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the current
matrix Z of transformations accumulated by DHSEQR, and on
exit Z has been updated; transformations are applied only to
the submatrix Z(ILOZ:IHIZ,ILO:IHI).
If WANTZ is .FALSE., Z is not referenced.```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).```

INFO

```          INFO is INTEGER
= 0:  successful exit
> 0:  If INFO = i, DLAHQR failed to compute all the
eigenvalues ILO to IHI in a total of 30 iterations
per eigenvalue; elements i+1:ihi of WR and WI
contain those eigenvalues which have been
successfully computed.

If INFO > 0 and WANTT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the
eigenvalues of the upper Hessenberg matrix rows
and columns ILO through INFO of the final, output
value of H.

If INFO > 0 and WANTT is .TRUE., then on exit
(*)       (initial value of H)*U  = U*(final value of H)
where U is an orthogonal matrix.    The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.

If INFO > 0 and WANTZ is .TRUE., then on exit
(final value of Z)  = (initial value of Z)*U
where U is the orthogonal matrix in (*)
(regardless of the value of WANTT.)```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```     02-96 Based on modifications by
David Day, Sandia National Laboratory, USA

12-04 Further modifications by
Ralph Byers, University of Kansas, USA
This is a modified version of DLAHQR from LAPACK version 3.0.
It is (1) more robust against overflow and underflow and
(2) adopts the more conservative Ahues & Tisseur stopping
criterion (LAWN 122, 1997).```

Definition at line 205 of file dlahqr.f.

### subroutine slahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, integer info)

SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

```    SLAHQR is an auxiliary routine called by SHSEQR to update the
eigenvalues and Schur decomposition already computed by SHSEQR, by
dealing with the Hessenberg submatrix in rows and columns ILO to
IHI.```
Parameters

WANTT

```          WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.```

WANTZ

```          WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.```

N

```          N is INTEGER
The order of the matrix H.  N >= 0.```

ILO

`          ILO is INTEGER`

IHI

```          IHI is INTEGER
It is assumed that H is already upper quasi-triangular in
rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
ILO = 1). SLAHQR works primarily with the Hessenberg
submatrix in rows and columns ILO to IHI, but applies
transformations to all of H if WANTT is .TRUE..
1 <= ILO <= max(1,IHI); IHI <= N.```

H

```          H is REAL array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO is zero and if WANTT is .TRUE., H is upper
quasi-triangular in rows and columns ILO:IHI, with any
2-by-2 diagonal blocks in standard form. If INFO is zero
and WANTT is .FALSE., the contents of H are unspecified on
exit.  The output state of H if INFO is nonzero is given
below under the description of INFO.```

LDH

```          LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).```

WR

`          WR is REAL array, dimension (N)`

WI

```          WI is REAL array, dimension (N)
The real and imaginary parts, respectively, of the computed
eigenvalues ILO to IHI are stored in the corresponding
elements of WR and WI. If two eigenvalues are computed as a
complex conjugate pair, they are stored in consecutive
elements of WR and WI, say the i-th and (i+1)th, with
WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
eigenvalues are stored in the same order as on the diagonal
of the Schur form returned in H, with WR(i) = H(i,i), and, if
H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).```

ILOZ

`          ILOZ is INTEGER`

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.```

Z

```          Z is REAL array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the current
matrix Z of transformations accumulated by SHSEQR, and on
exit Z has been updated; transformations are applied only to
the submatrix Z(ILOZ:IHIZ,ILO:IHI).
If WANTZ is .FALSE., Z is not referenced.```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).```

INFO

```          INFO is INTEGER
= 0:   successful exit
> 0:   If INFO = i, SLAHQR failed to compute all the
eigenvalues ILO to IHI in a total of 30 iterations
per eigenvalue; elements i+1:ihi of WR and WI
contain those eigenvalues which have been
successfully computed.

If INFO > 0 and WANTT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the
eigenvalues of the upper Hessenberg matrix rows
and columns ILO through INFO of the final, output
value of H.

If INFO > 0 and WANTT is .TRUE., then on exit
(*)       (initial value of H)*U  = U*(final value of H)
where U is an orthogonal matrix.    The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.

If INFO > 0 and WANTZ is .TRUE., then on exit
(final value of Z)  = (initial value of Z)*U
where U is the orthogonal matrix in (*)
(regardless of the value of WANTT.)```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```     02-96 Based on modifications by
David Day, Sandia National Laboratory, USA

12-04 Further modifications by
Ralph Byers, University of Kansas, USA
This is a modified version of SLAHQR from LAPACK version 3.0.
It is (1) more robust against overflow and underflow and
(2) adopts the more conservative Ahues & Tisseur stopping
criterion (LAWN 122, 1997).```

Definition at line 205 of file slahqr.f.

### subroutine zlahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) w, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, integer info)

ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

```    ZLAHQR is an auxiliary routine called by CHSEQR to update the
eigenvalues and Schur decomposition already computed by CHSEQR, by
dealing with the Hessenberg submatrix in rows and columns ILO to
IHI.```
Parameters

WANTT

```          WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.```

WANTZ

```          WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.```

N

```          N is INTEGER
The order of the matrix H.  N >= 0.```

ILO

`          ILO is INTEGER`

IHI

```          IHI is INTEGER
It is assumed that H is already upper triangular in rows and
columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
ZLAHQR works primarily with the Hessenberg submatrix in rows
and columns ILO to IHI, but applies transformations to all of
H if WANTT is .TRUE..
1 <= ILO <= max(1,IHI); IHI <= N.```

H

```          H is COMPLEX*16 array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO is zero and if WANTT is .TRUE., then H
is upper triangular in rows and columns ILO:IHI.  If INFO
is zero and if WANTT is .FALSE., then the contents of H
are unspecified on exit.  The output state of H in case
INF is positive is below under the description of INFO.```

LDH

```          LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).```

W

```          W is COMPLEX*16 array, dimension (N)
The computed eigenvalues ILO to IHI are stored in the
corresponding elements of W. If WANTT is .TRUE., the
eigenvalues are stored in the same order as on the diagonal
of the Schur form returned in H, with W(i) = H(i,i).```

ILOZ

`          ILOZ is INTEGER`

IHIZ

```          IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.```

Z

```          Z is COMPLEX*16 array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the current
matrix Z of transformations accumulated by CHSEQR, and on
exit Z has been updated; transformations are applied only to
the submatrix Z(ILOZ:IHIZ,ILO:IHI).
If WANTZ is .FALSE., Z is not referenced.```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).```

INFO

```          INFO is INTEGER
= 0:   successful exit
> 0:   if INFO = i, ZLAHQR failed to compute all the
eigenvalues ILO to IHI in a total of 30 iterations
per eigenvalue; elements i+1:ihi of W contain
those eigenvalues which have been successfully
computed.

If INFO > 0 and WANTT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the
eigenvalues of the upper Hessenberg matrix
rows and columns ILO through INFO of the final,
output value of H.

If INFO > 0 and WANTT is .TRUE., then on exit
(*)       (initial value of H)*U  = U*(final value of H)
where U is an orthogonal matrix.    The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.

If INFO > 0 and WANTZ is .TRUE., then on exit
(final value of Z)  = (initial value of Z)*U
where U is the orthogonal matrix in (*)
(regardless of the value of WANTT.)```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```     02-96 Based on modifications by
David Day, Sandia National Laboratory, USA

12-04 Further modifications by
Ralph Byers, University of Kansas, USA
This is a modified version of ZLAHQR from LAPACK version 3.0.
It is (1) more robust against overflow and underflow and
(2) adopts the more conservative Ahues & Tisseur stopping
criterion (LAWN 122, 1997).```

Definition at line 193 of file zlahqr.f.

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