# hseqr - Man Page

hseqr: Hessenberg eig, QR iteration

## Synopsis

### Functions

subroutine chseqr (job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info)
CHSEQR
subroutine dhseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
DHSEQR
subroutine shseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
SHSEQR
subroutine zhseqr (job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info)
ZHSEQR

## Function Documentation

### subroutine chseqr (character job, character compz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, integer info)

CHSEQR

Purpose:

```    CHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.

Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*T*(QZ)**H.```
Parameters

JOB

```          JOB is CHARACTER*1
= 'E':  compute eigenvalues only;
= 'S':  compute eigenvalues and the Schur form T.```

COMPZ

```          COMPZ is CHARACTER*1
= 'N':  no Schur vectors are computed;
= 'I':  Z is initialized to the unit matrix and the matrix Z
of Schur vectors of H is returned;
= 'V':  Z must contain an unitary matrix Q on entry, and
the product Q*Z is returned.```

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 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to CGEBAL, and then passed to ZGEHRD
when the matrix output by CGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
If N = 0, then ILO = 1 and IHI = 0.```

H

```          H is COMPLEX array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and JOB = 'S', H contains the upper
triangular matrix T from the Schur decomposition (the
Schur form). If INFO = 0 and JOB = 'E', the contents of
H are unspecified on exit.  (The output value of H when
INFO > 0 is given under the description of INFO below.)

Unlike earlier versions of CHSEQR, this subroutine may
explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
or j = IHI+1, IHI+2, ... N.```

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. If JOB = 'S', 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).```

Z

```          Z is COMPLEX array, dimension (LDZ,N)
If COMPZ = 'N', Z is not referenced.
If COMPZ = 'I', on entry Z need not be set and on exit,
if INFO = 0, Z contains the unitary matrix Z of the Schur
vectors of H.  If COMPZ = 'V', on entry Z must contain an
N-by-N matrix Q, which is assumed to be equal to the unit
matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
if INFO = 0, Z contains Q*Z.
Normally Q is the unitary matrix generated by CUNGHR
after the call to CGEHRD which formed the Hessenberg matrix
H. (The output value of Z when INFO > 0 is given under
the description of INFO below.)```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  if COMPZ = 'I' or
COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.```

WORK

```          WORK is COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of
the optimal value for LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= max(1,N)
is sufficient and delivers very good and sometimes
optimal performance.  However, LWORK as large as 11*N
may be required for optimal performance.  A workspace
query is recommended to determine the optimal workspace
size.

If LWORK = -1, then CHSEQR does a workspace query.
In this case, CHSEQR checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI.  The estimate is returned
in WORK(1).  No error message related to LWORK is
issued by XERBLA.  Neither H nor Z are accessed.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, CHSEQR failed to compute all of
the eigenvalues.  Elements 1:ilo-1 and i+1:n of W
contain those eigenvalues which have been
successfully computed.  (Failures are rare.)

If INFO > 0 and JOB = 'E', then on exit, the
remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.

If INFO > 0 and JOB   = 'S', then on exit

(*)  (initial value of H)*U  = U*(final value of H)

where U is a unitary matrix.  The final
value of  H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.

If INFO > 0 and COMPZ = 'V', then on exit

(final value of Z)  =  (initial value of Z)*U

where U is the unitary matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'I', then on exit
(final value of Z)  = U
where U is the unitary matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'N', then Z is not
accessed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

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

Further Details:

```             Default values supplied by
ILAENV(ISPEC,'CHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.

ISPEC=12: The CLAHQR vs CLAQR0 crossover point.
Default: 75. (Must be at least 11.)

ISPEC=13: Recommended deflation window size.
This depends on ILO, IHI and NS.  NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
The default for (IHI-ILO+1) <= 500 is NS.
The default for (IHI-ILO+1) >  500 is 3*NS/2.

ISPEC=14: Nibble crossover point. (See IPARMQ for
details.)  Default: 14% of deflation window
size.

ISPEC=15: Number of simultaneous shifts in a multishift
QR iteration.

If IHI-ILO+1 is ...

greater than      ...but less    ... the
or equal to ...      than        default is

1               30          NS =   2(+)
30               60          NS =   4(+)
60              150          NS =  10(+)
150              590          NS =  **
590             3000          NS =  64
3000             6000          NS = 128
6000             infinity      NS = 256

(+)  By default some or all matrices of this order
are passed to the implicit double shift routine
CLAHQR and this parameter is ignored.  See
ISPEC=12 above and comments in IPARMQ for
details.

(**)  The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.

ISPEC=16: Select structured matrix multiply.
If the number of simultaneous shifts (specified
by ISPEC=15) is less than 14, then the default
for ISPEC=16 is 0.  Otherwise the default for
ISPEC=16 is 2.```

References:

```  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.```

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 297 of file chseqr.f.

### subroutine dhseqr (character job, character compz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)

DHSEQR

Purpose:

```    DHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.

Optionally Z may be postmultiplied into an input orthogonal
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.```
Parameters

JOB

```          JOB is CHARACTER*1
= 'E':  compute eigenvalues only;
= 'S':  compute eigenvalues and the Schur form T.```

COMPZ

```          COMPZ is CHARACTER*1
= 'N':  no Schur vectors are computed;
= 'I':  Z is initialized to the unit matrix and the matrix Z
of Schur vectors of H is returned;
= 'V':  Z must contain an orthogonal matrix Q on entry, and
the product Q*Z is returned.```

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 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to DGEBAL, and then passed to ZGEHRD
when the matrix output by DGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
If N = 0, then ILO = 1 and IHI = 0.```

H

```          H is DOUBLE PRECISION array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and JOB = 'S', then H contains the
upper quasi-triangular matrix T from the Schur decomposition
(the Schur form); 2-by-2 diagonal blocks (corresponding to
complex conjugate pairs of eigenvalues) are returned in
standard form, with H(i,i) = H(i+1,i+1) and
H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
contents of H are unspecified on exit.  (The output value of
H when INFO > 0 is given under the description of INFO
below.)

Unlike earlier versions of DHSEQR, this subroutine may
explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
or j = IHI+1, IHI+2, ... N.```

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. 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 JOB = 'S', 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).```

Z

```          Z is DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N', Z is not referenced.
If COMPZ = 'I', on entry Z need not be set and on exit,
if INFO = 0, Z contains the orthogonal matrix Z of the Schur
vectors of H.  If COMPZ = 'V', on entry Z must contain an
N-by-N matrix Q, which is assumed to be equal to the unit
matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
if INFO = 0, Z contains Q*Z.
Normally Q is the orthogonal matrix generated by DORGHR
after the call to DGEHRD which formed the Hessenberg matrix
H. (The output value of Z when INFO > 0 is given under
the description of INFO below.)```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  if COMPZ = 'I' or
COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.```

WORK

```          WORK is DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of
the optimal value for LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= max(1,N)
is sufficient and delivers very good and sometimes
optimal performance.  However, LWORK as large as 11*N
may be required for optimal performance.  A workspace
query is recommended to determine the optimal workspace
size.

If LWORK = -1, then DHSEQR does a workspace query.
In this case, DHSEQR checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI.  The estimate is returned
in WORK(1).  No error message related to LWORK is
issued by XERBLA.  Neither H nor Z are accessed.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, DHSEQR failed to compute all of
the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed.  (Failures are rare.)

If INFO > 0 and JOB = 'E', then on exit, the
remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.

If INFO > 0 and JOB   = 'S', 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 quasi-triangular
in rows and columns INFO+1 through IHI.

If INFO > 0 and COMPZ = 'V', then on exit

(final value of Z)  =  (initial value of Z)*U

where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'I', then on exit
(final value of Z)  = U
where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'N', then Z is not
accessed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

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

Further Details:

```             Default values supplied by
ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.

ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
Default: 75. (Must be at least 11.)

ISPEC=13: Recommended deflation window size.
This depends on ILO, IHI and NS.  NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
The default for (IHI-ILO+1) <= 500 is NS.
The default for (IHI-ILO+1) >  500 is 3*NS/2.

ISPEC=14: Nibble crossover point. (See IPARMQ for
details.)  Default: 14% of deflation window
size.

ISPEC=15: Number of simultaneous shifts in a multishift
QR iteration.

If IHI-ILO+1 is ...

greater than      ...but less    ... the
or equal to ...      than        default is

1               30          NS =   2(+)
30               60          NS =   4(+)
60              150          NS =  10(+)
150              590          NS =  **
590             3000          NS =  64
3000             6000          NS = 128
6000             infinity      NS = 256

(+)  By default some or all matrices of this order
are passed to the implicit double shift routine
DLAHQR and this parameter is ignored.  See
ISPEC=12 above and comments in IPARMQ for
details.

(**)  The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.

ISPEC=16: Select structured matrix multiply.
If the number of simultaneous shifts (specified
by ISPEC=15) is less than 14, then the default
for ISPEC=16 is 0.  Otherwise the default for
ISPEC=16 is 2.```

References:

```  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.```

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 314 of file dhseqr.f.

### subroutine shseqr (character job, character compz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

SHSEQR

Purpose:

```    SHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.

Optionally Z may be postmultiplied into an input orthogonal
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.```
Parameters

JOB

```          JOB is CHARACTER*1
= 'E':  compute eigenvalues only;
= 'S':  compute eigenvalues and the Schur form T.```

COMPZ

```          COMPZ is CHARACTER*1
= 'N':  no Schur vectors are computed;
= 'I':  Z is initialized to the unit matrix and the matrix Z
of Schur vectors of H is returned;
= 'V':  Z must contain an orthogonal matrix Q on entry, and
the product Q*Z is returned.```

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 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL, and then passed to ZGEHRD
when the matrix output by SGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
If N = 0, then ILO = 1 and IHI = 0.```

H

```          H is REAL array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and JOB = 'S', then H contains the
upper quasi-triangular matrix T from the Schur decomposition
(the Schur form); 2-by-2 diagonal blocks (corresponding to
complex conjugate pairs of eigenvalues) are returned in
standard form, with H(i,i) = H(i+1,i+1) and
H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
contents of H are unspecified on exit.  (The output value of
H when INFO > 0 is given under the description of INFO
below.)

Unlike earlier versions of SHSEQR, this subroutine may
explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
or j = IHI+1, IHI+2, ... N.```

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. 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 JOB = 'S', 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).```

Z

```          Z is REAL array, dimension (LDZ,N)
If COMPZ = 'N', Z is not referenced.
If COMPZ = 'I', on entry Z need not be set and on exit,
if INFO = 0, Z contains the orthogonal matrix Z of the Schur
vectors of H.  If COMPZ = 'V', on entry Z must contain an
N-by-N matrix Q, which is assumed to be equal to the unit
matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
if INFO = 0, Z contains Q*Z.
Normally Q is the orthogonal matrix generated by SORGHR
after the call to SGEHRD which formed the Hessenberg matrix
H. (The output value of Z when INFO > 0 is given under
the description of INFO below.)```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  if COMPZ = 'I' or
COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.```

WORK

```          WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of
the optimal value for LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= max(1,N)
is sufficient and delivers very good and sometimes
optimal performance.  However, LWORK as large as 11*N
may be required for optimal performance.  A workspace
query is recommended to determine the optimal workspace
size.

If LWORK = -1, then SHSEQR does a workspace query.
In this case, SHSEQR checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI.  The estimate is returned
in WORK(1).  No error message related to LWORK is
issued by XERBLA.  Neither H nor Z are accessed.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, SHSEQR failed to compute all of
the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed.  (Failures are rare.)

If INFO > 0 and JOB = 'E', then on exit, the
remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.

If INFO > 0 and JOB   = 'S', 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 quasi-triangular
in rows and columns INFO+1 through IHI.

If INFO > 0 and COMPZ = 'V', then on exit

(final value of Z)  =  (initial value of Z)*U

where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'I', then on exit
(final value of Z)  = U
where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'N', then Z is not
accessed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

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

Further Details:

```             Default values supplied by
ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.

ISPEC=12: The SLAHQR vs SLAQR0 crossover point.
Default: 75. (Must be at least 11.)

ISPEC=13: Recommended deflation window size.
This depends on ILO, IHI and NS.  NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
The default for (IHI-ILO+1) <= 500 is NS.
The default for (IHI-ILO+1) >  500 is 3*NS/2.

ISPEC=14: Nibble crossover point. (See IPARMQ for
details.)  Default: 14% of deflation window
size.

ISPEC=15: Number of simultaneous shifts in a multishift
QR iteration.

If IHI-ILO+1 is ...

greater than      ...but less    ... the
or equal to ...      than        default is

1               30          NS =   2(+)
30               60          NS =   4(+)
60              150          NS =  10(+)
150              590          NS =  **
590             3000          NS =  64
3000             6000          NS = 128
6000             infinity      NS = 256

(+)  By default some or all matrices of this order
are passed to the implicit double shift routine
SLAHQR and this parameter is ignored.  See
ISPEC=12 above and comments in IPARMQ for
details.

(**)  The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.

ISPEC=16: Select structured matrix multiply.
If the number of simultaneous shifts (specified
by ISPEC=15) is less than 14, then the default
for ISPEC=16 is 0.  Otherwise the default for
ISPEC=16 is 2.```

References:

```  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.```

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 314 of file shseqr.f.

### subroutine zhseqr (character job, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, integer info)

ZHSEQR

Purpose:

```    ZHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.

Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*T*(QZ)**H.```
Parameters

JOB

```          JOB is CHARACTER*1
= 'E':  compute eigenvalues only;
= 'S':  compute eigenvalues and the Schur form T.```

COMPZ

```          COMPZ is CHARACTER*1
= 'N':  no Schur vectors are computed;
= 'I':  Z is initialized to the unit matrix and the matrix Z
of Schur vectors of H is returned;
= 'V':  Z must contain an unitary matrix Q on entry, and
the product Q*Z is returned.```

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 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to ZGEBAL, and then passed to ZGEHRD
when the matrix output by ZGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
If N = 0, then ILO = 1 and IHI = 0.```

H

```          H is COMPLEX*16 array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and JOB = 'S', H contains the upper
triangular matrix T from the Schur decomposition (the
Schur form). If INFO = 0 and JOB = 'E', the contents of
H are unspecified on exit.  (The output value of H when
INFO > 0 is given under the description of INFO below.)

Unlike earlier versions of ZHSEQR, this subroutine may
explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
or j = IHI+1, IHI+2, ... N.```

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. If JOB = 'S', 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).```

Z

```          Z is COMPLEX*16 array, dimension (LDZ,N)
If COMPZ = 'N', Z is not referenced.
If COMPZ = 'I', on entry Z need not be set and on exit,
if INFO = 0, Z contains the unitary matrix Z of the Schur
vectors of H.  If COMPZ = 'V', on entry Z must contain an
N-by-N matrix Q, which is assumed to be equal to the unit
matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
if INFO = 0, Z contains Q*Z.
Normally Q is the unitary matrix generated by ZUNGHR
after the call to ZGEHRD which formed the Hessenberg matrix
H. (The output value of Z when INFO > 0 is given under
the description of INFO below.)```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z.  if COMPZ = 'I' or
COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.```

WORK

```          WORK is COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of
the optimal value for LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= max(1,N)
is sufficient and delivers very good and sometimes
optimal performance.  However, LWORK as large as 11*N
may be required for optimal performance.  A workspace
query is recommended to determine the optimal workspace
size.

If LWORK = -1, then ZHSEQR does a workspace query.
In this case, ZHSEQR checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI.  The estimate is returned
in WORK(1).  No error message related to LWORK is
issued by XERBLA.  Neither H nor Z are accessed.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, ZHSEQR failed to compute all of
the eigenvalues.  Elements 1:ilo-1 and i+1:n of W
contain those eigenvalues which have been
successfully computed.  (Failures are rare.)

If INFO > 0 and JOB = 'E', then on exit, the
remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.

If INFO > 0 and JOB   = 'S', then on exit

(*)  (initial value of H)*U  = U*(final value of H)

where U is a unitary matrix.  The final
value of  H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.

If INFO > 0 and COMPZ = 'V', then on exit

(final value of Z)  =  (initial value of Z)*U

where U is the unitary matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'I', then on exit
(final value of Z)  = U
where U is the unitary matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = 'N', then Z is not
accessed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

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

Further Details:

```             Default values supplied by
ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.

ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point.
Default: 75. (Must be at least 11.)

ISPEC=13: Recommended deflation window size.
This depends on ILO, IHI and NS.  NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
The default for (IHI-ILO+1) <= 500 is NS.
The default for (IHI-ILO+1) >  500 is 3*NS/2.

ISPEC=14: Nibble crossover point. (See IPARMQ for
details.)  Default: 14% of deflation window
size.

ISPEC=15: Number of simultaneous shifts in a multishift
QR iteration.

If IHI-ILO+1 is ...

greater than      ...but less    ... the
or equal to ...      than        default is

1               30          NS =   2(+)
30               60          NS =   4(+)
60              150          NS =  10(+)
150              590          NS =  **
590             3000          NS =  64
3000             6000          NS = 128
6000             infinity      NS = 256

(+)  By default some or all matrices of this order
are passed to the implicit double shift routine
ZLAHQR and this parameter is ignored.  See
ISPEC=12 above and comments in IPARMQ for
details.

(**)  The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.

ISPEC=16: Select structured matrix multiply.
If the number of simultaneous shifts (specified
by ISPEC=15) is less than 14, then the default
for ISPEC=16 is 0.  Otherwise the default for
ISPEC=16 is 2.```

References:

```  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.```

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 297 of file zhseqr.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK