# hetrd_hb2st - Man Page

{he,sy}trd_hb2st: band to tridiagonal (2nd stage)

## Synopsis

### Functions

subroutine chetrd_hb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T
subroutine dsytrd_sb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T
subroutine ssytrd_sb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T
subroutine zhetrd_hb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

## Function Documentation

### subroutine chetrd_hb2st (character stage1, character vect, character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) hous, integer lhous, complex, dimension( * ) work, integer lwork, integer info)

CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

Purpose:

``` CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q**H * A * Q = T.```
Parameters

STAGE1

```          STAGE1 is CHARACTER*1
= 'N':  'No': to mention that the stage 1 of the reduction
from dense to band using the chetrd_he2hb routine
was not called before this routine to reproduce AB.
In other term this routine is called as standalone.
= 'Y':  'Yes': to mention that the stage 1 of the
reduction from dense to band using the chetrd_he2hb
routine has been called to produce AB (e.g., AB is
the output of chetrd_he2hb.```

VECT

```          VECT is CHARACTER*1
= 'N':  No need for the Housholder representation,
and thus LHOUS is of size max(1, 4*N);
= 'V':  the Householder representation is needed to
either generate or to apply Q later on,
then LHOUS is to be queried and computed.
(NOT AVAILABLE IN THIS RELEASE).```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.```

N

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

KD

```          KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'.  KD >= 0.```

AB

```          AB is COMPLEX array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = 'U') or the
first subdiagonal (if UPLO = 'L') are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.```

D

```          D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T.```

E

```          E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.```

HOUS

```          HOUS is COMPLEX array, dimension LHOUS, that
store the Householder representation.```

LHOUS

```          LHOUS is INTEGER
The dimension of the array HOUS. LHOUS = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a query is assumed; the routine
only calculates the optimal size of the HOUS array, returns
this value as the first entry of the HOUS array, and no error
message related to LHOUS is issued by XERBLA.
LHOUS = MAX(1, dimension) where
dimension = 4*N if VECT='N'
not available now if VECT='H'```

WORK

`          WORK is COMPLEX array, dimension LWORK.`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK = MAX(1, dimension) where
dimension   = (2KD+1)*N + KD*NTHREADS
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196```

Definition at line 228 of file chetrd_hb2st.F.

### subroutine dsytrd_sb2st (character stage1, character vect, character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) hous, integer lhous, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T

Purpose:

``` DSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric
tridiagonal form T by a orthogonal similarity transformation:
Q**T * A * Q = T.```
Parameters

STAGE1

```          STAGE1 is CHARACTER*1
= 'N':  'No': to mention that the stage 1 of the reduction
from dense to band using the dsytrd_sy2sb routine
was not called before this routine to reproduce AB.
In other term this routine is called as standalone.
= 'Y':  'Yes': to mention that the stage 1 of the
reduction from dense to band using the dsytrd_sy2sb
routine has been called to produce AB (e.g., AB is
the output of dsytrd_sy2sb.```

VECT

```          VECT is CHARACTER*1
= 'N':  No need for the Housholder representation,
and thus LHOUS is of size max(1, 4*N);
= 'V':  the Householder representation is needed to
either generate or to apply Q later on,
then LHOUS is to be queried and computed.
(NOT AVAILABLE IN THIS RELEASE).```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.```

N

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

KD

```          KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'.  KD >= 0.```

AB

```          AB is DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = 'U') or the
first subdiagonal (if UPLO = 'L') are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.```

D

```          D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T.```

E

```          E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.```

HOUS

```          HOUS is DOUBLE PRECISION array, dimension LHOUS, that
store the Householder representation.```

LHOUS

```          LHOUS is INTEGER
The dimension of the array HOUS. LHOUS = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a query is assumed; the routine
only calculates the optimal size of the HOUS array, returns
this value as the first entry of the HOUS array, and no error
message related to LHOUS is issued by XERBLA.
LHOUS = MAX(1, dimension) where
dimension = 4*N if VECT='N'
not available now if VECT='H'```

WORK

`          WORK is DOUBLE PRECISION array, dimension LWORK.`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK = MAX(1, dimension) where
dimension   = (2KD+1)*N + KD*NTHREADS
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196```

Definition at line 228 of file dsytrd_sb2st.F.

### subroutine ssytrd_sb2st (character stage1, character vect, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) hous, integer lhous, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T

Purpose:

``` SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric
tridiagonal form T by a orthogonal similarity transformation:
Q**T * A * Q = T.```
Parameters

STAGE1

```          STAGE1 is CHARACTER*1
= 'N':  'No': to mention that the stage 1 of the reduction
from dense to band using the ssytrd_sy2sb routine
was not called before this routine to reproduce AB.
In other term this routine is called as standalone.
= 'Y':  'Yes': to mention that the stage 1 of the
reduction from dense to band using the ssytrd_sy2sb
routine has been called to produce AB (e.g., AB is
the output of ssytrd_sy2sb.```

VECT

```          VECT is CHARACTER*1
= 'N':  No need for the Housholder representation,
and thus LHOUS is of size max(1, 4*N);
= 'V':  the Householder representation is needed to
either generate or to apply Q later on,
then LHOUS is to be queried and computed.
(NOT AVAILABLE IN THIS RELEASE).```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.```

N

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

KD

```          KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'.  KD >= 0.```

AB

```          AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = 'U') or the
first subdiagonal (if UPLO = 'L') are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.```

D

```          D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T.```

E

```          E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.```

HOUS

```          HOUS is REAL array, dimension LHOUS, that
store the Householder representation.```

LHOUS

```          LHOUS is INTEGER
The dimension of the array HOUS. LHOUS = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a query is assumed; the routine
only calculates the optimal size of the HOUS array, returns
this value as the first entry of the HOUS array, and no error
message related to LHOUS is issued by XERBLA.
LHOUS = MAX(1, dimension) where
dimension = 4*N if VECT='N'
not available now if VECT='H'```

WORK

`          WORK is REAL array, dimension LWORK.`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK = MAX(1, dimension) where
dimension   = (2KD+1)*N + KD*NTHREADS
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196```

Definition at line 228 of file ssytrd_sb2st.F.

### subroutine zhetrd_hb2st (character stage1, character vect, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) hous, integer lhous, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

Purpose:

``` ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q**H * A * Q = T.```
Parameters

STAGE1

```          STAGE1 is CHARACTER*1
= 'N':  'No': to mention that the stage 1 of the reduction
from dense to band using the zhetrd_he2hb routine
was not called before this routine to reproduce AB.
In other term this routine is called as standalone.
= 'Y':  'Yes': to mention that the stage 1 of the
reduction from dense to band using the zhetrd_he2hb
routine has been called to produce AB (e.g., AB is
the output of zhetrd_he2hb.```

VECT

```          VECT is CHARACTER*1
= 'N':  No need for the Housholder representation,
and thus LHOUS is of size max(1, 4*N);
= 'V':  the Householder representation is needed to
either generate or to apply Q later on,
then LHOUS is to be queried and computed.
(NOT AVAILABLE IN THIS RELEASE).```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.```

N

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

KD

```          KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'.  KD >= 0.```

AB

```          AB is COMPLEX*16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = 'U') or the
first subdiagonal (if UPLO = 'L') are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.```

LDAB

```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.```

D

```          D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T.```

E

```          E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.```

HOUS

```          HOUS is COMPLEX*16 array, dimension LHOUS, that
store the Householder representation.```

LHOUS

```          LHOUS is INTEGER
The dimension of the array HOUS. LHOUS = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a query is assumed; the routine
only calculates the optimal size of the HOUS array, returns
this value as the first entry of the HOUS array, and no error
message related to LHOUS is issued by XERBLA.
LHOUS = MAX(1, dimension) where
dimension = 4*N if VECT='N'
not available now if VECT='H'```

WORK

`          WORK is COMPLEX*16 array, dimension LWORK.`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK = MAX(1, dimension)
If LWORK = -1, or LHOUS=-1,
then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK = MAX(1, dimension) where
dimension   = (2KD+1)*N + KD*NTHREADS
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196```

Definition at line 228 of file zhetrd_hb2st.F.

## Author

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## Info

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