# hegv - Man Page

{he,sy}gv: eig, QR iteration

## Synopsis

### Functions

subroutine chegv (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)
CHEGV
subroutine dsygv (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)
DSYGV
subroutine ssygv (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)
SSYGV
subroutine zhegv (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)
ZHEGV

## Function Documentation

### subroutine chegv (integer itype, character jobz, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) w, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

CHEGV

Purpose:

``` CHEGV computes all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
Here A and B are assumed to be Hermitian and B is also
positive definite.```
Parameters

ITYPE

```          ITYPE is INTEGER
Specifies the problem type to be solved:
= 1:  A*x = (lambda)*B*x
= 2:  A*B*x = (lambda)*x
= 3:  B*A*x = (lambda)*x```

JOBZ

```          JOBZ is CHARACTER*1
= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangles of A and B are stored;
= 'L':  Lower triangles of A and B are stored.```

N

```          N is INTEGER
The order of the matrices A and B.  N >= 0.```

A

```          A is COMPLEX array, dimension (LDA, N)
On entry, the Hermitian matrix A.  If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A.  If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors.  The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.```

LDA

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

B

```          B is COMPLEX array, dimension (LDB, N)
On entry, the Hermitian positive definite matrix B.
If UPLO = 'U', the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**H*U or B = L*L**H.```

LDB

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

W

```          W is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.```

WORK

```          WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The length of the array WORK.  LWORK >= max(1,2*N-1).
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the blocksize for CHETRD returned by ILAENV.

If LWORK = -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.```

RWORK

`          RWORK is REAL array, dimension (max(1, 3*N-2))`

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  CPOTRF or CHEEV returned an error code:
<= N:  if INFO = i, CHEEV failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N:   if INFO = N + i, for 1 <= i <= N, then the leading
principal minor of order i of B is not positive.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 179 of file chegv.f.

### subroutine dsygv (integer itype, character jobz, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) w, double precision, dimension( * ) work, integer lwork, integer info)

DSYGV

Purpose:

``` DSYGV computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
Here A and B are assumed to be symmetric and B is also
positive definite.```
Parameters

ITYPE

```          ITYPE is INTEGER
Specifies the problem type to be solved:
= 1:  A*x = (lambda)*B*x
= 2:  A*B*x = (lambda)*x
= 3:  B*A*x = (lambda)*x```

JOBZ

```          JOBZ is CHARACTER*1
= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangles of A and B are stored;
= 'L':  Lower triangles of A and B are stored.```

N

```          N is INTEGER
The order of the matrices A and B.  N >= 0.```

A

```          A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the symmetric matrix A.  If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A.  If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors.  The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the symmetric positive definite matrix B.
If UPLO = 'U', the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T.```

LDB

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

W

```          W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.```

WORK

```          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The length of the array WORK.  LWORK >= max(1,3*N-1).
For optimal efficiency, LWORK >= (NB+2)*N,
where NB is the blocksize for DSYTRD returned by ILAENV.

If LWORK = -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.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  DPOTRF or DSYEV returned an error code:
<= N:  if INFO = i, DSYEV failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N:   if INFO = N + i, for 1 <= i <= N, then the leading
principal minor of order i of B is not positive.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 173 of file dsygv.f.

### subroutine ssygv (integer itype, character jobz, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) w, real, dimension( * ) work, integer lwork, integer info)

SSYGV

Purpose:

``` SSYGV computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
Here A and B are assumed to be symmetric and B is also
positive definite.```
Parameters

ITYPE

```          ITYPE is INTEGER
Specifies the problem type to be solved:
= 1:  A*x = (lambda)*B*x
= 2:  A*B*x = (lambda)*x
= 3:  B*A*x = (lambda)*x```

JOBZ

```          JOBZ is CHARACTER*1
= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangles of A and B are stored;
= 'L':  Lower triangles of A and B are stored.```

N

```          N is INTEGER
The order of the matrices A and B.  N >= 0.```

A

```          A is REAL array, dimension (LDA, N)
On entry, the symmetric matrix A.  If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A.  If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors.  The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.```

LDA

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

B

```          B is REAL array, dimension (LDB, N)
On entry, the symmetric positive definite matrix B.
If UPLO = 'U', the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**T*U or B = L*L**T.```

LDB

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

W

```          W is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The length of the array WORK.  LWORK >= max(1,3*N-1).
For optimal efficiency, LWORK >= (NB+2)*N,
where NB is the blocksize for SSYTRD returned by ILAENV.

If LWORK = -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.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  SPOTRF or SSYEV returned an error code:
<= N:  if INFO = i, SSYEV failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N:   if INFO = N + i, for 1 <= i <= N, then the leading
principal minor of order i of B is not positive.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 173 of file ssygv.f.

### subroutine zhegv (integer itype, character jobz, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) w, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZHEGV

Purpose:

``` ZHEGV computes all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
Here A and B are assumed to be Hermitian and B is also
positive definite.```
Parameters

ITYPE

```          ITYPE is INTEGER
Specifies the problem type to be solved:
= 1:  A*x = (lambda)*B*x
= 2:  A*B*x = (lambda)*x
= 3:  B*A*x = (lambda)*x```

JOBZ

```          JOBZ is CHARACTER*1
= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.```

UPLO

```          UPLO is CHARACTER*1
= 'U':  Upper triangles of A and B are stored;
= 'L':  Lower triangles of A and B are stored.```

N

```          N is INTEGER
The order of the matrices A and B.  N >= 0.```

A

```          A is COMPLEX*16 array, dimension (LDA, N)
On entry, the Hermitian matrix A.  If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A.  If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors.  The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.```

LDA

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

B

```          B is COMPLEX*16 array, dimension (LDB, N)
On entry, the Hermitian positive definite matrix B.
If UPLO = 'U', the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**H*U or B = L*L**H.```

LDB

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

W

```          W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.```

WORK

```          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The length of the array WORK.  LWORK >= max(1,2*N-1).
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the blocksize for ZHETRD returned by ILAENV.

If LWORK = -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.```

RWORK

`          RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))`

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  ZPOTRF or ZHEEV returned an error code:
<= N:  if INFO = i, ZHEEV failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N:   if INFO = N + i, for 1 <= i <= N, then the leading
principal minor of order i of B is not positive.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 179 of file zhegv.f.

## Author

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## Info

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