.TH "hegv" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
hegv \- {he,sy}gv: eig, QR iteration
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBchegv\fP (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)"
.br
.RI "\fBCHEGV\fP "
.ti -1c
.RI "subroutine \fBdsygv\fP (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)"
.br
.RI "\fBDSYGV\fP "
.ti -1c
.RI "subroutine \fBssygv\fP (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info)"
.br
.RI "\fBSSYGV\fP "
.ti -1c
.RI "subroutine \fBzhegv\fP (itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)"
.br
.RI "\fBZHEGV\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "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)"

.PP
\fBCHEGV\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIITYPE\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIJOBZ\fP 
.PP
.nf
          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is REAL array, dimension (max(1, 3*N-2))
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "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)"

.PP
\fBDSYGV\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIITYPE\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIJOBZ\fP 
.PP
.nf
          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "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)"

.PP
\fBSSYGV\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIITYPE\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIJOBZ\fP 
.PP
.nf
          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is REAL array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "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)"

.PP
\fBZHEGV\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIITYPE\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIJOBZ\fP 
.PP
.nf
          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max(1,N)\&.
.fi
.PP
.br
\fIW\fP 
.PP
.nf
          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.