.TH "hbgst" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
hbgst \- {hb,sb}gst: reduction to standard form, banded
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBchbgst\fP (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, rwork, info)"
.br
.RI "\fBCHBGST\fP "
.ti -1c
.RI "subroutine \fBdsbgst\fP (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info)"
.br
.RI "\fBDSBGST\fP "
.ti -1c
.RI "subroutine \fBssbgst\fP (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info)"
.br
.RI "\fBSSBGST\fP "
.ti -1c
.RI "subroutine \fBzhbgst\fP (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, rwork, info)"
.br
.RI "\fBZHBGST\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine chbgst (character vect, character uplo, integer n, integer ka, integer kb, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldbb, * ) bb, integer ldbb, complex, dimension( ldx, * ) x, integer ldx, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)"

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

.PP
.nf
 CHBGST reduces a complex Hermitian-definite banded generalized
 eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
 such that C has the same bandwidth as A\&.

 B must have been previously factorized as S**H*S by CPBSTF, using a
 split Cholesky factorization\&. A is overwritten by C = X**H*A*X, where
 X = S**(-1)*Q and Q is a unitary matrix chosen to preserve the
 bandwidth of A\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIKA\fP 
.PP
.nf
          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= 0\&.
.fi
.PP
.br
\fIKB\fP 
.PP
.nf
          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= KB >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka)\&.

          On exit, the transformed matrix X**H*A*X, stored in the same
          format as A\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KA+1\&.
.fi
.PP
.br
\fIBB\fP 
.PP
.nf
          BB is COMPLEX array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of
          B, as returned by CPBSTF, stored in the first kb+1 rows of
          the array\&.
.fi
.PP
.br
\fILDBB\fP 
.PP
.nf
          LDBB is INTEGER
          The leading dimension of the array BB\&.  LDBB >= KB+1\&.
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is COMPLEX array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X\&.
          If VECT = 'N', the array X is not referenced\&.
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
          LDX is INTEGER
          The leading dimension of the array X\&.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (N)
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is REAL array, dimension (N)
.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\&.
.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 dsbgst (character vect, character uplo, integer n, integer ka, integer kb, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) work, integer info)"

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

.PP
.nf
 DSBGST reduces a real symmetric-definite banded generalized
 eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
 such that C has the same bandwidth as A\&.

 B must have been previously factorized as S**T*S by DPBSTF, using a
 split Cholesky factorization\&. A is overwritten by C = X**T*A*X, where
 X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
 bandwidth of A\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIKA\fP 
.PP
.nf
          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= 0\&.
.fi
.PP
.br
\fIKB\fP 
.PP
.nf
          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= KB >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          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 ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka)\&.

          On exit, the transformed matrix X**T*A*X, stored in the same
          format as A\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KA+1\&.
.fi
.PP
.br
\fIBB\fP 
.PP
.nf
          BB is DOUBLE PRECISION array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of
          B, as returned by DPBSTF, stored in the first KB+1 rows of
          the array\&.
.fi
.PP
.br
\fILDBB\fP 
.PP
.nf
          LDBB is INTEGER
          The leading dimension of the array BB\&.  LDBB >= KB+1\&.
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is DOUBLE PRECISION array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X\&.
          If VECT = 'N', the array X is not referenced\&.
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
          LDX is INTEGER
          The leading dimension of the array X\&.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (2*N)
.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\&.
.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 ssbgst (character vect, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) work, integer info)"

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

.PP
.nf
 SSBGST reduces a real symmetric-definite banded generalized
 eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
 such that C has the same bandwidth as A\&.

 B must have been previously factorized as S**T*S by SPBSTF, using a
 split Cholesky factorization\&. A is overwritten by C = X**T*A*X, where
 X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
 bandwidth of A\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIKA\fP 
.PP
.nf
          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= 0\&.
.fi
.PP
.br
\fIKB\fP 
.PP
.nf
          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= KB >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          AB is REAL array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka)\&.

          On exit, the transformed matrix X**T*A*X, stored in the same
          format as A\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KA+1\&.
.fi
.PP
.br
\fIBB\fP 
.PP
.nf
          BB is REAL array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of
          B, as returned by SPBSTF, stored in the first KB+1 rows of
          the array\&.
.fi
.PP
.br
\fILDBB\fP 
.PP
.nf
          LDBB is INTEGER
          The leading dimension of the array BB\&.  LDBB >= KB+1\&.
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is REAL array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X\&.
          If VECT = 'N', the array X is not referenced\&.
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
          LDX is INTEGER
          The leading dimension of the array X\&.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (2*N)
.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\&.
.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 zhbgst (character vect, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, complex*16, dimension( ldx, * ) x, integer ldx, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)"

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

.PP
.nf
 ZHBGST reduces a complex Hermitian-definite banded generalized
 eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
 such that C has the same bandwidth as A\&.

 B must have been previously factorized as S**H*S by ZPBSTF, using a
 split Cholesky factorization\&. A is overwritten by C = X**H*A*X, where
 X = S**(-1)*Q and Q is a unitary matrix chosen to preserve the
 bandwidth of A\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIVECT\fP 
.PP
.nf
          VECT is CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X\&.
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIKA\fP 
.PP
.nf
          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= 0\&.
.fi
.PP
.br
\fIKB\fP 
.PP
.nf
          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'\&.  KA >= KB >= 0\&.
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
          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 ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka)\&.

          On exit, the transformed matrix X**H*A*X, stored in the same
          format as A\&.
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
          LDAB is INTEGER
          The leading dimension of the array AB\&.  LDAB >= KA+1\&.
.fi
.PP
.br
\fIBB\fP 
.PP
.nf
          BB is COMPLEX*16 array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of
          B, as returned by ZPBSTF, stored in the first kb+1 rows of
          the array\&.
.fi
.PP
.br
\fILDBB\fP 
.PP
.nf
          LDBB is INTEGER
          The leading dimension of the array BB\&.  LDBB >= KB+1\&.
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is COMPLEX*16 array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X\&.
          If VECT = 'N', the array X is not referenced\&.
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
          LDX is INTEGER
          The leading dimension of the array X\&.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (N)
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (N)
.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\&.
.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\&.