.TH "trevc3" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
trevc3 \- trevc3: eigenvectors of triangular Schur form, blocked
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBctrevc3\fP (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, rwork, lrwork, info)"
.br
.RI "\fBCTREVC3\fP "
.ti -1c
.RI "subroutine \fBdtrevc3\fP (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)"
.br
.RI "\fBDTREVC3\fP "
.ti -1c
.RI "subroutine \fBstrevc3\fP (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)"
.br
.RI "\fBSTREVC3\fP "
.ti -1c
.RI "subroutine \fBztrevc3\fP (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, rwork, lrwork, info)"
.br
.RI "\fBZTREVC3\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine ctrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer info)"

.PP
\fBCTREVC3\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CTREVC3 computes some or all of the right and/or left eigenvectors of
 a complex upper triangular matrix T\&.
 Matrices of this type are produced by the Schur factorization of
 a complex general matrix:  A = Q*T*Q**H, as computed by CHSEQR\&.

 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:

              T*x = w*x,     (y**H)*T = w*(y**H)

 where y**H denotes the conjugate transpose of the vector y\&.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal of T\&.

 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix\&. If Q is the unitary factor that reduces a matrix A to
 Schur form T, then Q*X and Q*Y are the matrices of right and left
 eigenvectors of A\&.

 This uses a Level 3 BLAS version of the back transformation\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP 
.PP
.nf
          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed using the matrices supplied in
                  VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed\&.
          The eigenvector corresponding to the j-th eigenvalue is
          computed if SELECT(j) = \&.TRUE\&.\&.
          Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix T\&. N >= 0\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is COMPLEX array, dimension (LDT,N)
          The upper triangular matrix T\&.  T is modified, but restored
          on exit\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is COMPLEX array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR)\&.
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues\&.
          Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
          LDVL is INTEGER
          The leading dimension of the array VL\&.
          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
          VR is COMPLEX array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR)\&.
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues\&.
          Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
          LDVR is INTEGER
          The leading dimension of the array VR\&.
          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
          MM is INTEGER
          The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors\&.
          If HOWMNY = 'A' or 'B', M is set to N\&.
          Each selected eigenvector occupies one column\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of array WORK\&. LWORK >= max(1,2*N)\&.
          For optimum performance, LWORK >= N + 2*N*NB, where NB is
          the optimal blocksize\&.

          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 (LRWORK)
.fi
.PP
.br
\fILRWORK\fP 
.PP
.nf
          LRWORK is INTEGER
          The dimension of array RWORK\&. LRWORK >= max(1,N)\&.

          If LRWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the RWORK array, returns
          this value as the first entry of the RWORK array, and no error
          message related to LRWORK 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
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow\&.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine dtrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer lwork, integer info)"

.PP
\fBDTREVC3\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DTREVC3 computes some or all of the right and/or left eigenvectors of
 a real upper quasi-triangular matrix T\&.
 Matrices of this type are produced by the Schur factorization of
 a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR\&.

 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:

    T*x = w*x,     (y**T)*T = w*(y**T)

 where y**T denotes the transpose of the vector y\&.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal blocks of T\&.

 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix\&. If Q is the orthogonal factor that reduces a matrix
 A to Schur form T, then Q*X and Q*Y are the matrices of right and
 left eigenvectors of A\&.

 This uses a Level 3 BLAS version of the back transformation\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP 
.PP
.nf
          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed by the matrices in VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed\&.
          If w(j) is a real eigenvalue, the corresponding real
          eigenvector is computed if SELECT(j) is \&.TRUE\&.\&.
          If w(j) and w(j+1) are the real and imaginary parts of a
          complex eigenvalue, the corresponding complex eigenvector is
          computed if either SELECT(j) or SELECT(j+1) is \&.TRUE\&., and
          on exit SELECT(j) is set to \&.TRUE\&. and SELECT(j+1) is set to
          \&.FALSE\&.\&.
          Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix T\&. N >= 0\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is DOUBLE PRECISION array, dimension (LDT,N)
          The upper quasi-triangular matrix T in Schur canonical form\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by DHSEQR)\&.
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues\&.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part, and the second the imaginary part\&.
          Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
          LDVL is INTEGER
          The leading dimension of the array VL\&.
          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by DHSEQR)\&.
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues\&.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part and the second the imaginary part\&.
          Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
          LDVR is INTEGER
          The leading dimension of the array VR\&.
          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
          MM is INTEGER
          The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors\&.
          If HOWMNY = 'A' or 'B', M is set to N\&.
          Each selected real eigenvector occupies one column and each
          selected complex eigenvector occupies two columns\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of array WORK\&. LWORK >= max(1,3*N)\&.
          For optimum performance, LWORK >= N + 2*N*NB, where NB is
          the optimal blocksize\&.

          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
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow\&.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine strevc3 (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer lwork, integer info)"

.PP
\fBSTREVC3\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 STREVC3 computes some or all of the right and/or left eigenvectors of
 a real upper quasi-triangular matrix T\&.
 Matrices of this type are produced by the Schur factorization of
 a real general matrix:  A = Q*T*Q**T, as computed by SHSEQR\&.

 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:

    T*x = w*x,     (y**T)*T = w*(y**T)

 where y**T denotes the transpose of the vector y\&.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal blocks of T\&.

 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix\&. If Q is the orthogonal factor that reduces a matrix
 A to Schur form T, then Q*X and Q*Y are the matrices of right and
 left eigenvectors of A\&.

 This uses a Level 3 BLAS version of the back transformation\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP 
.PP
.nf
          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed by the matrices in VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed\&.
          If w(j) is a real eigenvalue, the corresponding real
          eigenvector is computed if SELECT(j) is \&.TRUE\&.\&.
          If w(j) and w(j+1) are the real and imaginary parts of a
          complex eigenvalue, the corresponding complex eigenvector is
          computed if either SELECT(j) or SELECT(j+1) is \&.TRUE\&., and
          on exit SELECT(j) is set to \&.TRUE\&. and SELECT(j+1) is set to
          \&.FALSE\&.\&.
          Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix T\&. N >= 0\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is REAL array, dimension (LDT,N)
          The upper quasi-triangular matrix T in Schur canonical form\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is REAL array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by SHSEQR)\&.
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues\&.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part, and the second the imaginary part\&.
          Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
          LDVL is INTEGER
          The leading dimension of the array VL\&.
          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
          VR is REAL array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by SHSEQR)\&.
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues\&.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part and the second the imaginary part\&.
          Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
          LDVR is INTEGER
          The leading dimension of the array VR\&.
          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
          MM is INTEGER
          The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors\&.
          If HOWMNY = 'A' or 'B', M is set to N\&.
          Each selected real eigenvector occupies one column and each
          selected complex eigenvector occupies two columns\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (MAX(1,LWORK))
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of array WORK\&. LWORK >= max(1,3*N)\&.
          For optimum performance, LWORK >= N + 2*N*NB, where NB is
          the optimal blocksize\&.

          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
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow\&.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine ztrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer info)"

.PP
\fBZTREVC3\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZTREVC3 computes some or all of the right and/or left eigenvectors of
 a complex upper triangular matrix T\&.
 Matrices of this type are produced by the Schur factorization of
 a complex general matrix:  A = Q*T*Q**H, as computed by ZHSEQR\&.

 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:

              T*x = w*x,     (y**H)*T = w*(y**H)

 where y**H denotes the conjugate transpose of the vector y\&.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal of T\&.

 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix\&. If Q is the unitary factor that reduces a matrix A to
 Schur form T, then Q*X and Q*Y are the matrices of right and left
 eigenvectors of A\&.

 This uses a Level 3 BLAS version of the back transformation\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP 
.PP
.nf
          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed using the matrices supplied in
                  VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed\&.
          The eigenvector corresponding to the j-th eigenvalue is
          computed if SELECT(j) = \&.TRUE\&.\&.
          Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix T\&. N >= 0\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is COMPLEX*16 array, dimension (LDT,N)
          The upper triangular matrix T\&.  T is modified, but restored
          on exit\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          VL is COMPLEX*16 array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by ZHSEQR)\&.
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues\&.
          Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
          LDVL is INTEGER
          The leading dimension of the array VL\&.
          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
          VR is COMPLEX*16 array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by ZHSEQR)\&.
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues\&.
          Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
          LDVR is INTEGER
          The leading dimension of the array VR\&.
          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
          MM is INTEGER
          The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors\&.
          If HOWMNY = 'A' or 'B', M is set to N\&.
          Each selected eigenvector occupies one column\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of array WORK\&. LWORK >= max(1,2*N)\&.
          For optimum performance, LWORK >= N + 2*N*NB, where NB is
          the optimal blocksize\&.

          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 (LRWORK)
.fi
.PP
.br
\fILRWORK\fP 
.PP
.nf
          LRWORK is INTEGER
          The dimension of array RWORK\&. LRWORK >= max(1,N)\&.

          If LRWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the RWORK array, returns
          this value as the first entry of the RWORK array, and no error
          message related to LRWORK 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
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow\&.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|\&.
.fi
.PP
 
.RE
.PP

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