trevc(3)

Library Functions Manual

trevc(3)

NAME

trevc - trevc: eigenvectors of triangular Schur form, old

SYNOPSIS

Functions

subroutine ctrevc (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
CTREVC subroutine dtrevc (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info)
DTREVC subroutine strevc (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info)
STREVC subroutine ztrevc (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
ZTREVC

Detailed Description

Function Documentation

subroutine ctrevc (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, real, dimension( * ) rwork, integer info)

CTREVC

Purpose:

!>
!> CTREVC 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.
!>

Parameters

SIDE

!>          SIDE is CHARACTER*1
!>          = 'R':  compute right eigenvectors only;
!>          = 'L':  compute left eigenvectors only;
!>          = 'B':  compute both right and left eigenvectors.
!>

HOWMNY

!>          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.
!>

SELECT

!>          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'.
!>

!>          N is INTEGER
!>          The order of the matrix T. N >= 0.
!>

!>          T is COMPLEX array, dimension (LDT,N)
!>          The upper triangular matrix T.  T is modified, but restored
!>          on exit.
!>

LDT

!>          LDT is INTEGER
!>          The leading dimension of the array T. LDT >= max(1,N).
!>

!>          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'.
!>

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL.  LDVL >= 1, and if
!>          SIDE = 'L' or 'B', LDVL >= N.
!>

!>          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'.
!>

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR.  LDVR >= 1, and if
!>          SIDE = 'R' or 'B'; LDVR >= N.
!>

!>          MM is INTEGER
!>          The number of columns in the arrays VL and/or VR. MM >= M.
!>

!>          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.
!>

WORK

!>          WORK is COMPLEX array, dimension (2*N)
!>

RWORK

!>          RWORK is REAL array, dimension (N)
!>

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:

!>
!>  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|.
!>

Definition at line 216 of file ctrevc.f.

subroutine dtrevc (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 info)

DTREVC

Purpose:

!>
!> DTREVC 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**H)*T = w*(y**H)
!>
!> where y**H denotes the conjugate transpose of 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.
!>

Parameters

SIDE

!>          SIDE is CHARACTER*1
!>          = 'R':  compute right eigenvectors only;
!>          = 'L':  compute left eigenvectors only;
!>          = 'B':  compute both right and left eigenvectors.
!>

HOWMNY

!>          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.
!>

SELECT

!>          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'.
!>

!>          N is INTEGER
!>          The order of the matrix T. N >= 0.
!>

!>          T is DOUBLE PRECISION array, dimension (LDT,N)
!>          The upper quasi-triangular matrix T in Schur canonical form.
!>

LDT

!>          LDT is INTEGER
!>          The leading dimension of the array T. LDT >= max(1,N).
!>

!>          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'.
!>

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL.  LDVL >= 1, and if
!>          SIDE = 'L' or 'B', LDVL >= N.
!>

!>          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'.
!>

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR.  LDVR >= 1, and if
!>          SIDE = 'R' or 'B', LDVR >= N.
!>

!>          MM is INTEGER
!>          The number of columns in the arrays VL and/or VR. MM >= M.
!>

!>          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.
!>

WORK

!>          WORK is DOUBLE PRECISION array, dimension (3*N)
!>

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:

!>
!>  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|.
!>

Definition at line 220 of file dtrevc.f.

subroutine strevc (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 info)

STREVC

Purpose:

!>
!> STREVC 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**H)*T = w*(y**H)
!>
!> where y**H denotes the conjugate transpose of 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.
!>

Parameters

SIDE

!>          SIDE is CHARACTER*1
!>          = 'R':  compute right eigenvectors only;
!>          = 'L':  compute left eigenvectors only;
!>          = 'B':  compute both right and left eigenvectors.
!>

HOWMNY

!>          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.
!>

SELECT

!>          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'.
!>

!>          N is INTEGER
!>          The order of the matrix T. N >= 0.
!>

!>          T is REAL array, dimension (LDT,N)
!>          The upper quasi-triangular matrix T in Schur canonical form.
!>

LDT

!>          LDT is INTEGER
!>          The leading dimension of the array T. LDT >= max(1,N).
!>

!>          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'.
!>

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL.  LDVL >= 1, and if
!>          SIDE = 'L' or 'B', LDVL >= N.
!>

!>          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'.
!>

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR.  LDVR >= 1, and if
!>          SIDE = 'R' or 'B', LDVR >= N.
!>

!>          MM is INTEGER
!>          The number of columns in the arrays VL and/or VR. MM >= M.
!>

!>          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.
!>

WORK

!>          WORK is REAL array, dimension (3*N)
!>

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:

!>
!>  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|.
!>

Definition at line 220 of file strevc.f.

subroutine ztrevc (character side, character howmny, logical, dimension( * ) select, integer n, complex16, dimension( ldt, ) t, integer ldt, complex16, dimension( ldvl, ) vl, integer ldvl, complex16, dimension( ldvr, ) vr, integer ldvr, integer mm, integer m, complex16, dimension( ) work, double precision, dimension( * ) rwork, integer info)

ZTREVC

Purpose:

!>
!> ZTREVC 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.
!>

Parameters

SIDE

!>          SIDE is CHARACTER*1
!>          = 'R':  compute right eigenvectors only;
!>          = 'L':  compute left eigenvectors only;
!>          = 'B':  compute both right and left eigenvectors.
!>

HOWMNY

!>          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.
!>

SELECT

!>          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'.
!>

!>          N is INTEGER
!>          The order of the matrix T. N >= 0.
!>

!>          T is COMPLEX*16 array, dimension (LDT,N)
!>          The upper triangular matrix T.  T is modified, but restored
!>          on exit.
!>

LDT

!>          LDT is INTEGER
!>          The leading dimension of the array T. LDT >= max(1,N).
!>

!>          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'.
!>

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL.  LDVL >= 1, and if
!>          SIDE = 'L' or 'B', LDVL >= N.
!>

!>          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'.
!>

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR.  LDVR >= 1, and if
!>          SIDE = 'R' or 'B'; LDVR >= N.
!>

!>          MM is INTEGER
!>          The number of columns in the arrays VL and/or VR. MM >= M.
!>

!>          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.
!>

WORK

!>          WORK is COMPLEX*16 array, dimension (2*N)
!>

RWORK

!>          RWORK is DOUBLE PRECISION array, dimension (N)
!>

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:

!>
!>  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|.
!>

Definition at line 216 of file ztrevc.f.

Author

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