Functions

subroutine sgegs (JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO)
SGEGS computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgegv (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgees (JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS, WORK, LWORK, BWORK, INFO)
SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgeev (JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgges (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgges3 (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)
subroutine sggesx (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sggev (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sggev3 (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)
subroutine sggevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, BWORK, INFO)
SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

This is the group of real eigenvalue driver functions for GE matrices

Function Documentation

subroutine sgees (character JOBVS, character SORT, external SELECT, integer N, real, dimension( lda, * ) A, integer LDA, integer SDIM, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvs, * ) VS, integer LDVS, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO)

SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

 SGEES computes for an N-by-N real nonsymmetric matrix A, the
 eigenvalues, the real Schur form T, and, optionally, the matrix of
 Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
 Optionally, it also orders the eigenvalues on the diagonal of the
 real Schur form so that selected eigenvalues are at the top left.
 The leading columns of Z then form an orthonormal basis for the
 invariant subspace corresponding to the selected eigenvalues.
 A matrix is in real Schur form if it is upper quasi-triangular with
 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
 form
         [  a  b  ]
         [  c  a  ]
 where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

 

Parameters:

JOBVS

          JOBVS is CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed.

SORT

          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the Schur form.
          = 'N': Eigenvalues are not ordered;
          = 'S': Eigenvalues are ordered (see SELECT).

SELECT

          SELECT is LOGICAL FUNCTION of two REAL arguments
          SELECT must be declared EXTERNAL in the calling subroutine.
          If SORT = 'S', SELECT is used to select eigenvalues to sort
          to the top left of the Schur form.
          If SORT = 'N', SELECT is not referenced.
          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
          SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
          conjugate pair of eigenvalues is selected, then both complex
          eigenvalues are selected.
          Note that a selected complex eigenvalue may no longer
          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned); in this
          case INFO is set to N+2 (see INFO below).

N

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

A

          A is REAL array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten by its real Schur form T.

LDA

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

SDIM

          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                         for which SELECT is true. (Complex conjugate
                         pairs for which SELECT is true for either
                         eigenvalue count as 2.)

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues in the same order
          that they appear on the diagonal of the output Schur form T.
          Complex conjugate pairs of eigenvalues will appear
          consecutively with the eigenvalue having the positive
          imaginary part first.

VS

          VS is REAL array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
          vectors.
          If JOBVS = 'N', VS is not referenced.

LDVS

          LDVS is INTEGER
          The leading dimension of the array VS.  LDVS >= 1; if
          JOBVS = 'V', LDVS >= N.

WORK

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

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N).
          For good performance, LWORK must generally be larger.
          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.

BWORK

          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0: if INFO = i, and i is
             <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                   contain those eigenvalues which have converged; if
                   JOBVS = 'V', VS contains the matrix which reduces A
                   to its partially converged Schur form.
             = N+1: the eigenvalues could not be reordered because some
                   eigenvalues were too close to separate (the problem
                   is very ill-conditioned);
             = N+2: after reordering, roundoff changed values of some
                   complex eigenvalues so that leading eigenvalues in
                   the Schur form no longer satisfy SELECT=.TRUE.  This
                   could also be caused by underflow due to scaling.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine sgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, integer SDIM, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvs, * ) VS, integer LDVS, real RCONDE, real RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)

SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

 SGEESX computes for an N-by-N real nonsymmetric matrix A, the
 eigenvalues, the real Schur form T, and, optionally, the matrix of
 Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
 Optionally, it also orders the eigenvalues on the diagonal of the
 real Schur form so that selected eigenvalues are at the top left;
 computes a reciprocal condition number for the average of the
 selected eigenvalues (RCONDE); and computes a reciprocal condition
 number for the right invariant subspace corresponding to the
 selected eigenvalues (RCONDV).  The leading columns of Z form an
 orthonormal basis for this invariant subspace.
 For further explanation of the reciprocal condition numbers RCONDE
 and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
 these quantities are called s and sep respectively).
 A real matrix is in real Schur form if it is upper quasi-triangular
 with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
 the form
           [  a  b  ]
           [  c  a  ]
 where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

 

Parameters:

JOBVS

          JOBVS is CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed.

SORT

          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the Schur form.
          = 'N': Eigenvalues are not ordered;
          = 'S': Eigenvalues are ordered (see SELECT).

SELECT

          SELECT is a LOGICAL FUNCTION of two REAL arguments
          SELECT must be declared EXTERNAL in the calling subroutine.
          If SORT = 'S', SELECT is used to select eigenvalues to sort
          to the top left of the Schur form.
          If SORT = 'N', SELECT is not referenced.
          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
          SELECT(WR(j),WI(j)) is true; i.e., if either one of a
          complex conjugate pair of eigenvalues is selected, then both
          are.  Note that a selected complex eigenvalue may no longer
          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned); in this
          case INFO may be set to N+3 (see INFO below).

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for average of selected eigenvalues only;
          = 'V': Computed for selected right invariant subspace only;
          = 'B': Computed for both.
          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

N

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

A

          A is REAL array, dimension (LDA, N)
          On entry, the N-by-N matrix A.
          On exit, A is overwritten by its real Schur form T.

LDA

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

SDIM

          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                         for which SELECT is true. (Complex conjugate
                         pairs for which SELECT is true for either
                         eigenvalue count as 2.)

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)
          WR and WI contain the real and imaginary parts, respectively,
          of the computed eigenvalues, in the same order that they
          appear on the diagonal of the output Schur form T.  Complex
          conjugate pairs of eigenvalues appear consecutively with the
          eigenvalue having the positive imaginary part first.

VS

          VS is REAL array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
          vectors.
          If JOBVS = 'N', VS is not referenced.

LDVS

          LDVS is INTEGER
          The leading dimension of the array VS.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N.

RCONDE

          RCONDE is REAL
          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
          condition number for the average of the selected eigenvalues.
          Not referenced if SENSE = 'N' or 'V'.

RCONDV

          RCONDV is REAL
          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
          condition number for the selected right invariant subspace.
          Not referenced if SENSE = 'N' or 'E'.

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 dimension of the array WORK.  LWORK >= max(1,3*N).
          Also, if SENSE = 'E' or 'V' or 'B',
          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
          selected eigenvalues computed by this routine.  Note that
          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
          returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
          'B' this may not be large enough.
          For good performance, LWORK must generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates upper bounds on the optimal sizes of the
          arrays WORK and IWORK, returns these values as the first
          entries of the WORK and IWORK arrays, and no error messages
          related to LWORK or LIWORK are issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
          may not be large enough.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates upper bounds on the optimal sizes of
          the arrays WORK and IWORK, returns these values as the first
          entries of the WORK and IWORK arrays, and no error messages
          related to LWORK or LIWORK are issued by XERBLA.

BWORK

          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0: if INFO = i, and i is
             <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                   contain those eigenvalues which have converged; if
                   JOBVS = 'V', VS contains the transformation which
                   reduces A to its partially converged Schur form.
             = N+1: the eigenvalues could not be reordered because some
                   eigenvalues were too close to separate (the problem
                   is very ill-conditioned);
             = N+2: after reordering, roundoff changed values of some
                   complex eigenvalues so that leading eigenvalues in
                   the Schur form no longer satisfy SELECT=.TRUE.  This
                   could also be caused by underflow due to scaling.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine sgeev (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 SGEEV computes for an N-by-N real nonsymmetric matrix A, the
 eigenvalues and, optionally, the left and/or right eigenvectors.
 The right eigenvector v(j) of A satisfies
                  A * v(j) = lambda(j) * v(j)
 where lambda(j) is its eigenvalue.
 The left eigenvector u(j) of A satisfies
               u(j)**H * A = lambda(j) * u(j)**H
 where u(j)**H denotes the conjugate-transpose of u(j).
 The computed eigenvectors are normalized to have Euclidean norm
 equal to 1 and largest component real.

 

Parameters:

JOBVL

          JOBVL is CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.

JOBVR

          JOBVR is CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.

N

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

A

          A is REAL array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten.

LDA

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

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues.  Complex
          conjugate pairs of eigenvalues appear consecutively
          with the eigenvalue having the positive imaginary part
          first.

VL

          VL is REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order
          as their eigenvalues.
          If JOBVL = 'N', VL is not referenced.
          If the j-th eigenvalue is real, then u(j) = VL(:,j),
          the j-th column of VL.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
          u(j+1) = VL(:,j) - i*VL(:,j+1).

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL.  LDVL >= 1; if
          JOBVL = 'V', LDVL >= N.

VR

          VR is REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order
          as their eigenvalues.
          If JOBVR = 'N', VR is not referenced.
          If the j-th eigenvalue is real, then v(j) = VR(:,j),
          the j-th column of VR.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
          v(j+1) = VR(:,j) - i*VR(:,j+1).

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR.  LDVR >= 1; if
          JOBVR = 'V', LDVR >= N.

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 dimension of the array WORK.  LWORK >= max(1,3*N), and
          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  For good
          performance, LWORK must generally be larger.
          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:  if INFO = i, the QR algorithm failed to compute all the
                eigenvalues, and no eigenvectors have been computed;
                elements i+1:N of WR and WI contain eigenvalues which
                have converged.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine sgeevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, integer ILO, integer IHI, real, dimension( * ) SCALE, real ABNRM, real, dimension( * ) RCONDE, real, dimension( * ) RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 SGEEVX computes for an N-by-N real nonsymmetric matrix A, the
 eigenvalues and, optionally, the left and/or right eigenvectors.
 Optionally also, it computes a balancing transformation to improve
 the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
 SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
 (RCONDE), and reciprocal condition numbers for the right
 eigenvectors (RCONDV).
 The right eigenvector v(j) of A satisfies
                  A * v(j) = lambda(j) * v(j)
 where lambda(j) is its eigenvalue.
 The left eigenvector u(j) of A satisfies
               u(j)**H * A = lambda(j) * u(j)**H
 where u(j)**H denotes the conjugate-transpose of u(j).
 The computed eigenvectors are normalized to have Euclidean norm
 equal to 1 and largest component real.
 Balancing a matrix means permuting the rows and columns to make it
 more nearly upper triangular, and applying a diagonal similarity
 transformation D * A * D**(-1), where D is a diagonal matrix, to
 make its rows and columns closer in norm and the condition numbers
 of its eigenvalues and eigenvectors smaller.  The computed
 reciprocal condition numbers correspond to the balanced matrix.
 Permuting rows and columns will not change the condition numbers
 (in exact arithmetic) but diagonal scaling will.  For further
 explanation of balancing, see section 4.10.2 of the LAPACK
 Users' Guide.

 

Parameters:

BALANC

          BALANC is CHARACTER*1
          Indicates how the input matrix should be diagonally scaled
          and/or permuted to improve the conditioning of its
          eigenvalues.
          = 'N': Do not diagonally scale or permute;
          = 'P': Perform permutations to make the matrix more nearly
                 upper triangular. Do not diagonally scale;
          = 'S': Diagonally scale the matrix, i.e. replace A by
                 D*A*D**(-1), where D is a diagonal matrix chosen
                 to make the rows and columns of A more equal in
                 norm. Do not permute;
          = 'B': Both diagonally scale and permute A.
          Computed reciprocal condition numbers will be for the matrix
          after balancing and/or permuting. Permuting does not change
          condition numbers (in exact arithmetic), but balancing does.

JOBVL

          JOBVL is CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVL must = 'V'.

JOBVR

          JOBVR is CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVR must = 'V'.

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for eigenvalues only;
          = 'V': Computed for right eigenvectors only;
          = 'B': Computed for eigenvalues and right eigenvectors.
          If SENSE = 'E' or 'B', both left and right eigenvectors
          must also be computed (JOBVL = 'V' and JOBVR = 'V').

N

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

A

          A is REAL array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten.  If JOBVL = 'V' or
          JOBVR = 'V', A contains the real Schur form of the balanced
          version of the input matrix A.

LDA

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

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues.  Complex
          conjugate pairs of eigenvalues will appear consecutively
          with the eigenvalue having the positive imaginary part
          first.

VL

          VL is REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order
          as their eigenvalues.
          If JOBVL = 'N', VL is not referenced.
          If the j-th eigenvalue is real, then u(j) = VL(:,j),
          the j-th column of VL.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
          u(j+1) = VL(:,j) - i*VL(:,j+1).

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL.  LDVL >= 1; if
          JOBVL = 'V', LDVL >= N.

VR

          VR is REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order
          as their eigenvalues.
          If JOBVR = 'N', VR is not referenced.
          If the j-th eigenvalue is real, then v(j) = VR(:,j),
          the j-th column of VR.
          If the j-th and (j+1)-st eigenvalues form a complex
          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
          v(j+1) = VR(:,j) - i*VR(:,j+1).

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          JOBVR = 'V', LDVR >= N.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI are integer values determined when A was
          balanced.  The balanced A(i,j) = 0 if I > J and
          J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE

          SCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          when balancing A.  If P(j) is the index of the row and column
          interchanged with row and column j, and D(j) is the scaling
          factor applied to row and column j, then
          SCALE(J) = P(J),    for J = 1,...,ILO-1
                   = D(J),    for J = ILO,...,IHI
                   = P(J)     for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

ABNRM

          ABNRM is REAL
          The one-norm of the balanced matrix (the maximum
          of the sum of absolute values of elements of any column).

RCONDE

          RCONDE is REAL array, dimension (N)
          RCONDE(j) is the reciprocal condition number of the j-th
          eigenvalue.

RCONDV

          RCONDV is REAL array, dimension (N)
          RCONDV(j) is the reciprocal condition number of the j-th
          right eigenvector.

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 dimension of the array WORK.   If SENSE = 'N' or 'E',
          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
          For good performance, LWORK must generally be larger.
          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.

IWORK

          IWORK is INTEGER array, dimension (2*N-2)
          If SENSE = 'N' or 'E', not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the QR algorithm failed to compute all the
                eigenvalues, and no eigenvectors or condition numbers
                have been computed; elements 1:ILO-1 and i+1:N of WR
                and WI contain eigenvalues which have converged.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine sgegs (character JOBVSL, character JOBVSR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEGS computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 This routine is deprecated and has been replaced by routine SGGES.
 SGEGS computes the eigenvalues, real Schur form, and, optionally,
 left and or/right Schur vectors of a real matrix pair (A,B).
 Given two square matrices A and B, the generalized real Schur
 factorization has the form
 
   A = Q*S*Z**T,  B = Q*T*Z**T
 where Q and Z are orthogonal matrices, T is upper triangular, and S
 is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
 blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
 of eigenvalues of (A,B).  The columns of Q are the left Schur vectors
 and the columns of Z are the right Schur vectors.
 
 If only the eigenvalues of (A,B) are needed, the driver routine
 SGEGV should be used instead.  See SGEGV for a description of the
 eigenvalues of the generalized nonsymmetric eigenvalue problem
 (GNEP).

 

Parameters:

JOBVSL

          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors (returned in VSL).

JOBVSR

          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors (returned in VSR).

N

          N is INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the matrix A.
          On exit, the upper quasi-triangular matrix S from the
          generalized real Schur factorization.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the matrix B.
          On exit, the upper triangular matrix T from the generalized
          real Schur factorization.

LDB

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

ALPHAR

          ALPHAR is REAL array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of GNEP.

ALPHAI

          ALPHAI is REAL array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
          eigenvalue is real; if positive, then the j-th and (j+1)-st
          eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) = -ALPHAI(j).

BETA

          BETA is REAL array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

VSL

          VSL is REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', the matrix of left Schur vectors Q.
          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', the matrix of right Schur vectors Z.
          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

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 dimension of the array WORK.  LWORK >= max(1,4*N).
          For good performance, LWORK must generally be larger.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for SGEQRF, SORMQR, and SORGQR.)  Then compute:
          NB  -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR
          The optimal LWORK is  2*N + N*(NB+1).
          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.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  errors that usually indicate LAPACK problems:
                =N+1: error return from SGGBAL
                =N+2: error return from SGEQRF
                =N+3: error return from SORMQR
                =N+4: error return from SORGQR
                =N+5: error return from SGGHRD
                =N+6: error return from SHGEQZ (other than failed
                                                iteration)
                =N+7: error return from SGGBAK (computing VSL)
                =N+8: error return from SGGBAK (computing VSR)
                =N+9: error return from SLASCL (various places)

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine sgegv (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 This routine is deprecated and has been replaced by routine SGGEV.
 SGEGV computes the eigenvalues and, optionally, the left and/or right
 eigenvectors of a real matrix pair (A,B).
 Given two square matrices A and B,
 the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
 eigenvalues lambda and corresponding (non-zero) eigenvectors x such
 that
    A*x = lambda*B*x.
 An alternate form is to find the eigenvalues mu and corresponding
 eigenvectors y such that
    mu*A*y = B*y.
 These two forms are equivalent with mu = 1/lambda and x = y if
 neither lambda nor mu is zero.  In order to deal with the case that
 lambda or mu is zero or small, two values alpha and beta are returned
 for each eigenvalue, such that lambda = alpha/beta and
 mu = beta/alpha.
 The vectors x and y in the above equations are right eigenvectors of
 the matrix pair (A,B).  Vectors u and v satisfying
    u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B
 are left eigenvectors of (A,B).
 Note: this routine performs "full balancing" on A and B

 

Parameters:

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors (returned
                  in VL).

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors (returned
                  in VR).

N

          N is INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the matrix A.
          If JOBVL = 'V' or JOBVR = 'V', then on exit A
          contains the real Schur form of A from the generalized Schur
          factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only the diagonal
          blocks from the Schur form will be correct.  See SGGHRD and
          SHGEQZ for details.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the matrix B.
          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
          upper triangular matrix obtained from B in the generalized
          Schur factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only those elements of
          B corresponding to the diagonal blocks from the Schur form of
          A will be correct.  See SGGHRD and SHGEQZ for details.

LDB

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

ALPHAR

          ALPHAR is REAL array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue of
          GNEP.

ALPHAI

          ALPHAI is REAL array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
          eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) = -ALPHAI(j).

BETA

          BETA is REAL array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

VL

          VL is REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored
          in the columns of VL, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then u(j) = VL(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
             u(j) = VL(:,j) + i*VL(:,j+1)
          and
            u(j+1) = VL(:,j) - i*VL(:,j+1).
          Each eigenvector is scaled so that its largest component has
          abs(real part) + abs(imag. part) = 1, except for eigenvectors
          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

VR

          VR is REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors x(j) are stored
          in the columns of VR, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then x(j) = VR(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
            x(j) = VR(:,j) + i*VR(:,j+1)
          and
            x(j+1) = VR(:,j) - i*VR(:,j+1).
          Each eigenvector is scaled so that its largest component has
          abs(real part) + abs(imag. part) = 1, except for eigenvalues
          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

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 dimension of the array WORK.  LWORK >= max(1,8*N).
          For good performance, LWORK must generally be larger.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for SGEQRF, SORMQR, and SORGQR.)  Then compute:
          NB  -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;
          The optimal LWORK is:
              2*N + MAX( 6*N, N*(NB+1) ).
          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.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  errors that usually indicate LAPACK problems:
                =N+1: error return from SGGBAL
                =N+2: error return from SGEQRF
                =N+3: error return from SORMQR
                =N+4: error return from SORGQR
                =N+5: error return from SGGHRD
                =N+6: error return from SHGEQZ (other than failed
                                                iteration)
                =N+7: error return from STGEVC
                =N+8: error return from SGGBAK (computing VL)
                =N+9: error return from SGGBAK (computing VR)
                =N+10: error return from SLASCL (various calls)

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

  Balancing
  ---------
  This driver calls SGGBAL to both permute and scale rows and columns
  of A and B.  The permutations PL and PR are chosen so that PL*A*PR
  and PL*B*R will be upper triangular except for the diagonal blocks
  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
  possible.  The diagonal scaling matrices DL and DR are chosen so
  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
  one (except for the elements that start out zero.)
  After the eigenvalues and eigenvectors of the balanced matrices
  have been computed, SGGBAK transforms the eigenvectors back to what
  they would have been (in perfect arithmetic) if they had not been
  balanced.
  Contents of A and B on Exit
  -------- -- - --- - -- ----
  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
  both), then on exit the arrays A and B will contain the real Schur
  form[*] of the "balanced" versions of A and B.  If no eigenvectors
  are computed, then only the diagonal blocks will be correct.
  [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
      by Golub & van Loan, pub. by Johns Hopkins U. Press.

 

subroutine sgges (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO)

SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

 SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
 the generalized eigenvalues, the generalized real Schur form (S,T),
 optionally, the left and/or right matrices of Schur vectors (VSL and
 VSR). This gives the generalized Schur factorization
          (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
 Optionally, it also orders the eigenvalues so that a selected cluster
 of eigenvalues appears in the leading diagonal blocks of the upper
 quasi-triangular matrix S and the upper triangular matrix T.The
 leading columns of VSL and VSR then form an orthonormal basis for the
 corresponding left and right eigenspaces (deflating subspaces).
 (If only the generalized eigenvalues are needed, use the driver
 SGGEV instead, which is faster.)
 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
 usually represented as the pair (alpha,beta), as there is a
 reasonable interpretation for beta=0 or both being zero.
 A pair of matrices (S,T) is in generalized real Schur form if T is
 upper triangular with non-negative diagonal and S is block upper
 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
 to real generalized eigenvalues, while 2-by-2 blocks of S will be
 "standardized" by making the corresponding elements of T have the
 form:
         [  a  0  ]
         [  0  b  ]
 and the pair of corresponding 2-by-2 blocks in S and T will have a
 complex conjugate pair of generalized eigenvalues.

 

Parameters:

JOBVSL

          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);

SELCTG

          SELCTG is a LOGICAL FUNCTION of three REAL arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.
          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.

N

          N is INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

LDB

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

SDIM

          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

ALPHAR

          ALPHAR is REAL array, dimension (N)

ALPHAI

          ALPHAI is REAL array, dimension (N)

BETA

          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.
          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).

VSL

          VSL is REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

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 dimension of the array WORK.
          If N = 0, LWORK >= 1, else LWORK >= max(8*N,6*N+16).
          For good performance , LWORK must generally be larger.
          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.

BWORK

          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in STGSEN.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine sgges3 (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO)

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:

 SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
 the generalized eigenvalues, the generalized real Schur form (S,T),
 optionally, the left and/or right matrices of Schur vectors (VSL and
 VSR). This gives the generalized Schur factorization
          (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
 Optionally, it also orders the eigenvalues so that a selected cluster
 of eigenvalues appears in the leading diagonal blocks of the upper
 quasi-triangular matrix S and the upper triangular matrix T.The
 leading columns of VSL and VSR then form an orthonormal basis for the
 corresponding left and right eigenspaces (deflating subspaces).
 (If only the generalized eigenvalues are needed, use the driver
 SGGEV instead, which is faster.)
 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
 usually represented as the pair (alpha,beta), as there is a
 reasonable interpretation for beta=0 or both being zero.
 A pair of matrices (S,T) is in generalized real Schur form if T is
 upper triangular with non-negative diagonal and S is block upper
 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
 to real generalized eigenvalues, while 2-by-2 blocks of S will be
 "standardized" by making the corresponding elements of T have the
 form:
         [  a  0  ]
         [  0  b  ]
 and the pair of corresponding 2-by-2 blocks in S and T will have a
 complex conjugate pair of generalized eigenvalues.

 

Parameters:

JOBVSL

          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);

SELCTG

          SELCTG is a LOGICAL FUNCTION of three REAL arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.
          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.

N

          N is INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

LDB

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

SDIM

          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

ALPHAR

          ALPHAR is REAL array, dimension (N)

ALPHAI

          ALPHAI is REAL array, dimension (N)

BETA

          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.
          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).

VSL

          VSL is REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

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 dimension of the array WORK.
          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.

BWORK

          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in STGSEN.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

subroutine sggesx (character JOBVSL, character JOBVSR, character SORT, external SELCTG, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( 2 ) RCONDE, real, dimension( 2 ) RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)

SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

 SGGESX computes for a pair of N-by-N real nonsymmetric matrices
 (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
 optionally, the left and/or right matrices of Schur vectors (VSL and
 VSR).  This gives the generalized Schur factorization
      (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
 Optionally, it also orders the eigenvalues so that a selected cluster
 of eigenvalues appears in the leading diagonal blocks of the upper
 quasi-triangular matrix S and the upper triangular matrix T; computes
 a reciprocal condition number for the average of the selected
 eigenvalues (RCONDE); and computes a reciprocal condition number for
 the right and left deflating subspaces corresponding to the selected
 eigenvalues (RCONDV). The leading columns of VSL and VSR then form
 an orthonormal basis for the corresponding left and right eigenspaces
 (deflating subspaces).
 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
 usually represented as the pair (alpha,beta), as there is a
 reasonable interpretation for beta=0 or for both being zero.
 A pair of matrices (S,T) is in generalized real Schur form if T is
 upper triangular with non-negative diagonal and S is block upper
 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
 to real generalized eigenvalues, while 2-by-2 blocks of S will be
 "standardized" by making the corresponding elements of T have the
 form:
         [  a  0  ]
         [  0  b  ]
 and the pair of corresponding 2-by-2 blocks in S and T will have a
 complex conjugate pair of generalized eigenvalues.

 

Parameters:

JOBVSL

          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG).

SELCTG

          SELCTG is procedure) LOGICAL FUNCTION of three REAL arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.
          Note that a selected complex eigenvalue may no longer satisfy
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
          since ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned), in this
          case INFO is set to N+3.

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N' : None are computed;
          = 'E' : Computed for average of selected eigenvalues only;
          = 'V' : Computed for selected deflating subspaces only;
          = 'B' : Computed for both.
          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

N

          N is INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

LDB

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

SDIM

          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

ALPHAR

          ALPHAR is REAL array, dimension (N)

ALPHAI

          ALPHAI is REAL array, dimension (N)

BETA

          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.
          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).

VSL

          VSL is REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.

RCONDE

          RCONDE is REAL array, dimension ( 2 )
          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
          reciprocal condition numbers for the average of the selected
          eigenvalues.
          Not referenced if SENSE = 'N' or 'V'.

RCONDV

          RCONDV is REAL array, dimension ( 2 )
          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
          reciprocal condition numbers for the selected deflating
          subspaces.
          Not referenced if SENSE = 'N' or 'E'.

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 dimension of the array WORK.
          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
          LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
          LWORK >= max( 8*N, 6*N+16 ).
          Note that 2*SDIM*(N-SDIM) <= N*N/2.
          Note also that an error is only returned if
          LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
          this may not be large enough.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the bound on the optimal size of the WORK
          array and the minimum size of the IWORK array, returns these
          values as the first entries of the WORK and IWORK arrays, and
          no error message related to LWORK or LIWORK is issued by
          XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
          LIWORK >= N+6.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the bound on the optimal size of the
          WORK array and the minimum size of the IWORK array, returns
          these values as the first entries of the WORK and IWORK
          arrays, and no error message related to LWORK or LIWORK is
          issued by XERBLA.

BWORK

          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in STGSEN.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is
       EPS * norm((A, B)) / RCONDE( 1 ).
  An approximate (asymptotic) bound on the maximum angular error in
  the computed deflating subspaces is
       EPS * norm((A, B)) / RCONDV( 2 ).
  See LAPACK User's Guide, section 4.11 for more information.

 

subroutine sggev (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 SGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
 the generalized eigenvalues, and optionally, the left and/or right
 generalized eigenvectors.
 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
 singular. It is usually represented as the pair (alpha,beta), as
 there is a reasonable interpretation for beta=0, and even for both
 being zero.
 The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j).
 The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  u(j)**H * A  = lambda(j) * u(j)**H * B .
 where u(j)**H is the conjugate-transpose of u(j).

 

Parameters:

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

N

          N is INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten.

LDB

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

ALPHAR

          ALPHAR is REAL array, dimension (N)

ALPHAI

          ALPHAI is REAL array, dimension (N)

BETA

          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
          the j-th eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.
          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio
          alpha/beta.  However, ALPHAR and ALPHAI will be always less
          than and usually comparable with norm(A) in magnitude, and
          BETA always less than and usually comparable with norm(B).

VL

          VL is REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          u(j) = VL(:,j), the j-th column of VL. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

VR

          VR is REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          v(j) = VR(:,j), the j-th column of VR. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

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 dimension of the array WORK.  LWORK >= max(1,8*N).
          For good performance, LWORK must generally be larger.
          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.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: error return from STGEVC.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

subroutine sggev3 (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

 SGGEV3 computes for a pair of N-by-N real nonsymmetric matrices (A,B)
 the generalized eigenvalues, and optionally, the left and/or right
 generalized eigenvectors.
 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
 singular. It is usually represented as the pair (alpha,beta), as
 there is a reasonable interpretation for beta=0, and even for both
 being zero.
 The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j).
 The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  u(j)**H * A  = lambda(j) * u(j)**H * B .
 where u(j)**H is the conjugate-transpose of u(j).

 

Parameters:

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

N

          N is INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten.

LDB

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

ALPHAR

          ALPHAR is REAL array, dimension (N)

ALPHAI

          ALPHAI is REAL array, dimension (N)

BETA

          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
          the j-th eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.
          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio
          alpha/beta.  However, ALPHAR and ALPHAI will be always less
          than and usually comparable with norm(A) in magnitude, and
          BETA always less than and usually comparable with norm(B).

VL

          VL is REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          u(j) = VL(:,j), the j-th column of VL. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

VR

          VR is REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          v(j) = VR(:,j), the j-th column of VR. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
          Each eigenvector is scaled so the largest component has
          abs(real part)+abs(imag. part)=1.
          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

WORK

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

LWORK

          LWORK is INTEGER
          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.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: error return from STGEVC.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

subroutine sggevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, integer ILO, integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE, real ABNRM, real BBNRM, real, dimension( * ) RCONDE, real, dimension( * ) RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO)

SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
 the generalized eigenvalues, and optionally, the left and/or right
 generalized eigenvectors.
 Optionally also, it computes a balancing transformation to improve
 the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
 LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
 the eigenvalues (RCONDE), and reciprocal condition numbers for the
 right eigenvectors (RCONDV).
 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
 singular. It is usually represented as the pair (alpha,beta), as
 there is a reasonable interpretation for beta=0, and even for both
 being zero.
 The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j) .
 The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  u(j)**H * A  = lambda(j) * u(j)**H * B.
 where u(j)**H is the conjugate-transpose of u(j).

 

Parameters:

BALANC

          BALANC is CHARACTER*1
          Specifies the balance option to be performed.
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

N

          N is INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then A contains the first part of the real Schur
          form of the "balanced" versions of the input A and B.

LDA

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then B contains the second part of the real Schur
          form of the "balanced" versions of the input A and B.

LDB

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

ALPHAR

          ALPHAR is REAL array, dimension (N)

ALPHAI

          ALPHAI is REAL array, dimension (N)

BETA

          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
          the j-th eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.
          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio
          ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
          than and usually comparable with norm(A) in magnitude, and
          BETA always less than and usually comparable with norm(B).

VL

          VL is REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          u(j) = VL(:,j), the j-th column of VL. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
          Each eigenvector will be scaled so the largest component have
          abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

VR

          VR is REAL array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order as
          their eigenvalues. If the j-th eigenvalue is real, then
          v(j) = VR(:,j), the j-th column of VR. If the j-th and
          (j+1)-th eigenvalues form a complex conjugate pair, then
          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
          Each eigenvector will be scaled so the largest component have
          abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI are integer values such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

LSCALE

          LSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If PL(j) is the index of the
          row interchanged with row j, and DL(j) is the scaling
          factor applied to row j, then
            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                      = DL(j)  for j = ILO,...,IHI
                      = PL(j)  for j = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

RSCALE

          RSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If PR(j) is the index of the
          column interchanged with column j, and DR(j) is the scaling
          factor applied to column j, then
            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                      = DR(j)  for j = ILO,...,IHI
                      = PR(j)  for j = IHI+1,...,N
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

ABNRM

          ABNRM is REAL
          The one-norm of the balanced matrix A.

BBNRM

          BBNRM is REAL
          The one-norm of the balanced matrix B.

RCONDE

          RCONDE is REAL array, dimension (N)
          If SENSE = 'E' or 'B', the reciprocal condition numbers of
          the eigenvalues, stored in consecutive elements of the array.
          For a complex conjugate pair of eigenvalues two consecutive
          elements of RCONDE are set to the same value. Thus RCONDE(j),
          RCONDV(j), and the j-th columns of VL and VR all correspond
          to the j-th eigenpair.
          If SENSE = 'N' or 'V', RCONDE is not referenced.

RCONDV

          RCONDV is REAL array, dimension (N)
          If SENSE = 'V' or 'B', the estimated reciprocal condition
          numbers of the eigenvectors, stored in consecutive elements
          of the array. For a complex eigenvector two consecutive
          elements of RCONDV are set to the same value. If the
          eigenvalues cannot be reordered to compute RCONDV(j),
          RCONDV(j) is set to 0; this can only occur when the true
          value would be very small anyway.
          If SENSE = 'N' or 'E', RCONDV is not referenced.

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 dimension of the array WORK. LWORK >= max(1,2*N).
          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
          LWORK >= max(1,6*N).
          If SENSE = 'E', LWORK >= max(1,10*N).
          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
          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.

IWORK

          IWORK is INTEGER array, dimension (N+6)
          If SENSE = 'E', IWORK is not referenced.

BWORK

          BWORK is LOGICAL array, dimension (N)
          If SENSE = 'N', BWORK is not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                =N+2: error return from STGEVC.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

Further Details:

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity
  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.
  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is
       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by
       EPS * norm(ABNRM, BBNRM) / DIF(i).
  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.

 

Author

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