DTREVC3 computes some or all of the right and/or left eigenvectors of


 a real upper quasi-triangular matrix T.


 Matrices of this type are produced by the Schur factorization of


 a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.


 The right eigenvector x and the left eigenvector y of T corresponding


 to an eigenvalue w are defined by:


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


 where y**H denotes the conjugate transpose of y.


 The eigenvalues are not input to this routine, but are read directly


 from the diagonal blocks of T.


 This routine returns the matrices X and/or Y of right and left


 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an


 input matrix. If Q is the orthogonal factor that reduces a matrix


 A to Schur form T, then Q*X and Q*Y are the matrices of right and


 left eigenvectors of A.


 This uses a Level 3 BLAS version of the back transformation.

SIDE

          SIDE is CHARACTER*1


          = 'R':  compute right eigenvectors only;


          = 'L':  compute left eigenvectors only;


          = 'B':  compute both right and left eigenvectors.

HOWMNY

          HOWMNY is CHARACTER*1


          = 'A':  compute all right and/or left eigenvectors;


          = 'B':  compute all right and/or left eigenvectors,


                  backtransformed by the matrices in VR and/or VL;


          = 'S':  compute selected right and/or left eigenvectors,


                  as indicated by the logical array SELECT.

SELECT

          SELECT is LOGICAL array, dimension (N)


          If HOWMNY = 'S', SELECT specifies the eigenvectors to be


          computed.


          If w(j) is a real eigenvalue, the corresponding real


          eigenvector is computed if SELECT(j) is .TRUE..


          If w(j) and w(j+1) are the real and imaginary parts of a


          complex eigenvalue, the corresponding complex eigenvector is


          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and


          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to


          .FALSE..


          Not referenced if HOWMNY = 'A' or 'B'.

N

          N is INTEGER


          The order of the matrix T. N >= 0.

T

          T is DOUBLE PRECISION array, dimension (LDT,N)


          The upper quasi-triangular matrix T in Schur canonical form.

LDT

          LDT is INTEGER


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

VL

          VL is DOUBLE PRECISION array, dimension (LDVL,MM)


          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must


          contain an N-by-N matrix Q (usually the orthogonal matrix Q


          of Schur vectors returned by DHSEQR).


          On exit, if SIDE = 'L' or 'B', VL contains:


          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;


          if HOWMNY = 'B', the matrix Q*Y;


          if HOWMNY = 'S', the left eigenvectors of T specified by


                           SELECT, stored consecutively in the columns


                           of VL, in the same order as their


                           eigenvalues.


          A complex eigenvector corresponding to a complex eigenvalue


          is stored in two consecutive columns, the first holding the


          real part, and the second the imaginary part.


          Not referenced if SIDE = 'R'.

LDVL

          LDVL is INTEGER


          The leading dimension of the array VL.


          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.

VR

          VR is DOUBLE PRECISION array, dimension (LDVR,MM)


          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must


          contain an N-by-N matrix Q (usually the orthogonal matrix Q


          of Schur vectors returned by DHSEQR).


          On exit, if SIDE = 'R' or 'B', VR contains:


          if HOWMNY = 'A', the matrix X of right eigenvectors of T;


          if HOWMNY = 'B', the matrix Q*X;


          if HOWMNY = 'S', the right eigenvectors of T specified by


                           SELECT, stored consecutively in the columns


                           of VR, in the same order as their


                           eigenvalues.


          A complex eigenvector corresponding to a complex eigenvalue


          is stored in two consecutive columns, the first holding the


          real part and the second the imaginary part.


          Not referenced if SIDE = 'L'.

LDVR

          LDVR is INTEGER


          The leading dimension of the array VR.


          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.

MM

          MM is INTEGER


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

M

          M is INTEGER


          The number of columns in the arrays VL and/or VR actually


          used to store the eigenvectors.


          If HOWMNY = 'A' or 'B', M is set to N.


          Each selected real eigenvector occupies one column and each


          selected complex eigenvector occupies two columns.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))

LWORK

          LWORK is INTEGER


          The dimension of array WORK. LWORK >= max(1,3*N).


          For optimum performance, LWORK >= N + 2*N*NB, where NB is


          the optimal blocksize.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2017

  The algorithm used in this program is basically backward (forward)


  substitution, with scaling to make the the code robust against


  possible overflow.


  Each eigenvector is normalized so that the element of largest


  magnitude has magnitude 1; here the magnitude of a complex number


  (x,y) is taken to be |x| + |y|.