strsna.f.html

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      SUBROUTINE <a name="STRSNA.1"></a><a href="strsna.f.html#STRSNA.1">STRSNA</a>( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
     $                   LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK,
     $                   INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Modified to call <a name="SLACN2.9"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a> in place of <a name="SLACON.9"></a><a href="slacon.f.html#SLACON.1">SLACON</a>, 7 Feb 03, SJH.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      CHARACTER          HOWMNY, JOB
      INTEGER            INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      REAL               S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ),
     $                   VR( LDVR, * ), WORK( LDWORK, * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="STRSNA.25"></a><a href="strsna.f.html#STRSNA.1">STRSNA</a> estimates reciprocal condition numbers for specified
</span><span class="comment">*</span><span class="comment">  eigenvalues and/or right eigenvectors of a real upper
</span><span class="comment">*</span><span class="comment">  quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
</span><span class="comment">*</span><span class="comment">  orthogonal).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  T must be in Schur canonical form (as returned by <a name="SHSEQR.30"></a><a href="shseqr.f.html#SHSEQR.1">SHSEQR</a>), that is,
</span><span class="comment">*</span><span class="comment">  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
</span><span class="comment">*</span><span class="comment">  2-by-2 diagonal block has its diagonal elements equal and its
</span><span class="comment">*</span><span class="comment">  off-diagonal elements of opposite sign.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  JOB     (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies whether condition numbers are required for
</span><span class="comment">*</span><span class="comment">          eigenvalues (S) or eigenvectors (SEP):
</span><span class="comment">*</span><span class="comment">          = 'E': for eigenvalues only (S);
</span><span class="comment">*</span><span class="comment">          = 'V': for eigenvectors only (SEP);
</span><span class="comment">*</span><span class="comment">          = 'B': for both eigenvalues and eigenvectors (S and SEP).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  HOWMNY  (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'A': compute condition numbers for all eigenpairs;
</span><span class="comment">*</span><span class="comment">          = 'S': compute condition numbers for selected eigenpairs
</span><span class="comment">*</span><span class="comment">                 specified by the array SELECT.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SELECT  (input) LOGICAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
</span><span class="comment">*</span><span class="comment">          condition numbers are required. To select condition numbers
</span><span class="comment">*</span><span class="comment">          for the eigenpair corresponding to a real eigenvalue w(j),
</span><span class="comment">*</span><span class="comment">          SELECT(j) must be set to .TRUE.. To select condition numbers
</span><span class="comment">*</span><span class="comment">          corresponding to a complex conjugate pair of eigenvalues w(j)
</span><span class="comment">*</span><span class="comment">          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
</span><span class="comment">*</span><span class="comment">          set to .TRUE..
</span><span class="comment">*</span><span class="comment">          If HOWMNY = 'A', SELECT is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix T. N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  T       (input) REAL array, dimension (LDT,N)
</span><span class="comment">*</span><span class="comment">          The upper quasi-triangular matrix T, in Schur canonical form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDT     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array T. LDT &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  VL      (input) REAL array, dimension (LDVL,M)
</span><span class="comment">*</span><span class="comment">          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
</span><span class="comment">*</span><span class="comment">          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
</span><span class="comment">*</span><span class="comment">          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
</span><span class="comment">*</span><span class="comment">          must be stored in consecutive columns of VL, as returned by
</span><span class="comment">*</span><span class="comment">          <a name="SHSEIN.74"></a><a href="shsein.f.html#SHSEIN.1">SHSEIN</a> or <a name="STREVC.74"></a><a href="strevc.f.html#STREVC.1">STREVC</a>.
</span><span class="comment">*</span><span class="comment">          If JOB = 'V', VL is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDVL    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array VL.
</span><span class="comment">*</span><span class="comment">          LDVL &gt;= 1; and if JOB = 'E' or 'B', LDVL &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  VR      (input) REAL array, dimension (LDVR,M)
</span><span class="comment">*</span><span class="comment">          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
</span><span class="comment">*</span><span class="comment">          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
</span><span class="comment">*</span><span class="comment">          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
</span><span class="comment">*</span><span class="comment">          must be stored in consecutive columns of VR, as returned by
</span><span class="comment">*</span><span class="comment">          <a name="SHSEIN.86"></a><a href="shsein.f.html#SHSEIN.1">SHSEIN</a> or <a name="STREVC.86"></a><a href="strevc.f.html#STREVC.1">STREVC</a>.
</span><span class="comment">*</span><span class="comment">          If JOB = 'V', VR is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDVR    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array VR.
</span><span class="comment">*</span><span class="comment">          LDVR &gt;= 1; and if JOB = 'E' or 'B', LDVR &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  S       (output) REAL array, dimension (MM)
</span><span class="comment">*</span><span class="comment">          If JOB = 'E' or 'B', the reciprocal condition numbers of the
</span><span class="comment">*</span><span class="comment">          selected eigenvalues, stored in consecutive elements of the
</span><span class="comment">*</span><span class="comment">          array. For a complex conjugate pair of eigenvalues two
</span><span class="comment">*</span><span class="comment">          consecutive elements of S are set to the same value. Thus
</span><span class="comment">*</span><span class="comment">          S(j), SEP(j), and the j-th columns of VL and VR all
</span><span class="comment">*</span><span class="comment">          correspond to the same eigenpair (but not in general the
</span><span class="comment">*</span><span class="comment">          j-th eigenpair, unless all eigenpairs are selected).
</span><span class="comment">*</span><span class="comment">          If JOB = 'V', S is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SEP     (output) REAL array, dimension (MM)
</span><span class="comment">*</span><span class="comment">          If JOB = 'V' or 'B', the estimated reciprocal condition
</span><span class="comment">*</span><span class="comment">          numbers of the selected eigenvectors, stored in consecutive
</span><span class="comment">*</span><span class="comment">          elements of the array. For a complex eigenvector two
</span><span class="comment">*</span><span class="comment">          consecutive elements of SEP are set to the same value. If
</span><span class="comment">*</span><span class="comment">          the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
</span><span class="comment">*</span><span class="comment">          is set to 0; this can only occur when the true value would be
</span><span class="comment">*</span><span class="comment">          very small anyway.
</span><span class="comment">*</span><span class="comment">          If JOB = 'E', SEP is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  MM      (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of elements in the arrays S (if JOB = 'E' or 'B')
</span><span class="comment">*</span><span class="comment">           and/or SEP (if JOB = 'V' or 'B'). MM &gt;= M.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (output) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of elements of the arrays S and/or SEP actually
</span><span class="comment">*</span><span class="comment">          used to store the estimated condition numbers.
</span><span class="comment">*</span><span class="comment">          If HOWMNY = 'A', M is set to N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace) REAL array, dimension (LDWORK,N+6)
</span><span class="comment">*</span><span class="comment">          If JOB = 'E', WORK is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDWORK  (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array WORK.
</span><span class="comment">*</span><span class="comment">          LDWORK &gt;= 1; and if JOB = 'V' or 'B', LDWORK &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  IWORK   (workspace) INTEGER array, dimension (2*(N-1))
</span><span class="comment">*</span><span class="comment">          If JOB = 'E', IWORK is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0: successful exit
</span><span class="comment">*</span><span class="comment">          &lt; 0: if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The reciprocal of the condition number of an eigenvalue lambda is
</span><span class="comment">*</span><span class="comment">  defined as
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          S(lambda) = |v'*u| / (norm(u)*norm(v))
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where u and v are the right and left eigenvectors of T corresponding
</span><span class="comment">*</span><span class="comment">  to lambda; v' denotes the conjugate-transpose of v, and norm(u)
</span><span class="comment">*</span><span class="comment">  denotes the Euclidean norm. These reciprocal condition numbers always
</span><span class="comment">*</span><span class="comment">  lie between zero (very badly conditioned) and one (very well
</span><span class="comment">*</span><span class="comment">  conditioned). If n = 1, S(lambda) is defined to be 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  An approximate error bound for a computed eigenvalue W(i) is given by
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                      EPS * norm(T) / S(i)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where EPS is the machine precision.
</span><span class="comment">*</span><span class="comment">
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