ztgsna.f

          SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
         $                   LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
         $                   IWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          HOWMNY, JOB
          INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
    *     ..
    *     .. Array Arguments ..
          LOGICAL            SELECT( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   DIF( * ), S( * )
          COMPLEX*16         A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
         $                   VR( LDVR, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZTGSNA estimates reciprocal condition numbers for specified
    *  eigenvalues and/or eigenvectors of a matrix pair (A, B).
    *
    *  (A, B) must be in generalized Schur canonical form, that is, A and
    *  B are both upper triangular.
    *
    *  Arguments
    *  =========
    *
    *  JOB     (input) CHARACTER*1
    *          Specifies whether condition numbers are required for
    *          eigenvalues (S) or eigenvectors (DIF):
    *          = 'E': for eigenvalues only (S);
    *          = 'V': for eigenvectors only (DIF);
    *          = 'B': for both eigenvalues and eigenvectors (S and DIF).
    *
    *  HOWMNY  (input) CHARACTER*1
    *          = 'A': compute condition numbers for all eigenpairs;
    *          = 'S': compute condition numbers for selected eigenpairs
    *                 specified by the array SELECT.
    *
    *  SELECT  (input) LOGICAL array, dimension (N)
    *          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
    *          condition numbers are required. To select condition numbers
    *          for the corresponding j-th eigenvalue and/or eigenvector,
    *          SELECT(j) must be set to .TRUE..
    *          If HOWMNY = 'A', SELECT is not referenced.
    *
    *  N       (input) INTEGER
    *          The order of the square matrix pair (A, B). N >= 0.
    *
    *  A       (input) COMPLEX*16 array, dimension (LDA,N)
    *          The upper triangular matrix A in the pair (A,B).
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,N).
    *
    *  B       (input) COMPLEX*16 array, dimension (LDB,N)
    *          The upper triangular matrix B in the pair (A, B).
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B. LDB >= max(1,N).
    *
    *  VL      (input) COMPLEX*16 array, dimension (LDVL,M)
    *          IF JOB = 'E' or 'B', VL must contain left eigenvectors of
    *          (A, B), corresponding to the eigenpairs specified by HOWMNY
    *          and SELECT.  The eigenvectors must be stored in consecutive
    *          columns of VL, as returned by ZTGEVC.
    *          If JOB = 'V', VL is not referenced.
    *
    *  LDVL    (input) INTEGER
    *          The leading dimension of the array VL. LDVL >= 1; and
    *          If JOB = 'E' or 'B', LDVL >= N.
    *
    *  VR      (input) COMPLEX*16 array, dimension (LDVR,M)
    *          IF JOB = 'E' or 'B', VR must contain right eigenvectors of
    *          (A, B), corresponding to the eigenpairs specified by HOWMNY
    *          and SELECT.  The eigenvectors must be stored in consecutive
    *          columns of VR, as returned by ZTGEVC.
    *          If JOB = 'V', VR is not referenced.
    *
    *  LDVR    (input) INTEGER
    *          The leading dimension of the array VR. LDVR >= 1;
    *          If JOB = 'E' or 'B', LDVR >= N.
    *
    *  S       (output) DOUBLE PRECISION array, dimension (MM)
    *          If JOB = 'E' or 'B', the reciprocal condition numbers of the
    *          selected eigenvalues, stored in consecutive elements of the
    *          array.
    *          If JOB = 'V', S is not referenced.
    *
    *  DIF     (output) DOUBLE PRECISION array, dimension (MM)
    *          If JOB = 'V' or 'B', the estimated reciprocal condition
    *          numbers of the selected eigenvectors, stored in consecutive
    *          elements of the array.
    *          If the eigenvalues cannot be reordered to compute DIF(j),
    *          DIF(j) is set to 0; this can only occur when the true value
    *          would be very small anyway.
    *          For each eigenvalue/vector specified by SELECT, DIF stores
    *          a Frobenius norm-based estimate of Difl.
    *          If JOB = 'E', DIF is not referenced.
    *
    *  MM      (input) INTEGER
    *          The number of elements in the arrays S and DIF. MM >= M.
    *
    *  M       (output) INTEGER
    *          The number of elements of the arrays S and DIF used to store
    *          the specified condition numbers; for each selected eigenvalue
    *          one element is used. If HOWMNY = 'A', M is set to N.
    *
    *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK  (input) INTEGER
    *          The dimension of the array WORK. LWORK >= max(1,N).
    *          If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
    *
    *  IWORK   (workspace) INTEGER array, dimension (N+2)
    *          If JOB = 'E', IWORK is not referenced.
    *
    *  INFO    (output) INTEGER
    *          = 0: Successful exit
    *          < 0: If INFO = -i, the i-th argument had an illegal value
    *
    *  Further Details
    *  ===============
    *
    *  The reciprocal of the condition number of the i-th generalized
    *  eigenvalue w = (a, b) is defined as
    *
    *          S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))
    *
    *  where u and v are the right and left eigenvectors of (A, B)
    *  corresponding to w; |z| denotes the absolute value of the complex
    *  number, and norm(u) denotes the 2-norm of the vector u. The pair
    *  (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the
    *  matrix pair (A, B). If both a and b equal zero, then (A,B) is
    *  singular and S(I) = -1 is returned.
    *
    *  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(A, B) / S(I),
    *
    *  where EPS is the machine precision.
    *
    *  The reciprocal of the condition number of the right eigenvector u
    *  and left eigenvector v corresponding to the generalized eigenvalue w
    *  is defined as follows. Suppose
    *
    *                   (A, B) = ( a   *  ) ( b  *  )  1
    *                            ( 0  A22 ),( 0 B22 )  n-1
    *                              1  n-1     1 n-1
    *
    *  Then the reciprocal condition number DIF(I) is
    *
    *          Difl[(a, b), (A22, B22)]  = sigma-min( Zl )
    *
    *  where sigma-min(Zl) denotes the smallest singular value of
    *
    *         Zl = [ kron(a, In-1) -kron(1, A22) ]
    *              [ kron(b, In-1) -kron(1, B22) ].
    *
    *  Here In-1 is the identity matrix of size n-1 and X' is the conjugate
    *  transpose of X. kron(X, Y) is the Kronecker product between the
    *  matrices X and Y.
    *
    *  We approximate the smallest singular value of Zl with an upper
    *  bound. This is done by ZLATDF.
    *
    *  An approximate error bound for a computed eigenvector VL(i) or
    *  VR(i) is given by
    *
    *                      EPS * norm(A, B) / DIF(i).
    *
    *  See ref. [2-3] for more details and further references.
    *
    *  Based on contributions by
    *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
    *     Umea University, S-901 87 Umea, Sweden.
    *
    *  References
    *  ==========
    *
    *  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
    *      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
    *      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
    *      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
    *
    *  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
    *      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
    *      Estimation: Theory, Algorithms and Software, Report
    *      UMINF - 94.04, Department of Computing Science, Umea University,
    *      S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
    *      To appear in Numerical Algorithms, 1996.
    *
    *  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
    *      for Solving the Generalized Sylvester Equation and Estimating the
    *      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
    *      Department of Computing Science, Umea University, S-901 87 Umea,
    *      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
    *      Note 75.
    *      To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          INTEGER            IDIFJB
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, IDIFJB = 3 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY, SOMCON, WANTBH, WANTDF, WANTS
          INTEGER            I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
          DOUBLE PRECISION   BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
          COMPLEX*16         YHAX, YHBX
    *     ..
    *     .. Local Arrays ..
          COMPLEX*16         DUMMY( 1 ), DUMMY1( 1 )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH, DLAPY2, DZNRM2
          COMPLEX*16         ZDOTC
          EXTERNAL           LSAME, DLAMCH, DLAPY2, DZNRM2, ZDOTC
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLABAD, XERBLA, ZGEMV, ZLACPY, ZTGEXC, ZTGSYL
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DCMPLX, MAX
    *     ..
    *     .. Executable Statements ..
    *
    *     Decode and test the input parameters
    *
          WANTBH = LSAME( JOB, 'B' )
          WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
          WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
    *
          SOMCON = LSAME( HOWMNY, 'S' )
    *
          INFO = 0
          LQUERY = ( LWORK.EQ.-1 )
    *
          IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
             INFO = -1
          ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -4
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -6
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -8
          ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
             INFO = -10
          ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
             INFO = -12
          ELSE
    *
    *        Set M to the number of eigenpairs for which condition numbers
    *        are required, and test MM.
    *
             IF( SOMCON ) THEN
                M = 0
                DO 10 K = 1, N
                   IF( SELECT( K ) )
         $            M = M + 1
       10       CONTINUE
             ELSE
                M = N
             END IF
    *
             IF( N.EQ.0 ) THEN
                LWMIN = 1
             ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
                LWMIN = 2*N*N
             ELSE
                LWMIN = N
             END IF
             WORK( 1 ) = LWMIN
    *
             IF( MM.LT.M ) THEN
                INFO = -15
             ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
                INFO = -18
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZTGSNA', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Get machine constants
    *
          EPS = DLAMCH( 'P' )
          SMLNUM = DLAMCH( 'S' ) / EPS
          BIGNUM = ONE / SMLNUM
          CALL DLABAD( SMLNUM, BIGNUM )
          KS = 0
          DO 20 K = 1, N
    *
    *        Determine whether condition numbers are required for the k-th
    *        eigenpair.
    *
             IF( SOMCON ) THEN
                IF( .NOT.SELECT( K ) )
         $         GO TO 20
             END IF
    *
             KS = KS + 1
    *
             IF( WANTS ) THEN
    *
    *           Compute the reciprocal condition number of the k-th
    *           eigenvalue.
    *
                RNRM = DZNRM2( N, VR( 1, KS ), 1 )
                LNRM = DZNRM2( N, VL( 1, KS ), 1 )
                CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), A, LDA,
         $                  VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
                YHAX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
                CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), B, LDB,
         $                  VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
                YHBX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
                COND = DLAPY2( ABS( YHAX ), ABS( YHBX ) )
                IF( COND.EQ.ZERO ) THEN
                   S( KS ) = -ONE
                ELSE
                   S( KS ) = COND / ( RNRM*LNRM )
                END IF
             END IF
    *
             IF( WANTDF ) THEN
                IF( N.EQ.1 ) THEN
                   DIF( KS ) = DLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) )
                ELSE
    *
    *              Estimate the reciprocal condition number of the k-th
    *              eigenvectors.
    *
    *              Copy the matrix (A, B) to the array WORK and move the
    *              (k,k)th pair to the (1,1) position.
    *
                   CALL ZLACPY( 'Full', N, N, A, LDA, WORK, N )
                   CALL ZLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
                   IFST = K
                   ILST = 1
    *
                   CALL ZTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ),
         $                      N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR )
    *
                   IF( IERR.GT.0 ) THEN
    *
    *                 Ill-conditioned problem - swap rejected.
    *
                      DIF( KS ) = ZERO
                   ELSE
    *
    *                 Reordering successful, solve generalized Sylvester
    *                 equation for R and L,
    *                            A22 * R - L * A11 = A12
    *                            B22 * R - L * B11 = B12,
    *                 and compute estimate of Difl[(A11,B11), (A22, B22)].
    *
                      N1 = 1
                      N2 = N - N1
                      I = N*N + 1
                      CALL ZTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ),
         $                         N, WORK, N, WORK( N1+1 ), N,
         $                         WORK( N*N1+N1+I ), N, WORK( I ), N,
         $                         WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY,
         $                         1, IWORK, IERR )
                   END IF
                END IF
             END IF
    *
       20 CONTINUE
          WORK( 1 ) = LWMIN
          RETURN
    *
    *     End of ZTGSNA
    *
          END