stgsna.f

          SUBROUTINE STGSNA( 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( * )
          REAL               A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
         $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  STGSNA estimates reciprocal condition numbers for specified
    *  eigenvalues and/or eigenvectors of a matrix pair (A, B) in
    *  generalized real Schur canonical form (or of any matrix pair
    *  (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where
    *  Z' denotes the transpose of Z.
    *
    *  (A, B) must be in generalized real Schur form (as returned by SGGES),
    *  i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
    *  blocks. B is 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 eigenpair corresponding to a real eigenvalue w(j),
    *          SELECT(j) must be set to .TRUE.. To select condition numbers
    *          corresponding to a complex conjugate pair of eigenvalues w(j)
    *          and w(j+1), either SELECT(j) or SELECT(j+1) or both, 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) REAL array, dimension (LDA,N)
    *          The upper quasi-triangular matrix A in the pair (A,B).
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,N).
    *
    *  B       (input) REAL 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) REAL 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 STGEVC.
    *          If JOB = 'V', VL is not referenced.
    *
    *  LDVL    (input) INTEGER
    *          The leading dimension of the array VL. LDVL >= 1.
    *          If JOB = 'E' or 'B', LDVL >= N.
    *
    *  VR      (input) REAL 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 ov VR, as returned by STGEVC.
    *          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) REAL array, dimension (MM)
    *          If JOB = 'E' or 'B', the reciprocal condition numbers of the
    *          selected eigenvalues, stored in consecutive elements of the
    *          array. For a complex conjugate pair of eigenvalues two
    *          consecutive elements of S are set to the same value. Thus
    *          S(j), DIF(j), and the j-th columns of VL and VR all
    *          correspond to the same eigenpair (but not in general the
    *          j-th eigenpair, unless all eigenpairs are selected).
    *          If JOB = 'V', S is not referenced.
    *
    *  DIF     (output) REAL array, dimension (MM)
    *          If JOB = 'V' or 'B', the estimated reciprocal condition
    *          numbers of the selected eigenvectors, stored in consecutive
    *          elements of the array. For a complex eigenvector two
    *          consecutive elements of DIF are set to the same value. 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.
    *          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 real
    *          eigenvalue one element is used, and for each selected complex
    *          conjugate pair of eigenvalues, two elements are used.
    *          If HOWMNY = 'A', M is set to N.
    *
    *  WORK    (workspace/output) REAL 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 >= 2*N*(N+2)+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   (workspace) INTEGER array, dimension (N + 6)
    *          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 a generalized eigenvalue
    *  w = (a, b) is defined as
    *
    *       S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))
    *
    *  where u and v are the left and right 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 (= u'Av/u'Bv)
    *  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 DIF(i) of right eigenvector u
    *  and left eigenvector v corresponding to the generalized eigenvalue w
    *  is defined as follows:
    *
    *  a) If the i-th eigenvalue w = (a,b) is real
    *
    *     Suppose U and V are orthogonal transformations such that
    *
    *                U'*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
    *                                        ( 0  S22 ),( 0 T22 )  n-1
    *                                          1  n-1     1 n-1
    *
    *     Then the reciprocal condition number DIF(i) is
    *
    *                Difl((a, b), (S22, T22)) = sigma-min( Zl ),
    *
    *     where sigma-min(Zl) denotes the smallest singular value of the
    *     2(n-1)-by-2(n-1) matrix
    *
    *         Zl = [ kron(a, In-1)  -kron(1, S22) ]
    *              [ kron(b, In-1)  -kron(1, T22) ] .
    *
    *     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
    *     Kronecker product between the matrices X and Y.
    *
    *     Note that if the default method for computing DIF(i) is wanted
    *     (see SLATDF), then the parameter DIFDRI (see below) should be
    *     changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
    *     See STGSYL for more details.
    *
    *  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
    *
    *     Suppose U and V are orthogonal transformations such that
    *
    *                U'*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
    *                                       ( 0    S22 ),( 0    T22) n-2
    *                                         2    n-2     2    n-2
    *
    *     and (S11, T11) corresponds to the complex conjugate eigenvalue
    *     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
    *     that
    *
    *         U1'*S11*V1 = ( s11 s12 )   and U1'*T11*V1 = ( t11 t12 )
    *                      (  0  s22 )                    (  0  t22 )
    *
    *     where the generalized eigenvalues w = s11/t11 and
    *     conjg(w) = s22/t22.
    *
    *     Then the reciprocal condition number DIF(i) is bounded by
    *
    *         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
    *
    *     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
    *     Z1 is the complex 2-by-2 matrix
    *
    *              Z1 =  [ s11  -s22 ]
    *                    [ t11  -t22 ],
    *
    *     This is done by computing (using real arithmetic) the
    *     roots of the characteristical polynomial det(Z1' * Z1 - lambda I),
    *     where Z1' denotes the conjugate transpose of Z1 and det(X) denotes
    *     the determinant of X.
    *
    *     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
    *     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
    *
    *              Z2 = [ kron(S11', In-2)  -kron(I2, S22) ]
    *                   [ kron(T11', In-2)  -kron(I2, T22) ]
    *
    *     Note that if the default method for computing DIF is wanted (see
    *     SLATDF), then the parameter DIFDRI (see below) should be changed
    *     from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
    *     for more details.
    *
    *  For each eigenvalue/vector specified by SELECT, DIF stores a
    *  Frobenius norm-based estimate of Difl.
    *
    *  An approximate error bound for the i-th 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 ..
          INTEGER            DIFDRI
          PARAMETER          ( DIFDRI = 3 )
          REAL               ZERO, ONE, TWO, FOUR
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0,
         $                   FOUR = 4.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
          INTEGER            I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2
          REAL               ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND,
         $                   EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM,
         $                   TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV,
         $                   UHBVI
    *     ..
    *     .. Local Arrays ..
          REAL               DUMMY( 1 ), DUMMY1( 1 )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          REAL               SDOT, SLAMCH, SLAPY2, SNRM2
          EXTERNAL           LSAME, SDOT, SLAMCH, SLAPY2, SNRM2
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SGEMV, SLACPY, SLAG2, STGEXC, STGSYL, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN, SQRT
    *     ..
    *     .. 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
                PAIR = .FALSE.
                DO 10 K = 1, N
                   IF( PAIR ) THEN
                      PAIR = .FALSE.
                   ELSE
                      IF( K.LT.N ) THEN
                         IF( A( K+1, K ).EQ.ZERO ) THEN
                            IF( SELECT( K ) )
         $                     M = M + 1
                         ELSE
                            PAIR = .TRUE.
                            IF( SELECT( K ) .OR. SELECT( K+1 ) )
         $                     M = M + 2
                         END IF
                      ELSE
                         IF( SELECT( N ) )
         $                  M = M + 1
                      END IF
                   END IF
       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 + 2 ) + 16
             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( 'STGSNA', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Get machine constants
    *
          EPS = SLAMCH( 'P' )
          SMLNUM = SLAMCH( 'S' ) / EPS
          KS = 0
          PAIR = .FALSE.
    *
          DO 20 K = 1, N
    *
    *        Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block.
    *
             IF( PAIR ) THEN
                PAIR = .FALSE.
                GO TO 20
             ELSE
                IF( K.LT.N )
         $         PAIR = A( K+1, K ).NE.ZERO
             END IF
    *
    *        Determine whether condition numbers are required for the k-th
    *        eigenpair.
    *
             IF( SOMCON ) THEN
                IF( PAIR ) THEN
                   IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
         $            GO TO 20
                ELSE
                   IF( .NOT.SELECT( K ) )
         $            GO TO 20
                END IF
             END IF
    *
             KS = KS + 1
    *
             IF( WANTS ) THEN
    *
    *           Compute the reciprocal condition number of the k-th
    *           eigenvalue.
    *
                IF( PAIR ) THEN
    *
    *              Complex eigenvalue pair.
    *
                   RNRM = SLAPY2( SNRM2( N, VR( 1, KS ), 1 ),
         $                SNRM2( N, VR( 1, KS+1 ), 1 ) )
                   LNRM = SLAPY2( SNRM2( N, VL( 1, KS ), 1 ),
         $                SNRM2( N, VL( 1, KS+1 ), 1 ) )
                   CALL SGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
         $                     WORK, 1 )
                   TMPRR = SDOT( N, WORK, 1, VL( 1, KS ), 1 )
                   TMPRI = SDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
                   CALL SGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1,
         $                     ZERO, WORK, 1 )
                   TMPII = SDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
                   TMPIR = SDOT( N, WORK, 1, VL( 1, KS ), 1 )
                   UHAV = TMPRR + TMPII
                   UHAVI = TMPIR - TMPRI
                   CALL SGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
         $                     WORK, 1 )
                   TMPRR = SDOT( N, WORK, 1, VL( 1, KS ), 1 )
                   TMPRI = SDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
                   CALL SGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1,
         $                     ZERO, WORK, 1 )
                   TMPII = SDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
                   TMPIR = SDOT( N, WORK, 1, VL( 1, KS ), 1 )
                   UHBV = TMPRR + TMPII
                   UHBVI = TMPIR - TMPRI
                   UHAV = SLAPY2( UHAV, UHAVI )
                   UHBV = SLAPY2( UHBV, UHBVI )
                   COND = SLAPY2( UHAV, UHBV )
                   S( KS ) = COND / ( RNRM*LNRM )
                   S( KS+1 ) = S( KS )
    *
                ELSE
    *
    *              Real eigenvalue.
    *
                   RNRM = SNRM2( N, VR( 1, KS ), 1 )
                   LNRM = SNRM2( N, VL( 1, KS ), 1 )
                   CALL SGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
         $                     WORK, 1 )
                   UHAV = SDOT( N, WORK, 1, VL( 1, KS ), 1 )
                   CALL SGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
         $                     WORK, 1 )
                   UHBV = SDOT( N, WORK, 1, VL( 1, KS ), 1 )
                   COND = SLAPY2( UHAV, UHBV )
                   IF( COND.EQ.ZERO ) THEN
                      S( KS ) = -ONE
                   ELSE
                      S( KS ) = COND / ( RNRM*LNRM )
                   END IF
                END IF
             END IF
    *
             IF( WANTDF ) THEN
                IF( N.EQ.1 ) THEN
                   DIF( KS ) = SLAPY2( A( 1, 1 ), B( 1, 1 ) )
                   GO TO 20
                END IF
    *
    *           Estimate the reciprocal condition number of the k-th
    *           eigenvectors.
                IF( PAIR ) THEN
    *
    *              Copy the  2-by 2 pencil beginning at (A(k,k), B(k, k)).
    *              Compute the eigenvalue(s) at position K.
    *
                   WORK( 1 ) = A( K, K )
                   WORK( 2 ) = A( K+1, K )
                   WORK( 3 ) = A( K, K+1 )
                   WORK( 4 ) = A( K+1, K+1 )
                   WORK( 5 ) = B( K, K )
                   WORK( 6 ) = B( K+1, K )
                   WORK( 7 ) = B( K, K+1 )
                   WORK( 8 ) = B( K+1, K+1 )
                   CALL SLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA,
         $                     DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI )
                   ALPRQT = ONE
                   C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA )
                   C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI
                   ROOT1 = C1 + SQRT( C1*C1-4.0*C2 )
                   ROOT2 = C2 / ROOT1
                   ROOT1 = ROOT1 / TWO
                   COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) )
                END IF
    *
    *           Copy the matrix (A, B) to the array WORK and swap the
    *           diagonal block beginning at A(k,k) to the (1,1) position.
    *
                CALL SLACPY( 'Full', N, N, A, LDA, WORK, N )
                CALL SLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
                IFST = K
                ILST = 1
    *
                CALL STGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), N,
         $                   DUMMY, 1, DUMMY1, 1, IFST, ILST,
         $                   WORK( N*N*2+1 ), LWORK-2*N*N, 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
                   IF( WORK( 2 ).NE.ZERO )
         $            N1 = 2
                   N2 = N - N1
                   IF( N2.EQ.0 ) THEN
                      DIF( KS ) = COND
                   ELSE
                      I = N*N + 1
                      IZ = 2*N*N + 1
                      CALL STGSYL( 'N', DIFDRI, 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 ),
         $                         WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR )
    *
                      IF( PAIR )
         $               DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ),
         $                           COND )
                   END IF
                END IF
                IF( PAIR )
         $         DIF( KS+1 ) = DIF( KS )
             END IF
             IF( PAIR )
         $      KS = KS + 1
    *
       20 CONTINUE
          WORK( 1 ) = LWMIN
          RETURN
    *
    *     End of STGSNA
    *
          END