sggbal.f.html

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      SUBROUTINE <a name="SGGBAL.1"></a><a href="sggbal.f.html#SGGBAL.1">SGGBAL</a>( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
     $                   RSCALE, WORK, 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">     .. Scalar Arguments ..
</span>      CHARACTER          JOB
      INTEGER            IHI, ILO, INFO, LDA, LDB, N
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      REAL               A( LDA, * ), B( LDB, * ), LSCALE( * ),
     $                   RSCALE( * ), WORK( * )
<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="SGGBAL.20"></a><a href="sggbal.f.html#SGGBAL.1">SGGBAL</a> balances a pair of general real matrices (A,B).  This
</span><span class="comment">*</span><span class="comment">  involves, first, permuting A and B by similarity transformations to
</span><span class="comment">*</span><span class="comment">  isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
</span><span class="comment">*</span><span class="comment">  elements on the diagonal; and second, applying a diagonal similarity
</span><span class="comment">*</span><span class="comment">  transformation to rows and columns ILO to IHI to make the rows
</span><span class="comment">*</span><span class="comment">  and columns as close in norm as possible. Both steps are optional.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Balancing may reduce the 1-norm of the matrices, and improve the
</span><span class="comment">*</span><span class="comment">  accuracy of the computed eigenvalues and/or eigenvectors in the
</span><span class="comment">*</span><span class="comment">  generalized eigenvalue problem A*x = lambda*B*x.
</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 the operations to be performed on A and B:
</span><span class="comment">*</span><span class="comment">          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
</span><span class="comment">*</span><span class="comment">                  and RSCALE(I) = 1.0 for i = 1,...,N.
</span><span class="comment">*</span><span class="comment">          = 'P':  permute only;
</span><span class="comment">*</span><span class="comment">          = 'S':  scale only;
</span><span class="comment">*</span><span class="comment">          = 'B':  both permute and scale.
</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 matrices A and B.  N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) REAL array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment">          On entry, the input matrix A.
</span><span class="comment">*</span><span class="comment">          On exit,  A is overwritten by the balanced matrix.
</span><span class="comment">*</span><span class="comment">          If JOB = 'N', A is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A. LDA &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input/output) REAL array, dimension (LDB,N)
</span><span class="comment">*</span><span class="comment">          On entry, the input matrix B.
</span><span class="comment">*</span><span class="comment">          On exit,  B is overwritten by the balanced matrix.
</span><span class="comment">*</span><span class="comment">          If JOB = 'N', B is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDB     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array B. LDB &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ILO     (output) INTEGER
</span><span class="comment">*</span><span class="comment">  IHI     (output) INTEGER
</span><span class="comment">*</span><span class="comment">          ILO and IHI are set to integers such that on exit
</span><span class="comment">*</span><span class="comment">          A(i,j) = 0 and B(i,j) = 0 if i &gt; j and
</span><span class="comment">*</span><span class="comment">          j = 1,...,ILO-1 or i = IHI+1,...,N.
</span><span class="comment">*</span><span class="comment">          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LSCALE  (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          Details of the permutations and scaling factors applied
</span><span class="comment">*</span><span class="comment">          to the left side of A and B.  If P(j) is the index of the
</span><span class="comment">*</span><span class="comment">          row interchanged with row j, and D(j)
</span><span class="comment">*</span><span class="comment">          is the scaling factor applied to row j, then
</span><span class="comment">*</span><span class="comment">            LSCALE(j) = P(j)    for J = 1,...,ILO-1
</span><span class="comment">*</span><span class="comment">                      = D(j)    for J = ILO,...,IHI
</span><span class="comment">*</span><span class="comment">                      = P(j)    for J = IHI+1,...,N.
</span><span class="comment">*</span><span class="comment">          The order in which the interchanges are made is N to IHI+1,
</span><span class="comment">*</span><span class="comment">          then 1 to ILO-1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RSCALE  (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          Details of the permutations and scaling factors applied
</span><span class="comment">*</span><span class="comment">          to the right side of A and B.  If P(j) is the index of the
</span><span class="comment">*</span><span class="comment">          column interchanged with column j, and D(j)
</span><span class="comment">*</span><span class="comment">          is the scaling factor applied to column j, then
</span><span class="comment">*</span><span class="comment">            LSCALE(j) = P(j)    for J = 1,...,ILO-1
</span><span class="comment">*</span><span class="comment">                      = D(j)    for J = ILO,...,IHI
</span><span class="comment">*</span><span class="comment">                      = P(j)    for J = IHI+1,...,N.
</span><span class="comment">*</span><span class="comment">          The order in which the interchanges are made is N to IHI+1,
</span><span class="comment">*</span><span class="comment">          then 1 to ILO-1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace) REAL array, dimension (lwork)
</span><span class="comment">*</span><span class="comment">          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
</span><span class="comment">*</span><span class="comment">          at least 1 when JOB = 'N' or 'P'.
</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">  See R.C. WARD, Balancing the generalized eigenvalue problem,
</span><span class="comment">*</span><span class="comment">                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      REAL               ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0 )
      REAL               THREE, SCLFAC
      PARAMETER          ( THREE = 3.0E+0, SCLFAC = 1.0E+1 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            I, ICAB, IFLOW, IP1, IR, IRAB, IT, J, JC, JP1,
     $                   K, KOUNT, L, LCAB, LM1, LRAB, LSFMAX, LSFMIN,
     $                   M, NR, NRP2
      REAL               ALPHA, BASL, BETA, CAB, CMAX, COEF, COEF2,
     $                   COEF5, COR, EW, EWC, GAMMA, PGAMMA, RAB, SFMAX,
     $                   SFMIN, SUM, T, TA, TB, TC
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      LOGICAL            <a name="LSAME.121"></a><a href="lsame.f.html#LSAME.1">LSAME</a>
      INTEGER            ISAMAX
      REAL               SDOT, <a name="SLAMCH.123"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>
      EXTERNAL           <a name="LSAME.124"></a><a href="lsame.f.html#LSAME.1">LSAME</a>, ISAMAX, SDOT, <a name="SLAMCH.124"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           SAXPY, SSCAL, SSWAP, <a name="XERBLA.127"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, INT, LOG10, MAX, MIN, REAL, SIGN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test the input parameters
</span><span class="comment">*</span><span class="comment">
</span>      INFO = 0
      IF( .NOT.<a name="LSAME.137"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( JOB, <span class="string">'N'</span> ) .AND. .NOT.<a name="LSAME.137"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( JOB, <span class="string">'P'</span> ) .AND.
     $    .NOT.<a name="LSAME.138"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( JOB, <span class="string">'S'</span> ) .AND. .NOT.<a name="LSAME.138"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( JOB, <span class="string">'B'</span> ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL <a name="XERBLA.148"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="SGGBAL.148"></a><a href="sggbal.f.html#SGGBAL.1">SGGBAL</a>'</span>, -INFO )
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span>      IF( N.EQ.0 ) THEN
         ILO = 1
         IHI = N
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span>      IF( N.EQ.1 ) THEN
         ILO = 1
         IHI = N
         LSCALE( 1 ) = ONE
         RSCALE( 1 ) = ONE
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span>      IF( <a name="LSAME.168"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( JOB, <span class="string">'N'</span> ) ) THEN
         ILO = 1
         IHI = N
         DO 10 I = 1, N
            LSCALE( I ) = ONE
            RSCALE( I ) = ONE
   10    CONTINUE
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span>      K = 1
      L = N
      IF( <a name="LSAME.180"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( JOB, <span class="string">'S'</span> ) )
     $   GO TO 190
<span class="comment">*</span><span class="comment">
</span>      GO TO 30
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Permute the matrices A and B to isolate the eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Find row with one nonzero in columns 1 through L
</span><span class="comment">*</span><span class="comment">
</span>   20 CONTINUE
      L = LM1
      IF( L.NE.1 )
     $   GO TO 30
<span class="comment">*</span><span class="comment">
</span>      RSCALE( 1 ) = ONE
      LSCALE( 1 ) = ONE
      GO TO 190
<span class="comment">*</span><span class="comment">
</span>   30 CONTINUE
      LM1 = L - 1
      DO 80 I = L, 1, -1
         DO 40 J = 1, LM1
            JP1 = J + 1
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 50
   40    CONTINUE
         J = L
         GO TO 70
<span class="comment">*</span><span class="comment">
</span>   50    CONTINUE
         DO 60 J = JP1, L
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 80
   60    CONTINUE
         J = JP1 - 1
<span class="comment">*</span><span class="comment">
</span>   70    CONTINUE
         M = L
         IFLOW = 1
         GO TO 160
   80 CONTINUE
      GO TO 100
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Find column with one nonzero in rows K through N
</span><span class="comment">*</span><span class="comment">
</span>   90 CONTINUE
      K = K + 1
<span class="comment">*</span><span class="comment">
</span>  100 CONTINUE
12

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