1
       2
       3
       4
       5
       6
       7
       8
       9
      10
      11
      12
      13
      14
      15
      16
      17
      18
      19
      20
      21
      22
      23
      24
      25
      26
      27
      28
      29
      30
      31
      32
      33
      34
      35
      36
      37
      38
      39
      40
      41
      42
      43
      44
      45
      46
      47
      48
      49
      50
      51
      52
      53
      54
      55
      56
      57
      58
      59
      60
      61
      62
      63
      64
      65
      66
      67
      68
      69
      70
      71
      72
      73
      74
      75
      76
      77
      78
      79
      80
      81
      82
      83
      84
      85
      86
      87
      88
      89
      90
      91
      92
      93
      94
      95
      96
      97
      98
      99
     100
     101
     102
     103
     104
     105
     106
     107
     108
     109
     110
     111
     112
     113
     114
     115
     116
     117
     118
     119
     120
     121
     122
     123
     124
     125
     126
     127
     128
     129
     130
     131
     132
     133
     134
     135
     136
     137
     138
     139
     140
     141
     142
     143
     144
     145
     146
     147
     148
     149
     150
     151
     152
     153
     154
     155
     156
     157
     158
     159
     160
     161
     162
     163
     164
     165
     166
     167
     168
     169
     170
     171
     172
     173
     174
     175
     176
     177
     178
     179
     180
     181
     182
     183
     184
     185
     186
     187
     188
     189
     190
     191
     192
     193
     194
     195
     196
     197
     198
     199
     200
     201
     202
     203
     204
     205
     206
     207
     208
     209
     210
     211
     212
     213
     214
     215
     216
     217
     218
     219
     220
     221
     222
     223
     224
     225
     226
     227
     228
     229
     230
     231
     232
     233
     234
     235
     236
     237
     238
     239
     240
     241
     242
     243
     244
     245
     246
     247
     248
     249
     250
     251
     252
     253
     254
     255
     256
     257
     258
     259
     260
     261
     262
     263
     264
     265
     266
     267
     268
     269
     270
     271
     272
     273
     274
     275
     276
     277
     278
     279
     280
     281
     282
     283
     284
     285
     286
     287
     288
     289
     290
     291
     292
     293
     294
     295
     296
     297
     298
     299
     300
     301
     302
     303
     304
     305
     306
     307
     308
     309
     310
     311
     312
     313
     314
     315
     316
     317
     318
     319
     320
     321
     322
     323
     324
     325
     326
     327
     328
     329
     330
     331
     332
     333
     334
     335
     336
     337
     338
     339
     340
     341
     342
     343
     344
     345
     346
     347
     348
     349
     350
     351
     352
     353
     354
     355
     356
     357
     358
     359
     360
     361
     362
     363
     364
     365
     366
     367
     368
     369
     370
     371
     372
     373
     374
     375
     376
     377
     378
     379
     380
     381
     382
     383
     384
     385
     386
     387
     388
     389
     390
     391
     392
     393
     394
     395
     396
     397
     398
     399
     400
     401
     402
     403
     404
     405
     406
     407
     408
     409
     410
     411
     412
     413
     414
     415
     416
     417
     418
     419
     420
     421
     422
     423
     424
     425
     426
     427
     428
     429
     430
     431
     432
     433
     434
     435
     436
     437
     438
     439
     440
     441
     442
     443
     444
     445
     446
     447
     448
     449
     450
     451
     452
     453
     454
     455
     456
     457
     458
     459
     460
     461
     462
     463
     464
     465
     466
     467
     468
     469
     470
      SUBROUTINE SGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
     $                   RSCALE, WORK, INFO )
*
*  -- LAPACK routine (version 3.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          JOB
      INTEGER            IHI, ILO, INFO, LDA, LDB, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * ), LSCALE( * ),
     $                   RSCALE( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  SGGBAL balances a pair of general real matrices (A,B).  This
*  involves, first, permuting A and B by similarity transformations to
*  isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
*  elements on the diagonal; and second, applying a diagonal similarity
*  transformation to rows and columns ILO to IHI to make the rows
*  and columns as close in norm as possible. Both steps are optional.
*
*  Balancing may reduce the 1-norm of the matrices, and improve the
*  accuracy of the computed eigenvalues and/or eigenvectors in the
*  generalized eigenvalue problem A*x = lambda*B*x.
*
*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          Specifies the operations to be performed on A and B:
*          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
*                  and RSCALE(I) = 1.0 for i = 1,...,N.
*          = 'P':  permute only;
*          = 'S':  scale only;
*          = 'B':  both permute and scale.
*
*  N       (input) INTEGER
*          The order of the matrices A and B.  N >= 0.
*
*  A       (input/output) REAL array, dimension (LDA,N)
*          On entry, the input matrix A.
*          On exit,  A is overwritten by the balanced matrix.
*          If JOB = 'N', A is not referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,N).
*
*  B       (input/output) REAL array, dimension (LDB,N)
*          On entry, the input matrix B.
*          On exit,  B is overwritten by the balanced matrix.
*          If JOB = 'N', B is not referenced.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B. LDB >= max(1,N).
*
*  ILO     (output) INTEGER
*  IHI     (output) INTEGER
*          ILO and IHI are set to integers such that on exit
*          A(i,j) = 0 and B(i,j) = 0 if i > j and
*          j = 1,...,ILO-1 or i = IHI+1,...,N.
*          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*
*  LSCALE  (output) REAL array, dimension (N)
*          Details of the permutations and scaling factors applied
*          to the left side of A and B.  If P(j) is the index of the
*          row interchanged with row j, and D(j)
*          is the scaling factor applied to row j, then
*            LSCALE(j) = P(j)    for J = 1,...,ILO-1
*                      = D(j)    for J = ILO,...,IHI
*                      = P(j)    for J = IHI+1,...,N.
*          The order in which the interchanges are made is N to IHI+1,
*          then 1 to ILO-1.
*
*  RSCALE  (output) REAL array, dimension (N)
*          Details of the permutations and scaling factors applied
*          to the right side of A and B.  If P(j) is the index of the
*          column interchanged with column j, and D(j)
*          is the scaling factor applied to column j, then
*            LSCALE(j) = P(j)    for J = 1,...,ILO-1
*                      = D(j)    for J = ILO,...,IHI
*                      = P(j)    for J = IHI+1,...,N.
*          The order in which the interchanges are made is N to IHI+1,
*          then 1 to ILO-1.
*
*  WORK    (workspace) REAL array, dimension (lwork)
*          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
*          at least 1 when JOB = 'N' or 'P'.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  See R.C. WARD, Balancing the generalized eigenvalue problem,
*                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
*
*  =====================================================================
*
*     .. Parameters ..
      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 )
*     ..
*     .. Local Scalars ..
      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
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SDOT, SLAMCH
      EXTERNAL           LSAME, ISAMAX, SDOT, SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           SAXPY, SSCAL, SSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABSINTLOG10MAXMIN, REAL, SIGN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      IF.NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
     $    .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX1, N ) ) THEN
         INFO = -4
      ELSE IF( LDB.LT.MAX1, N ) ) THEN
         INFO = -6
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SGGBAL'-INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         ILO = 1
         IHI = N
         RETURN
      END IF
*
      IF( N.EQ.1 ) THEN
         ILO = 1
         IHI = N
         LSCALE( 1 ) = ONE
         RSCALE( 1 ) = ONE
         RETURN
      END IF
*
      IF( LSAME( JOB, 'N' ) ) THEN
         ILO = 1
         IHI = N
         DO 10 I = 1, N
            LSCALE( I ) = ONE
            RSCALE( I ) = ONE
   10    CONTINUE
         RETURN
      END IF
*
      K = 1
      L = N
      IF( LSAME( JOB, 'S' ) )
     $   GO TO 190
*
      GO TO 30
*
*     Permute the matrices A and B to isolate the eigenvalues.
*
*     Find row with one nonzero in columns 1 through L
*
   20 CONTINUE
      L = LM1
      IF( L.NE.1 )
     $   GO TO 30
*
      RSCALE( 1 ) = ONE
      LSCALE( 1 ) = ONE
      GO TO 190
*
   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
*
   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
*
   70    CONTINUE
         M = L
         IFLOW = 1
         GO TO 160
   80 CONTINUE
      GO TO 100
*
*     Find column with one nonzero in rows K through N
*
   90 CONTINUE
      K = K + 1
*
  100 CONTINUE
      DO 150 J = K, L
         DO 110 I = K, LM1
            IP1 = I + 1
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 120
  110    CONTINUE
         I = L
         GO TO 140
  120    CONTINUE
         DO 130 I = IP1, L
            IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
     $         GO TO 150
  130    CONTINUE
         I = IP1 - 1
  140    CONTINUE
         M = K
         IFLOW = 2
         GO TO 160
  150 CONTINUE
      GO TO 190
*
*     Permute rows M and I
*
  160 CONTINUE
      LSCALE( M ) = I
      IF( I.EQ.M )
     $   GO TO 170
      CALL SSWAP( N-K+1, A( I, K ), LDA, A( M, K ), LDA )
      CALL SSWAP( N-K+1, B( I, K ), LDB, B( M, K ), LDB )
*
*     Permute columns M and J
*
  170 CONTINUE
      RSCALE( M ) = J
      IF( J.EQ.M )
     $   GO TO 180
      CALL SSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
      CALL SSWAP( L, B( 1, J ), 1, B( 1, M ), 1 )
*
  180 CONTINUE
      GO TO ( 2090 )IFLOW
*
  190 CONTINUE
      ILO = K
      IHI = L
*
      IF( LSAME( JOB, 'P' ) ) THEN
         DO 195 I = ILO, IHI
            LSCALE( I ) = ONE
            RSCALE( I ) = ONE
  195    CONTINUE
         RETURN
      END IF
*
      IF( ILO.EQ.IHI )
     $   RETURN
*
*     Balance the submatrix in rows ILO to IHI.
*
      NR = IHI - ILO + 1
      DO 200 I = ILO, IHI
         RSCALE( I ) = ZERO
         LSCALE( I ) = ZERO
*
         WORK( I ) = ZERO
         WORK( I+N ) = ZERO
         WORK( I+2*N ) = ZERO
         WORK( I+3*N ) = ZERO
         WORK( I+4*N ) = ZERO
         WORK( I+5*N ) = ZERO
  200 CONTINUE
*
*     Compute right side vector in resulting linear equations
*
      BASL = LOG10( SCLFAC )
      DO 240 I = ILO, IHI
         DO 230 J = ILO, IHI
            TB = B( I, J )
            TA = A( I, J )
            IF( TA.EQ.ZERO )
     $         GO TO 210
            TA = LOG10ABS( TA ) ) / BASL
  210       CONTINUE
            IF( TB.EQ.ZERO )
     $         GO TO 220
            TB = LOG10ABS( TB ) ) / BASL
  220       CONTINUE
            WORK( I+4*N ) = WORK( I+4*N ) - TA - TB
            WORK( J+5*N ) = WORK( J+5*N ) - TA - TB
  230    CONTINUE
  240 CONTINUE
*
      COEF = ONE / REAL2*NR )
      COEF2 = COEF*COEF
      COEF5 = HALF*COEF2
      NRP2 = NR + 2
      BETA = ZERO
      IT = 1
*
*     Start generalized conjugate gradient iteration
*
  250 CONTINUE
*
      GAMMA = SDOT( NR, WORK( ILO+4*N ), 1, WORK( ILO+4*N ), 1 ) +
     $        SDOT( NR, WORK( ILO+5*N ), 1, WORK( ILO+5*N ), 1 )
*
      EW = ZERO
      EWC = ZERO
      DO 260 I = ILO, IHI
         EW = EW + WORK( I+4*N )
         EWC = EWC + WORK( I+5*N )
  260 CONTINUE
*
      GAMMA = COEF*GAMMA - COEF2*( EW**2+EWC**2 ) - COEF5*( EW-EWC )**2
      IFGAMMA.EQ.ZERO )
     $   GO TO 350
      IF( IT.NE.1 )
     $   BETA = GAMMA / PGAMMA
      T = COEF5*( EWC-THREE*EW )
      TC = COEF5*( EW-THREE*EWC )
*
      CALL SSCAL( NR, BETA, WORK( ILO ), 1 )
      CALL SSCAL( NR, BETA, WORK( ILO+N ), 1 )
*
      CALL SAXPY( NR, COEF, WORK( ILO+4*N ), 1, WORK( ILO+N ), 1 )
      CALL SAXPY( NR, COEF, WORK( ILO+5*N ), 1, WORK( ILO ), 1 )
*
      DO 270 I = ILO, IHI
         WORK( I ) = WORK( I ) + TC
         WORK( I+N ) = WORK( I+N ) + T
  270 CONTINUE
*
*     Apply matrix to vector
*
      DO 300 I = ILO, IHI
         KOUNT = 0
         SUM = ZERO
         DO 290 J = ILO, IHI
            IF( A( I, J ).EQ.ZERO )
     $         GO TO 280
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( J )
  280       CONTINUE
            IF( B( I, J ).EQ.ZERO )
     $         GO TO 290
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( J )
  290    CONTINUE
         WORK( I+2*N ) = REAL( KOUNT )*WORK( I+N ) + SUM
  300 CONTINUE
*
      DO 330 J = ILO, IHI
         KOUNT = 0
         SUM = ZERO
         DO 320 I = ILO, IHI
            IF( A( I, J ).EQ.ZERO )
     $         GO TO 310
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( I+N )
  310       CONTINUE
            IF( B( I, J ).EQ.ZERO )
     $         GO TO 320
            KOUNT = KOUNT + 1
            SUM = SUM + WORK( I+N )
  320    CONTINUE
         WORK( J+3*N ) = REAL( KOUNT )*WORK( J ) + SUM
  330 CONTINUE
*
      SUM = SDOT( NR, WORK( ILO+N ), 1, WORK( ILO+2*N ), 1 ) +
     $      SDOT( NR, WORK( ILO ), 1, WORK( ILO+3*N ), 1 )
      ALPHA = GAMMA / SUM
*
*     Determine correction to current iteration
*
      CMAX = ZERO
      DO 340 I = ILO, IHI
         COR = ALPHA*WORK( I+N )
         IFABS( COR ).GT.CMAX )
     $      CMAX = ABS( COR )
         LSCALE( I ) = LSCALE( I ) + COR
         COR = ALPHA*WORK( I )
         IFABS( COR ).GT.CMAX )
     $      CMAX = ABS( COR )
         RSCALE( I ) = RSCALE( I ) + COR
  340 CONTINUE
      IF( CMAX.LT.HALF )
     $   GO TO 350
*
      CALL SAXPY( NR, -ALPHA, WORK( ILO+2*N ), 1, WORK( ILO+4*N ), 1 )
      CALL SAXPY( NR, -ALPHA, WORK( ILO+3*N ), 1, WORK( ILO+5*N ), 1 )
*
      PGAMMA = GAMMA
      IT = IT + 1
      IF( IT.LE.NRP2 )
     $   GO TO 250
*
*     End generalized conjugate gradient iteration
*
  350 CONTINUE
      SFMIN = SLAMCH( 'S' )
      SFMAX = ONE / SFMIN
      LSFMIN = INTLOG10( SFMIN ) / BASL+ONE )
      LSFMAX = INTLOG10( SFMAX ) / BASL )
      DO 360 I = ILO, IHI
         IRAB = ISAMAX( N-ILO+1, A( I, ILO ), LDA )
         RAB = ABS( A( I, IRAB+ILO-1 ) )
         IRAB = ISAMAX( N-ILO+1, B( I, ILO ), LDB )
         RAB = MAX( RAB, ABS( B( I, IRAB+ILO-1 ) ) )
         LRAB = INTLOG10( RAB+SFMIN ) / BASL+ONE )
         IR = LSCALE( I ) + SIGN( HALF, LSCALE( I ) )
         IR = MINMAX( IR, LSFMIN ), LSFMAX, LSFMAX-LRAB )
         LSCALE( I ) = SCLFAC**IR
         ICAB = ISAMAX( IHI, A( 1, I ), 1 )
         CAB = ABS( A( ICAB, I ) )
         ICAB = ISAMAX( IHI, B( 1, I ), 1 )
         CAB = MAX( CAB, ABS( B( ICAB, I ) ) )
         LCAB = INTLOG10( CAB+SFMIN ) / BASL+ONE )
         JC = RSCALE( I ) + SIGN( HALF, RSCALE( I ) )
         JC = MINMAX( JC, LSFMIN ), LSFMAX, LSFMAX-LCAB )
         RSCALE( I ) = SCLFAC**JC
  360 CONTINUE
*
*     Row scaling of matrices A and B
*
      DO 370 I = ILO, IHI
         CALL SSCAL( N-ILO+1, LSCALE( I ), A( I, ILO ), LDA )
         CALL SSCAL( N-ILO+1, LSCALE( I ), B( I, ILO ), LDB )
  370 CONTINUE
*
*     Column scaling of matrices A and B
*
      DO 380 J = ILO, IHI
         CALL SSCAL( IHI, RSCALE( J ), A( 1, J ), 1 )
         CALL SSCAL( IHI, RSCALE( J ), B( 1, J ), 1 )
  380 CONTINUE
*
      RETURN
*
*     End of SGGBAL
*
      END