LAPACK 3.3.1 Linear Algebra PACKage

# zgbequb.f

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```00001       SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
00002      \$                    AMAX, INFO )
00003 *
00004 *     -- LAPACK routine (version 3.2)                                 --
00005 *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
00006 *     -- Jason Riedy of Univ. of California Berkeley.                 --
00007 *     -- November 2008                                                --
00008 *
00009 *     -- LAPACK is a software package provided by Univ. of Tennessee, --
00010 *     -- Univ. of California Berkeley and NAG Ltd.                    --
00011 *
00012       IMPLICIT NONE
00013 *     ..
00014 *     .. Scalar Arguments ..
00015       INTEGER            INFO, KL, KU, LDAB, M, N
00016       DOUBLE PRECISION   AMAX, COLCND, ROWCND
00017 *     ..
00018 *     .. Array Arguments ..
00019       DOUBLE PRECISION   C( * ), R( * )
00020       COMPLEX*16         AB( LDAB, * )
00021 *     ..
00022 *
00023 *  Purpose
00024 *  =======
00025 *
00026 *  ZGBEQUB computes row and column scalings intended to equilibrate an
00027 *  M-by-N matrix A and reduce its condition number.  R returns the row
00028 *  scale factors and C the column scale factors, chosen to try to make
00029 *  the largest element in each row and column of the matrix B with
00030 *  elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
00032 *
00033 *  R(i) and C(j) are restricted to be a power of the radix between
00034 *  SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
00035 *  of these scaling factors is not guaranteed to reduce the condition
00036 *  number of A but works well in practice.
00037 *
00038 *  This routine differs from ZGEEQU by restricting the scaling factors
00039 *  to a power of the radix.  Baring over- and underflow, scaling by
00040 *  these factors introduces no additional rounding errors.  However, the
00041 *  scaled entries' magnitured are no longer approximately 1 but lie
00043 *
00044 *  Arguments
00045 *  =========
00046 *
00047 *  M       (input) INTEGER
00048 *          The number of rows of the matrix A.  M >= 0.
00049 *
00050 *  N       (input) INTEGER
00051 *          The number of columns of the matrix A.  N >= 0.
00052 *
00053 *  KL      (input) INTEGER
00054 *          The number of subdiagonals within the band of A.  KL >= 0.
00055 *
00056 *  KU      (input) INTEGER
00057 *          The number of superdiagonals within the band of A.  KU >= 0.
00058 *
00059 *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
00060 *          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
00061 *          The j-th column of A is stored in the j-th column of the
00062 *          array AB as follows:
00063 *          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
00064 *
00065 *  LDAB    (input) INTEGER
00066 *          The leading dimension of the array A.  LDAB >= max(1,M).
00067 *
00068 *  R       (output) DOUBLE PRECISION array, dimension (M)
00069 *          If INFO = 0 or INFO > M, R contains the row scale factors
00070 *          for A.
00071 *
00072 *  C       (output) DOUBLE PRECISION array, dimension (N)
00073 *          If INFO = 0,  C contains the column scale factors for A.
00074 *
00075 *  ROWCND  (output) DOUBLE PRECISION
00076 *          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
00077 *          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
00078 *          AMAX is neither too large nor too small, it is not worth
00079 *          scaling by R.
00080 *
00081 *  COLCND  (output) DOUBLE PRECISION
00082 *          If INFO = 0, COLCND contains the ratio of the smallest
00083 *          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
00084 *          worth scaling by C.
00085 *
00086 *  AMAX    (output) DOUBLE PRECISION
00087 *          Absolute value of largest matrix element.  If AMAX is very
00088 *          close to overflow or very close to underflow, the matrix
00089 *          should be scaled.
00090 *
00091 *  INFO    (output) INTEGER
00092 *          = 0:  successful exit
00093 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00094 *          > 0:  if INFO = i,  and i is
00095 *                <= M:  the i-th row of A is exactly zero
00096 *                >  M:  the (i-M)-th column of A is exactly zero
00097 *
00098 *  =====================================================================
00099 *
00100 *     .. Parameters ..
00101       DOUBLE PRECISION   ONE, ZERO
00102       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
00103 *     ..
00104 *     .. Local Scalars ..
00105       INTEGER            I, J, KD
00106       DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
00107      \$                   LOGRDX
00108       COMPLEX*16         ZDUM
00109 *     ..
00110 *     .. External Functions ..
00111       DOUBLE PRECISION   DLAMCH
00112       EXTERNAL           DLAMCH
00113 *     ..
00114 *     .. External Subroutines ..
00115       EXTERNAL           XERBLA
00116 *     ..
00117 *     .. Intrinsic Functions ..
00118       INTRINSIC          ABS, MAX, MIN, LOG, REAL, DIMAG
00119 *     ..
00120 *     .. Statement Functions ..
00121       DOUBLE PRECISION   CABS1
00122 *     ..
00123 *     .. Statement Function definitions ..
00124       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00125 *     ..
00126 *     .. Executable Statements ..
00127 *
00128 *     Test the input parameters.
00129 *
00130       INFO = 0
00131       IF( M.LT.0 ) THEN
00132          INFO = -1
00133       ELSE IF( N.LT.0 ) THEN
00134          INFO = -2
00135       ELSE IF( KL.LT.0 ) THEN
00136          INFO = -3
00137       ELSE IF( KU.LT.0 ) THEN
00138          INFO = -4
00139       ELSE IF( LDAB.LT.KL+KU+1 ) THEN
00140          INFO = -6
00141       END IF
00142       IF( INFO.NE.0 ) THEN
00143          CALL XERBLA( 'ZGBEQUB', -INFO )
00144          RETURN
00145       END IF
00146 *
00147 *     Quick return if possible.
00148 *
00149       IF( M.EQ.0 .OR. N.EQ.0 ) THEN
00150          ROWCND = ONE
00151          COLCND = ONE
00152          AMAX = ZERO
00153          RETURN
00154       END IF
00155 *
00156 *     Get machine constants.  Assume SMLNUM is a power of the radix.
00157 *
00158       SMLNUM = DLAMCH( 'S' )
00159       BIGNUM = ONE / SMLNUM
00160       RADIX = DLAMCH( 'B' )
00162 *
00163 *     Compute row scale factors.
00164 *
00165       DO 10 I = 1, M
00166          R( I ) = ZERO
00167    10 CONTINUE
00168 *
00169 *     Find the maximum element in each row.
00170 *
00171       KD = KU + 1
00172       DO 30 J = 1, N
00173          DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
00174             R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) )
00175    20    CONTINUE
00176    30 CONTINUE
00177       DO I = 1, M
00178          IF( R( I ).GT.ZERO ) THEN
00179             R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
00180          END IF
00181       END DO
00182 *
00183 *     Find the maximum and minimum scale factors.
00184 *
00185       RCMIN = BIGNUM
00186       RCMAX = ZERO
00187       DO 40 I = 1, M
00188          RCMAX = MAX( RCMAX, R( I ) )
00189          RCMIN = MIN( RCMIN, R( I ) )
00190    40 CONTINUE
00191       AMAX = RCMAX
00192 *
00193       IF( RCMIN.EQ.ZERO ) THEN
00194 *
00195 *        Find the first zero scale factor and return an error code.
00196 *
00197          DO 50 I = 1, M
00198             IF( R( I ).EQ.ZERO ) THEN
00199                INFO = I
00200                RETURN
00201             END IF
00202    50    CONTINUE
00203       ELSE
00204 *
00205 *        Invert the scale factors.
00206 *
00207          DO 60 I = 1, M
00208             R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
00209    60    CONTINUE
00210 *
00211 *        Compute ROWCND = min(R(I)) / max(R(I)).
00212 *
00213          ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
00214       END IF
00215 *
00216 *     Compute column scale factors.
00217 *
00218       DO 70 J = 1, N
00219          C( J ) = ZERO
00220    70 CONTINUE
00221 *
00222 *     Find the maximum element in each column,
00223 *     assuming the row scaling computed above.
00224 *
00225       DO 90 J = 1, N
00226          DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
00227             C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) )
00228    80    CONTINUE
00229          IF( C( J ).GT.ZERO ) THEN
00230             C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
00231          END IF
00232    90 CONTINUE
00233 *
00234 *     Find the maximum and minimum scale factors.
00235 *
00236       RCMIN = BIGNUM
00237       RCMAX = ZERO
00238       DO 100 J = 1, N
00239          RCMIN = MIN( RCMIN, C( J ) )
00240          RCMAX = MAX( RCMAX, C( J ) )
00241   100 CONTINUE
00242 *
00243       IF( RCMIN.EQ.ZERO ) THEN
00244 *
00245 *        Find the first zero scale factor and return an error code.
00246 *
00247          DO 110 J = 1, N
00248             IF( C( J ).EQ.ZERO ) THEN
00249                INFO = M + J
00250                RETURN
00251             END IF
00252   110    CONTINUE
00253       ELSE
00254 *
00255 *        Invert the scale factors.
00256 *
00257          DO 120 J = 1, N
00258             C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
00259   120    CONTINUE
00260 *
00261 *        Compute COLCND = min(C(J)) / max(C(J)).
00262 *
00263          COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
00264       END IF
00265 *
00266       RETURN
00267 *
00268 *     End of ZGBEQUB
00269 *
00270       END
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