*> \brief \b ZGBEQUB * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGBEQUB + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, * AMAX, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, KL, KU, LDAB, M, N * DOUBLE PRECISION AMAX, COLCND, ROWCND * .. * .. Array Arguments .. * DOUBLE PRECISION C( * ), R( * ) * COMPLEX*16 AB( LDAB, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGBEQUB computes row and column scalings intended to equilibrate an *> M-by-N matrix A and reduce its condition number. R returns the row *> scale factors and C the column scale factors, chosen to try to make *> the largest element in each row and column of the matrix B with *> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most *> the radix. *> *> R(i) and C(j) are restricted to be a power of the radix between *> SMLNUM = smallest safe number and BIGNUM = largest safe number. Use *> of these scaling factors is not guaranteed to reduce the condition *> number of A but works well in practice. *> *> This routine differs from ZGEEQU by restricting the scaling factors *> to a power of the radix. Baring over- and underflow, scaling by *> these factors introduces no additional rounding errors. However, the *> scaled entries' magnitured are no longer approximately 1 but lie *> between sqrt(radix) and 1/sqrt(radix). *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] KL *> \verbatim *> KL is INTEGER *> The number of subdiagonals within the band of A. KL >= 0. *> \endverbatim *> *> \param[in] KU *> \verbatim *> KU is INTEGER *> The number of superdiagonals within the band of A. KU >= 0. *> \endverbatim *> *> \param[in] AB *> \verbatim *> AB is DOUBLE PRECISION array, dimension (LDAB,N) *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1. *> The j-th column of A is stored in the j-th column of the *> array AB as follows: *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array A. LDAB >= max(1,M). *> \endverbatim *> *> \param[out] R *> \verbatim *> R is DOUBLE PRECISION array, dimension (M) *> If INFO = 0 or INFO > M, R contains the row scale factors *> for A. *> \endverbatim *> *> \param[out] C *> \verbatim *> C is DOUBLE PRECISION array, dimension (N) *> If INFO = 0, C contains the column scale factors for A. *> \endverbatim *> *> \param[out] ROWCND *> \verbatim *> ROWCND is DOUBLE PRECISION *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and *> AMAX is neither too large nor too small, it is not worth *> scaling by R. *> \endverbatim *> *> \param[out] COLCND *> \verbatim *> COLCND is DOUBLE PRECISION *> If INFO = 0, COLCND contains the ratio of the smallest *> C(i) to the largest C(i). If COLCND >= 0.1, it is not *> worth scaling by C. *> \endverbatim *> *> \param[out] AMAX *> \verbatim *> AMAX is DOUBLE PRECISION *> Absolute value of largest matrix element. If AMAX is very *> close to overflow or very close to underflow, the matrix *> should be scaled. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: if INFO = i, and i is *> <= M: the i-th row of A is exactly zero *> > M: the (i-M)-th column of A is exactly zero *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * *> \ingroup complex16GBcomputational * * ===================================================================== SUBROUTINE ZGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, INFO ) * * -- LAPACK computational routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2011 * * .. Scalar Arguments .. INTEGER INFO, KL, KU, LDAB, M, N DOUBLE PRECISION AMAX, COLCND, ROWCND * .. * .. Array Arguments .. DOUBLE PRECISION C( * ), R( * ) COMPLEX*16 AB( LDAB, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. INTEGER I, J, KD DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, $ LOGRDX COMPLEX*16 ZDUM * .. * .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH * .. * .. External Subroutines .. EXTERNAL XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGBEQUB', -INFO ) RETURN END IF * * Quick return if possible. * IF( M.EQ.0 .OR. N.EQ.0 ) THEN ROWCND = ONE COLCND = ONE AMAX = ZERO RETURN END IF * * Get machine constants. Assume SMLNUM is a power of the radix. * SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM RADIX = DLAMCH( 'B' ) LOGRDX = LOG(RADIX) * * Compute row scale factors. * DO 10 I = 1, M R( I ) = ZERO 10 CONTINUE * * Find the maximum element in each row. * KD = KU + 1 DO 30 J = 1, N DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M ) R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) ) 20 CONTINUE 30 CONTINUE DO I = 1, M IF( R( I ).GT.ZERO ) THEN R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX ) END IF END DO * * Find the maximum and minimum scale factors. * RCMIN = BIGNUM RCMAX = ZERO DO 40 I = 1, M RCMAX = MAX( RCMAX, R( I ) ) RCMIN = MIN( RCMIN, R( I ) ) 40 CONTINUE AMAX = RCMAX * IF( RCMIN.EQ.ZERO ) THEN * * Find the first zero scale factor and return an error code. * DO 50 I = 1, M IF( R( I ).EQ.ZERO ) THEN INFO = I RETURN END IF 50 CONTINUE ELSE * * Invert the scale factors. * DO 60 I = 1, M R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM ) 60 CONTINUE * * Compute ROWCND = min(R(I)) / max(R(I)). * ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) END IF * * Compute column scale factors. * DO 70 J = 1, N C( J ) = ZERO 70 CONTINUE * * Find the maximum element in each column, * assuming the row scaling computed above. * DO 90 J = 1, N DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M ) C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) ) 80 CONTINUE IF( C( J ).GT.ZERO ) THEN C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX ) END IF 90 CONTINUE * * Find the maximum and minimum scale factors. * RCMIN = BIGNUM RCMAX = ZERO DO 100 J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) 100 CONTINUE * IF( RCMIN.EQ.ZERO ) THEN * * Find the first zero scale factor and return an error code. * DO 110 J = 1, N IF( C( J ).EQ.ZERO ) THEN INFO = M + J RETURN END IF 110 CONTINUE ELSE * * Invert the scale factors. * DO 120 J = 1, N C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM ) 120 CONTINUE * * Compute COLCND = min(C(J)) / max(C(J)). * COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) END IF * RETURN * * End of ZGBEQUB * END