SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
$ AMAX, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
* -- Jason Riedy of Univ. of California Berkeley. --
* -- November 2008 --
*
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley and NAG Ltd. --
*
IMPLICIT NONE
* ..
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
* ..
*
* Purpose
* =======
*
* DGBEQUB 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 DGEEQU 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).
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* KL (input) INTEGER
* The number of subdiagonals within the band of A. KL >= 0.
*
* KU (input) INTEGER
* The number of superdiagonals within the band of A. KU >= 0.
*
* AB (input) 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)
*
* LDAB (input) INTEGER
* The leading dimension of the array A. LDAB >= max(1,M).
*
* R (output) DOUBLE PRECISION array, dimension (M)
* If INFO = 0 or INFO > M, R contains the row scale factors
* for A.
*
* C (output) DOUBLE PRECISION array, dimension (N)
* If INFO = 0, C contains the column scale factors for A.
*
* ROWCND (output) 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.
*
* COLCND (output) 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.
*
* AMAX (output) 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.
*
* INFO (output) 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
*
* =====================================================================
*
* .. 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
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, LOG
* ..
* .. 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( 'DGBEQUB', -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 ), ABS( 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 ), ABS( 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 DGBEQUB
*
END