SUBROUTINE ZGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, KL, KU, LDAB, M, N DOUBLE PRECISION AMAX, COLCND, ROWCND * .. * .. Array Arguments .. DOUBLE PRECISION C( * ), R( * ) COMPLEX*16 AB( LDAB, * ) * .. * * Purpose * ======= * * ZGBEQU computes row and column scalings intended to equilibrate an * M-by-N band 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 absolute value 1. * * R(i) and C(j) are restricted to be 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. * * 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) COMPLEX*16 array, dimension (LDAB,N) * The band matrix A, stored 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(m,j+kl). * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KL+KU+1. * * 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 COMPLEX*16 ZDUM * .. * .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH * .. * .. External Subroutines .. EXTERNAL XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DIMAG, MAX, MIN * .. * .. 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( 'ZGBEQU', -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. * SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM * * 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 * * 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. * KD = KU + 1 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 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 ZGBEQU * END