SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N DOUBLE PRECISION AMAX, SCOND * .. * .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), S( * ) * .. * * Purpose * ======= * * DPBEQU computes row and column scalings intended to equilibrate a * symmetric positive definite band matrix A and reduce its condition * number (with respect to the two-norm). S contains the scale factors, * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This * choice of S puts the condition number of B within a factor N of the * smallest possible condition number over all possible diagonal * scalings. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangular of A is stored; * = 'L': Lower triangular of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 0. * * AB (input) DOUBLE PRECISION array, dimension (LDAB,N) * The upper or lower triangle of the symmetric band matrix A, * stored in the first KD+1 rows of the array. The j-th column * of A is stored in the j-th column of the array AB as follows: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). * * LDAB (input) INTEGER * The leading dimension of the array A. LDAB >= KD+1. * * S (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, S contains the scale factors for A. * * SCOND (output) DOUBLE PRECISION * If INFO = 0, S contains the ratio of the smallest S(i) to * the largest S(i). If SCOND >= 0.1 and AMAX is neither too * large nor too small, it is not worth scaling by S. * * 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, the i-th diagonal element is nonpositive. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER I, J DOUBLE PRECISION SMIN * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBEQU', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF * IF( UPPER ) THEN J = KD + 1 ELSE J = 1 END IF * * Initialize SMIN and AMAX. * S( 1 ) = AB( J, 1 ) SMIN = S( 1 ) AMAX = S( 1 ) * * Find the minimum and maximum diagonal elements. * DO 10 I = 2, N S( I ) = AB( J, I ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 10 CONTINUE * IF( SMIN.LE.ZERO ) THEN * * Find the first non-positive diagonal element and return. * DO 20 I = 1, N IF( S( I ).LE.ZERO ) THEN INFO = I RETURN END IF 20 CONTINUE ELSE * * Set the scale factors to the reciprocals * of the diagonal elements. * DO 30 I = 1, N S( I ) = ONE / SQRT( S( I ) ) 30 CONTINUE * * Compute SCOND = min(S(I)) / max(S(I)) * SCOND = SQRT( SMIN ) / SQRT( AMAX ) END IF RETURN * * End of DPBEQU * END