SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
REAL AMAX, SCOND
* ..
* .. Array Arguments ..
REAL AB( LDAB, * ), S( * )
* ..
*
* Purpose
* =======
*
* SPBEQU 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) REAL 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) REAL array, dimension (N)
* If INFO = 0, S contains the scale factors for A.
*
* SCOND (output) REAL
* 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) REAL
* 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 ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, J
REAL 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( 'SPBEQU', -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 SPBEQU
*
END