dlansb.f

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*> \brief \b DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
*                        WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM, UPLO
*       INTEGER            K, LDAB, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLANSB  returns the value of the one norm,  or the Frobenius norm, or
*> the  infinity norm,  or the element of  largest absolute value  of an
*> n by n symmetric band matrix A,  with k super-diagonals.
*> \endverbatim
*>
*> \return DLANSB
*> \verbatim
*>
*>    DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
*>             (
*>             ( norm1(A),         NORM = '1', 'O' or 'o'
*>             (
*>             ( normI(A),         NORM = 'I' or 'i'
*>             (
*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
*>
*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies the value to be returned in DLANSB as described
*>          above.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          band matrix A is supplied.
*>          = 'U':  Upper triangular part is supplied
*>          = 'L':  Lower triangular part is supplied
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.  When N = 0, DLANSB is
*>          set to zero.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of super-diagonals or sub-diagonals of the
*>          band matrix A.  K >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangle of the symmetric band matrix A,
*>          stored in the first K+1 rows of AB.  The j-th column of A is
*>          stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= K+1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
*>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
*>          WORK is not referenced.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleOTHERauxiliary
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION dlansb( NORM, UPLO, N, K, AB, LDAB,
     $                 WORK )
*
*  -- LAPACK auxiliary routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
      IMPLICIT NONE
*     .. Scalar Arguments ..
      CHARACTER          norm, uplo
      INTEGER            k, ldab, n
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   ab( ldab, * ), work( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   one, zero
      parameter( one = 1.0d+0, zero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            i, j, l
      DOUBLE PRECISION   absa, sum, value
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   ssq( 2 ), colssq( 2 )
*     ..
*     .. External Functions ..
      LOGICAL            lsame, disnan
      EXTERNAL           lsame, disnan
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlassq, dcombssq
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, sqrt
*     ..
*     .. Executable Statements ..
*
      IF( n.EQ.0 ) THEN
         VALUE = zero
      ELSE IF( lsame( norm, 'M' ) ) THEN
*
*        Find max(abs(A(i,j))).
*
         VALUE = zero
         IF( lsame( uplo, 'U' ) ) THEN
            DO 20 j = 1, n
               DO 10 i = max( k+2-j, 1 ), k + 1
                  sum = abs( ab( i, j ) )
                  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
   10          CONTINUE
   20       CONTINUE
         ELSE
            DO 40 j = 1, n
               DO 30 i = 1, min( n+1-j, k+1 )
                  sum = abs( ab( i, j ) )
                  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
   30          CONTINUE
   40       CONTINUE
         END IF
      ELSE IF( ( lsame( norm, 'I' ) ) .OR. ( lsame( norm, 'O' ) ) .OR.
     $         ( norm.EQ.'1' ) ) THEN
*
*        Find normI(A) ( = norm1(A), since A is symmetric).
*
         VALUE = zero
         IF( lsame( uplo, 'U' ) ) THEN
            DO 60 j = 1, n
               sum = zero
               l = k + 1 - j
               DO 50 i = max( 1, j-k ), j - 1
                  absa = abs( ab( l+i, j ) )
                  sum = sum + absa
                  work( i ) = work( i ) + absa
   50          CONTINUE
               work( j ) = sum + abs( ab( k+1, j ) )
   60       CONTINUE
            DO 70 i = 1, n
               sum = work( i )
               IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
   70       CONTINUE
         ELSE
            DO 80 i = 1, n
               work( i ) = zero
   80       CONTINUE
            DO 100 j = 1, n
               sum = work( j ) + abs( ab( 1, j ) )
               l = 1 - j
               DO 90 i = j + 1, min( n, j+k )
                  absa = abs( ab( l+i, j ) )
                  sum = sum + absa
                  work( i ) = work( i ) + absa
   90          CONTINUE
               IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
  100       CONTINUE
         END IF
      ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
*
*        Find normF(A).
*        SSQ(1) is scale
*        SSQ(2) is sum-of-squares
*        For better accuracy, sum each column separately.
*
         ssq( 1 ) = zero
         ssq( 2 ) = one
*
*        Sum off-diagonals
*
         IF( k.GT.0 ) THEN
            IF( lsame( uplo, 'U' ) ) THEN
               DO 110 j = 2, n
                  colssq( 1 ) = zero
                  colssq( 2 ) = one
                  CALL dlassq( min( j-1, k ), ab( max( k+2-j, 1 ), j ),
     $                         1, colssq( 1 ), colssq( 2 ) )
                  CALL dcombssq( ssq, colssq )
  110          CONTINUE
               l = k + 1
            ELSE
               DO 120 j = 1, n - 1
                  colssq( 1 ) = zero
                  colssq( 2 ) = one
                  CALL dlassq( min( n-j, k ), ab( 2, j ), 1,
     $                         colssq( 1 ), colssq( 2 ) )
                  CALL dcombssq( ssq, colssq )
  120          CONTINUE
               l = 1
            END IF
            ssq( 2 ) = 2*ssq( 2 )
         ELSE
            l = 1
         END IF
*
*        Sum diagonal
*
         colssq( 1 ) = zero
         colssq( 2 ) = one
         CALL dlassq( n, ab( l, 1 ), ldab, colssq( 1 ), colssq( 2 ) )
         CALL dcombssq( ssq, colssq )
         VALUE = ssq( 1 )*sqrt( ssq( 2 ) )
      END IF
*
      dlansb = VALUE
      RETURN
*
*     End of DLANSB
*
      END