dlansb.f

          DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
         $                 WORK )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          NORM, UPLO
          INTEGER            K, LDAB, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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.
    *
    *  Description
    *  ===========
    *
    *  DLANSB returns the value
    *
    *     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.
    *
    *  Arguments
    *  =========
    *
    *  NORM    (input) CHARACTER*1
    *          Specifies the value to be returned in DLANSB as described
    *          above.
    *
    *  UPLO    (input) 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
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.  When N = 0, DLANSB is
    *          set to zero.
    *
    *  K       (input) INTEGER
    *          The number of super-diagonals or sub-diagonals of the
    *          band matrix A.  K >= 0.
    *
    *  AB      (input) 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).
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= K+1.
    *
    *  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
    *          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
    *          WORK is not referenced.
    *
    * =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO
          PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, J, L
          DOUBLE PRECISION   ABSA, SCALE, SUM, VALUE
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLASSQ
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. 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
                      VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
       10          CONTINUE
       20       CONTINUE
             ELSE
                DO 40 J = 1, N
                   DO 30 I = 1, MIN( N+1-J, K+1 )
                      VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
       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
                   VALUE = MAX( VALUE, WORK( I ) )
       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
                   VALUE = MAX( VALUE, SUM )
      100       CONTINUE
             END IF
          ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
    *
    *        Find normF(A).
    *
             SCALE = ZERO
             SUM = ONE
             IF( K.GT.0 ) THEN
                IF( LSAME( UPLO, 'U' ) ) THEN
                   DO 110 J = 2, N
                      CALL DLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
         $                         1, SCALE, SUM )
      110          CONTINUE
                   L = K + 1
                ELSE
                   DO 120 J = 1, N - 1
                      CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
         $                         SUM )
      120          CONTINUE
                   L = 1
                END IF
                SUM = 2*SUM
             ELSE
                L = 1
             END IF
             CALL DLASSQ( N, AB( L, 1 ), LDAB, SCALE, SUM )
             VALUE = SCALE*SQRT( SUM )
          END IF
    *
          DLANSB = VALUE
          RETURN
    *
    *     End of DLANSB
    *
          END