slansb()

real function slansb	(	character	norm,
		character	uplo,
		integer	n,
		integer	k,
		real, dimension( ldab, * )	ab,
		integer	ldab,
		real, dimension( * )	work
	)

SLANSB 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.

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Purpose:

 SLANSB  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.

Returns

SLANSB

    SLANSB = ( 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.

Parameters

[in]	NORM	NORM is CHARACTER*1 Specifies the value to be returned in SLANSB as described above.
[in]	UPLO	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
[in]	N	N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANSB is set to zero.
[in]	K	K is INTEGER The number of super-diagonals or sub-diagonals of the band matrix A. K >= 0.
[in]	AB	AB is REAL 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).
[in]	LDAB	LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.
[out]	WORK	WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 127 of file slansb.f.

*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          NORM, UPLO
      INTEGER            K, LDAB, N
*     ..
*     .. Array Arguments ..
      REAL               AB( LDAB, * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      parameter( one = 1.0e+0, zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, L
      REAL               ABSA, SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           slassq
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, SISNAN
      EXTERNAL           lsame, sisnan
*     ..
*     .. 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. sisnan( 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. sisnan( 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. sisnan( 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. sisnan( sum ) ) 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 slassq( 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 slassq( 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 slassq( n, ab( l, 1 ), ldab, scale, sum )
         VALUE = scale*sqrt( sum )
      END IF
*
      slansb = VALUE
      RETURN
*
*     End of SLANSB
*

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