dlansp()

double precision function dlansp	(	character	norm,
		character	uplo,
		integer	n,
		double precision, dimension( * )	ap,
		double precision, dimension( * )	work
	)

DLANSP 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 matrix supplied in packed form.

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Purpose:

 DLANSP  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 real symmetric matrix A,  supplied in packed form.

Returns

DLANSP

    DLANSP = ( 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 DLANSP as described above.
[in]	UPLO	UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is supplied. = 'U': Upper triangular part of A is supplied = 'L': Lower triangular part of A is supplied
[in]	N	N is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANSP is set to zero.
[in]	AP	AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
[out]	WORK	WORK is DOUBLE PRECISION 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 113 of file dlansp.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            N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), WORK( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      parameter( one = 1.0d+0, zero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, K
      DOUBLE PRECISION   ABSA, SCALE, SUM, VALUE
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlassq
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      EXTERNAL           lsame, disnan
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, 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
            k = 1
            DO 20 j = 1, n
               DO 10 i = k, k + j - 1
                  sum = abs( ap( i ) )
                  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
   10          CONTINUE
               k = k + j
   20       CONTINUE
         ELSE
            k = 1
            DO 40 j = 1, n
               DO 30 i = k, k + n - j
                  sum = abs( ap( i ) )
                  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
   30          CONTINUE
               k = k + n - j + 1
   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
         k = 1
         IF( lsame( uplo, 'U' ) ) THEN
            DO 60 j = 1, n
               sum = zero
               DO 50 i = 1, j - 1
                  absa = abs( ap( k ) )
                  sum = sum + absa
                  work( i ) = work( i ) + absa
                  k = k + 1
   50          CONTINUE
               work( j ) = sum + abs( ap( k ) )
               k = k + 1
   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( ap( k ) )
               k = k + 1
               DO 90 i = j + 1, n
                  absa = abs( ap( k ) )
                  sum = sum + absa
                  work( i ) = work( i ) + absa
                  k = k + 1
   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).
*
         scale = zero
         sum = one
         k = 2
         IF( lsame( uplo, 'U' ) ) THEN
            DO 110 j = 2, n
               CALL dlassq( j-1, ap( k ), 1, scale, sum )
               k = k + j
  110       CONTINUE
         ELSE
            DO 120 j = 1, n - 1
               CALL dlassq( n-j, ap( k ), 1, scale, sum )
               k = k + n - j + 1
  120       CONTINUE
         END IF
         sum = 2*sum
         k = 1
         DO 130 i = 1, n
            IF( ap( k ).NE.zero ) THEN
               absa = abs( ap( k ) )
               IF( scale.LT.absa ) THEN
                  sum = one + sum*( scale / absa )**2
                  scale = absa
               ELSE
                  sum = sum + ( absa / scale )**2
               END IF
            END IF
            IF( lsame( uplo, 'U' ) ) THEN
               k = k + i + 1
            ELSE
               k = k + n - i + 1
            END IF
  130    CONTINUE
         VALUE = scale*sqrt( sum )
      END IF
*
      dlansp = VALUE
      RETURN
*
*     End of DLANSP
*

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