*> \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/ * *> \htmlonly *> Download DLANSB + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * 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 * * .. 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, SCALE, SUM, VALUE * .. * .. External Subroutines .. EXTERNAL DLASSQ * .. * .. External Functions .. LOGICAL LSAME, DISNAN EXTERNAL LSAME, DISNAN * .. * .. 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). * 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