*> \brief \b SLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLANTB + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * REAL FUNCTION SLANTB( NORM, UPLO, DIAG, N, K, AB, * LDAB, WORK ) * * .. Scalar Arguments .. * CHARACTER DIAG, NORM, UPLO * INTEGER K, LDAB, N * .. * .. Array Arguments .. * REAL AB( LDAB, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLANTB 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 triangular band matrix A, with ( k + 1 ) diagonals. *> \endverbatim *> *> \return SLANTB *> \verbatim *> *> SLANTB = ( 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 SLANTB as described *> above. *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> Specifies whether the matrix A is upper or lower triangular. *> = 'U': Upper triangular *> = 'L': Lower triangular *> \endverbatim *> *> \param[in] DIAG *> \verbatim *> DIAG is CHARACTER*1 *> Specifies whether or not the matrix A is unit triangular. *> = 'N': Non-unit triangular *> = 'U': Unit triangular *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. When N = 0, SLANTB is *> set to zero. *> \endverbatim *> *> \param[in] K *> \verbatim *> K is INTEGER *> The number of super-diagonals of the matrix A if UPLO = 'U', *> or the number of sub-diagonals of the matrix A if UPLO = 'L'. *> K >= 0. *> \endverbatim *> *> \param[in] AB *> \verbatim *> AB is REAL array, dimension (LDAB,N) *> The upper or lower triangular 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). *> Note that when DIAG = 'U', the elements of the array AB *> corresponding to the diagonal elements of the matrix A are *> not referenced, but are assumed to be one. *> \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 REAL array, dimension (MAX(1,LWORK)), *> where LWORK >= N when NORM = 'I'; 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 realOTHERauxiliary * * ===================================================================== REAL FUNCTION SLANTB( NORM, UPLO, DIAG, 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 DIAG, 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 .. LOGICAL UDIAG INTEGER I, J, L REAL SUM, VALUE * .. * .. Local Arrays .. REAL SSQ( 2 ), COLSSQ( 2 ) * .. * .. External Functions .. LOGICAL LSAME, SISNAN EXTERNAL LSAME, SISNAN * .. * .. External Subroutines .. EXTERNAL SLASSQ, SCOMBSSQ * .. * .. 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))). * IF( LSAME( DIAG, 'U' ) ) THEN VALUE = ONE IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = MAX( K+2-J, 1 ), K 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 = 2, 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 VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 60 J = 1, N DO 50 I = MAX( K+2-J, 1 ), K + 1 SUM = ABS( AB( I, J ) ) IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM 50 CONTINUE 60 CONTINUE ELSE DO 80 J = 1, N DO 70 I = 1, MIN( N+1-J, K+1 ) SUM = ABS( AB( I, J ) ) IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM 70 CONTINUE 80 CONTINUE END IF END IF ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN * * Find norm1(A). * VALUE = ZERO UDIAG = LSAME( DIAG, 'U' ) IF( LSAME( UPLO, 'U' ) ) THEN DO 110 J = 1, N IF( UDIAG ) THEN SUM = ONE DO 90 I = MAX( K+2-J, 1 ), K SUM = SUM + ABS( AB( I, J ) ) 90 CONTINUE ELSE SUM = ZERO DO 100 I = MAX( K+2-J, 1 ), K + 1 SUM = SUM + ABS( AB( I, J ) ) 100 CONTINUE END IF IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM 110 CONTINUE ELSE DO 140 J = 1, N IF( UDIAG ) THEN SUM = ONE DO 120 I = 2, MIN( N+1-J, K+1 ) SUM = SUM + ABS( AB( I, J ) ) 120 CONTINUE ELSE SUM = ZERO DO 130 I = 1, MIN( N+1-J, K+1 ) SUM = SUM + ABS( AB( I, J ) ) 130 CONTINUE END IF IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM 140 CONTINUE END IF ELSE IF( LSAME( NORM, 'I' ) ) THEN * * Find normI(A). * VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN DO 150 I = 1, N WORK( I ) = ONE 150 CONTINUE DO 170 J = 1, N L = K + 1 - J DO 160 I = MAX( 1, J-K ), J - 1 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 160 CONTINUE 170 CONTINUE ELSE DO 180 I = 1, N WORK( I ) = ZERO 180 CONTINUE DO 200 J = 1, N L = K + 1 - J DO 190 I = MAX( 1, J-K ), J WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 190 CONTINUE 200 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN DO 210 I = 1, N WORK( I ) = ONE 210 CONTINUE DO 230 J = 1, N L = 1 - J DO 220 I = J + 1, MIN( N, J+K ) WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 220 CONTINUE 230 CONTINUE ELSE DO 240 I = 1, N WORK( I ) = ZERO 240 CONTINUE DO 260 J = 1, N L = 1 - J DO 250 I = J, MIN( N, J+K ) WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 250 CONTINUE 260 CONTINUE END IF END IF DO 270 I = 1, N SUM = WORK( I ) IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM 270 CONTINUE 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. * IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN SSQ( 1 ) = ONE SSQ( 2 ) = N IF( K.GT.0 ) THEN DO 280 J = 2, N COLSSQ( 1 ) = ZERO COLSSQ( 2 ) = ONE CALL SLASSQ( MIN( J-1, K ), $ AB( MAX( K+2-J, 1 ), J ), 1, $ COLSSQ( 1 ), COLSSQ( 2 ) ) CALL SCOMBSSQ( SSQ, COLSSQ ) 280 CONTINUE END IF ELSE SSQ( 1 ) = ZERO SSQ( 2 ) = ONE DO 290 J = 1, N COLSSQ( 1 ) = ZERO COLSSQ( 2 ) = ONE CALL SLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ), J ), $ 1, COLSSQ( 1 ), COLSSQ( 2 ) ) CALL SCOMBSSQ( SSQ, COLSSQ ) 290 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN SSQ( 1 ) = ONE SSQ( 2 ) = N IF( K.GT.0 ) THEN DO 300 J = 1, N - 1 COLSSQ( 1 ) = ZERO COLSSQ( 2 ) = ONE CALL SLASSQ( MIN( N-J, K ), AB( 2, J ), 1, $ COLSSQ( 1 ), COLSSQ( 2 ) ) CALL SCOMBSSQ( SSQ, COLSSQ ) 300 CONTINUE END IF ELSE SSQ( 1 ) = ZERO SSQ( 2 ) = ONE DO 310 J = 1, N COLSSQ( 1 ) = ZERO COLSSQ( 2 ) = ONE CALL SLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1, $ COLSSQ( 1 ), COLSSQ( 2 ) ) CALL SCOMBSSQ( SSQ, COLSSQ ) 310 CONTINUE END IF END IF VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) ) END IF * SLANTB = VALUE RETURN * * End of SLANTB * END