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DOUBLE PRECISION FUNCTION ZLANHB( NORM, UPLO, N, K, AB, LDAB,
$ WORK ) * * -- LAPACK auxiliary routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER NORM, UPLO INTEGER K, LDAB, N * .. * .. Array Arguments .. DOUBLE PRECISION WORK( * ) COMPLEX*16 AB( LDAB, * ) * .. * * Purpose * ======= * * ZLANHB 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 hermitian band matrix A, with k super-diagonals. * * Description * =========== * * ZLANHB returns the value * * ZLANHB = ( 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 ZLANHB 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 * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. When N = 0, ZLANHB is * set to zero. * * K (input) INTEGER * The number of super-diagonals or sub-diagonals of the * band matrix A. K >= 0. * * AB (input) COMPLEX*16 array, dimension (LDAB,N) * The upper or lower triangle of the hermitian 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 the imaginary parts of the diagonal elements need * not be set and are assumed to be zero. * * 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 Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL ZLASSQ * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, 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 VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 10 CONTINUE VALUE = MAX( VALUE, ABS( DBLE( AB( K+1, J ) ) ) ) 20 CONTINUE ELSE DO 40 J = 1, N VALUE = MAX( VALUE, ABS( DBLE( AB( 1, J ) ) ) ) DO 30 I = 2, 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 hermitian). * 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( DBLE( 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( DBLE( 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 ZLASSQ( 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 ZLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE, $ SUM ) 120 CONTINUE L = 1 END IF SUM = 2*SUM ELSE L = 1 END IF DO 130 J = 1, N IF( DBLE( AB( L, J ) ).NE.ZERO ) THEN ABSA = ABS( DBLE( AB( L, J ) ) ) IF( SCALE.LT.ABSA ) THEN SUM = ONE + SUM*( SCALE / ABSA )**2 SCALE = ABSA ELSE SUM = SUM + ( ABSA / SCALE )**2 END IF END IF 130 CONTINUE VALUE = SCALE*SQRT( SUM ) END IF * ZLANHB = VALUE RETURN * * End of ZLANHB * END |