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DOUBLE PRECISION FUNCTION ZLANHB( NORM, UPLO, N, K, AB, LDAB,
$ WORK )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. 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 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 (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
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