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zlanhf.f(3) LAPACK zlanhf.f(3)

zlanhf.f -


DOUBLE PRECISION function zlanhf (NORM, TRANSR, UPLO, N, A, WORK)
 
ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.

ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.
Purpose:
 ZLANHF  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 complex Hermitian matrix A in RFP format.
Returns:
ZLANHF
    ZLANHF = ( 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.
Parameters:
NORM
          NORM is CHARACTER
            Specifies the value to be returned in ZLANHF as described
            above.
TRANSR
          TRANSR is CHARACTER
            Specifies whether the RFP format of A is normal or
            conjugate-transposed format.
            = 'N':  RFP format is Normal
            = 'C':  RFP format is Conjugate-transposed
UPLO
          UPLO is CHARACTER
            On entry, UPLO specifies whether the RFP matrix A came from
            an upper or lower triangular matrix as follows:
UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
UPLO = 'L' or 'l' RFP A came from a lower triangular matrix
N
          N is INTEGER
            The order of the matrix A.  N >= 0.  When N = 0, ZLANHF is
            set to zero.
A
          A is COMPLEX*16 array, dimension ( N*(N+1)/2 );
            On entry, the matrix A in RFP Format.
            RFP Format is described by TRANSR, UPLO and N as follows:
            If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
            K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
            TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A
            as defined when TRANSR = 'N'. The contents of RFP A are
            defined by UPLO as follows: If UPLO = 'U' the RFP A
            contains the ( N*(N+1)/2 ) elements of upper packed A
            either in normal or conjugate-transpose Format. If
            UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) elements
            of lower packed A either in normal or conjugate-transpose
            Format. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When
            TRANSR is 'N' the LDA is N+1 when N is even and is N when
            is odd. See the Note below for more details.
            Unchanged on exit.
WORK
          WORK is DOUBLE PRECISION array, dimension (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.
Date:
September 2012
Further Details:
  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugate-transpose of the first three columns of AP upper. For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first three columns of AP lower. The upper triangle A(0:2,0:2) consists of conjugate-transpose of the last three columns of AP lower. To denote conjugate we place -- above the element. This covers the case N even and TRANSR = 'N'.
RFP A RFP A
-- -- -- 03 04 05 33 43 53 -- -- 13 14 15 00 44 54 -- 23 24 25 10 11 55
33 34 35 20 21 22 -- 00 44 45 30 31 32 -- -- 01 11 55 40 41 42 -- -- -- 02 12 22 50 51 52
Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- transpose of RFP A above. One therefore gets:
RFP A RFP A
-- -- -- -- -- -- -- -- -- -- 03 13 23 33 00 01 02 33 00 10 20 30 40 50 -- -- -- -- -- -- -- -- -- -- 04 14 24 34 44 11 12 43 44 11 21 31 41 51 -- -- -- -- -- -- -- -- -- -- 05 15 25 35 45 55 22 53 54 55 22 32 42 52
We next consider Standard Packed Format when N is odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugate-transpose of the first two columns of AP upper. For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first three columns of AP lower. The upper triangle A(0:1,1:2) consists of conjugate-transpose of the last two columns of AP lower. To denote conjugate we place -- above the element. This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
-- -- 02 03 04 00 33 43 -- 12 13 14 10 11 44
22 23 24 20 21 22 -- 00 33 34 30 31 32 -- -- 01 11 44 40 41 42
Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- transpose of RFP A above. One therefore gets:
RFP A RFP A
-- -- -- -- -- -- -- -- -- 02 12 22 00 01 00 10 20 30 40 50 -- -- -- -- -- -- -- -- -- 03 13 23 33 11 33 11 21 31 41 51 -- -- -- -- -- -- -- -- -- 04 14 24 34 44 43 44 22 32 42 52
Definition at line 247 of file zlanhf.f.

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