SRC/zlanhf.f(3)

NAME
SRC/zlanhf.f

SYNOPSIS

  Functions/Subroutines
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.

Function/Subroutine Documentation

  double precision function zlanhf (character norm, character transr, character uplo, integer n, complex*16, dimension( 0: * ) a, double precision, dimension( 0: * ) 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.

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.

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 245 of file zlanhf.f.

Author

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