ZLANHF(3)      LAPACK routine of NEC Numeric Library Collection      ZLANHF(3)



NAME
       ZLANHF

SYNOPSIS
       DOUBLE PRECISION FUNCTION ZLANHF (NORM, TRANSR, UPLO, N, A, WORK)



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.




ARGUMENTS
           NORM      (input)
                     NORM is CHARACTER
                       Specifies the value to be returned in ZLANHF as described
                       above.

           TRANSR    (input)
                     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      (input)
                     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         (input)
                     N is INTEGER
                       The order of the matrix A.  N >= 0.  When N = 0, ZLANHF is
                       set to zero.

           A         (input)
                     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      (output)
                     WORK is DOUBLE PRECISION array, dimension (LWORK),
                       where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
                       WORK is not referenced.






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



LAPACK routine                  31 October 2017                      ZLANHF(3)